Action Theory Evolution

Action Theory Ev olution Iv an Jos ´ e V arzinczak Merak a Institute CSIR Pretoria, South Africa ivan.varzin czak@meraka. org.za Abstract Like an y other logical theory , domain descriptions in rea soning ab out actions may ev olv e, and th us need revision methods to ade- quately accommo date new informatio n ab out the b ehavior of actions. The pr esent w ork is ab out changing action domain descr iptions in prop ositional dynamic lo gic. Its contribution is threefold: ﬁrst we revisit the semantics o f a ction theory contraction that has b een do ne in previo us work, giving more r o bust op erator s that expre ss minimal change based on a notion of distance b etw een Kripke-models . Second we give algorithms for syntactical actio n theor y contraction a nd estab- lish their correctness w.r.t. our semantics. Finally we state postulates for action theory con traction and assess the behavior of our operato rs w.r.t. them. Moreov er, w e also addr ess the revision co un terpart of action theo ry change, showing that it b eneﬁts from our semantics for contraction. 1 Con ten ts 1 In tro duction 4 2 Logical Preliminaries 6 2.1 Action Theories in Dynamic Logic . . . . . . . . . . . . . . . 6 2.2 Essen tial A toms . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Prime V aluations . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Closeness b et w een Mo dels . . . . . . . . . . . . . . . . . . . . 1 3 3 Seman tics of Action Theory Change 13 3.1 Mo del Contrac tion of Ex ecutabilit y La ws . . . . . . . . . . . 1 4 3.2 Mo del Contrac tion of Eﬀect L a w s . . . . . . . . . . . . . . . . 1 6 3.3 Mo del Contrac tion of S tatic Laws . . . . . . . . . . . . . . . . 20 4 Syn tactic Op erators for Con traction of La ws 21 4.1 Con tracting Executabilit y Laws . . . . . . . . . . . . . . . . . 22 4.2 Con tracting Eﬀect Laws . . . . . . . . . . . . . . . . . . . . . 23 4.3 Con tracting Static La ws . . . . . . . . . . . . . . . . . . . . . 27 5 Correctness of the Op e ra tors 28 5.1 Tw o C oun ter-Examples . . . . . . . . . . . . . . . . . . . . . 28 5.2 Mo dular Theories . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Correctness Under Mo du larit y . . . . . . . . . . . . . . . . . 31 6 Assessmen t of Postulates for Change 32 6.1 Con traction or Erasur e? . . . . . . . . . . . . . . . . . . . . . 32 6.2 The P ostulates . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7 A Semantics for Ac tion Theory Revision 37 7.1 Revising a Mo del by a S tatic La w . . . . . . . . . . . . . . . 38 7.2 Revising a Mo del by an Eﬀect La w . . . . . . . . . . . . . . . 40 7.3 Revising a Mo del by an Executabilit y La w . . . . . . . . . . . 41 7.4 Revising Sets of Mo d els . . . . . . . . . . . . . . . . . . . . . 42 8 Related W ork 43 9 Commen ts 46 9.1 Other Distance Notions . . . . . . . . . . . . . . . . . . . . . 46 9.2 Inducing Executabilit y . . . . . . . . . . . . . . . . . . . . . . 47 10 Concluding Remarks 49 A Pro of of Theorem 5.2 56 B Pro of of Theorem 5.3 61 3 1 In tro duction Consider an in telligen t agent designed to p erform rationally in a dynamic w orld, and su pp ose she should reason ab ou t the dy n amics of an automatic coﬀee mac hin e (Figure 1). S u pp ose, for examp le, that th e agen t b eliev es that coﬀee is al w a ys a hot b ev erage. Sup p ose no w that some da y she ge ts a coﬀee an d observ es that it is cold. In suc h a case, the agent must change her b eliefs about the relat ion b et w een the p rop ositions “I h old a coﬀee” and “I hold a hot b everag e”. This example is an instance of the problem of c hanging prop ositional b elief bases and is largely addr essed in the literat ure ab out b elief change [15 ] and b elief up date [31]. Figure 1: The coﬀee d eliv er er agen t. Next, let our agen t b eliev e that wh enev er she buys a coﬀee f rom the mac h ine, sh e gets a hot b ev erage. This means th at in ev ery state of the w orld that follo ws the execution of bu ying a coﬀee, the agen t p ossesses a hot b eve rage. Then, in a situation wh ere the machine is r unnin g out of cups, after b uying, the coﬀe e runs thr ough the shelf and th e agen t does not hold a hot b ev erage in h er hands. Imagine no w that the agen t nev er consid er ed any relatio n b et w een b uying a coﬀee on the mac hine and its service a v ailabilit y , in the sense that the agen t al w a ys b eliev ed that buying do es n ot prev en t other u sers from u sing the mac hine. Neve rtheless, someda y our agen t is queuing to buy a coﬀee and observes that j u st after the agen t b efore h er has b ough t, the mac h ine w en t out of ord er (ma yb e due to a lack of coﬀee p owder). Completing our agen t’s struggle in disco v erin g the intricac ies of a coﬀee mac h ine, supp ose she alw ays b eliev ed that if she has a tok en, then it is 4 p ossible to buy coﬀe e, pro vided that some other conditions lik e b eing close enough to the b utton, h aving a fr ee h and, etc, are satisﬁed. Ho wev er, dur ing a blac ko ut, the agen t, ev en with a tok en , do es n ot m anage to buy her coﬀee. The last thr ee examples illustrate situations where c hanging the b eliefs ab out the b ehavior of th e action of buy in g coﬀee is mandatory . In the ﬁr s t one, buying coﬀe e, once b eliev ed to b e deterministic, has no w to b e seen as nondeterministic, or alternativ ely to ha v e a d iﬀeren t outcome in a more sp eciﬁc con text (e.g. if there is no cup in the machine). In the second example, buying a coﬀee is n o w kno wn to ha ve side-eﬀects (ramiﬁcations) one was n ot aw are of. Fin ally , in the last example, the executa bilit y of the action u nder concern is questioned in the light of n ew information sho w ing a con text that was n ot kno wn to pr eclude its execution. Suc h cases of theory change are v ery imp ortant when one deals with logica l descriptions of dynamic domains: i t ma y alw a ys happ en that one disco v ers that an action actually has a b eha vior that is diﬀerent from that one has alw a ys b eliev ed it had. Up to n o w , theory change has b een studied mainly for kn o wledge bases in classical logics, b oth in terms of revision and up d ate. Sin ce the w ork b y F uhrm ann [14], only in a few recen t studies has it b een considered in the realm of m o dal logics, viz. in epistemic logic [19] an d in dyn amic logics [21]. Recen tly some studies hav e inv estigated revision of b eliefs ab out facts of the w orld [47, 28] or the agen t’s goals [46]. In our scenario, this would concern for instance the truth of token in a giv en state: th e agen t b eliev es that she has a toke n, but is actually wrong ab out that. Then sh e might su b sequen tly b e forced to revise her beliefs ab out the current state of aﬀairs or c hange her goals according to what she can p erform in th at state. S uc h b elief revision op erations do n ot mo d ify the agen t’s b eliefs ab ou t the action laws . In opp osition to that, here w e are intereste d exactly in suc h mo diﬁcations. Starting with Baral and Lob o’s work [4 ], some recen t studies h a v e b een done on that issu e [12, 13] for d omain d escriptions in action languages [16]. W e here tak e a s tep further in this d irection and prop ose a metho d based on th at given by Herzig et al. [21] that is more robu st by in tegrating a notion of minimal change and complying w ith p ostulates of th eory c h ange. The pr esen t text is structur ed as follo ws: in S ection 2 w e establish the formal bac kgrou n d that will b e u sed throughout this work. Secti ons 3 – 5 are the core of the work: in S ection 3 w e present the cen tral deﬁn itions for a seman tics of act ion theory c h ange, Section 4 is dev oted to its synta ctical coun terpart while Section 5 to the pro of of its corresp ond ence with the se- man tics. In Section 6 w e discuss some p ostulates f or con traction/erasure and 5 then p resen t a seman tics for action theory r evision (S ection 7). In Section 8 w e add ress existing w ork in th e ﬁ eld. After making some commen ts on our metho d (Sectio n 9), w e ﬁnish with some conclusions an d future directio ns of researc h. 2 Logical Preliminarie s F ollo win g the tradition in the reasoning ab out actions (RAA) comm unity , w e consider ac tion theories to b e ﬁnite collections of statemen ts that ha v e the particular f orm: • if c ontext , then eﬀe ct after e v ery exe cution of action (eﬀect laws); • if pr e c ondition , then action exe cu table (executabilit y la ws). Statemen ts mentio ning n o action at all r epresen t la ws ab out the un derlying structure of the world, i.e., its p ossible states (static laws). Sev eral logical framewo rks h a v e been pr op osed to formalize suc h state- men ts. Among the most prominent on es are the Situation Calculus [39, 45], the family of Action Languages [16, 30, 17], the Fluent Calculus [49, 50], and the dynamic log ic-based app roac h es [10, 6, 57]. Here w e opt to formalize action theories us in g a v ers ion of Pr op ositional Dynamic Logic ( PDL ) [20]. 2.1 Action Theories in Dynamic Logic Let Act = { a 1 , a 2 , . . . } b e the set of all atomic action c onstants of a giv en domain. An example of atomic action is buy . T o eac h atomic action a th ere is asso ciated a mo dal op erator [ a ]. 1 Prop = { p 1 , p 2 , . . . } denotes the set of all pr op ositional c onstants , also called ﬂuents or atoms . Examp les of those are token (“the agent has a tok en ”) and c oﬀe e (“the agent holds a coﬀee”) . T he set of all literals is Lit = { ℓ 1 , ℓ 2 , . . . } , where eac h ℓ i is either p or ¬ p , for some p ∈ Prop . If ℓ = ¬ p , then we id entify ¬ ℓ with p . By | ℓ | we denote the atom in ℓ . W e use small Greek letters ϕ, ψ, . . . to denote Bo ole an formulas . They are recursively deﬁn ed in th e usual w a y: ϕ ::= p | ⊤ | ⊥ | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | ϕ ↔ ϕ 1 W e here supp ose t h at our m ultimod al logic is indep endently axiomatized [32], i.e., the logic is a fusion an d there is no interaction b etw een th e mo dal op erators. This is a requirement to achiev e mod ularit y of action theories [25] (see further). 6 Fml is the set of all Bo olean formulas. An example of a Bo olean form ula is c oﬀe e → hot . A prop ositional v aluation v is a maximal ly c onsistent set of literals. W e denote by v  ϕ the fact that v satisﬁes a p rop ositional form ula ϕ . By val ( ϕ ) w e d en ote the set of all v aluations satisfying ϕ . | = CPL denotes the classical consequence relation. Cn ( ϕ ) den otes all logical consequences of ϕ in classical prop ositional logic. If ϕ is a pr op ositional form ula, atm ( ϕ ) d enotes the set of element ary atoms actual ly o ccurring in ϕ . F or example, atm ( ¬ p 1 ∧ ( ¬ p 1 ∨ p 2 )) = { p 1 , p 2 } . F or ϕ a Bo olean formula, IP ( ϕ ) denotes the set of its prime impli- c ants [43], i.e., the w eak est terms (conjunctions of literals) that im p ly ϕ . As an example, IP ( p 1 ⊕ p 2 ) = { p 1 ∧ ¬ p 2 , ¬ p 1 ∧ p 2 } . F or more on prime implican ts, their prop erties and ho w to compute them see the c hapter b y Marquis [36]. By π we denote a prime imp lican t, and given ℓ and π , ℓ ∈ π abbreviates ‘ ℓ is a literal of π ’. W e den ote complex formula s (p ossibly w ith m o dal op erators) by Φ, Ψ , . . . They are recurs iv ely d eﬁned in the follo wing w a y: Φ ::= ϕ | [ a ] Φ | ¬ Φ | Φ ∧ Φ | Φ ∨ Φ | Φ → Φ | Φ ↔ Φ h a i is the dual op erator of [ a ], deﬁned as h a i Φ = def ¬ [ a ] ¬ Φ . An example of a complex formula is ¬ c oﬀe e → [ buy ] c oﬀe e . The seman tics is th at of PDL w ith ou t the ∗ op erator, wh ic h amounts to multimodal logic K n [42]. In th e follo w ing we will refer to PDL bu t our underlying logical formalism is essentia lly the simpler multimod al logic K n , whic h turns out to b e exp ressiv e enough for our p urp oses here. Deﬁnition 2.1 ( PDL -mo del) A PDL -mo d el is a tuple M = h W , R i wher e W is a set of valuations (also c al le d p ossible world s), and R maps action c onstants a to ac c essibility r e lations R a ⊆ W × W. As an example, for Act = { a 1 , a 2 } and Prop = { p 1 , p 2 } , we ha v e the PDL -mo del M = h W , R i , where W = {{ p 1 , p 2 } , { p 1 , ¬ p 2 } , {¬ p 1 , p 2 }} , R ( a 1 ) =  ( { p 1 , p 2 } , { p 1 , ¬ p 2 } ) , ( { p 1 , p 2 } , {¬ p 1 , p 2 } ) , ( { p 1 , ¬ p 2 } , { p 1 , ¬ p 2 } ) , ( { p 1 , ¬ p 2 } , {¬ p 1 , p 2 } )  R ( a 2 ) = { ( { p 1 , p 2 } , {¬ p 1 , p 2 } ) , ( {¬ p 1 , p 2 } , {¬ p 1 , p 2 } ) } Figure 2 giv es a graph ical r epresen tation of M . 2 2 Notice th at our notion of PDL - model do es not follow the standard notion from mo dal 7 M : p 1 , p 2 p 1 , ¬ p 2 ¬ p 1 , p 2 a 1 a 1 a 2 a 1 a 1 a 2 Figure 2: Example of a PD L -mo del for Act = { a 1 , a 2 } , and Prop = { p 1 , p 2 } . Deﬁnition 2.2 (T ruth conditions) Given a PD L -mo del M = h W , R i , • | = M w p (p is true at world w of mo del M ) if w  p (the valuation w satisﬁes p, i . e., p ∈ w ); • | = M w [ a ] Φ if | = M w ′ Φ for ev e ry w ′ s.t. ( w, w ′ ) ∈ R a ; • | = M w Φ ∧ Ψ if | = M w Φ and | = M w Ψ ; • | = M w Φ ∨ Ψ if | = M w Φ or | = M w Ψ , or b oth; • | = M w ¬ Φ i f 6| = M w Φ , i.e., not | = M w Φ ; • truth c onditions for the other c onne ctives ar e as usu al. By M we will d enote a set of PDL -mo dels. A PDL -mo del M is a mo del of Φ (denoted | = M Φ ) if an d only if for all w ∈ W , | = M w Φ . In the mo del dep icted in Figure 2, w e hav e | = M p 1 → [ a 2 ] p 2 and | = M p 1 ∨ p 2 . Deﬁnition 2.3 (Global c onsequence) M is a mo del of a set of formulas Σ (note d | = M Σ ) if and only if | = M Φ for e v ery Φ ∈ Σ . A formula Φ is a consequence of a set of global axioms Σ in the class of al l PD L -mo dels (note d Σ | = PDL Φ ) if and only if for every PDL -mo del M , if | = M Σ , then | = M Φ . logics: here n o tw o worlds satisfy the same va luation. This is a p ragmatic choice (see Sec- tion 4). Nevertheless, all w e are ab out to state in the sequel can b e straigh tforw ard ly form ulated for standard PDL mo dels as well. 8 With PDL we can state laws describing the b ehavior of actions. One wa y of doing this is by stating some formulas as global axioms. 3 As usually done in the RAA comm u nit y , w e her e distinguish three t yp es of la ws. Th e ﬁrst kind of statement s are sta tic laws , whic h are Bo olean formulas that m ust hold in ev ery p ossible state of the world. Deﬁnition 2.4 (Static L aw) A static la w is a formula ϕ ∈ Fml . An example of a static law is c oﬀe e → hot , sa ying that if the agen t h olds a coﬀee, then s h e holds a hot b eve rage. The set of all static la ws of a domain is denoted by S ⊆ Fml . In our example w e will h a ve S = { c oﬀe e → hot } . The second kind of act ion la w w e consider is giv en by th e eﬀe ct laws . These are form ulas relating an action to its eﬀects, w hic h can b e conditional. Deﬁnition 2.5 (Eﬀect L aw) A n eﬀect law for action a is of the f orm ϕ → [ a ] ψ , wher e ϕ, ψ ∈ Fm l . The consequent ψ is th e eﬀe ct whic h alw a ys obtai ns when action a is exe- cuted in a state wh ere the an tece dent ϕ holds. If a is a nondeterministic action, then th e consequent ψ is typica lly a disjun ction. An example of an eﬀect la w is ¬ c oﬀe e → [ buy ] c oﬀe e , sa ying that in a situation wh ere the agen t has no coﬀee, after b uying, the agen t h as a coﬀee. If ψ is inconsisten t, then w e h av e a sp ecial kind of eﬀect law that w e call an inexe cutability law . F or example, w e could also ha v e ¬ token → [ buy ] ⊥ , expressing that buy cannot b e executed if the agen t h as n o tok en. The set of eﬀect la ws of a domain is denoted by E . I n our coﬀee machine scenario, we could hav e for example: E =  ¬ c oﬀe e → [ buy ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥  Finally , we also deﬁne exe cutability laws , whic h stipulate the con text where an action is guarante ed to b e executable. In PD L , the op erator h a i is used to exp ress executabilit y . h a i⊤ th us reads “the execution of a is p ossi- ble”. Deﬁnition 2.6 (Executability La w) An executabilit y la w for action a is of the form ϕ → h a i⊤ , wher e ϕ ∈ Fml . 3 An alternative to that is given by Castilho et al. [6], with laws b eing stated with the aid of an extra universal mo dality and lo cal consequen ce b eing thus considered. 9 F or instance, token → h buy i⊤ s a y s that buying can b e executed wheneve r the agent has a token. The set of all executabilit y la ws of a giv en domain is denoted b y X . In our scenario example w e w ould ha ve X = { token → h buy i⊤} . With ou r thr ee b asic types of la ws, we are able to deﬁ ne action theories: Deﬁnition 2.7 (Action Theory) Given a domain and any (p ossibly empty) sets of laws S , E , and X , T = S ∪ E ∪ X is an actio n theory . F or giv en action a , E a (resp. X a ) w ill denote th e set of only those eﬀect (resp. executabilit y) laws ab out a . T a = S ∪ E a ∪ X a is then the action theory for a . 4 F or the sak e of clarit y , we abstract here from the f rame and ramiﬁca- tion problems, and su pp ose the agen t’s theory a lready en tails all the rele- v an t frame axioms. W e could ha v e used any su itable solution to the frame problem, lik e e.g. the dep endence relati on [6], which is used in the w ork of Herzig e t al. [21], or a k in d of successor state axioms in a sligh tly m o diﬁed setting [11]. T o m ak e the presentat ion more clear to the r eader, h ere we do not b other with a solution to the frame problem and ju st assume all frame axioms ca n b e inferr ed from the theory . Actually w e can supp ose that all in tended frame axioms are auto matically reco v er ed and stated in th e the- ory , more sp eciﬁc, in the set of eﬀect la ws. 5 Hence the action theory of our example will b e: T =        c oﬀe e → hot , token → h buy i⊤ , ¬ c oﬀe e → [ buy ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot        (W e ha v e not stated the frame axiom ¬ token → [ buy ] ¬ token b ecause it can b e trivially deduced from the inexecutabilit y la w ¬ token → [ bu y ] ⊥ .) Figure 3 b elo w sho ws a PDL -mo del for the theory T . Giv en an action theory T , s ometimes it will b e useful to consider mo dels whose p ossible wo rlds are al l the p ossible w orlds allo wed by T : 4 Notice that for a 1 , a 2 ∈ A ct , a 1 6 = a 2 , the intuition is indeed that T a 1 and T a 2 o verla p only on S , i.e., the only laws th at are common to b oth T a 1 and T a 2 are th e laws ab out the structure of th e worl d. This req uirement is someho w related with th e logic being indep endently axiomatized (see ab ov e). 5 F rame axioms are a sp ecial type of eﬀect law, having t he form ℓ → [ a ] ℓ , for ℓ ∈ L it . 10 M : t , c , h ¬ t , c , h t , ¬ c , h t , ¬ c , ¬ h b b b Figure 3: A m o del for our coﬀee mac hine scenario: b , t , c , and h stand for, resp ectiv ely , buy , token , c oﬀe e , and hot . Deﬁnition 2.8 (Big Mo de l) L et T = S ∪ E ∪ X b e an action the ory. M big = h W big , R big i is the b ig mo del of T if and only if: • W big = val ( S ) ; and • R big = S a ∈ A c t R a s.t. R a = { ( w , w ′ ) : for al l ϕ → [ a ] ψ ∈ E a , if | = M w ϕ, then | = M w ′ ψ } . Figure 4 b elo w sho ws the big mo d el of T . M : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b Figure 4: The big mod el for the coﬀee mac hin e scenario. 2.2 Essen tial A toms An atom p is essential to a form ula ϕ if and only if p ∈ atm ( ϕ ′ ) f or ev ery ϕ ′ suc h that | = CPL ϕ ↔ ϕ ′ . F or instance, p 1 is essen tial to ¬ p 1 ∧ ( ¬ p 1 ∨ p 2 ). Give n ϕ , atm !( ϕ ) denotes the set of essen tial atoms of ϕ . (If ϕ is not con tingent, i.e., ϕ is a tautol ogy or a contradictio n, then atm !( ϕ ) = ∅ .) 11 Giv en ϕ a Boolean formula, ϕ ∗ is the set of all formulas ϕ ′ suc h that ϕ | = CPL ϕ ′ and atm ( ϕ ′ ) ⊆ atm !( ϕ ). F o r instance, p 1 ∨ p 2 / ∈ p 1 ∗ , as p 1 | = CPL p 1 ∨ p 2 but atm ( p 1 ∨ p 2 ) 6⊆ atm !( p 1 ). C learly , atm ( V ϕ ∗ ) = atm !( V ϕ ∗ ), moreo ver when ev er | = CPL ϕ ↔ ϕ ′ is the case, then atm !( ϕ ) = atm !( ϕ ′ ) and also ϕ ∗ = ϕ ′ ∗ . Theorem 2.1 (Least atom-set theorem [41 ]) Given ϕ a pr op ositional formula, | = CPL ϕ ↔ V ϕ ∗ , and for ev e ry ϕ ′ s.t. | = CPL ϕ ↔ ϕ ′ , atm ( ϕ ∗ ) ⊆ atm ( ϕ ′ ) . A pro of of this theorem is giv en b y Ma kinson [35] and we do not state it here. Essentiall y , the theorem establishes that for every form ula ϕ , there is a unique least set of elementa ry atoms suc h that ϕ ma y equiv alen tly b e expressed using only letters fr om that set. 6 Hence, Cn ( ϕ ) = Cn ( ϕ ∗ ). 2.3 Prime V aluations Giv en a v aluation v , v ′ ⊆ v is a subvaluation . Giv en a set of v aluations W , a subv aluation v ′ satisﬁes a prop ositional formula ϕ m o dulo W (noted v ′  W ϕ ) if and only if v  ϕ for all v ∈ W such that v ′ ⊆ v . W e sa y that a su b v aluation v e ssential ly satisﬁes ϕ (mo du lo W ), noted v  ! W ϕ , if and only if v  W ϕ and {| ℓ | : ℓ ∈ v } ⊆ atm !( ϕ ). If v  ! W ϕ , w e call v an essential subvaluation of ϕ (mo dulo W ). Deﬁnition 2.9 (Prime Subv aluat ion) L et ϕ b e a pr op ositional formula and W a set of valuations. A subvaluation v is a pr ime subv aluation of ϕ (mo dulo W) if and only i f v  ! W ϕ and ther e is no v ′ ⊆ v s.t. v ′  ! W ϕ . Our notion of prime sub v aluation is closely related to V eltman’s deﬁni- tion of b asis for a formula [54]. 7 A p rime subv aluation of a f orm ula ϕ is th us one of the w eak est states of truth in w hic h ϕ is true. Hence, prime subv aluations are j ust another wa y of seeing pr ime implicant s [43] of ϕ . By b ase ( ϕ, W ) we will den ote the set of all prime su b v aluations of ϕ mo dulo W . Theorem 2.2 L e t ϕ ∈ Fml and W b e a set of valuations. Then for al l w ∈ W, w  ϕ if and only if w  W v ∈ b ase ( ϕ, W ) V ℓ ∈ v ℓ . 6 The dual n otion, i.e., t hat of redundant atoms is also addressed in the literature [22], with similar purp oses. 7 The author is ind ebted t o A ndreas H erzig for p ointing this out. 12 Pro of: Right to left direction is straight forw ard. F or the left to righ t d i- rection, if w  ϕ , then w  ϕ ∗ . Let w ′ ⊆ w b e the least sub s et of w still satisfying ϕ ∗ . Clearly , w ′ is a prime su b v aluation of ϕ mo d ulo W , and then b ecause w  V ℓ ∈ w ′ ℓ , the result follo ws. 2.4 Closeness b et w een Mo dels When con tracting a formula from a mo d el, we will p erform a c hange in its structure. Because there can b e sev eral diﬀerent w a y s of mo difying a mo d el (not all of them minimal), w e need a notion of d istance b et w een mo dels to iden tify those that are closest to th e original one. As we are going to see in more depth in wh at follo ws, changing a mo del amoun ts to mo difying its p ossible w orlds or its accessibilit y rela- tion. Hence, the d istance b et w een t w o PDL -mo d els will dep end up on the distance b et w een their s ets of w orlds and accessibilit y r elations. These here will b e based on the symmetric diﬀer enc e b et w een sets, d eﬁ ned as X ˙ − Y = ( X \ Y ) ∪ ( Y \ X ). Deﬁnition 2.10 (Closeness betw een PDL -Models) L et M = h W , R i b e a mo del. Then M ′ = h W ′ , R ′ i is at least as close to M as M ′′ = h W ′′ , R ′′ i , note d M ′  M M ′′ , if and only if • either W ˙ − W ′ ⊆ W ˙ − W ′′ • or W ˙ − W ′ = W ˙ − W ′′ and R ˙ − R ′ ⊆ R ˙ − R ′′ Although simple, this notion of closeness is suﬃcien t for our pur p oses here, as w e will see in the sequel. Notice that other d istance notions could ha v e b een consider ed as we ll, lik e e.g. the c ar dinality of sym m etric diﬀer- ences. (See S ection 9 for a discu s sion on this.) 3 Seman tics of Action Theory Change When admitting the p ossibilit y of a la w Φ failing, one must ensure that Φ b ecomes inv alid, i.e., not true in at least one mo d el of the dynamic domain. Because there can b e lots of suc h mo dels, w e ma y hav e a set M of mo d els in whic h Φ is (p oten tiall y) v alid. Th us con tracting Φ amounts to making it no longer v alid in this set of mo d els. W hat are the op er ations that must b e carried out to ac hiev e that? Throwing mo dels out of M do es not w ork, since Φ will keep on b eing v alid in all mo dels of the remaining set. Thus one should add new mo dels to M . Wh ich mod els? W ell, mod els in which Φ is 13 not true. But not any of suc h mo dels: taking mo d els falsifying Φ that are to o d iﬀeren t fr om our original mo d els will certainly violate minimal c hange. Hence, w e sh all take some mo del M ∈ M as basis and manipu late it to get a new mo del M ′ in which Φ is not tru e. In dynamic logic, th e remov al of a law Φ from a mo del M = h W , R i means mo difyin g the p ossible worlds or the accessibilit y relation in M so that Φ b ecomes false. Suc h an op eration giv es as result a set M − Φ of m o dels eac h of wh ich is no longer a mo del of Φ . But if there are several candidates, which ones sh ould we choose? W e s hall tak e those mo dels that are minimal mo d iﬁ cations of the original M , i.e., those m inimal w .r.t.  M . Note that there can b e more than one M ′ that is minimal. Hence, b ecause adding ju st one of these new mo dels is en ough to in v alidate Φ , we tak e all p ossible combinatio ns M ∪ { M ′ } of exp an d ing our original set of mo d els by one of these minimal mo dels. Th e r esult will b e a set of sets of mo dels . In eac h set of mod els there will b e one M ′ falsifying Φ . 3.1 Mo del Con traction of Execut abilit y Laws T o contrac t an executabilit y la w ϕ → h a i⊤ f r om one mo del, one intuitiv ely r emoves arr ows lea v in g ϕ -wo rlds. In order to succeed in the op eration, we ha v e to guarantee that in the r esulting mo del th ere will b e at least one ϕ -w orld with no departing a -arro w. Deﬁnition 3.1 L et M = h W , R i . M ′ = h W ′ , R ′ i ∈ M − ϕ →h a i⊤ if and only if • W ′ = W • R ′ ⊆ R • If ( w , w ′ ) ∈ R \ R ′ , then | = M w ϕ • Ther e is w ∈ W ′ s.t. 6| = M ′ w ϕ → h a i⊤ Observe that M − ϕ →h a i⊤ 6 = ∅ if and only if ϕ is satisﬁable in W . Moreo v er, M ∈ M − ϕ →h a i⊤ if and only if 6| = M ϕ → h a i⊤ . T o get minimal change, we w an t su c h an op eration to b e minimal w.r.t. the original mo del: one should remo ve a minimum set of arrows su ﬃcien t to get the desired result. Deﬁnition 3.2 c ontr act ( M , ϕ → h a i⊤ ) = S min { M − ϕ →h a i⊤ ,  M } 14 And no w we deﬁne the set s of p ossible models resulting from th e con- traction of an executabilit y la w in a set of mo d els: Deﬁnition 3.3 L et M b e a set of mo dels, and ϕ → h a i⊤ an exe cutability law. Then M − ϕ →h a i⊤ = {M ′ : M ′ = M∪ { M ′ } , M ′ ∈ c ontr act ( M , ϕ → h a i⊤ ) , M ∈ M} In our running example, consider M = { M } , wh ere M is the mo del in Figure 4. When the agent disco v ers that ev en with a tok en she do es n ot manage to buy a coﬀee any more, she has to c hange her mo dels in order to admit (new) m o dels with state s wh ere token is the case but from w h ic h there is n o buy -transition at all. Because ha ving just one su c h world in eac h new mo d el is enough, taking those resulting mo dels whose accessibil- it y relatio ns are maximal guarantee s minimal c h an ge. Hence w e will hav e M − token →h buy i⊤ = {M ∪ { M ′ 1 } , M ∪ { M ′ 2 } , M ∪ { M ′ 3 }} , where eac h M ′ i is depicted in Figure 5. M ′ 1 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b M ′ 2 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b M ′ 3 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b Figure 5: Mo dels r esu lting fr om contrac ting token → h buy i⊤ in the mo del M of Figure 4. Clearly , if ϕ is not satisﬁed in M , i.e., | = M ¬ ϕ for all M ∈ M , then the con traction of ϕ → h a i⊤ do es not succeed. In this c ase, ¬ ϕ should b e con tracted fr om the set of mo d els (see fur th er in this section). 15 3.2 Mo del Con traction of Eﬀect La ws When the agen t discov ers that there ma y b e cases where after buying she gets n o h ot b eve rage, she must e.g. giv e up the b elief token → [ buy ] hot in her set of mo dels. Th is means that token ∧ h buy i¬ hot shall now b e admitted in at least one wo rld of some of her new mod els of b eliefs. Hence, to con tract an eﬀect law ϕ → [ a ] ψ from a giv en mo del, int uitiv ely we ha v e to add arr ows lea ving ϕ -w orlds to w orlds satisfying ¬ ψ . The challe nge in suc h an op eration is how to guarant ee min imal c hange. In our example, when con tracting token → [ buy ] hot in the mo del of Figure 4 we add arro ws from token -w orlds to ¬ hot -worlds. Because c oﬀe e → hot , and th en ¬ hot → ¬ c oﬀe e , this should also giv e h buy i¬ c oﬀe e in some token -w orld ( ¬ c oﬀe e is r elevant to ¬ hot , i.e., to h av e ¬ hot w e m ust ha v e ¬ c oﬀe e ). Th is means that if we allo w for h buy i¬ hot in some token -world, we also hav e to allo w for h buy i¬ c oﬀe e in th at same world. Hence, in our example one can add arro w s from token -wo rlds to ¬ hot ∧ ¬ c oﬀe e ∧ to ken -w orlds, as w ell as to ¬ hot ∧ ¬ c oﬀe e ∧ ¬ token (Figure 6). F or instance, one can add a buy -arro w from { token , ¬ c oﬀe e , ¬ hot } to one of these candidates (Figure 7). M : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b Figure 6: Candid ate wo rlds to r eceiv e arro ws from token -w orlds. Notice that adding the arr o w to { token , ¬ c oﬀe e , ¬ hot } itself w ould make us lose the eﬀect ¬ token , tr u e after ev ery execution of buy in the original mo del ( | = M token → [ buy ] ¬ token ). How d o we pr eserv e th is law wh ile allo wing for the new transition to a ¬ hot -wo rld? That is, ho w d o we get rid of the eﬀect hot without losing eﬀects that are not relev ant for th at? W e here dev elop an app r oac h for this issue. When addin g a new arro w lea ving a w orld w we in tuitiv ely wa n t to preserve as many eﬀe cts as we had b efore doing so. T o ac hiev e this, it is enough to pr eserve old eﬀects only in w (b ecause the remaining structure of 16 M : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b b b Figure 7: T w o candid ate n ew buy -arro ws to falsify token → [ buy ] hot in M . the mo del remains u nc hanged after adding the new arr o w). Of co urse, w e cannot p reserv e eﬀects that are in consistent with ¬ ψ (those will all b e lost). So, it suﬃ ces to preserv e only the eﬀect s that are consisten t with ¬ ψ . T o ac h iev e that w e m ust obs er ve w hat is true in w and in the target world w ′ : • What change s fr om w to w ′ ( w ′ \ w ) must b e what is obliged to d o so: either b ecause that is n ecessary to h a ving ¬ ψ in w ′ or b ecause that is necessary to ha vin g another eﬀect (indep enden t of ¬ ψ ) in w ′ that we w an t to preserve. • What do es not c hange from w to w ′ ( w ∩ w ′ ) should b e wh at is al- lo wed to do so: certain literals are nev er p reserv ed (lik e token in o ur example), then wh en p oin ting the arrow to a world wh ere it do es not c hange w.r.t. th e lea ving world ( ¬ hot ∧ ¬ c oﬀe e ∧ token in our example), w e lose eﬀects that held in w b efore adding the arr o w . This means that the only things allo w ed to c hange in the candidate target w orld m ust b e those that are forced to c hange, either b y some non-related la w o r b ecause of ha ving ¬ ψ mo dulo a set of states W . In other w ords, w e w an t the literals that c hange to b e at most those that are su ﬃcien t to get ¬ ψ mo dulo W , while pr eserving th e maximum of eﬀects. Every change ou tsid e that is not an in tended one. S imilarly , we wan t the literals that are preserved in the target world to b e at most those that are usu ally preserved in a give n set of mo dels. Ev ery pr eserv ation outside those ma y mak e us lose some la w. This lo oks like prime implican ts, and that is wh ere prim e subv aluations pla y their role: the w orlds to w hic h the n ew arro w will p oint are those whose diﬀerence w.r.t. the dep arting wo rld are literals that are relev ant and whose similarit y w.r.t. it are literals that we kn o w d o not c hange. 17 Deﬁnition 3.4 (Relev an t T arget W orlds) L et M = h W , R i b e a mo del, w, w ′ ∈ W, M a set of mo dels such that M ∈ M , and ϕ → [ a ] ψ an eﬀe ct law. Then w ′ is a relev ant target wo rld of w w.r.t. ϕ → [ a ] ψ for M in M if and only if • | = M w ϕ , 6| = M w ′ ψ • for al l ℓ ∈ w ′ \ w – either ther e i s v ∈ b ase ( ¬ ψ , W ) s.t. v ⊆ w ′ and ℓ ∈ v – or ther e is ψ ′ ∈ Fm l s.t. ther e is v ′ ∈ b ase ( ψ ′ , W ) s.t. v ′ ⊆ w ′ , ℓ ∈ v ′ , and f or every M i ∈ M , | = M i w [ a ] ψ ′ • for al l ℓ ∈ w ∩ w ′ – either ther e i s v ∈ b ase ( ¬ ψ , W ) s.t. v ⊆ w ′ and ℓ ∈ v – or ther e is M i ∈ M such that 6| = M i w [ a ] ¬ ℓ By R elT ar get ( w , ϕ → [ a ] ψ , M , M ) we denote the set of al l r elevant tar get worlds of w w.r.t. ϕ → [ a ] ψ for M in M . Note that we n eed the set of mod els M (and here we can sup p ose it con tains all mo dels of the theory w e w an t to c hange) b ecause preserving eﬀects dep ends on what other eﬀects hold in the other mo dels that inte rest us. W e need to tak e them in to account in the lo cal op eration of c hanging one mo del: 8 Deﬁnition 3.5 L et M = h W , R i , and M b e such that M ∈ M . Then M ′ = h W ′ , R ′ i ∈ M − ϕ → [ a ] ψ if and only if • W ′ = W • R ⊆ R ′ • If ( w , w ′ ) ∈ R ′ \ R, then w ′ ∈ R elT ar get ( w , ϕ → [ a ] ψ , M , M ) • Ther e is w ∈ W ′ s.t. 6| = M ′ w ϕ → [ a ] ψ 8 The reason we do not need M in the d eﬁnition of the lo cal (one mo del) contraction of executability law s M − ϕ →h a i⊤ is that when removing arro ws there is n o wa y of losing eﬀects, as every eﬀect law that held in the world from whic h an arro w has b een remov ed remains true in the same w orld in the resulting mo del. 18 Observe that M − ϕ → [ a ] ψ 6 = ∅ if and only if ϕ and ¬ ψ are b oth satisﬁable in W . Moreo v er, M ∈ M − ϕ → [ a ] ψ if and only if 6| = M ϕ → [ a ] ψ . Because ha ving just one w orld where the la w is no longer true in eac h mo del is enou gh , taking those resulting mo dels wh ose accessibilit y relations are minimal w .r .t. the original one guaran tees minimal change. Deﬁnition 3.6 c ontr act ( M , ϕ → [ a ] ψ ) = S min { M − ϕ → [ a ] ψ ,  M } No w we can deﬁn e the p ossible sets of mo dels resu lting f r om con tracting an eﬀect la w f rom a s et of mo d els: Deﬁnition 3.7 L et M b e a set of mo dels, and ϕ → [ a ] ψ an eﬀe ct law. Then M − ϕ → [ a ] ψ = {M ′ : M ′ = M ∪{ M ′ } , M ′ ∈ c ontr act ( M , ϕ → [ a ] ψ ) , M ∈ M} T aking agai n M = { M } , wh ere M is the mo d el in Figure 4, after con- tracting token → [ buy ] hot f rom M we get M − token → [ buy ] hot = {M ∪{ M ′ 1 } , M∪ { M ′ 2 } , M ∪ { M ′ 3 }} , w here all M ′ i s are as d epicted in Figure 8. M ′ 1 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b b M ′ 2 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b b M ′ 3 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b b Figure 8: Models resulting from con tracting token → [ buy ] hot in the mod el M of Figure 4. 19 In b oth cases wh ere ϕ is n ot s atisﬁable in M or ψ is v alid in M , of course our op erator do es not succeed in falsifying ϕ → [ a ] ψ (cf. end of Section 3.1 ). In tuitiv ely , pr ior to doing that w e hav e to c hange our set of p ossible states. This is what w e add ress in the n ext section. 3.3 Mo del Con traction of Static Laws When contract ing a static la w from a mo del, w e wa nt to admit the existence of at least one (new) p ossible state falsifying it. This means that in tuitiv ely w e should add new worlds to the original mod el. This is quite easy . A v ery delicate issue ho wev er is what to do with th e ac cessibilit y relation: should new arr o ws lea ve/ arriv e at the new wo rld? If n o arrow lea v es th e n ew add ed w orld, we may lose some executabilit y la w. If some arro w lea v es it, then w e ma y lose some eﬀect la w, th e same holding if w e add an arro w p ointing to the new w orld. On the other hand, if no arro w arriv es at th e n ew wo rld, what ab out th e intuition? Is it int uitiv e to h av e an u nreac hable state? All this discussion shows ho w drastic a c hange in the static la ws may b e: it is a change in the u nderlying structure (p ossible states) of the world! Changing it ma y hav e as consequence the loss of an eﬀect la w or an exe- cutabilit y law. What we can do is c ho ose whic h laws we accept to lose and p ostp one their change (by the other op erators). F ollo wing th e tradition in the RAA comm u n it y whic h states that executabilit y la ws are, in general, more diﬃcult to formalize than eﬀect la w s , and hence are more lik ely to b e incorr ect, here w e prefer not to c hange th e accessibilit y relation, whic h means preservin g eﬀect la ws an d p ostp onin g correction of executabilit y la ws, if needed. (cf. S ections 4.3 and 10 b elo w). Deﬁnition 3.8 L et M = h W , R i . M ′ = h W ′ , R ′ i ∈ M − ϕ if and only if • W ⊆ W ′ • R = R ′ • Ther e is w ∈ W ′ s.t. 6| = M ′ w ϕ Notice that we ha v e M − ϕ = ∅ if and only if ϕ is a tautolo gy . Moreo ver, M ∈ M − ϕ if and only if 6| = M ϕ . The minimal mo diﬁcations of one mo del are deﬁ n ed as us u al: Deﬁnition 3.9 c ontr act ( M , ϕ ) = S min { M − ϕ ,  M } 20 And now w e d eﬁne the s ets of m o dels resu lting from contrac ting a static la w from a giv en set of mo dels: Deﬁnition 3.10 L et M b e a set of mo dels, and ϕ a static law. Then M − ϕ = {M ′ : M ′ = M ∪ { M ′ } , M ′ ∈ c ontr act ( M , ϕ ) , M ∈ M} In our s cenario examp le, if M = { M } , where M is the m o del in Figure 4, then cont racting c oﬀe e → hot from M wo uld give us M − c oﬀee → hot = {M ∪ { M ′ 1 } , M ∪ { M ′ 2 }} , w here eac h M ′ i is as depicted in Figure 9. M ′ 1 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h t , c , ¬ h b b b M ′ 2 : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h ¬ t , c , ¬ h b b b Figure 9: Mo dels r esu lting from contrac ting c oﬀe e → hot in the mod el M of Figure 4. Notice that b y not mo d ifying the accessibilit y r elation all the eﬀect la w s are p reserv ed with minimal c hange. Moreo v er, our approac h is in line with in tuition: wh en learning that a new state is n o w p ossible, w e do n ot nec- essarily kno w all the b eha vior of the actions in the new added state. W e ma y exp ect some action la ws to hold in the new state (see Sectio n 10 for an alternativ e solution), but, with the information w e disp ose, not touc hing the accessibilit y r elation is the safest w ay of con tracting static la ws. 4 Syn tactic Op erators for Con traction of La ws No w that we ha v e deﬁned the semanti cs of our theory c h an ge, w e tur n our atte nt ion to th e deﬁnition of syn tactic op erators for changing sets of form ulas. As Neb el [40] sa ys, “[. . . ] ﬁ nite bases us u ally represen t [. . . ] la w s, and when w e are f orced to c hange the theory w e w ould like to sta y as close as p ossib le to the original [. . . ] b ase.” Hence, b esides the deﬁnition of syn tactical op erators, w e should also guaran tee that they p erform minimal c hange. 21 By T − Φ w e denote in the sequel the r esult of cont racting a la w Φ f rom the set of la ws T . 4.1 Con tracting Executabilit y La ws F or the case of contrac ting ϕ → h a i⊤ from an action theory , ﬁrst w e ha v e to ensure that the action a is still executable in all those con texts wh er e ¬ ϕ is the case. Second, in order to get minimalit y , we must make a executable in some cont exts where ϕ is true, viz. all ϕ -wo rlds bu t one. This means that w e can hav e sev eral action theories as outcome. Algorithm 1 giv es a syn tactical op erator to ac hiev e this. Algorithm 1 Erasure of an executabilit y la w input: T , ϕ → h a i⊤ output: T − ϕ →h a i⊤ 1: T − ϕ →h a i⊤ : = ∅ 2: if T | = PDL ϕ → h a i⊤ t hen 3: for all π ∈ IP ( S ∧ ϕ ) do 4: for all A ⊆ atm ( π ) do 5: ϕ A : = V p i ∈ atm ( π ) p i ∈ A p i ∧ V p i ∈ atm ( π ) p i / ∈ A ¬ p i 6: if S 6| = CPL ( π ∧ ϕ A ) → ⊥ t hen 7: T ′ : = ( T \ X a ) ∪ { ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } ! 8: T − ϕ →h a i⊤ : = T − ϕ →h a i⊤ ∪ {T ′ } 9: else 10: T − ϕ →h a i⊤ : = {T } return T − ϕ →h a i⊤ Observe that from the ﬁniteness o f T and that of atm ( π ), for an y π ∈ IP ( S ∧ ϕ ), and the decidabilit y of PDL [20] and of classical pr op ositional logic, it follo ws th at Algo rithm 1 termin ates. In our runn in g example, con tracting the executabilit y la w token → h buy i⊤ from th e action theory T w ould give us T − token →h buy i⊤ = {T ′ 1 , T ′ 2 , T ′ 3 } , 22 where: T ′ 1 =            c oﬀe e → hot , ¬ c oﬀe e → [ bu y ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot , ( token ∧ ¬ c oﬀe e ∧ hot ) → h buy i⊤ , ( token ∧ ¬ c oﬀe e ∧ ¬ hot ) → h buy i⊤            T ′ 2 =            c oﬀe e → hot , ¬ c oﬀe e → [ bu y ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot , ( token ∧ c oﬀe e ∧ hot ) → h buy i⊤ , ( token ∧ ¬ c oﬀe e ∧ ¬ hot ) → h buy i⊤            T ′ 3 =            c oﬀe e → hot , ¬ c oﬀe e → [ bu y ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot , ( token ∧ c oﬀe e ∧ hot ) → h buy i⊤ , ( token ∧ ¬ c oﬀe e ∧ hot ) → h buy i⊤            No w the kno wledge engineer has only to c ho ose whic h theory is more in line with her intuitions and implement the change s (cf. Figure 5). 4.2 Con tracting Eﬀect La ws When contrac ting ϕ → [ a ] ψ from a theory T , in tuitiv ely we should con tract some eﬀect la ws that p reclude ¬ ψ in ta rget w orlds. In order to cop e with minimalit y , w e must c hange only those laws that are relev ant to ϕ → [ a ] ψ . Let E ϕ,ψ a denote a minimum subset of E a suc h that S , E ϕ,ψ a | = PDL ϕ → [ a ] ψ . In the case the theory is mo du lar [25] (see furth er), such a set alwa ys exists. Moreo ver, note that th ere can b e more than one suc h a set, in whic h case w e denote them ( E ϕ,ψ a ) 1 , . . . , ( E ϕ,ψ a ) n . Let E − a = [ 1 ≤ i ≤ n ( E ϕ,ψ a ) i The laws in E − a will s er ve as guid elines to get rid of ϕ → [ a ] ψ in the theory . The ﬁrst thing w e must do is to ensu r e that acti on a still has eﬀect ψ in all those conte xts in which ϕ do es not hold. This m eans we sh all wea k en the laws in E ϕ,ψ a sp ecializing th em to ¬ ϕ . No w, w e need to preserve all old eﬀects in all ϕ -w orlds but one. T o ac hieve that w e sp ecialize the ab o v e 23 la w s to eac h p ossib le v aluation (maximal conjunction of literals) satisfying ϕ b u t one. Then, in th e left ϕ -v aluation, we must ensure that action a has either its old eﬀects or ¬ ψ as outcome. W e ac hieve that b y w eak enin g the c onse quent of the la ws in E − a . Finally , in order to get min imal change , we m ust ens u re that all literals in this ϕ -v aluation that are not forced to c hange in ¬ ψ -worlds sh ould b e p reserv ed. W e do this by stating an eﬀect law of the form ( ϕ k ∧ ℓ ) → [ a ]( ψ ∨ ℓ ), wh ere ϕ k is the ab o v e ϕ -v aluation. Th e r eason this is needed is clear: there can b e several ¬ ψ -v aluations, and as far as we w an t at most one to b e reac hable fr om the ϕ k -w orld, we should force it to b e the one wh ose d iﬀerence to this ϕ k -v aluation is min imal. Again, the result will b e a set of action theories. Algorithm 2 b elow giv es the op erator. The reader is in vited to c hec k that Algorithm 2 a lw a ys te rminates (cf. Section 4.1). F or an example of execution of the alg orithm, supp ose we wa n t to con- tract the eﬀect la w token → [ buy ] hot fr om our theory T . W e ﬁ rst determin e the min im um sets of eﬀect la ws that together with S en tail token → [ bu y ] hot . They are ( E token , hot buy ) 1 =  c oﬀe e → [ buy ] c oﬀe e , ¬ c oﬀe e → [ buy ] c oﬀe e  ( E token , hot buy ) 2 =  hot → [ buy ] hot , ¬ c oﬀe e → [ buy ] c oﬀe e  No w for eac h con text where token is the case, w e weak en the eﬀect la ws in E − buy = ( E token , hot buy ) 1 ∪ ( E token , hot buy ) 2 . Giv en S = { c oﬀe e → hot } , such con texts are token ∧ c oﬀe e ∧ hot , token ∧ ¬ c oﬀe e ∧ ¬ hot and token ∧ ¬ c oﬀe e ∧ hot . F or token ∧ c oﬀe e ∧ hot : w e replace in T the laws fr om E − buy b y    ( c oﬀe e ∧ ¬ ( token ∧ c oﬀe e ∧ hot )) → [ buy ] c oﬀe e , ( hot ∧ ¬ ( token ∧ c oﬀe e ∧ hot )) → [ buy ] hot , ( ¬ c oﬀe e ∧ ¬ ( token ∧ c oﬀe e ∧ hot )) → [ buy ] c oﬀe e    so that we preserve their eﬀects in all p ossible cont exts but token ∧ c oﬀe e ∧ hot . No w , in order to pr eserv e some eﬀects in token ∧ c oﬀe e ∧ hot -con texts while allo wing for r eac hable ¬ hot -worlds, w e add the la ws:  ( token ∧ c oﬀe e ∧ hot ) → [ buy ]( c oﬀe e ∨ ¬ hot ) , ( token ∧ c oﬀe e ∧ hot ) → [ buy ]( hot ∨ ¬ c oﬀe e )  24 Algorithm 2 Cont raction of an eﬀect la w input: T , ϕ → [ a ] ψ output: T − ϕ → [ a ] ψ 1: T − ϕ → [ a ] ψ : = ∅ 2: if T | = PDL ϕ → [ a ] ψ then 3: for all π ∈ IP ( S ∧ ϕ ) do 4: for all A ⊆ atm ( π ) do 5: ϕ A : = V p i ∈ atm ( π ) p i ∈ A p i ∧ V p i ∈ atm ( π ) p i / ∈ A ¬ p i 6: if S 6| = CPL ( π ∧ ϕ A ) → ⊥ t hen 7: for all π ′ ∈ IP ( S ∧ ¬ ψ ) do 8: T ′ : =    ( T \ E − a ) ∪ { ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → [ a ] ψ i : ϕ i → [ a ] ψ i ∈ E − a } ∪ { ( ϕ i ∧ π ∧ ϕ A ) → [ a ]( ψ i ∨ π ′ ) : ϕ i → [ a ] ψ i ∈ E − a }    9: for all L ⊆ Lit do 10: if S | = CPL ( π ∧ ϕ A ) → V ℓ ∈ L ℓ and S 6| = CPL ( π ′ ∧ V ℓ ∈ L ℓ ) → ⊥ then 11: for all ℓ ∈ L do 12: if T 6| = PDL ( π ∧ ϕ A ∧ ℓ ) → [ a ] ¬ ℓ or ℓ ∈ π ′ then 13: T ′ : = T ′ ∪ { ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) } 14: T − ϕ → [ a ] ψ : = T − ϕ → [ a ] ψ ∪ {T ′ } 15: else 16: T − ϕ → [ a ] ψ : = {T } return T − ϕ → [ a ] ψ 25 No w , we searc h all p ossible com bin ations of la ws from E buy that apply on token ∧ c oﬀe e ∧ hot con texts and ﬁ nd token → [ buy ] ¬ token . Because ¬ token m ust b e true after every execution of b u y , we d o not state the la w ( token ∧ c oﬀe e ∧ hot ) → [ buy ]( hot ∨ token ), and end up with the theory: T ′ 1 =                    c oﬀe e → hot , token → h buy i⊤ , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , ( c oﬀe e ∧ ¬ ( token ∧ c oﬀe e ∧ hot )) → [ bu y ] c oﬀe e , ( hot ∧ ¬ ( token ∧ c oﬀe e ∧ hot )) → [ buy ] hot , ( ¬ c oﬀe e ∧ ¬ ( token ∧ c oﬀe e ∧ hot )) → [ buy ] c oﬀe e , ( token ∧ c oﬀe e ∧ hot ) → [ buy ]( c oﬀe e ∨ ¬ hot ) , ( token ∧ c oﬀe e ∧ hot ) → [ buy ]( hot ∨ ¬ c oﬀe e )                    On the other hand , if in our language w e also had an atom p with the same theory T , then we shou ld add a la w ( token ∧ c oﬀe e ∧ hot ∧ p ) → [ buy ]( hot ∨ p ) to meet min imal change b y preserving eﬀects th at are n ot relev an t to ¬ ψ . The execution for con texts token ∧ ¬ c oﬀe e ∧ ¬ hot and token ∧ ¬ c oﬀe e ∧ hot are analogous and the algorithm ends with T − token → [ buy ] hot = {T ′ 1 , T ′ 2 , T ′ 3 } , where: T ′ 2 =                c oﬀe e → hot , token → h buy i⊤ , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , ( c oﬀe e ∧ ¬ ( token ∧ ¬ c oﬀe e ∧ ¬ hot )) → [ bu y ] c oﬀe e , ( hot ∧ ¬ ( token ∧ ¬ c oﬀe e ∧ ¬ hot )) → [ buy ] hot , ( ¬ c oﬀe e ∧ ¬ ( token ∧ ¬ c oﬀe e ∧ ¬ hot )) → [ buy ] c oﬀe e , ( token ∧ ¬ c oﬀe e ∧ ¬ hot ) → [ buy ]( c oﬀe e ∨ ¬ hot )                T ′ 3 =                    c oﬀe e → hot , token → h buy i⊤ , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , ( c oﬀe e ∧ ¬ ( token ∧ ¬ c oﬀe e ∧ hot )) → [ buy ] c oﬀe e , ( hot ∧ ¬ ( token ∧ ¬ c oﬀe e ∧ hot )) → [ buy ] hot , ( ¬ c oﬀe e ∧ ¬ ( token ∧ ¬ c oﬀe e ∧ hot )) → [ buy ] c oﬀe e , ( token ∧ ¬ c oﬀe e ∧ hot ) → [ buy ]( hot ∨ ¬ c oﬀe e ) , ( token ∧ ¬ c oﬀe e ∧ hot ) → [ buy ]( c oﬀe e ∨ ¬ hot )                    Lo oking at Figur e 8, we can see the corresp ondence b et w een th ese the- ories and their resp ectiv e mo dels. 26 4.3 Con tracting Static La ws Finally , in order to con tract a static la w from a theory , we can use an y con tractio n/erasure op er ator ⊖ for classical logic. Because con tracting static la w s means admitting new p ossible states (cf. the seman tics), ju st mo d ifying the set S of static laws ma y not b e enough for the dynamic logic case. Since w e in general do n ot necessarily kno w the b eha vior of the actions in a new disco v ered state of the w orld, a careful approac h is to c hange the th eory so that all ac tion la ws remain the s ame in the cont exts where the con tracted la w is the case. In our example, if when cont racting the law c oﬀe e → hot w e are not sur e wh ether b u y is still executable or n ot, th en we sh ould w eak en our executabilit y la ws sp ecializing them to the con text c oﬀe e → hot , and mak e buy a p riori inexecutable in all ¬ ( c oﬀe e → hot ) con texts. The op erator give n in Algorithm 3 form alizes this. Algorithm 3 Cont raction of a static law input: T , ϕ output: T − ϕ 1: T − ϕ : = ∅ 2: if S | = CPL ϕ the n 3: for all S − ∈ S ⊖ ϕ do 4: T ′ : =    (( T \ S ) ∪ S − ) \ X a ∪ { ( ϕ i ∧ ϕ ) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } ∪ {¬ ϕ → [ a ] ⊥}    5: T − ϕ : = T − ϕ ∪ {T ′ } 6: else 7: T − ϕ : = {T } return T − ϕ In our running example, con tracting the la w c oﬀe e → hot from T p ro- duces T − c oﬀee → hot = {T ′ 1 , T ′ 2 } , where 27 T ′ 1 =            ¬ ( ¬ token ∧ c oﬀe e ∧ ¬ hot ) , ( token ∧ c oﬀe e → hot ) → h buy i⊤ , ¬ c oﬀe e → [ buy ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot , ( c oﬀe e ∧ ¬ hot ) → [ buy ] ⊥            T ′ 2 =            ¬ ( token ∧ c oﬀe e ∧ ¬ hot ) , ( token ∧ c oﬀe e → hot ) → h buy i⊤ , ¬ c oﬀe e → [ buy ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot , ( c oﬀe e ∧ ¬ hot ) → [ buy ] ⊥            Observe that the eﬀect la ws are not aﬀected by th e c hange: as far as we do not pronounce ourselv es ab out the executabilit y of some action in th e new added world, all the eﬀect laws r emain tr ue in it. If the kno wledge engineer is not happ y with ( c oﬀe e ∧ ¬ hot ) → [ buy ] ⊥ , she can contract this formula from the theo ry using Al gorithm 2. Ideally , b esides stating th at buy is executable in the con text c oﬀe e ∧ ¬ hot , we should w an t to sp ecify its outcome in this conte xt as w ell. F or example, we could w an t ( c oﬀe e ∧ ¬ hot ) → h buy i hot to b e true in the result. This wo uld require theory r evision . S ee Section 7 for th e seman tics of suc h an op eration. 5 Correctness of the Op erators W e here address th e correctness of our algorithms w.r.t. our seman tics for con tractio n. 5.1 Tw o Coun t er-Examples Let the theory T = { p 1 → h a i⊤ , ( ¬ p 1 ∨ p 2 ) → [ a ] ⊥ , [ a ] ¬ p 2 } and consider its mo del M depicted in Figure 10. (Notice that T | = PDL ¬ ( p 1 ∧ p 2 ).) When con tracting p 1 → [ a ] ¬ p 2 in M , w e get M ′ in Figure 10. No w con tracting p 1 → [ a ] ¬ p 2 from T using Algorithm 2 give s T − p 1 → [ a ] ¬ p 2 = {T ′ } , where T ′ =    p 1 → h a i⊤ , ( ¬ p 1 ∨ p 2 ) → [ a ] ⊥ , ( p 1 ∧ ¬ p 2 ) → [ a ]( ¬ p 2 ∨ p 2 ) , ( p 1 ∧ ¬ p 2 ) → [ a ]( ¬ p 2 ∨ p 1 )    Notice that the formula ( p 1 ∧ ¬ p 2 ) → [ a ]( ¬ p 2 ∨ p 1 ) is pu t in T ′ b y Algorithm 2 b ecause there is { p 1 } ⊆ Lit suc h that S 6| = CPL ( p 1 ∧ p 2 ) → ⊥ and T 6| = PDL 28 M : p 1 , ¬ p 2 ¬ p 1 , ¬ p 2 ¬ p 1 , p 2 a a M ′ : p 1 , ¬ p 2 ¬ p 1 , ¬ p 2 ¬ p 1 , p 2 a a a Figure 10: A mod el M of T and the result M ′ of contract ing p 1 → [ a ] ¬ p 2 in it. ( p 1 ∧ ¬ p 2 ) → [ a ] ¬ p 1 . Clearly 6| = M ′ T ′ and n o theory in T − p 1 → [ a ] ¬ p 2 has M ′ as mo del. This means that the contrac tion op erators are not correct. This issu e arises b ecause Algorithm 2 tries to allo w an arro w from the p 1 ∧ ¬ p 2 -w orld to a p 2 -w orld that is closest to it, viz. { p 1 , p 2 } , but has n o wa y of knowing that suc h a world do es not exist. A remedy for that is replacing the test T 6⊢ PDL ( π ′ ∧ V ℓ ∈ L ℓ ) → ⊥ for S 6⊢ CPL ( π ′ ∧ V ℓ ∈ L ℓ ) → ⊥ , bu t that w ould increase ev en more the complexit y of the algorithm. A b etter option w ould b e to ha ve S ‘complete enough’ to allo w the algorithm to determine the wo rlds to wh ic h a new tr ansition could exist. The other wa y round , it do es not hold in general that the mo dels of eac h T ′ ∈ T − Φ result from the seman tic contrac tion of mo d els of T by Φ . T o see this supp ose that there is on ly one atom p and one actio n a , and consider the action theory T = { p → [ a ] ⊥ , h a i⊤} . The only mo del of T is M = h{{¬ p }} , { ( { ¬ p } , {¬ p } ) }i in Figur e 11. M : ¬ p a M ′ : ¬ p p a Figure 11: Incompleteness of con traction: a mo d el M of T and a mo del M ′ of the theory r esulting from con tracting p → h a i⊤ from T . By d eﬁnition, c ontr act ( M , p → h a i⊤ ) = { M } . On the other hand, T − p →h a i⊤ is the singleton {T ′ } su c h that T ′ = { p → [ a ] ⊥ , ¬ p → h a i⊤} . Then 29 M ′ = h{{¬ p } , { p }} , ( {¬ p } , {¬ p } ) i in Figure 11 is a mo del of th e con tracted theory . Clearly , M ′ do es not result from the seman tic co n traction of p → h a i⊤ from M : while ¬ p is v alid in the con traction of the mo dels of T , it is not v alid in the mo d els of T ′ . This means that the op erators are not complete. This problem occurs b ecause, in our exa mple, the w orlds that are for- bidden by T , e.g. { p } , are not preserved as such in T ′ . When con tracting an executabilit y or an eﬀect la w, w e are not supp osed to c hange the p ossib le w orlds of a th eory (cf. Section 3). F ortunately correctness of the algorithms w.r.t. our seman tics can b e guaran teed for th ose theories wh ose S is maximal, i.e. , th e set of static la ws in S alone determine what wo rlds are authorized in the mo dels of the theory . This is the pr inciple of mo dularity [25] and w e b rieﬂy review it in the next section. 5.2 Mo dular Theories Deﬁnition 5.1 (Mo dularity [25]) An action th e ory T is modu lar if and only if for eve ry ϕ ∈ Fml , if T | = PDL ϕ , then S | = CPL ϕ . F or an example of a n on-mo dular theory , sup p ose th at the action theory T of our coﬀee mac hine scenario were stated as T =        c oﬀe e → hot , h buy i⊤ , ¬ c oﬀe e → [ buy ] c oﬀe e , token → [ buy ] ¬ token , ¬ token → [ buy ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot        The mo d iﬁed law is u nderlined: w e hav e (in this case wr on gly) stated that the agen t can alw a ys buy at the mac hine. Th en T | = PDL token and S 6| = CPL token . As th e un derlying multimod al logic is indep endentl y axiomatized (see Section 2.1), we can use the algorithms giv en by Herzig and V arzinczak [25] to c hec k whether an action theory s atisﬁes the principle of mo dularit y . Whenev er this is not the case, th e algo rithms return the Boolean formu- las entai led b y th e theory that are not consequences of S alone. F or the theory T ab o v e, they w ou ld return { token } : as we stated h buy i⊤ , from th is and ¬ token → [ buy ] ⊥ w e get T | = PDL token . Because S 6| = CPL token , token is what is called an implicit static law [23] of T . 9 9 Implicit static law s are very closely related to veridical p arado xes [44]. It turns ou t 30 Mo dular theories hav e in teresting pr op erties. F or e xample, consistency can b e c hec ked by just c hec king consistency of the static la ws in S : if T is mo dular, then T | = PDL ⊥ if and only if S | = CPL ⊥ . Deduction of eﬀect la w s do es not need th e executabilit y ones and vice v ersa. Deduction of an eﬀect of a sequence of actions a 1 ; . . . ; a n (prediction) do es not need to tak e in to account the eﬀect la ws for ac tions other than a 1 , . . . , a n . Th is applies in particular to plan v alidation when deciding whether h a 1 ; . . . ; a n i ϕ is the case. Similar notions to mo dularity hav e b een inv estigated in th e literature on regulation consistency [7], Situation C alculus [2, 24], DL on tologies [8, 26] and also in dynamic logic [56]. F or more details on mo d ularit y in action theories, see th e work b y V arzinczak [51 ]. Theorem 5.1 T is mo dular if and only if the bi g mo del of T is a mo del of T . Pro of: Let M big = h W big , R big i b e the big mo d el of T . ( ⇒ ): By deﬁn ition, M big is suc h that | = M big S ∧ E . It remains to sh o w that | = M big X . Let ϕ i → h a i⊤ ∈ X a , and let w ∈ W big b e such that | = M big w ϕ i . T h ere- fore for all ϕ j ∈ Fm l suc h that T | = PDL ϕ j → [ a ] ⊥ , w e m ust h a ve 6| = M big w ϕ j , b ecause T | = PDL ¬ ( ϕ i ∧ ϕ j ), and as T is mo d ular, S | = CPL ¬ ( ϕ i ∧ ϕ j ), and h ence | = M big ¬ ( ϕ i ∧ ϕ j ). Then by the construction of M big , th ere is some w ′ ∈ W big suc h that | = M big w ′ ψ for all ϕ → [ a ] ψ ∈ E a suc h that | = M big w ϕ . Thus R a ( w ) 6 = ∅ and | = M big ϕ i → h a i⊤ . ( ⇐ ): Supp ose T is not mo dular. Then there must b e some ϕ ∈ Fml suc h that T | = PDL ϕ and S 6| = CPL ϕ . This means that there is v ∈ val ( S ) suc h that v 6  ϕ . As v ∈ W big (b ecause W big con tains all p ossible v aluations of S ), M big is not a mo d el of T . 5.3 Correctness Under Mo dularity The follo wing theorem establishes that the semant ic con traction of a formula Φ fr om the set of mo d els of an action th eory T pr o duces mo dels of some con tracted theory in T − Φ . Theorem 5.2 L e t T b e mo dular, and Φ b e a la w. F or a l l M ′ ∈ M − Φ such that | = M T for every M ∈ M , ther e is T ′ ∈ T − Φ such that | = M ′ T ′ for every M ′ ∈ M ′ . that they are not alw a ys intuitive. F or a deep d iscussion on implicit static la ws, see t he article by Herzig and V arzinczak [27]. 31 Pro of: See App endix A. The next theorem establishes the other w a y round: mo dels of th eories in T − Φ are all mo dels of the seman tic con tractio n of Φ from mo d els of T . Theorem 5.3 L e t T b e mo dular, Φ a law, and T ′ ∈ T − Φ . F or al l M ′ such that | = M ′ T ′ , ther e is M ′ ∈ M − Φ such that M ′ ∈ M ′ and | = M T for every M ∈ M . Pro of: See App endix B. With these tw o theorems one gets correctness of the op erators: Corollary 5.1 L e t T b e mo dular, Φ a law, and T ′ ∈ T − Φ . Then T ′ | = PDL Ψ if and o nly if | = M ′ Ψ for every M ′ ∈ M ′ such tha t M ′ ∈ M − Φ for so me M such that | = M T for al l M ∈ M . Pro of: ( ⇒ ): Let M ′ b e such that | = M ′ T ′ . By Theorem 5.3, there is M ′ ∈ M − Φ suc h that M ′ ∈ M ′ for some M suc h th at | = M T for all M ∈ M . F rom this and T ′ | = PDL Ψ , we h a v e | = M ′ Ψ . ( ⇐ ): Su pp ose T ′ 6| = PDL Ψ . (W e sho w that there is some mo del M ′ ∈ M ′ suc h that M ′ ∈ M − Φ for some M w ith | = M T for all M ∈ M , and 6| = M ′ Ψ .) Giv en that T is mo dular, by Lemma B.1 T ′ is mo dular, to o. Then , b y Lemma B.3, there is M ′ = h val ( S ′ ) , R ′ i suc h th at 6| = M ′ Ψ . Clearly | = M ′ T ′ , and from Lemma B.4 the result follo ws. 6 Assessmen t of P ostulate s for Change Do our action theory c h ange op erators satisfy the classical p ostulates for c hange? Before answ ering this q u estion, one should ask: do our op erators b ehav e lik e r evision or up date o p erators? W e here add ress this issue and then sho w which p ostulates for theory change are satisﬁed by our d eﬁnitions. 6.1 Con traction or Erasure? The distinction b etw een revision/con tractio n and up date/erasure for classi- cal th eories is historically contro ve rsial in th e literature. The same is true for the case of mo dal theories describing acti ons and th eir eﬀects. W e her e 32 rephrase Katsuno and Mendelzon’s deﬁnitions [31] in our terms so th at w e can see to wh ic h one our m etho d is closer. In Katsun o and Mendelzon’s view, con tracting a la w Φ from an action theory T in tu itiv ely means that the description of the p ossible b eha vior of the dynamic w orld T must be adjusted to the p ossibilit y of Φ b eing false . This amount s to selecting from the mo dels of ¬ Φ those that are closest to mo dels of T and allo w them as mo d els of the result. In co n trast, u p date metho d s sel ect, for eac h mo d el M of T , the set of mo dels of Φ that are closest to M . Erasing Φ f rom T means adding mo dels to T ; for eac h m o del M , w e add all those mo dels closest to M in whic h Φ is false. Hence, from our constructions so far it seems that our op erators are closer to u p date than to revision. Moreo ver, acco rding to Katsuno and Mendelzon’s view [31], our c hange op erators would also b e classiﬁed as u p date b ecause we make mo diﬁcations in eac h mo del indep end en tly , i.e., withou t c hanging other mo dels. 10 Besides that, in our setting a diﬀerent ordering on the resulting mo dels is in duced b y eac h mo del of T (see Deﬁnitions 3.3, 3.7 and 3.10), which acco rding to Katsuno and Mendelzon is a typica l pr op ert y of an up d ate/erasure metho d . Nev ertheless, things get qu ite diﬀeren t when it comes to the p ostulates for theory change . 6.2 The P ostulates W e here analyze the beh avior of our action theory c h ange op erators w.r.t. Katsuno and Mendelzon’s p ostulates and v ariants. Let T = S ∪ E ∪ X denote an action theory an d Φ d enote a law. Monotonicit y Pos tulate: T | = PDL T ′ , f or all T ′ ∈ T − Φ . This p ostulate is our v ersion of Katsuno and Mendelzon’s (C1) and (E1) p ostulates for contract ion and erasure, resp ectiv ely , and is satisﬁed b y our c hange op erators. The p ro of is in L emm a A.1. Suc h a p ostulate is not sat- isﬁed by the op erators p rop osed b y Herzig et al. [21]: there when removing e.g. an executabilit y la w ϕ → h a i⊤ one ma y make ϕ → [ a ] ⊥ v alid in all mo dels of the resu lting theory . Preserv ation P ostulate: If T 6| = PDL Φ , then | = PDL T ↔ T ′ , for all T ′ ∈ T − Φ . 10 Even if when contracting an eﬀect la w from one particular mo del we need to chec k the other mod els of the theory , those are not mo diﬁed. 33 This is K atsuno and Mendelzon’s (C2) p ostulate. Our op erators satisfy it as f ar as wheneve r T 6| = PDL Φ , then the models of the resulting theory are exactly the mo dels of T , b ecause th ese are the minimal mo d els falsifying Φ . The corresp onding v ersion of Katsuno and Mendelzon’s (E2) p ostulate ab out erasu r e, i.e., if T | = PDL ¬ Φ , th en | = PDL T ↔ T ′ , f or all T ′ ∈ T − Φ , is clearly also satisﬁed b y our op erators as a sp ecial case of the p ostulate ab o v e. Satisfaction of (C2) indicates th at our op erators are closer to con tractio n than to erasur e. Success P ostulate: If T 6| = PDL ⊥ and 6| = PDL Φ , then T ′ 6| = PDL Φ , f or all T ′ ∈ T − Φ . This p ostulate is our v ersion of Katsuno and Mendelzon’s (C3) and (E3) p ostulates. If Φ is a pr op ositional ϕ ∈ Fml , our op erators satisfy it, as long as the classical prop ositional change op erator satisﬁes it. F or the general case, ho w ev er, as stated th e p ostulate is not alw a ys satisﬁed. This is shown by the follo win g example: let T = {¬ p , h a i⊤ , p → [ a ] ⊥} . Note that T is mod ular and consistent. Now, con tracting the (co n tingen t) form ula p → h a i⊤ from T give s us T ′ = T . Clearly T ′ | = PDL p → h a i⊤ . This h ap p ens b ecause, despite not b eing a ta utology , p → h a i⊤ is a ‘trivial’ form ula w.r.t. T : s ince ¬ p is v alid in all T -mo d els, p → h a i⊤ is trivially true in these mo dels. F ortunately , for all those f ormulas that are non-trivial consequences of the theory , our op erators guarant ee success of con traction: Theorem 6.1 L e t T b e c onsistent, and Φ b e an exe cutability or an eﬀe ct law such that S 6| = PDL Φ . If T is mo dular, then T ′ 6| = PDL Φ for ev e ry T ′ ∈ T − Φ . Pro of: Supp ose there is T ′ ∈ T − Φ suc h that T ′ | = PDL Φ . As T is mo dular, Corollary 5.1 giv es u s | = M ′ Φ for ev ery M ′ ∈ M ′ suc h that M ′ ∈ M − Φ , where M = { M : | = M T and M = h val ( S ) , R i} . If | = M ′ Φ f or ev ery M ′ ∈ M ′ , then ev en for M ′′ ∈ M ′ \ M we h a v e | = M ′′ Φ . But M ′′ ∈ M − Φ for some M ∈ M , and by deﬁn ition 6| = M ′′ Φ . Hence M − Φ = ∅ , and th en the truth of Φ in M do es not dep en d on R a . Then, w h ether Φ has the form ϕ → h a i⊤ or ϕ → [ a ] ψ , for ϕ, ψ ∈ Fml , this h olds only if S | = CPL ¬ ϕ (see Deﬁnitions 3.1 and 3.5), in wh ic h case we get S | = PDL Φ . Equiv alences P ostulate: If | = PDL T 1 ↔ T 2 and | = PDL Φ 1 ↔ Φ 2 , then | = PDL T ′ 1 ↔ T ′ 2 , f or T ′ 1 ∈ ( T 1 ) − Φ 2 and T ′ 2 ∈ ( T 2 ) − Φ 1 . This p ostulate corresp onds to Katsun o and Mendelzon’s (C4) and (E4) p ostulates. Under m o dularit y and the assump tion that the prop ositional c hange op erator satisﬁes (C 4)/(E4), our op erations satisfy this p ostulate: 34 Theorem 6.2 L e t T 1 and T 2 b e mo dular. If | = PDL T 1 ↔ T 2 and | = PDL Φ 1 ↔ Φ 2 , then for e ach T ′ 1 ∈ ( T 1 ) − Φ 2 ther e is T ′ 2 ∈ ( T 2 ) − Φ 1 such that | = PDL T ′ 1 ↔ T ′ 2 , and vic e-ve rsa. Pro of: The pr o of follo ws straigh t from our results: since | = PDL T 1 ↔ T 2 and | = PDL Φ 1 ↔ Φ 2 , they ha v e pairwise the same mod els. Hence, giv en M such that | = M T 1 and | = M T 2 , the seman tic contrac tion of Φ 1 and that of Φ 2 from M ha v e the same op erations on M . As T 1 and T 2 are mo dular, Corollary 5.1 guaran tees w e get the same syntact ical results. Moreo v er , as the cl assical op erator ⊖ satisﬁes (C4)/( E4), if follo ws that | = PDL T ′ 1 ↔ T ′ 2 . Reco very P ostulate: T ′ ∪ { Φ } | = PDL T , for all T ′ ∈ T − Φ . This is the action theory counterpart of Katsun o and Mendelzon’s (C5) and (E5) p ostulates. Again w e rely on mo d u larit y in order to satisfy it. Theorem 6.3 L e t T b e mo dular. T ′ ∪ { Φ } | = PDL T , f or al l T ′ ∈ T − Φ . Pro of: If T 6| = PDL Φ , b ecause our op erators satisfy the preserv ation p ostulate, T ′ = T , and then the resu lt follo ws b y mon otonicit y . Let T | = PDL Φ , and let M ′ denote the set of all mod els of T ′ . As T is mo dular, b y Corollary 5.1 eve ry M ′ ∈ M ′ is suc h that either | = M ′ T (and then | = M ′ Φ ) or M ′ ∈ c ontr act ( M , Φ ) (and then M ′ ∈ M − Φ ) for some M suc h that | = M T . Let M ′′ denote the set of all models of T ′ ∪ { Φ } . Clearly M ′′ ⊆ M ′ , b y monotonicit y . Moreo v er, ev ery M ′′ ∈ M ′′ is such that | = M ′′ Φ , hence M ′′ / ∈ M − Φ for e v ery M such that | = M T , and then M ′′ / ∈ c ontr act ( M , Φ ), for an y M m o del of T . Thus M ′′ is a mo del of T a nd then T ′ ∪ { Φ } | = PDL T . Let W T − Φ denote the d isj unction of all T ′ in T − Φ . Disjunctiv e rule : ( T 1 ∨ T 2 ) − Φ is equiv alen t to W ( T 1 ) − Φ ∨ W ( T 2 ) − Φ . This is our version of (E8) erasure p ostulate b y K atsun o and Mendelzon. Clearly our s yn tactica l op erators do not man age to con tract a la w from a disjunction of theories T 1 ∨ T 2 . Nev ertheless, by pr o ving th at it holds in the semantics, from the c orrectness o f our op erators, w e get an equiv alent op eration. Again the fact that the th eories und er concern are mo du lar giv es us the r esu lt. 35 Theorem 6.4 L e t T 1 and T 2 b e mo dular, and Φ b e a law. Then | = PDL _ ( T 1 ∨ T 2 ) − Φ ↔ ( _ ( T 1 ) − Φ ∨ _ ( T 2 ) − Φ ) Pro of: ( ⇐ ): Let M ′ b e such that | = M ′ W ( T 1 ) − Φ ∨ W ( T 2 ) − Φ . T hen | = M ′ W ( T 1 ) − Φ or | = M ′ W ( T 2 ) − Φ . S upp ose | = M ′ W ( T 1 ) − Φ (the other case is analogous). Then there is ( T 1 ) ′ ∈ ( T 1 ) − Φ suc h that | = M ′ ( T 1 ) ′ . Then by Corollary 5.1, there is M ′ ∈ M − Φ suc h that M ′ ∈ M ′ , for M a set of mo dels of T 1 . Then M ′ is a mo del resulting from contract ing Φ from mo dels of T 1 , and then M ′ also results from con tracting Φ in mo dels of T 1 ∨ T 2 , viz. those mo d els of T 1 . Then b y Corollary 5.1, there is ( T 1 ∨ T 2 ) ′ ∈ ( T 1 ∨ T 2 ) − Φ suc h that | = M ′ ( T 1 ∨ T 2 ) ′ , and then | = M ′ W ( T 1 ∨ T 2 ) − Φ . ( ⇒ ): L et M ′ b e such that | = M ′ W ( T 1 ∨ T 2 ) − Φ . Then there is ( T 1 ∨ T 2 ) ′ ∈ ( T 1 ∨ T 2 ) − Φ suc h that | = M ′ ( T 1 ∨ T 2 ) ′ . By Corollary 5.1, there is M ′ ∈ M − Φ suc h that M ′ ∈ M ′ , f or M a set of mo d els of T 1 ∨ T 2 . Then M ′ is a mo del resulting from con tracting Φ from mo dels of T 1 ∨ T 2 . Hence M ′ results from con tracting Φ fr om mod els of T 1 or from models of T 2 . Su p p ose the former is the case (the second is analogous). T hen by C orollary 5.1 there is ( T 1 ) ′ ∈ ( T 1 ) − Φ suc h that | = M ′ ( T 1 ) ′ , and then | = M ′ W ( T 1 ) − Φ . W e ha v e th us shown that our constru ctions satisfy (E8) p ostulate. Nev- ertheless there is no evidence whether it is really exp ected here. This sup- p orts our p osition that our op erators’ b eha v ior is closer to con tractio n than to erasure. T o ﬁ nish up w e state a new p ostulate: Preserv ation of mo dularity: If T is mo dular, then ev ery T ′ ∈ T − Φ is mo dular. Changing a mo d ular theory should n ot mak e it nonmo dular. Th is is not a standard p ostulate, bu t we think that as a go o d pr op ert y mo d ularit y should b e p reserv ed across changing an action theory . If so, this m eans that wh ether a theory is m o dular or not can b e c hec k ed once for all and one do es not need to care ab out it d uring the future ev olution of the action theory , i.e., when o ther c h anges will b e made on it. O ur op erators sat isfy this p ostulate and the pro of is giv en in App endix B. 36 7 A Seman tics for A c tion Th eory Revision So f ar we h a v e analyzed the case of con traction: wh en evo lving a theory one r ealizes that it is too strong and hence it has to b e w eak ened. Let ’s no w take a lo ok at the other w a y round, i.e., the theory is too liberal and the agen t disco v ers new laws ab out the w orld that should b e added to her b eliefs, wh ic h amoun ts to strengthening them. Supp ose the actio n theory of our scenario example w ere in itially stated as follo ws: T =    c oﬀe e → hot , token → h buy i⊤ , ¬ c oﬀe e → [ buy ] c oﬀe e , ¬ token → [ bu y ] ⊥ , c oﬀe e → [ buy ] c oﬀe e , hot → [ buy ] hot    Then the big-mo del of T is as sho w n in Figure 12. M : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b b b b Figure 12: Mo del of the new initial actio n domain description. Lo oking at mo del M in Figure 12 we can see that, for example, the agen t does not know that she loses h er tok en ev ery time she b uys coﬀee at the mac hine. Th is is a new la w that she should incorp orate to her kno wledge base at some s tage of her action theory evol ution. Con trary to con traction, where w e wa n t the negation of some law to b ecome satisﬁable , in r evision w e w an t to mak e a new la w v alid . This means that one has to eliminate all cases satisfying its n egation. This dep icts the d ualit y b et ween revision and contract ion: wh er eas in the latter one in v alidates a formula b y m aking its n egation satisﬁable, in the former one mak es a formula v alid b y f orcing its negation to b e un satisﬁable pr ior to adding the new la w to the theory . The idea b ehind our seman tics is as follo ws: we initially hav e a set of mo dels M in whic h a gi v en form ula Φ is (p oten tially) not v alid, i.e., Φ is 37 (p ossibly) not true in eve ry mo del in M . In the result we wa n t to ha v e only mo dels of Φ . Adding Φ -mo dels to M is of no help. Moreo ver, adding mo dels makes us lose la ws: the corresp onding resu lting theory would b e more lib eral. One solution amoun ts to d eleting fr om M those mo dels that are n ot Φ - mo dels. Of cour se r emo v in g only some of them d o es not s olv e the problem, w e must d elete ev ery suc h a mo d el. By d oing that, all resulting m o dels will b e mo dels of Φ . (This corresp onds to the ory exp ansion , when the resulting theory is satisﬁable.) Ho wev er, if M c on tains no mod el of Φ , w e will end up w ith ∅ . C onsequence: th e resulting theory is inconsisten t. (Th is is the main revision p roblem.) In this case the solution is to su bstitute eac h mo del M in M b y its ne ar est mo diﬁc ation M ⋆ Φ that mak es Φ true. This lets us to kee p as close as p ossib le to the original mo dels we had. But, wh at if for one mo del in M there are sev eral min im al (incomparable) mo diﬁcations of it v alidating Φ ? In that ca se w e will consider all of them. The result will also b e a list of mo dels M ⋆ Φ , all b eing m o dels of Φ . Before deﬁ ning r evision of s ets of mo dels, w e pr esen t what mo d iﬁ cations of (individu al) mo d els are. 7.1 Revising a Mo del by a Static La w Supp ose that our coﬀee deliv erer agen t disco v ers that the only hot b ev erage that is served on the mac hine is coﬀee. In this case, she migh t wa n t to revise her b eliefs with the new static la w ¬ c oﬀe e → ¬ hot : sh e cannot hold a hot b eve rage that is n ot a coﬀee. Considering th e mo del depicted in Figure 12, one sees th at the f orm ula ¬ c oﬀe e ∧ hot is satisﬁable. As we do n ot wan t this, the ﬁ rst step is to r emove all w orlds in whic h ¬ c oﬀe e ∧ hot is true. The second step is to guarante e that all the remaining w orlds satisfy the n ew la w. S uc h an issue has b een largely add ressed in the literature on prop ositional b elief base revision and up d ate [15, 55 , 31, 22]. Here w e can ac hiev e that with a seman tics similar to that of classical revision op erators: basically one ca n c h ange the set of p ossible v aluations, b y r emo v in g or add ing worlds. In our example, remo vin g the p ossible w orld s { t , ¬ c , h } and {¬ t , ¬ c , h } w ould do th e job (there is no n eed to add new v aluations since the new incoming la w is s atisﬁed in at least one w orld of the resulting mo d el). The delicate p oin t in remo ving w orlds is that this ma y h a v e as conse- quence the loss of some executabilit y la w s: in th e example, if there w ere some arro w p oin ting fr om some world w to sa y {¬ t , ¬ c , h } , then removing the latter from the mod el w ould mak e the ac tion under concern n o longer 38 executable in w , if it wa s the on ly arro w lab eled by that actio n lea ving it. F rom a seman tic p oint of view, this is in tuitiv e: if the state of th e w orld to whic h we could mov e is n o longer p ossible, then w e d o not hav e a transition to that state anymore. Hence, if that transition was the only one we had, it is natural to lose it. Similarly , one could ask what to do w ith the accessibilit y relation if new w orlds are added , i.e., wh en e xpansion is not p ossible. F ollo w ing the discussion in S ection 3.3, w e here p refer not to systematically add new arr o w s to the accessibilit y relation, and p ostp one correction of executa bilit y la ws, if needed. This appr oac h is d ebatable, bu t with the information w e h a v e at hand, this is the safest wa y of c hanging static la ws. The seman tics for r evision of one mo d el by a stati c law is as follo ws: Deﬁnition 7.1 L et M = h W , R i . M ′ = h W ′ , R ′ i ∈ M ⋆ ϕ if and only if: • W ′ = ( W \ v al ( ¬ ϕ )) ∪ val ( ϕ ) • R ′ ⊆ R Clearly | = M ′ ϕ for eac h M ′ ∈ M ⋆ ϕ . The minimal mo dels resulting from revising a mo del M by ϕ are those closest to M w.r.t.  M : Deﬁnition 7.2 L et M b e a mo del and ϕ a static law. r evi se ( M , ϕ ) = S min { M ⋆ ϕ ,  M } . In the example of mo del M in Figure 12, r evise ( M , ¬ c oﬀe e → ¬ hot ) is the singleton { M ′ } , where M ′ is as shown in Figure 13. M ′ : t , c , h ¬ t , c , h ¬ t , ¬ c , ¬ h t , ¬ c , ¬ h b b b b Figure 13: Mo del resu lting fr om revising the mo del M in Figure 12 with ¬ c oﬀe e → ¬ hot . 39 7.2 Revising a Mo del by an Eﬀect La w Let’s supp ose n o w that our agen t ev entually disco v er s that after buying coﬀee she do es not ke ep her toke n. This means that her theory s h ould n ow b e revised by the new eﬀect la w token → [ buy ] ¬ token . Lo oking at mo del M in Figure 12, this amount s to guarantee ing that the formula token ∧ h buy i token is satisﬁable in n on e of its w orld s. T o do that, we h av e to lo ok at all the w orlds satisfying this form ula (if an y) and • either make token false in eac h of these worlds, • or make h buy i token f alse in all of them. If w e c hose the ﬁrst option, w e will essent ially ﬂip the truth v alue of literal token in the resp ectiv e w orlds, whic h changes the set of v aluations of the mo del. I f we c hose the latter, w e will basically remo v e buy -arro w s leading to token -w orlds . In that case, a c hange in the acce ssibilit y relation will b e mad e. In our example, we ha v e that the p ossib le w orlds { token , c oﬀe e , hot } , { token , ¬ c oﬀe e , hot } an d { token , ¬ c oﬀe e , ¬ hot } satisfy token ∧ h buy i token and all they hav e to change. Flipping token in all these w orlds to ¬ token would do the job, but would also ha v e as consequen ce the introd uction of a new static la w : ¬ token would no w b e v alid, i.e. , the agen t n ev er h as a tok en. Here we think that changing action la ws sh ould not h a v e as side eﬀect a c hange in the static la ws. Giv en their sp ecial status, th ese should change only if explicitly r equ ired (see ab o v e). In this case, eac h w orld satisfying token ∧ h buy i token h as to b e c hanged so that h buy i token is no longer true in it. In our example, w e should remo v e the arro ws ( { token , c oﬀe e , hot } , { token , c oﬀe e , hot } ), ( { token , ¬ c oﬀe e , hot } , { token , c oﬀe e , hot } ) and ( { token , ¬ c oﬀe e , ¬ hot } , { token , c oﬀe e , hot } ). The seman tics of one mod el r evision for the case of a new eﬀect la w is: Deﬁnition 7.3 L et M = h W , R i . M ′ = h W ′ , R ′ i ∈ M ⋆ ϕ → [ a ] ψ if and only i f: • W ′ = W • R ′ ⊆ R • If ( w , w ′ ) ∈ R \ R ′ , then | = M w ϕ • | = M ′ ϕ → [ a ] ψ 40 The minimal mo dels resulting from th e r evision of a mo del M by a new eﬀect la w are those that are closest to M w.r.t.  M : Deﬁnition 7.4 L et M b e a mo del and ϕ → [ a ] ψ an eﬀe ct law. r evise ( M , ϕ → [ a ] ψ ) = S min { M ⋆ ϕ → [ a ] ψ ,  M } . T aking again M as in Fig ure 12, r evise ( M , token → [ buy ] ¬ token ) will b e the sin gleton { M ′ } (Figure 14). M ′ : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b Figure 14: Mo del resu lting fr om revising the mo del M in Figure 12 with the new eﬀect la w token → [ buy ] ¬ token . 7.3 Revising a Mo del by an Executabilit y La w Let us now sup p ose that in some stage it has b een decided to gran t free coﬀee to ev eryb o dy . F ace d with this information, the agen t will no w revise her la w s to reﬂect the fact that buy can also b e executed in ¬ token -con texts: ¬ token → h bu y i⊤ is a new executabilit y la w (and h ence we will hav e h buy i⊤ in all n ew mod els of the agen t’s b eliefs). Considering again th e m o del in Figure 12 , we ob s erv e that ¬ ( ¬ token → h buy i⊤ ) is satisﬁable in M . Hence we must th ro w ¬ token ∧ [ buy ] ⊥ a w a y to ensure the new formula b ecomes true. T o remo v e ¬ token ∧ [ bu y ] ⊥ we ha v e to look at all worlds satisfying it and mo dify M so that they no lo nger satisfy that form ula. Giv en w orlds {¬ token , ¬ c oﬀe e , ¬ hot } and {¬ token , ¬ c oﬀe e , hot } , we ha v e t w o options: c hange the inte rpretation of token or add new arr o w s lea v in g these w orlds. A ques- tion that arises is ‘wh at c hoice is more drastic: c hange a w orld or an ar- ro w’ ? Again, here we thin k that changing the w orld’s con ten t (the v alua- tion) is more dr astic, as the existence of such a world w as f oreseen by some static la w and is hence assumed to b e as it is, unless w e ha v e enough in- formation supp orting the con trary , in whic h case w e explicitly c hange the 41 static la ws (see ab ov e). Th us w e shall add a new buy -arro w from eac h of {¬ token , ¬ c oﬀe e , ¬ hot } and {¬ token , ¬ c oﬀe e , hot } . Ha vin g agreed on that, the issue no w is: wh ic h wo rlds should the new arro ws p oin t to? Recalling the reasoning dev elop ed in S ection 3.2, in order to comply w ith minim al c hange, the new arro ws shall p oin t to w orlds th at are relev an t ta rgets of eac h of the ¬ token -w orlds in question. In our example, {¬ token , c oﬀe e , hot } is th e only relev an t target world h er e: the tw o other ¬ token -w orlds v iolate the eﬀect c oﬀe e of buy , wh ile the thr ee token -wo rlds w ould mak e us viola te the frame axiom ¬ token → [ bu y ] ¬ token . The seman tics for one model revision b y a new executabilit y la w is as follo ws : Deﬁnition 7.5 L et M = h W , R i . M ′ = h W ′ , R ′ i ∈ M ⋆ ϕ →h a i⊤ if and only if: • W ′ = W • R ⊆ R ′ • If ( w, w ′ ) ∈ R ′ \ R, then w ′ ∈ R elT ar get ( w , ϕ → [ a ] ⊥ , M , M ) • | = M ′ ϕ → h a i⊤ The minim al mo d els resulting from revising a m o del M by a n ew exe- cutabilit y la w are those close st to M w.r.t.  M : Deﬁnition 7.6 L et M b e a mo del and ϕ → h a i⊤ b e an exe cu tability law. r evise ( M , ϕ → h a i⊤ ) = S min { M ⋆ ϕ →h a i⊤ ,  M } . In our runnin g exa mple, r evise ( M , ¬ token → h buy i⊤ ) is th e singleton { M ′ } , where M ′ is as shown in Figure 15. 7.4 Revising Sets of Mo dels Up u n til now w e h a v e seen what the revision of single mo d els means. This is needed when expansion by the new la w is not p ossible d u e to inconsistency . W e h ere give a uniﬁ ed deﬁn ition of revision of a set of mod els M by a new la w Φ : Deﬁnition 7.7 L et M b e a set of mo dels and Φ a law. Then M ⋆ Φ = ( M \ { M : 6| = M Φ } , if ther e is M ∈ M s.t. | = M Φ S M ∈M r evise ( M , Φ ) , otherwise 42 M ′ : t , c , h ¬ t , c , h t , ¬ c , h ¬ t , ¬ c , ¬ h ¬ t , ¬ c , h t , ¬ c , ¬ h b b b b b b b b b Figure 15: The result of revising mod el M in Figure 12 b y the new exe- cutabilit y la w ¬ token → h buy i⊤ . Observe that Deﬁnition 7.7 comprises b oth exp ansion and r evi si on : in the ﬁrst one, simple addition of the new la w give s a satisﬁable theory; in the latter a deep er change is n eeded to get rid of inconsistency . 8 Related W ork T o the b est of our kn o w ledge, the ﬁrs t work on up dating an action domain description is that b y Li and P ereira [33] in a narrativ e-based action de- scription language [16 ]. Con trary to u s, h o w ev er, th ey mainly inv estigate the problem of up dating the narrative w ith n ew observ ed facts and (p ossi- bly) with o ccurrences of actions that explain those facts. This amoun ts to up d ating a given state/co nﬁguration of the world (in our terms, w hat is true in a p ossible world) and fo cus ing on the mo dels of the narr ativ e in wh ic h some actions to ok place (in our terms , the mo dels of the action theory w ith a p articular sequence of action executions). Clearly the mo dels of th e action la w s remain the same. Baral and Lob o [4] in tro duce extensions of action languages that allo w for some causal la ws to b e stated as defeasible. Their work is similar to ours in that they also allo w for w eak enin g of la ws: in their setting, eﬀect prop ositions can b e replaced by w hat they call defeasible (wea k ened versions of ) eﬀect prop ositions. Our appr oac h is d iﬀeren t from theirs in the wa y executabilit y la ws are d ealt with. Here exe cutabilit y la w s are explicit and w e are also able to con tract them. This feature is imp ortan t when the qualiﬁcation problem [37] is considered: we ma y alw a ys discov er con texts that preclude the executio n of a giv en action (cf. the Introd uction). Lib eratore [34] prop oses a framew ork for reasoning ab out actions in 43 whic h it is p ossible to express a giv en s eman tics of b elief u p date, lik e Winslett’s [55] and Katsuno and Mendelzon’s [31]. This means it is th e formalism, essen- tially an actio n description language, that is used to describ e up dates (the c hange of prop ositions from one state of th e world to another) by expressing them as la ws in th e actio n theory . The main diﬀerence b et w een Lib eratore’s work and Li and P ereira’s is that, despite not b eing concerned, at least a priori, w ith c hanging action la w s, Lib eratore’s fr amew ork allo ws for ab ductiv ely in tro ducing in the action theory n ew eﬀect prop ositions (eﬀect la ws, in our te rms) that consisten tly explain the o ccurrence of an ev en t. The wo rk by Eiter et al. [12 , 13] is s imilar to ours in that they also prop ose a f ramew ork that is orien ted to u p dating action la ws. Th ey m ainly in v estigat e the case wh ere e.g. a new eﬀect la w is add ed to the description (and th en has to b e true in all mo d els of the mo diﬁed theory). This problem is the du al of cont raction and is then closer to our deﬁn ition of r evision (cf. Section 7). In Eiter et al. ’s fr amework, action theories are describ ed in a v arian t of a narr ativ e-based action description language. Lik e in the presen t w ork, the semant ics is also in terms of tr an s ition systems: directed graphs ha v- ing arr o ws (action occur r ences) linkin g n o des (conﬁgur ations of the world). Con trary to us , ho w ev er, the minimalit y condition on the outcome of the up d ate is in terms of in clus ion of sets of la ws, wh ic h means the appr oac h is more syntax oriente d. In their setting, du ring an up date an action theory T is seen as comp osed of tw o pieces, T u and T m , where T u stands for the part of T that is n ot supp osed to change and T m con tains the la ws that may b e mo d iﬁed. In our terms, when con tr acting a static la w w e would ha v e T m = S ∪ X a , when con tracting an executabilit y T m = X a , and w hen con tracting eﬀects la ws T m = E − a . T h e diﬀerence here is that in our approac h it is alwa ys clear what la w s should not c hange in a giv en type of con tractio n, and T u and T m do not need to b e explicitly sp eciﬁed p rior to th e up date. Their approac h and ours can b oth b e describ ed as c onstr aint-b ase d up- date, in that the theory c h ange is carried out relativ e to some restrictions (a set of la ws that w e w an t to hold in the r esult). In our framew ork, f or example, all changes in the action la ws are relativ e to the set of static la ws S (and that is why we concen trate on mo dels of T h a ving val ( S ) as worlds). When c hanging a law, we w ant to keep the same set of state s. T he diﬀer- ence w.r.t. Eiter et al. ’s appr oac h is that there it is also p ossible to up d ate a theory relativ ely to e.g. executabilit y la ws: when expan d ing T with a new 44 eﬀect la w, one ma y w an t to constrain the c hange s o that the actio n under concern is guarantee d to b e executable in the result. 11 As shown in the referred w ork, this ma y r equire the w ithdra w al of some static law. Hence, in Eiter e t al. ’s framew ork, static la w s do not ha v e the same status as in ours. Herzig et al. [21] d eﬁne a metho d for action theory con tractio n that, despite the similarit y with the curr ent work and the common underlying motiv ations, is more limited than the p resen t constructions. First, with the referred appr oac h we do not get minimal change. F or example, in the referred w ork the op erator for cont racting executabilit y la w s is su c h that in the resulting theory the mo d iﬁed set of executabilitie s is giv en by X − a = { ( ϕ i ∧ ¬ ϕ ) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } whic h, acc ording to its seman tics, giv es th eories among whose models are those resulting from removing arr ows from al l ϕ -worlds. A s imilar commen t can b e made w.r .t. con traction of eﬀect la ws. Second, Herzig et al. ’s con traction met ho d do es not satisfy most of the p ostulates f or theory change that w e hav e addressed in Section 6. Besides not satisfying the monotonicit y p ostulate, it do es not satisfy the pr eserv ation one. T o witness, supp ose we ha v e a language with only one atom p , and th e mo del M depicted in Figure 16. M : p ¬ p a a a M ′ : p ¬ p a a a a Figure 16: Counter-exa mple to preserv ation in the metho d of contract ion b y Herzig et al. [21]. Then | = M p → [ a ] ¬ p and 6| = M [ a ] ¬ p . Now the cont raction op erator deﬁned there is su c h that when remo ving [ a ] ¬ p from M yields the mo del M ′ in Figure 16 suc h that R ′ a = W × W . Th en 6| = M ′ p → [ a ] ¬ p , i.e., the eﬀect la w p → [ a ] ¬ p is n ot preserv ed. 11 W e could simulate t hat in our approach with tw o successive mo diﬁcations of T : ﬁrst adding the eﬀect law and then an execut abilit y law (cf. S ection 7). 45 9 Commen ts In this section we mak e some comments ab out p ossible mo diﬁcations or impro v emen ts in our constructions so f ar. 9.1 Other Distance Notions Here we ha v e u sed a m o del distance b ased on sy m metric diﬀerences b etw een sets. T his distance is quite close to Winslett’s [55] notion of closeness b e- t w een in terp r etations in the P ossible Mo dels Approac h (PMA). Instead of it, how ev er, w e could ha v e considered other distance n otions as we ll, like e.g. Dalal’s [9] distance, Hamming distance [18], or weigh ted distance. Due to space limitations, we do not deve lop a throu gh comparison among all these distances here. W e nevertheless d o s ho w that with a cardinalit y-based distance, for example, w e ma y not alwa ys get the in tended r esult. Let c ar d ( X ) d en ote the n um b er of elemen ts in set X . Then supp ose our closeness b et w een PDL -mo dels w as deﬁn ed as follo ws: Deﬁnition 9.1 (Cardinality-based closeness b et w een PDL -Mo dels) L et M = h W , R i b e a mo del. Then M ′ = h W ′ , R ′ i is at least as close to M as M ′′ = h W ′′ , R ′′ i , note d M ′ ≤ M M ′′ , if and only if • either c ar d ( W ˙ − W ′ ) ≤ c ar d ( W ˙ − W ′′ ) • or c ar d ( W ˙ − W ′ ) = c ar d ( W ˙ − W ′′ ) and c ar d ( R ˙ − R ′ ) ≤ c ar d ( R ˙ − R ′′ ) Suc h a notion of distance is closely related to Dalal’s [9] closeness. Since when con tracting a static law ϕ from a mod el M we usu ally add one new p ossible w orld, it is easy to see th at with this cardinalit y-based distance we get the same resu lt in c ontr act ( M , ϕ ) as with th e d istance fr om Deﬁnition 2.10. When it comes to con traction of action laws, and then changing the ac- cessibilit y r elations, ho w ev er, this cardinalit y-based distance do es n ot seem to ﬁt with the intuitio ns. T o witness, c onsider the mo del M in Figure 17, whic h satisﬁes the executa bilit y la w p 1 → h a i⊤ . Then, M − p 1 →h a i⊤ = { M ′ , M ′′ } , where M ′ and M ′′ are as depicted in Figure 18. Note that M ′′ is an inte nded con tracted mo del. Ho wev er, with the cardinalit y -b ased distance ab o v e we will get { M } − p 1 →h a i⊤ = {{ M , M ′ }} . W e do not hav e { M , M ′′ } in the result since M ′ ≤ M M ′′ : in M ′ only one arro w has b een r emo v ed, while in M ′′ two . 46 M : p 1 , p 2 p 1 , ¬ p 2 ¬ p 1 , ¬ p 2 a a a Figure 17: A mo d el M satisfying p 1 → h a i⊤ . M ′ : p 1 , p 2 p 1 , ¬ p 2 ¬ p 1 , ¬ p 2 a a M ′′ : p 1 , p 2 p 1 , ¬ p 2 ¬ p 1 , ¬ p 2 a Figure 18: Mo dels resulting fr om con tracting p 1 → h a i⊤ in the mo del M of Figure 17. 9.2 Inducing Executabilit y Regarding the semantics for contrac ting static la ws, w e could tr y to go further and at least mak e a guess ab out w hat executabilit y la ws w e sh ould preserve . Before d oing that, w e need a deﬁnition. Deﬁnition 9.2 (Closeness betw e e n V aluations) L et v b e a pr op ositional valuation. The valuation v ′ is as close to v as v ′′ , note d v ′  v v ′′ , if and only if v ˙ − v ′ ⊆ v ˙ − v ′′ . So the distance b et w een v aluations v 1 and v 2 is th e set of literals on whic h they diﬀer: v 1 ˙ − v 2 = { ℓ : v 1  ℓ an d v 2 6  ℓ } ∪ { ℓ : v 2  ℓ an d v 1 6  ℓ } . Our argument now is as follo ws: when adding a n ew w orld, w e can lo ok at its con tents and see wh at happ en s in worlds that are similar to it (b y 47 similar here we mean the p ossible wo rlds that are closest to it). A priori and in tuitive ly w e ca n exp ect that if w e put a new arro w lea ving the new w orld, it will neither p oin t to a w orld that is the target of no other w orld, nor p oin t to a world that is not closest to it. It is reasonable to exp ect th at in the n ew w orld a give n action m a y ha v e a b eha vior that is quite similar to that wh ic h it has in th e w orlds that are closest to the n ew one. Hence w e select the worlds whose distance to the new one is minimal, lo ok at where the arrows lea vin g them p oint to, and then p oin t the n ew arro w th er e. With a similar argument, we can decide wh ic h arro ws targeting the new w orld add to the mo d el. The d eﬁnition b elo w formalizes this. Deﬁnition 9.3 L et M = h W , R i . M ′ = h W ′ , R ′ i ∈ M − ϕ if and only if • W ⊆ W ′ • R ⊆ R ′ • If w ′ ∈ W ′ \ W , then R ′ ( w ′ ) ⊆ R ( w ) and R ′− 1 ( w ′ ) ⊆ R − 1 ( w ) , wher e w ∈ S min { W ,  w ′ } • Ther e is w ∈ W ′ s.t. 6| = M ′ w ϕ With th is new deﬁnition, what w e do is su pp ose that some of the kno wn la w s for the other wo rlds can still b e tr ue in the new sta te, by analogy to the other p ossible s tates. In a similar wa y , when facing a new situation, we ma y w onder ho w w e got there. Again, by a nalogy with kno wn states, we could exp ect that we get to the new state coming from a state th at usu ally pro du ces something similar to wh at w e ha v e no w in f ron t of u s. In this case w e h a v e a kind of ab du ction-lik e reasoning that m a y of course b e w rong bu t that is n ot illegal. Although intuitiv e, at least in its motiv ation, adopting Deﬁnition 9.3 could h a v e some undesirable side eﬀect s. F or example, if in the semant ics w e decide to add new arro ws p ointi ng fr om and to the new added world, then our corresp ondin g op erator ma y not satisfy the monotonicit y p ostulate. T o see, let T =  p 1 ⊗ p 2 , p 1 → [ a ] p 2 , p 1 → h a i⊤ , p 2 → [ a ] ⊥  The only mo d el of T is M = h W , R i such that W = {{ p 1 , ¬ p 2 } , {¬ p 1 , p 2 }} and R = { ( { p 1 , ¬ p 2 } , {¬ p 1 , p 2 } ) } (Figure 19). 48 M : p 1 , ¬ p 2 ¬ p 1 , p 2 a M ′ : p 1 , ¬ p 2 ¬ p 1 , p 2 p 1 , p 2 a a a Figure 19: Counte r-example to monotonicit y wh en ad d ing arr o ws to/from new wo rlds in the semantic s of static la w con traction. M denotes the orig- inal mo del of T , while M ′ sho ws the new added w orld and the candidate arro ws to add to R a . If we contract p 1 → ¬ p 2 from T , in th e semant ic r esult w e ha v e only the mo del M ′ in Figure 19 su c h th at | = M ′ p 1 → h a ih a i p 2 . Then, we would hav e T ′ | = PDL p 1 → h a ih a i p 2 , and then T 6| = PDL T ′ . The very iss ue with suc h a seman tic c haracterization ho w ev er would b e ho w to capture it at the syntacti c lev el: what synta x op erator for c h ange should w e ha ve in ord er to capture this closeness b etw een p ossib le worlds? More imp ortan tly , since we m a y b e wrong ab out a guess regarding the exe- cutabilit y or an eﬀect of a giv en action, h o w can it b e rolled bac k in the n ew theory? These are op en questions that we lea v e for fu rther inv estigation. 10 Concluding Remarks In this w ork w e h a v e giv en a seman tics for action theory c hange in terms of distances b et w een mod els th at captures the notion of minim al c hange. W e ha v e giv en algorithms to co n tract a formula f r om a theory that terminate and are correct w.r.t. the s eman tics (Corollary 5.1 ). W e ha v e sho wn the imp ortance that m o dularit y has in this result and in others. Under m o dularit y , our op er ators satisfy all the p ostulates for contrac tion. This sup p orts the thesis that our mod ularit y notion is fr uitful. By forcing form ulas to b e explicitly stated in their resp ectiv e mo dules (and th us p ossibly m aking them in ferable in ind ep endently d iﬀerent wa ys), mo dularity in tuitiv ely could b e seen to d iminish elab oration tolerance [38]. 49 F or instance, when contrac ting a Bo olean form ula ϕ in a n on-mo dular the- ory , it seems reasonable to exp ect not to c h ange the set of static la ws S , while the theory b eing m o dular s u rely forces c hanging such a m o dule. It is not diﬃcult, ho w ev er, to concei v e non-mo dular theories in whic h con tractio n of a form ula ϕ ma y demand a change in S as w ell. As an example, su p p ose S = { ϕ 1 → ϕ 2 } in an action theory fr om whose dyn amic part w e (implicitly) infer ¬ ϕ 2 . In this case, con tracting ¬ ϕ 1 while k eeping ¬ ϕ 2 w ould necessarily ask for a change in S . W e p oint out n ev ertheless that in b oth cases (mo du lar and n on-mo dular) the extra wo rk in changing other mo du les sta y s in th e mec hanical lev el, i.e., in the algorithms that carry out the mo diﬁcation, and do es not augmen t in a signiﬁcan t wa y the amount of work the kno wledge engineer is exp ected to d o. Moreo ver, considering the evo lution of the theory , i.e., futu re mo diﬁcations one should p erform in it, mo dularit y has to b e c hec ked/ensured only once, since it is pr eserv ed by our op er ators (cf. L emm a B.1). While terminating, our algo rithms come with a considerable computa- tional cost: th e entai lmen t test in K n with global axioms is kno w n to b e psp ace -complete. Although this ma y b e acceptable (theory c h ange can b e carried out oﬄine), the computation of IP ( . ) might result in exp onentia l gro wth. W e hav e also extended V arzinczak’s stud ies [52] by d eﬁning a seman tics for action theory r evision based on minimal m o diﬁcations of mo dels. F or the corr esp onding revision algorithms, the r eader is referred to the w ork by V arzinczak [53]. One of our o ngoing researc h es is on assessing our revision op erators’ b eha vior w.r.t. the A GM p ostulates for r evision [1]. Another issue that d riv es our futu r e researc h on the s ub ject is ho w to con tract n ot only la ws bu t any PD L -form ula. As deﬁned, the ord er of appli- cation of our op erators matter in the ﬁ n al result: if we con tract ϕ and then ϕ → [ a ] ψ from a theory T , the result ma y not b e the same as contract ing ϕ → [ a ] ψ ﬁr st and then remo ving ϕ . This problem w ould not app ear in a more general fr amew ork in whic h an y formula could b e contrac ted: remo ving ϕ ∧ ( ϕ → [ a ] ψ ) should giv e the same result as ( ϕ → [ a ] ψ ) ∧ ϕ . Deﬁnitions 3.1, 3.5 and 3.8 app ear to b e imp ortan t for b etter under - standing the problem of co nt racting general formulas: basically the set of mo diﬁcations to p erform in a giv en mo del in order to force it to falsify a general form ula will compr ise r emov al/a ddition of arro ws /w orlds. The def- inition of a general r evision/con traction method will then b eneﬁt from our constructions. 50 Giv en the connection b et w een m ultimo dal logics and Description Log- ics [3], we b eliev e that the deﬁnitions here giv en ma y also con tribute to on tolog y ev olution and d ebugging in DLs. Ac kno wledgemen ts The author is grateful to Arin a Britz and Ken Halland for pro ofreadin g an earlier version of th is article. Th eir commen ts help ed v ery muc h in improv- ing the text. References [1] C. Alchourr´ on, P . G¨ ardenfors, and D. Makinson. On the logic of theory c hange: Partial meet con traction and revision functions. J . of Symb olic L o gic , 50: 510–53 0, 1985. [2] E. Amir. (De)comp osition of situation calculus theories. In P r o c. 17th Natl. Conf. on Artiﬁcial Intel ligenc e (AA A I’2000) , pages 456–46 3, Austin, 2000. AAAI Press/MIT Press. [3] F. Baader, D. Calv anese, D. McGuinness, D. Nardi, and P . Pat el- Sc hneider, editors. Description L o gic Handb o ok . Cam bridge Unive rsit y Press, 2003. [4] C. Baral and J. Lob o. Defeasible sp eciﬁcations in action theories. In M.E. P ollac k, editor, Pr o c. 15th Intl. Joint Conf. on Artiﬁcial Intel li- genc e (IJCAI’97) , pages 1441– 1446, Nago ya , 1997. Morgan Kaufmann Publishers. [5] G. Brewk a, S. Coradesc hi, A. P erini, and P . T ra verso, editors. Pr o c. 17th Eur. Conf. on A rtiﬁcial Intel lig e nc e (ECAI’06) , Riv a del Garda, 2006. IOS Press. [6] M. Castilho, O. Gasqu et, and A. Herzig. F ormalizing action and c hange in m o dal logic I: the frame problem. J. of L o gic and Computation , 9(5):7 01–735 , 199 9. [7] L. Cholvy . Ch ec kin g regulation consistency b y using SOL-resolution. In Pr o c. 7th Intl. Conf. on AI and L aw , p ages 73–79, Oslo, 1999. 51 [8] B. Cuenca Grau, B. P arsia, E. Sirin, and A. Kaly anpur. Mo d ularit y and web on tolog ies. In P . D ohert y , J. Mylop oulos, and C. W elty , edi- tors, P r o c. 10th Intl. Conf. on Know le dge R epr esentation and R e ason- ing (KR’2006) , pages 198–208 , Lak e District, 2006. Morgan K aufmann Publishers. [9] M. Dalal. In v estigat ions in to a theory of knowledge base revision: pre- liminary rep ort. In Sm ith and Mitc hell [48], p ages 475–479. [10] G. De Giacomo and M. Lenzerini. PDL-based framework for reasoning ab out actions. In M. Gori and G. So da, editors, Pr o c . 4th Congr esss of the Italian Asso ciation for A rtiﬁcial Intel lige nc e (IA*AI’95) , num b er 992 in LNAI, pages 103–114. S pringer-V erlag, 199 5. [11] R. Demolom b e, A. Herzig, and I. V arzinczak. Regression in mo dal logic. J. of A pplie d Non-Classic al L o gi cs (JANCL) , 13(2):16 5–185 , 200 3. [12] T. Eiter, E. Erdem, M. Fink, and J. Senko. Up dating action domain descriptions. In Kaelbling and Saﬃotti [29], pages 418–423. [13] T. Eiter, E. E r dem, M. Fink , and J. Senko. Resolving conﬂicts in action descriptions. In Brewk a et al. [5 ], pages 367–3 71. [14] A. F uh rmann. On the mo dal logic of theory c h ange. In Th e L o gic of The ory Change , pages 259–281, 1989. [15] P . G¨ ard enfors. Know le dge in Flux: M o deling the Dynamics of E pistemic States . MIT Press, Cambridge, MA, 1988. [16] M. Gelfond and V. Lifschitz. Representing action and change by logic programs. Journal of L o gic Pr o gr amming , 17(2/3&4) :301–32 1, 1993. [17] E. Giun chiglia , G. Kartha, and V. Lifsc hitz. Representing action: in- determinacy and ramiﬁcatio ns. A rtiﬁcial Intel ligenc e , 95(2):409–4 38, 1997. [18] R.W. Hamming. Er ror detecting and er r or correcting co d es. Bel l Sys- tem T e c hnic al Journal , 26(2):14 7–160 , 195 0. [19] S. Hanss on. A T extb o ok of Belief Dynamics: The ory Change and Datab ase Up dating . K luw er Academic Publishers, 1999. [20] D. Harel, J. Tiuryn, and D. Kozen. Dynamic L o gic . MIT Press, Cam- bridge, MA, 2000. 52 [21] A. Herzig, L. P errussel, and I . V arzinczak. Elab orating d omain descrip- tions. In Brewk a et al. [5 ], p ages 397–4 01. [22] A. Herzig and O. Riﬁ. Prop ositional b elief base up date and minimal c hange. Artiﬁcial Intel ligenc e , 115(1):107 –138, 1999. [23] A. Herzig and I. V arzinczak. Domain descriptions sh ould b e mo dular. In R. L´ op ez d e M´ an taras and L. S aitta, editors, P r o c. 16th Eur. Conf. on A rtiﬁcial Intel ligenc e (ECAI’04) , pages 348–352, V alencia, 2004. IOS Press. [24] A. Herzig and I. V arzinczak. Cohesion, coupling and the meta-theory of actions. In Kaelbling and Saﬃotti [29 ], p ages 442–4 47. [25] A. Herzig and I. V arzinczak. On the mo du larity of th eories. In R. Sc hmidt, I. P r att-Hartmann, M. Reynolds, and H. W ansing, edi- tors, A dvanc es in Mo dal L o gic , v olume 5, pages 93–109 . King’s College Publications, 2005. Selected pap ers of AiML 2004 (also a v ailable at http://w ww.aiml.n et/volumes/volume5 ). [26] A. Herzig and I. V arzinczak. A mo du larit y ap p roac h for a fragment of ALC . In M. Fisher, W. v an der Ho ek, B. Konev, and A. Lisitsa, editors, Pr o c. 10th Eur. Conf. on L o gics in Artiﬁcial Intel ligenc e (JELIA’2006) , n um b er 4160 in LNAI, pages 216–228. Springer-V erlag, 2006. [27] A. Herzig and I. V arzinczak. Metatheory of actions: b eyo nd consistency . Artiﬁcial Intel ligenc e , 171:951–98 4, 20 07. [28] Y. Jin and M. Th ielsc h er. Iterated b elief r evision, revised. In K aelbling and Saﬃotti [29], pages 478–483. [29] L. Kaelbling and A. Saﬃotti, editors. Pr o c. 19th Intl. Joint Conf. on Artiﬁcial Intel ligenc e (IJCAI’05) , Edinburgh, 2005. Morgan Kaufmann Publishers. [30] N. Kartha and V. Lifsc hitz. Actions with indirect eﬀects (preliminary rep ort). In J. Do yle, E. Sandew all, and P . T orasso, edito rs, Pr o c. 4th Intl. Conf. on Know le dge R epr esentation and R e asoning (KR’94) , pages 341–3 50, Bonn, 1994. Morgan Kauf mann P ublishers. [31] H. Katsuno and A. Mendelzon. On the diﬀerence b et ween up dating a kno wledge b ase and revising it. In P . G¨ ardenfors, editor, Belief r evision , pages 183–20 3. Cam bridge Unive rsit y P r ess, 1992. 53 [32] M. Krac ht and F. W olter. Prop erties of ind ep endently axiomatiza ble bimo dal logics. J. of Symb olic L o gic , 56(4 ):1469– 1485, 1991. [33] R. Li and L.M. P ereira. What is b eliev ed is wh at is explained. In H. Shr ob e an d T. Senator, editors, Pr o c . 13th Natl. Conf. on Ar ti- ﬁcial Intel lig enc e (AAA I’ 96) , pages 550 –555, Po rtland, 1996. AAAI Press/MIT Press. [34] P . Lib eratore. A fr amew ork for b elief up date. In P r o c. 7th Eur. Conf. on L o gics in Artiﬁcial Intel ligenc e (JELIA’2000) , pages 361–3 75, 2000. [35] D. Makinson. F riendliness and sympathy in logic. In J .-Y. Beziau, editor, L o gic a Universalis . Spr in ger-V erlag, 2nd edition, 2007. [36] P . Marquis. Consequen ce ﬁn ding algo rithms. In D. Gabbay and P h. Smets, editors, Algorithm s f or Defe asible and U nc ertain R e asoning, in S. Mor al, J. Kohlas (Eds), Handb o ok of Defe asible R e asoning and Un- c ertainty Management Systems , vol ume 5, c hapter 2, pages 41– 145. Klu w er Academic Pub lishers, 2000. [37] J. McCarthy . Ep istemologic al problems of artiﬁcial in tellig ence. In N. Sridh aran, editor, Pr o c. 5th Intl. Joint Conf. on Artiﬁcial Intel- ligenc e (IJCAI’77) , pages 1038 –1044, Cam bridge, MA, 1977 . Morga n Kaufmann Publish ers. [38] J. McCarth y . Elab oration tolerance. In P r o c. Common Sense’98 , London, 1998 . Av ailable at http://w ww- formal.st anford.edu/jmc/elaboration.html . [39] J. McCarthy an d P . Ha ye s. S ome ph ilosophical problems from the standp oint of artiﬁ cial intelli gence. In B. Meltzer and D. Mitchie, ed- itors, Machine Intel ligenc e , v olume 4, pages 463–502. Edin burgh Uni- v ersit y Press, 1969. [40] B. Neb el. A kn o wledge lev el analysis of b elief r evision. In R. Brac hman, H. Lev esqu e, and R. Reiter, editors, Pr o c . Intl. Conf. on Know le dge R epr esentation and R e asoning (KR’89) , pages 301–311, T oronto , 1989. Morgan Kaufmann Pub lishers. [41] R. Parikh. Beli efs, b elief revision, and splitting languages. In L. Moss, editor, L o g i c, L anguage and Computation , v olume 2 of CSLI L e ctur e Notes , pages 266–278. C SLI P ublications, 1999. 54 [42] S. Po pk orn. First Steps i n Mo dal L o gic . Cambridge Univ ersit y Press, 1994. [43] W. V. O. Quin e. Th e pr oblem of simplifying truth functions. Americ an Mathematic al Monthly , 59:521–531 , 1952. [44] W. V. O. Quin e. P aradox. Scientiﬁc Am eric an , pages 84–96, 196 2. [45] R. Reiter. The frame problem in the situation ca lculus: A simple so- lution (sometimes) and a completeness result for goal regression. In V. Lifs chitz, editor, Artiﬁcial Intel lige nc e and Mathematic al Th e ory o f Computation : Pap ers in H onor of John McCarthy , pages 359–38 0. Aca- demic Press, San Dieg o, 1991. [46] S. Shapiro, Y. Lesp´ er an ce, and H. Lev esque. Goal c hange. In Kaelbling and Saﬃotti [29], pages 582–588. [47] S. Shapiro, M. P agnucco , Y. Lesp´ eran ce, and H. Leve sque. Iterated b elief c hange in the situation calculus. In T. Cohn, F. Giunc higlia, and B. Selman, editors, Pr o c. 7th Intl. Conf. on K now le dge R e pr ese ntation and R e asoning (KR’2000) , pages 527–538 , Brec k enridge, 2000. Morgan Kaufmann Publish ers. [48] R. Smith and T. Mitc hell, editors. Pr o c. 7th Natl. Conf. on Artiﬁcial Intel ligenc e (AAAI’88) , St. Paul, 1988. Morgan Kaufman n Publish ers. [49] M. Th ielsc h er. Comp uting ramiﬁ cations by p ostpr o cessing. In C. Mel- lish, editor, Pr o c. 14th Intl. Joint Conf. on A rtiﬁcial Intel ligenc e (IJ- CAI’95) , pages 1994–2000 , Mon treal, 1995. Morgan Kaufmann Pub- lishers. [50] M. T h ielsc h er. Ramiﬁcation and causalit y . Artiﬁcial Intel ligenc e , 89(1– 2):317 –364, 1997. [51] I. V arzinczak. W hat i s a go o d domain description? E valuating and r evising action the ories in dyna mic lo g ic . PhD thesis, Un iversit ´ e P aul Sabatier, T oulouse, 200 6. [52] I. V arzincza k. Action theory cont raction and minimal c hange. In G. Brewk a and J. Lang, editors, Pr o c. 11th Intl. Conf. on Know l- e dge R epr esentation and R e asoning (KR’2008) , pages 651–661, Sydn ey , 2008. AAAI Press/MIT Press. 55 [53] I. V arzincza k. Action theory r evision in dynamic lo gic. In Workshop on Nonmono tonic R e asoning (NMR’08) , Sydn ey , 2008. [54] F. V eltman. Making counterfactual assu m ptions. Journal of Semantics , 22(2): 159–18 0, 20 05. [55] M.-A. Winslett. Reasoning ab out action using a p ossible mo d els ap- proac h. In Smith and Mitc hell [48], pages 89–93. [56] D. Zhang, S. Chopra, and N. F o o. Consistency of action descriptions. In M. Ishizuk a and A. Sattar, editors, Pr o c. 7th Paciﬁc Rim Intl. Conf. on Artiﬁcial Intel ligenc e : T r ends in Artiﬁcial Intel ligenc e , num b er 2417 in LNCS, pages 70–79. Springer-V erlag, 2002. [57] D. Z hang and N. F oo. EPDL: A logic for causal r easoning. In B. Neb el, editor, Pr o c. 17th Intl. Joint Conf. on A rtiﬁcial Intel ligenc e (IJCAI’01) , pages 131–138, S eattle, 2001. Morgan Kauf m ann Pu blish- ers. A Pro of of Theorem 5.2 L et T b e mo dular, and Φ b e a law. F or al l M ′ ∈ M − Φ such that | = M T for every M ∈ M , ther e is T ′ ∈ T − Φ such that | = M ′ T ′ for every M ′ ∈ M ′ . Lemma A.1 T | = PDL T ′ . Pro of: Let T b e an a ction theory , and T ′ ∈ T − Φ , for Φ a la w. W e analyze eac h case. Let Φ b e of the form ϕ → h a i⊤ , for some ϕ ∈ Fml . Then T ′ is suc h th at T ′ = ( T \ X a ) ∪ { ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } where π ∈ IP ( S ∧ ϕ ) and ϕ A = V p i ∈ atm ( π ) p i ∈ A p i ∧ V p i ∈ atm ( π ) p i / ∈ A ¬ p i , for some A ⊆ atm ( π ). Let M = h W , R i b e su c h that | = M T . It is enough to show th at M is a mo del of the new la ws. F or every ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → h a i⊤ , for eve ry w ∈ W , if | = M w ϕ i ∧ ¬ ( π ∧ ϕ A ), then | = M w ϕ i . Because T | = PDL ϕ i → h a i⊤ , | = M ϕ i → h a i⊤ , and then R a ( w ) 6 = ∅ . Hence | = M T ′ . 56 Let now Φ ha v e the form ϕ → [ a ] ψ , for ϕ, ψ ∈ Fml . Then T ′ is suc h that T ′ = ( T \ E − a ) ∪ { ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → [ a ] ψ i : ϕ i → [ a ] ψ i ∈ E − a } ∪ { ( ϕ i ∧ π ∧ ϕ A ) → [ a ]( ψ i ∨ π ′ ) : ϕ i → [ a ] ψ i ∈ E − a } ∪    ℓ ∈ L, for some L ⊆ Lit s.t. ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) : S 6⊢ ( π ′ ∧ V ℓ ∈ L ℓ ) → ⊥ , and ℓ ∈ π ′ or T 6⊢ PDL ( π ∧ ϕ A ∧ ℓ ) → [ a ] ¬ ℓ    where E − a = S 1 ≤ i ≤ n ( E ϕ,ψ a ) i , π ∈ IP ( S ∧ ϕ ), ϕ A = V p i ∈ atm ( π ) p i ∈ A p i ∧ V p i ∈ atm ( π ) p i / ∈ A ¬ p i , for some A ⊆ atm ( π ), and π ′ ∈ IP ( S ∧ ¬ ψ ). Let M = h W , R i b e su c h that | = M T . It is enough to show th at M is a mo del of the add ed la ws. Giv en ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → [ a ] ψ i , for ev ery w ∈ W , if | = M w ϕ i ∧ ¬ ( π ∧ ϕ A ), then | = M w ϕ i . Because T | = PDL ϕ i → [ a ] ψ i , | = M ϕ i → [ a ] ψ i , and then | = M w ′ ψ i for ev ery w ′ ∈ W suc h that ( w , w ′ ) ∈ R a . F or ( ϕ i ∧ π ∧ ϕ A ) → [ a ]( ψ i ∨ π ′ ), for ev ery w ∈ W , if | = M w ϕ i ∧ π ∧ ϕ A , then again | = M w ′ ψ i for eve ry w ′ ∈ W suc h that ( w , w ′ ) ∈ R a . No w , giv en ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ), for every w ∈ W , if | = M w π ∧ ϕ A ∧ ℓ , then | = M w π , and then | = M w ϕ . Since T | = PDL ϕ → [ a ] ψ , we ha v e | = M ϕ → [ a ] ψ , and then | = M w ′ ψ for ev ery w ′ ∈ W such that ( w , w ′ ) ∈ R a . Hence | = M T ′ . Let Φ b e a prop ositional ϕ . Then T ′ is such that T ′ = (( T \ S ) ∪ S − ) \ X a ∪ { ( ϕ i ∧ ϕ ) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } ∪ {¬ ϕ → [ a ] ⊥} for some S − ∈ S ⊖ ϕ . Let M = h W , R i b e su c h that | = M T . It su ﬃ ces to s ho w that M satisﬁes the added laws. Since we assu me ⊖ b eha ves lik e a classical contrac tion op erator, lik e e.g. Katsuno and Mendelzon’s [31], w e h a v e | = CPL S → S − , and then, b ecause | = M S , w e hav e | = M S − . 57 No w giv en ( ϕ i ∧ ϕ ) → h a i⊤ , for every w ∈ W , if | = M w ϕ i ∧ ϕ , then | = M w ϕ i , and b ecause | = M ϕ i → h a i⊤ , w e ha v e R a ( w ) 6 = ∅ . Finally , f or ¬ ϕ → [ a ] ⊥ , b ecause | = M ϕ , M trivially satisﬁes ¬ ϕ → [ a ] ⊥ . Hence, | = M T ′ . Pro of of Theorem 5.2 Let M = { M : | = M T } , and M ′ ∈ M − Φ . W e sho w that there is T ′ ∈ T − Φ suc h that | = M ′ T ′ for ev ery M ′ ∈ M ′ . By d eﬁnition, eac h M ′ ∈ M ′ is such that either | = M ′ T or 6| = M ′ Φ . Because T − Φ 6 = ∅ , th ere must b e T ′ ∈ T − Φ . If | = M ′ T , by Lemma A.1 | = M ′ T ′ and we are done. Let’s then su pp ose that 6| = M ′ Φ . W e analyze eac h case. Let Φ h a v e the form ϕ → h a i⊤ for some ϕ ∈ Fml . T hen M ′ = h W ′ , R ′ i , where W ′ = W , R ′ = R \ R ϕ a , with R ϕ a = { ( w , w ′ ) : | = M w ϕ and ( w , w ′ ) ∈ R a } , for some M ∈ M . Let u ∈ W ′ b e such that 6| = M ′ u ϕ → h a i⊤ , i.e., | = M ′ u ϕ and R ′ a ( u ) = ∅ . Because u  ϕ , there m ust b e v ∈ b ase ( ϕ, W ′ ) su c h that v ⊆ u . Let π = V ℓ ∈ v ℓ . Clearly π is a prime implicant of S ∧ ϕ . Let also ϕ A = V ℓ ∈ u \ v ℓ , and consider T ′ = ( T \ X a ) ∪ { ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } (Clearly , T ′ is a theory p ro duced by Algorithm 1.) It is enough to show that M ′ is a mo del of the new add ed la ws. Giv en ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → h a i⊤ ∈ T ′ , for eve ry w ∈ W ′ , if | = M ′ w ϕ i ∧ ¬ ( π ∧ ϕ A ), then | = M ′ w ϕ i , from wh at it follo ws | = M w ϕ i . Because | = M ϕ i → h a i⊤ , there is w ′ ∈ W suc h that w ′ ∈ R a ( w ). W e need to show that ( w, w ′ ) ∈ R ′ a . If 6| = M w ϕ , then R ϕ a = ∅ , an d ( w , w ′ ) ∈ R ′ a . If | = M w ϕ , either w = u , and then from | = M ′ u π ∧ ϕ A w e conclude | = M ′ u ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → h a i⊤ , or w 6 = u and then we must h a v e ( w, w ′ ) ∈ R ′ a , otherwise there is S ϕ a ⊂ R ϕ a suc h that R ˙ − ( R \ S ϕ a ) ⊂ R ˙ − ( R \ R ϕ a ), and then M ′′ = h W ′ , R \ S ϕ a i is su c h that 6| = M ′′ ϕ → h a i⊤ and M ′′  M M ′ , a con tradiction b ecause M ′ is m inimal w .r.t.  M . Th us ( w , w ′ ) ∈ R ′ a , and then | = M ′ w h a i⊤ . Hence | = M ′ T ′ . 58 No w let Φ b e of the form ϕ → [ a ] ψ , for ϕ, ψ b oth Boolean. Then M ′ = h W ′ , R ′ i , where W ′ = W , R ′ = R ∪ R ϕ, ¬ ψ a , with R ϕ, ¬ ψ a = { ( w , w ′ ) : w ′ ∈ R elT ar get ( w , ϕ → [ a ] ψ , M , M ) } for some M = h W , R i ∈ M . Let u ∈ W ′ b e such th at 6| = M ′ u ϕ → [ a ] ψ . Then ther e is u ′ ∈ W ′ suc h that ( u, u ′ ) ∈ R ′ a and 6| = M ′ u ′ ψ . Because u  ϕ , there is v ∈ b ase ( ϕ, W ′ ) suc h that v ⊆ u , and as u ′  ¬ ψ , there must b e v ′ ∈ b ase ( ¬ ψ, W ′ ) suc h that v ′ ⊆ u ′ . Let π = V ℓ ∈ v ℓ , ϕ A = V ℓ ∈ u \ v ℓ , and π ′ = V ℓ ∈ v ′ ℓ . Clearly π (resp. π ′ ) is a prime implican t of S ∧ ϕ (resp . S ∧ ¬ ψ ). No w let E − a = S 1 ≤ i ≤ n ( E ϕ,ψ a ) i and let the th eory T ′ = ( T \ E − a ) ∪ { ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → [ a ] ψ i : ϕ i → [ a ] ψ i ∈ E − a } ∪ { ( ϕ i ∧ π ∧ ϕ A ) → [ a ]( ψ i ∨ π ′ ) : ϕ i → [ a ] ψ i ∈ E − a } ∪    ℓ ∈ L, for some L ⊆ Lit s.t. ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) : S 6⊢ ( π ′ ∧ V ℓ ∈ L ℓ ) → ⊥ , and ℓ ∈ π ′ or T 6⊢ PDL ( π ∧ ϕ A ∧ ℓ ) → [ a ] ¬ ℓ    (Clearly , T ′ is a theory p ro duced by Algorithm 2.) In order to sho w that M ′ is a mo del of T ′ , it is enou gh to sho w that it is a model of th e added la ws. Giv en ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → [ a ] ψ i ∈ T ′ , for ev ery w ∈ W ′ , if | = M ′ w ϕ i ∧ ¬ ( π ∧ ϕ A ), then | = M ′ w ϕ i , and then | = M w ϕ i . Because | = M ϕ i → [ a ] ψ i , | = M w ′ ψ i for all w ′ ∈ W suc h that ( w, w ′ ) ∈ R a . W e need to sh ow that R ′ a ( w ) = R a ( w ). If 6| = M w ϕ , then R ϕ, ¬ ψ a = ∅ , and then R ′ a ( w ) = R a ( w ). If | = M w ϕ , then either w = u , and from | = M ′ u π ∧ ϕ A w e conclude | = M ′ u ( ϕ i ∧ ¬ ( π ∧ ϕ A )) → [ a ] ψ i , or w 6 = u , and then we must h a v e R ϕ, ¬ ψ a = ∅ , otherwise there w ould b e S ϕ, ¬ ψ a ⊂ R ϕ, ¬ ψ a suc h that R ˙ − ( R ∪ S ϕ, ¬ ψ a ) ⊂ R ˙ − ( R ∪ R ϕ, ¬ ψ a ), and then M ′′ = h W ′ , R ∪ S ϕ, ¬ ψ a i would b e such that 6| = M ′′ ϕ → [ a ] ψ and M ′′  M M ′ , a con tradiction since M ′ is minimal w.r.t.  M . Hence R ′ a ( w ) = R a ( w ), and | = M ′ w ′ ψ i for all w ′ suc h that ( w , w ′ ) ∈ R ′ a . No w , giv en ( ϕ i ∧ π ∧ ϕ A ) → [ a ]( ψ i ∨ π ′ ), for ev ery w ∈ W ′ , if | = M ′ w ϕ i ∧ π ∧ ϕ A , then | = M ′ w ϕ i , and then | = M w ϕ i . Because, | = M ϕ i → [ a ] ψ i , w e ha v e | = M w ′ ψ i for all 59 w ′ ∈ W suc h that ( w , w ′ ) ∈ R a , and then | = M ′ w ′ ψ i for ev ery w ′ ∈ W ′ suc h that ( w, w ′ ) ∈ R ′ a \ R ϕ, ¬ ψ a . No w, giv en ( w , w ′ ) ∈ R ϕ, ¬ ψ a , | = M ′ w ′ π ′ , and the result follo ws . No w , for eac h ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ), for ev ery w ∈ W ′ , if | = M ′ w π ∧ ϕ A ∧ ℓ , then | = M ′ w ϕ , and then | = M w ϕ . Because | = M ϕ → [ a ] ψ , we ha v e | = M w ′ ψ for every w ′ ∈ W suc h that ( w, w ′ ) ∈ R a , and then | = M ′ w ′ ψ for all w ′ ∈ W ′ suc h that ( w, w ′ ) ∈ R ′ a \ R ϕ, ¬ ψ a . It remains to sho w that | = M ′ w ′ ℓ for ev ery w ′ ∈ W ′ suc h that ( w, w ′ ) ∈ R ϕ, ¬ ψ a . Since M ′ is m inimal, it is enough to show that | = M ′ u ′ ℓ for ev ery ℓ ∈ Lit s u c h that | = M ′ u π ∧ ϕ A ∧ ℓ . If ℓ ∈ π ′ , the result follo ws. Otherwise, su pp ose 6| = M ′ u ′ ℓ . Then • either ¬ ℓ ∈ π ′ , then π ′ and ℓ are unsatisﬁable, and in this case Al- gorithm 2 has not pu t the la w ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) in T ′ , a con tradiction; • or ¬ ℓ ∈ u ′ \ v ′ . In this case, ther e is a v aluation u ′′ = ( u ′ \ {¬ ℓ } ) ∪ { ℓ } suc h that u ′′ 6  ψ . W e must ha v e u ′′ ∈ W ′ , otherwise there will b e L ′ = { ℓ i : ℓ i ∈ u ′′ } such that T | = PDL ( π ′ ∧ V ℓ i ∈ L ′ ℓ i ) → ⊥ , and , b ecause T is mo d ular, S | = CPL ( π ′ ∧ V ℓ i ∈ L ′ ℓ i ) → ⊥ , and then Algorithm 2 has not put the law ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) in T ′ , a contradict ion. T hen u ′′ ∈ W ′ , and moreo v er u ′′ / ∈ R ϕ, ¬ ψ a ( u ), otherwise M ′ is n ot minimal. As u ′′ \ u ⊂ u ′ \ u , the only reason why u ′′ / ∈ R ϕ, ¬ ψ a ( u ) is that there is ℓ ′ ∈ u ∩ u ′′ suc h that | = M i V ℓ j ∈ u ℓ j → [ a ] ¬ ℓ ′ for ev ery M i ∈ M if and only if ℓ ′ / ∈ v ′ for an y v ′ ∈ b ase ( ¬ ψ , W ′ ) suc h that v ′ ⊆ u ′′ . Clearly ℓ ′ = ℓ , and b ecause ℓ / ∈ π ′ , w e ha v e | = M i V ℓ j ∈ u ℓ j → [ a ] ¬ ℓ for ev ery M i ∈ M . Th en T | = PDL ( π ∧ ϕ A ∧ ℓ ) → [ a ] ¬ ℓ , and then Algorithm 2 has not p ut the la w ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) in T ′ , a con tradiction. Hence we ha v e | = M ′ w ′ ψ ∨ ℓ f or every w ′ ∈ W ′ suc h that ( w , w ′ ) ∈ R ′ a . Putting the ab o v e r esults together, we get | = M ′ T ′ . Let now Φ b e some prop ositional ϕ . Then M ′ = h W ′ , R ′ i , where W ⊆ W ′ , R ′ = R , is minimal w.r.t.  M , i.e., W ′ is a minim um sup erset of W such that there is u ∈ W ′ with u 6  ϕ . Because w e ha v e assumed the syn tacti cal classical con traction op erator is sound and complete w.r.t. its semant ics and is moreo v er minimal, then there must b e S − ∈ S ⊖ ϕ su c h that W ′ = val ( S − ). Hence | = M ′ S − . 60 Because R ′ = R , ev ery eﬀect la w of T remains tru e in M ′ . No w , let T ′ = (( T \ S ) ∪ S − ) \ X a ∪ { ( ϕ i ∧ ϕ ) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } ∪ {¬ ϕ → [ a ] ⊥} (Clearly , T ′ is a theory p ro duced by Algorithm 3.) F or ev ery ( ϕ i ∧ ϕ ) → h a i⊤ ∈ T ′ and ev ery w ∈ W ′ , if | = M ′ w ϕ i ∧ ϕ , then R a ( w ) 6 = ∅ , b ecause | = M w ϕ i → h a i⊤ . Giv en ¬ ϕ → [ a ] ⊥ , for ev ery w ∈ W ′ , if | = M ′ w ¬ ϕ , th en w = u , and R a ( w ) = ∅ . Putting all these results together, w e ha v e | = M ′ T ′ . B Pro of of Theorem 5.3 L et T b e mo dular, Φ a law, and T ′ ∈ T − Φ . F or al l M ′ such that | = M ′ T ′ , ther e is M ′ ∈ M − Φ such that M ′ ∈ M ′ and | = M T for every M ∈ M . Lemma B .1 L et Φ b e a law. If T is mo dular, then every T ′ ∈ T − Φ is mo du- lar. Pro of: Let Φ b e nonclassical, and s upp ose there is T ′ ∈ T − Φ suc h that T ′ is n ot mo du lar. Then there is some ϕ ′ ∈ Fm l suc h that T ′ | = PDL ϕ ′ and S ′ 6| = CPL ϕ ′ , wher e S ′ is the set of static la ws in T ′ . By Lemm a A.1 , T | = PDL T ′ , and then we hav e T | = PDL ϕ ′ . Because Φ is nonclassical, S ′ = S . T h us S 6| = CPL ϕ ′ , and hence T is not mo dular. Let no w Φ b e some ϕ ∈ Fml . Then T ′ = (( T \ S ) ∪ S − ) \ X a ∪ { ( ϕ i ∧ ϕ ) → h a i⊤ : ϕ i → h a i⊤ ∈ X a } ∪ {¬ ϕ → [ a ] ⊥} for some S − ∈ S ⊖ ϕ . Supp ose T is mo d u lar, and let ϕ ′ ∈ Fml b e s u c h that T ′ | = PDL ϕ ′ and S − 6| = CPL ϕ ′ . As S − 6| = CPL ϕ ′ , there is v ∈ v al ( S − ) such th at v 6  ϕ ′ . If v ∈ val ( S ), then S 6| = CPL ϕ ′ , and as T is mo d u lar, T 6| = PDL ϕ ′ . By Lemma A.1, T | = PDL T ′ , and w e 61 ha v e T ′ 6| = PDL ϕ ′ , a con tradiction. Hence v / ∈ val ( S ). Moreo ver, w e must h a v e v 6  ϕ , otherwise ⊖ h as not wo rk ed as exp ected. Let M = h W , R i b e s uc h that | = M T ′ . (W e extend M to another mod el of T ′ .) Let M ′ = h W ′ , R ′ i b e su ch that W ′ = W ∪ { v } an d R ′ = R . T o sho w that M ′ is a mo del of T ′ , it suﬃces to show that v satisﬁes every la w in T ′ . As v ∈ val ( S − ), | = M ′ v S − . Giv en ¬ ϕ → [ a ] ⊥ ∈ T ′ , as v 6  ϕ and R ′ a ( v ) = ∅ , | = M ′ v ¬ ϕ → [ a ] ⊥ . No w, for eve ry ϕ i → [ a ] ψ i ∈ T ′ , if | = M ′ v ϕ i , then w e trivially h av e | = M ′ v ′ ψ i for ev ery v ′ suc h that ( v , v ′ ) ∈ R ′ a . Finally , given ( ϕ i ∧ ϕ ) → h a i⊤ ∈ T ′ , as v 6  ϕ , the formula trivially holds in v . Hence | = M ′ T ′ , and b ecause there is v ∈ W ′ suc h that 6| = M ′ v ϕ ′ , we ha v e T ′ 6| = PDL ϕ ′ , a con tradiction. Hence for all ϕ ′ ∈ Fm l suc h that T ′ | = PDL ϕ ′ , S − | = CPL ϕ ′ , and then T ′ is mo dular. Lemma B .2 If M big = h W big , R big i is a mo del of T , then for e v ery M = h W , R i su ch that | = M T ther e is a minimal (w.r.t. set inclusion) extension R ′ ⊆ R big \ R such that M ′ = h val ( S ) , R ∪ R ′ i is a mo del of T . Pro of: Let M big = h W big , R big i b e a mod el of T , and let M = h W , R i b e suc h that | = M T . Consider M ′ = h val ( S ) , R i . If | = M ′ T , we ha v e R ′ = ∅ ⊆ R big \ R that is minimal. Supp ose then 6| = M ′ T . W e exte nd M ′ to a mo del of T that is a minimal extension of M . As 6| = M ′ T , there is v ∈ val ( S ) \ W suc h that 6| = M ′ v T . Then there is Φ ∈ T su c h that 6| = M ′ v Φ . If Φ is some ϕ ∈ Fm l , as v ∈ W big , M big is not a mo d el of T . If Φ is of the form ϕ → [ a ] ψ , for ϕ, ψ ∈ Fml , th ere is v ′ ∈ val ( S ) suc h that ( v , v ′ ) ∈ R a and v ′ 6  ψ , a con tradiction since R a ( v ) = ∅ . Let now Φ h a v e the form ϕ → h a i⊤ for some ϕ ∈ Fml . Th en | = M ′ v ϕ . As v ∈ W big , if 6| = M big v ϕ → h a i⊤ , then 6| = M big T . Hence, R big a ( v ) 6 = ∅ . Th us taking any ( v , v ′ ) ∈ R big a giv es us a minimal R ′ = { ( v , v ′ ) } suc h that M ′′ = h val ( S ) , R ∪ R ′ i is a mo del of T . Lemma B .3 L et T b e mo dular, and Φ b e a law. Then T | = PDL Φ if and only if every M ′ = h val ( S ) , R ′ i such that | = h W , R i T and R ⊆ R ′ is a mo del of Φ . Pro of: ( ⇒ ): Straigh tforw ard, as T | = PDL Φ implies | = M Φ for every M such that | = M T , in particular for th ose that are extensions of some mo del of T . 62 ( ⇐ ): Sup p ose T 6| = PDL Φ . Th en there is M = h W , R i suc h that | = M T and 6| = M Φ . As T is mo du lar, the big mo del M big = h W big , R big i of T is a m o del of T . Then b y Lemma B.2 there is a minimal extension R ′ of R w.r.t. R big suc h that M ′ = h val ( S ) , R ∪ R ′ i is a mo del of T . Bec ause 6| = M Φ , there is w ∈ W suc h that 6| = M w Φ . If Φ is s ome prop ositional ϕ ∈ Fml or an eﬀect la w, an y extension M ′ of M is such that 6| = M ′ w Φ . If Φ is of the form ϕ → h a i⊤ , then | = M w ϕ and R a ( w ) = ∅ . As any extension of M is su c h that ( u, v ) ∈ R ′ if and only if u ∈ val ( S ) \ W , only worlds other than those in W get a new lea ving arro w. Thus ( R ∪ R ′ ) a ( w ) = ∅ , and then 6| = M ′ w Φ . Lemma B .4 L et T b e mo dular, Φ a law, and T ′ ∈ T − Φ . If M ′ = h val ( S ′ ) , R ′ i is a mo del of T ′ , then ther e is M = { M : M = h val ( S ) , R i and | = M T } such that M ′ ∈ M ′ for some M ′ ∈ M − Φ . Pro of: Let M ′ = h val ( S ′ ) , R ′ i b e suc h that | = M ′ T ′ . If | = M ′ T , the result follo ws . Let’s su pp ose then 6| = M ′ T . W e analyze eac h case. Let Φ b e of the form ϕ → h a i⊤ , for some ϕ ∈ Fm l . Let M = { M : M = h val ( S ) , R i} . As T is modu lar, b y Lemmas B.2 and B.3, M is non-emp t y and con tains only mo dels of T . Supp ose M ′ is n ot a minimal mo del of T ′ , i.e., there is M ′′ suc h that M ′′  M M ′ for some M ∈ M . Then M ′ and M ′′ diﬀer only in the executabilit y of a in a give n ϕ -w orld , viz. a π ∧ ϕ A -con text, for some π ∈ IP ( S ∧ ϕ ) and ϕ A = V p i ∈ atm ( π ) p i ∈ A p i ∧ V p i ∈ atm ( π ) p i / ∈ A ¬ p i suc h that A ⊆ atm ( π ). Because 6| = M ′ ( π ∧ ϕ A ) → h a i⊤ , we m ust hav e | = M ′′ ( π ∧ ϕ A ) → h a i⊤ and then | = M ′′ T . Hence M ′ is minimal w.r.t.  M . When cont racting executabilit y la ws, S ′ = S . Hence taking the right R and a minimal R ϕ a suc h that M = h val ( S ) , R i and R ′ = R \ R ϕ a , for some R ϕ a ⊆ { ( w , w ′ ) : | = M w ϕ and ( w , w ′ ) ∈ R a } , w e constru ct M ′ = M ∪ { M ′ } ∈ M − ϕ →h a i⊤ . Let Φ b e of th e form ϕ → [ a ] ψ , for ϕ, ψ ∈ Fml . Let M = { M : M = h val ( S ) , R i} . As T is modu lar, b y Lemmas B.2 and B.3, M is non-emp t y and con tains only mo dels of T . W e claim that M ′ has only one arro w linking a ϕ -w orld, viz. a con text ϕ i ∧ π ∧ ϕ A for some π ∈ IP ( S ∧ ϕ ) and ϕ A = V p i ∈ atm ( π ) p i ∈ A p i ∧ V p i ∈ atm ( π ) p i / ∈ A ¬ p i , 63 suc h that A ⊆ atm ( π ), to a π ′ -w orld, where π ′ ∈ IP ( S ∧ ¬ ψ ). Th e p ro of is as follo ws: giv en ℓ ∈ Lit su c h th at ℓ holds in this ϕ i ∧ π ∧ ϕ A -w orld • if ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) / ∈ T ′ , then ℓ / ∈ π ′ and T | = PDL ( π ∧ ϕ A ∧ ℓ ) → [ a ] ¬ ℓ . Then this wo rld h as only ¬ ℓ -successors. • if ( π ∧ ϕ A ∧ ℓ ) → [ a ]( ψ ∨ ℓ ) ∈ T ′ , then every π ′ -successor is an ℓ -world. By su ccessiv ely applying this reasoning to eac h ℓ that holds in this ϕ i ∧ π ∧ ϕ A - w orld, we will end u p w ith only one π ′ -successor. Supp ose n ow that M ′ is not a minimal mod el of T ′ , i.e., there is M ′′ suc h that | = M ′′ T ′ and M ′′  M M ′ for some M ∈ M . Then M ′ and M ′′ diﬀer only in the eﬀects on that ϕ i ∧ π ∧ ϕ A -w orld: M ′′ has no arrow linking it to a π ′ -w orld. Then we hav e | = M ′′ ( ϕ i ∧ π ∧ ϕ A ) → [ a ] ψ i , and then | = M ′′ T . Hence M ′ is a minimal mo del of T ′ w.r.t.  M . When con tracting eﬀect la ws, S ′ = S . Thus taking the right R and a minimal R ϕ,ψ a suc h that M = h val ( S ) , R i and R ′ = R ∪ R ϕ,ψ a , for some R ϕ,ψ a ⊆ { ( w, w ′ ) : | = M w ϕ and w ′ ∈ R elT ar g e t ( w , ϕ → [ a ] ψ , M , M ) } , we con- struct M ′ = M ∪ { M ′ } ∈ M − ϕ → [ a ] ψ . Let n ow Φ b e ϕ for some ϕ ∈ Fml . Since T is mo d ular, by Lemmas B.2 and B.3 there is M = h val ( S ) , R i s u c h that | = M T . W e kn o w val ( S ) ⊆ val ( S − ). Because ¬ ϕ → [ a ] ⊥ ∈ T ′ , R ′ a ( v ) = ∅ for ev ery ¬ ϕ -w orld v added in M ′ . Hence, b ecause ⊖ is minimal, taking M = { M } giv es us the resu lt. Pro of of Theorem 5.3 F rom the h yp othesis that T is mo dular and Lemm a B.1, it follo ws that T ′ is mo dular, to o. Then M ′ = h val ( S ′ ) , R i is a mo d el of T ′ , by Lemma B.3. F rom this and Lemma B.4 the resu lt follo ws. 64

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