Labeled Natural Deduction Systems for a Family of Tense Logics

Labeled Natural Deduction Systems f or a F amily of T ense Logics (Extended V ersion) Luca V igan ` o Marco V olpe Department of Computer Science, Univ ersity of V erona, Italy { luca.vigano, marco.v olpe } @univr .it Abstract W e give labeled natural deduction systems for a family of tense logics e xtendin g the basic linear tense logic Kl . W e pr ove that our s ystems ar e sound and complete with r espect to the usual Kripk e semantics, and that the y possess a number of useful normalization p r op erties (in particular , derivations r educe to a n ormal form that en joys a subformula p r o perty). W e a lso discuss how to extend our systems to capture richer logics lik e (fragments of) L TL . 1 Intr oduction Hilbert-style s ystems, although unifo rm, are difﬁcult to u se in pr actice, especially in compar ison with the m ore “natu ral” Gentzen-style systems such as natural deduction (ND), sequent, and tableaux systems. Howe ver , devising Gentzen-style sys- tems for modal, relev ance, and other non- classical logics often requires considerable ingenuity , as well as trading uniformity for simplicity and usability . A solution to this problem is to employ labeling techniques, which provide a general framework for presenting different logics in a uniform way in terms of Gentzen-style systems. The intuition is that labelin g (also called pre ﬁxing, annotatin g or subscriptin g) allows one to explicitly encode a dditional informa tion, o f a semantic or proof -theoretical natur e, that is otherwise implicit in th e lo gic on e wants to ca pture. So, for instance, in stead of a mod al formula A , w e can consid er the labeled formula (lwff) x : A , which intuitively means that A holds at the world denoted by x within the unde rlying Kripke semantics. W e can also use labels to specify how worlds are related in a particular Kripke model, e.g. the r elation al formula (rwf f) x < y states that the world y is accessible from x . Labeled deduction systems have bee n giv en for sev eral non-classical lo gics, e. g. [1, 3, 6 , 7, 8, 1 1, 12, 13, 16, 19], an d research has focu sed no t only on the design of sy stems for sp eciﬁc logics, but also, more g enerally , on the ch aracterization of the classes of lo gics that can be fo rmalized this way . General pro perties and lim itations of labeling techniq ues have also been investigated. For example, [19] hig hlights an important trade- off between limitations and properties, which can be rough ly summarized as follows. Assume that we have a set of rules f or r easoning abou t the introd uction and elimination o f modal ope rators in lwffs x : A such as the following rules for  , wher e we expre ss x :  A as the m etalev el implication x < y = ⇒ y : A for an arbitrary y accessible f rom x ( y is fr esh , i.e. it is different fro m x and doe s not occur in any assumption on which y : A depends other than x < y ) : [ x < y ] . . . . y : A x :  A  I ( y fresh) x :  A x < y y : A  E . Assume also that we reason on the semantic inform ation provided by labeling using Horn-style r elational rules x 1 < y 1 . . . x n < y n x 0 < y 0 where the x i and y i are lab els, and n ≥ 0 (so that the ru le has no premises when n = 0 ). While restricting o ur systems to such Horn rules allo ws us to present only a s ubset of all possible non-classical logics, we can still capture se veral of the most common mo dal and relev ance logics, and , mor e importan tly , labe ling provides an efﬁcient general m ethod for establishing the metatheoretica l proper ties of these lo gics, including their completeness, decidability , and computational com plexity . This method relies on the separation between the sub -system for reaso ning about lwffs and the sub-system f or reasoning abo ut rwffs: der i vations of lwf fs c an d epend on derivations of rwffs (e.g. v ia the  ru les), but rwffs depen d only on rwffs (via the Horn rules). In this pap er , we giv e labeled natural ded uction systems for a family of tense logics extend ing the basic linear tense log ic Kl [ 15]. Our starting poin t is [19] but it shou ld be imm ediately clear that Horn rules do not sufﬁce: even a minimal tense logic like Kl req uires its time points to be con nected, i.e. for any two po ints x an d y eithe r x = y , or x is befor e y , o r y is bef ore x . It is straightfor ward to see that such a prop erty can not be ca ptured by a Horn rule like the o ne above; rather, we need non -atomic rwffs, in par ticular disjunction ( ⊔ ) of relatio ns, and mo re complex ru les built using a fu ll ﬁrst-order languag e, such as the axiom ∀ x.y . x < y ⊔ x = y ⊔ y < x c onn . A similar situation occurs if we wish to impose irreﬂexi vity of our worlds. And that’ s not all: as shown in [19] (in the case of modal logics, but the same arguments ap ply h ere, mutatis mu tandis), if we move to such a ﬁrst-orde r lang uage and wish to retain completen ess of the resulting systems, then we need to aband on the strict separation between the sub-system for lwf fs and that for r wffs (and let deriv ations of rwffs depen d also on lwffs). As we will see in m ore detail below , this is best achie ved by introdu cing a so-called universal falsum , so that a con tradiction in a world can be propagated not only to any other world but also to the re lational stru cture to derive any rwff; and, vice versa, from a co ntradiction in the relatio nal sub-system we can obtain any l wff. The main co ntributions, and the structure, of this pa per a re thus the following. I n Section 2, we give a brief presen tation of the syntax and seman tics, and of a standard axiomatiza tion, of Kl . In Section 3, we give a labeled natu ral deduction system N ( K l ) fo r Kl , which we show to be sound and complete (extending th e completeness proofs gi ven for modal lo gics in [ 19]). Then, in Section 4, we show that N ( K l ) possesses a n umber of useful normalizatio n prop erties; in particular, deriv ations reduce to a normal f orm that en joys a subfo rmula p roperty . In Sectio n 5, we extend N ( K l ) to capture some inter esting extensions of Kl , and in Section 6 we discu ss how to extend our systems to cap ture r icher lo gics like (fr agments of) L TL . W e conclu de, in Section 7, by co mparing with related work an d discussing future work . Detailed pro ofs an d examples ar e giv en in an appendix. 2 The basic linear tense logic Kl 2.1 Syn tax Deﬁnition 1 Given a set P of pr opositional variables, the set of well-f ormed Kl formulas is deﬁn ed by the following Backus-Naur -form pr esentation, wher e p ∈ P : A ::= p |⊥| A ⊃ A | G A | H A . T ruth of a tense formula is relati ve to a world in a model, so, intuitiv ely , G A ho lds at a world iff A always holds in the future, and H A holds at a world iff A always hold s in th e past. W e will formalize this standard seman tics below , but in or der to giv e a labeled ND system fo r Kl , we exten d the syn tax with labels and re lational sym bols that capture the worlds and the accessibility relation between them. Deﬁnition 2 Let L be a set of labels and let x and y b e labels in L . If A is a well-formed Kl formu la, then x : A is a labeled well-forme d formula (labeled formula or lwf f, for short). The set of well-fo rmed relational formulas (r elatio nal formulas or rwf fs, for short) is deﬁn ed as follows: ρ ::= x < y | x = y | ∅ | ρ ⊐ ρ | ∀ x. ρ . W e wr ite ϕ to denote a gen eric formu la (lwff or rwff). W e say th at an lwff x : A is atomic when A is atomic, i.e. A is a propo sitional v ariable or A is ⊥ . An rwff ρ is atomic wh en it do es not contain any con nectiv e or quantiﬁers, i.e. ρ is ∅ or ρ has the form x < y or x = y . The grade of an lwf f or rwff is the nu mber of o ccurrences of c onnectives ( ⊃ or ⊐ ), operato rs ( G or H ), a nd quantiﬁer s ( ∀ ) . Finally , given a set of lwffs Γ an d a set of rwffs ∆ , we call the or dered pair (Γ , ∆) a pr oof context . The giv en syntax uses a m inimal set of conn ecti ves, operator s, and qu antiﬁers. As usua l, we can intr oduce abbr e viation s and use, e .g., ∼ , ∧ , ∨ and ¬ , ⊓ , ⊔ , f or th e negation , the c onjunction , and th e disjunction in th e labeled lang uage an d in the 2 relational one, respec ti vely . For in stance, ∼ A ≡ A ⊃⊥ an d ρ ′ ⊔ ρ ′′ ≡ ( ρ ′ ⊐ ∅ ) ⊐ ρ ′′ . W e can also deﬁne ⊤ ≡ ∼⊥ , other quantiﬁers, e. g. ∃ x. ρ ≡ ¬ ∀ x. ¬ ρ , and other temp oral o perators, e. g. F A ≡ ∼ G ∼ A to express that A holds some time in the future. 2.2 Seman tics Deﬁnition 3 A Kl frame is a pair ( W , ≺ ) , wher e W is a non-emp ty set of wo rlds and ≺ ⊆ W × W is a binary r elation that satisﬁes the pr operties of irr e ﬂexivity , transitivity and connectedness, i.e . for all ( x, y ) ∈ W 2 we have x = y or ( x, y ) ∈≺ o r ( y , x ) ∈≺ . A Kl model is a triple ( W , ≺ , V ) , wher e ( W , ≺ ) is a Kl frame and the valuation V is a fu nction that maps an element o f W a nd a pr o positional variable to a truth value ( 0 or 1 ). In order to giv e a semantics for our labeled system, we need to deﬁne explicitly an interpretation of labels as worlds. Deﬁnition 4 Given a set o f la bels L and a model M = ( W , ≺ , V ) , an inter pretation is a function λ : L → W that maps every label in L to a world in W . Given a model M and an interpretation λ o n it, truth for an rwf f or lwff ϕ is the smallest r elation | = M ,λ satisfying: | = M ,λ x < y iff ( λ ( x ) , λ ( y )) ∈≺ ; | = M ,λ x = y iff λ ( x ) = λ ( y ) ; | = M ,λ ρ 1 ⊐ ρ 2 iff | = M ,λ ρ 1 implies | = M ,λ ρ 2 ; | = M ,λ ∀ x. ρ iff for all y , | = M ,λ ρ [ y /x ] ; | = M ,λ x : p iff V ( λ ( x ) , p ) = 1 ; | = M ,λ x : A ⊃ B i ff | = M ,λ x : A implie s | = M ,λ x : B ; | = M ,λ x : G A iff for all y , | = M ,λ x < y implies | = M ,λ y : A ; | = M ,λ x : H A iff for all y , | = M ,λ y < x implies | = M ,λ y : A . Hence, 2 M ,λ x : ⊥ and 2 M ,λ ∅ . When | = M ,λ ϕ , we say th at ϕ is true in M acco r ding to the interpretation λ . By extension: | = M ,λ Γ iff | = M ,λ x : A for all x : A ∈ Γ ; | = M ,λ ∆ iff | = M ,λ ρ for all ρ ∈ ∆ ; | = M ,λ (Γ , ∆) iff | = M ,λ Γ and | = M ,λ ∆ ; Γ , ∆ | = M ,λ ϕ iff | = M ,λ (Γ , ∆) implies | = M ,λ ϕ . T ruth for lwf fs and rwffs b uilt using other connectives or operators can be deﬁned in the usual manner . 1 2.3 An axiomatization of Kl Sev eral different Hilbert-style axiomatiza tions hav e been giv en for the logic Kl ; the following one is tak en from [15]: ( G1 ) G ( A ⊃ B ) ⊃ ( G A ⊃ G B ) ( G2 ) ∼ H ∼ G A ⊃ A ( G3 ) G A ⊃ GG A ( G4 ) [ G ( A ∨ B ) ∧ G ( A ∨ G B ) ∧ G ( G A ∨ B )] ⊃ ( G A ∨ G B ) ( Ne c G ) If ⊢ A then ⊢ G A ( Ne c H ) If ⊢ A then ⊢ H A ( MP ) If ⊢ A and ⊢ A ⊃ B then ⊢ B The axiom ( G1 ) is stand ard for modal and temporal logics, while ( G2 ) sets the dual relation between G and H , ( G3 ) expresses the transitivity and ( G4 ) the connected ness of G . F or brevity , we have omitted the sym metric axiom s ( H1 ) - ( H4 ) that are ob tained by replacin g e very G by H and vice versa. Moreover , e very classical tautolog y is a tautolo gy , and ther e are rules for modus ponens and necessitation for both G an d H . 1 Note that truth for lwf fs is related to the standard truth relation for modal logics by observing that | = M x : A if f | = M x A . 3 [ x : A ⊃⊥ ] . . . . y : ⊥ x : A RAA ⊥ [ x : A ] . . . . x : B x : A ⊃ B ⊃ I x : A ⊃ B x : A x : B ⊃ E [ x < y ] . . . . y : A x : G A G I ∗ x : G A x < y y : A G E [ y < x ] . . . . y : A x : H A H I ∗ x : H A y < x y : A H E [ ρ ⊐ ∅ ] . . . . ∅ ρ RAA ∅ [ ρ 1 ] . . . . ρ 2 ρ 1 ⊐ ρ 2 ⊐ I ρ 1 ⊐ ρ 2 ρ 1 ρ 2 ⊐ E ρ ∀ x. ρ ∀ I ∗ ∀ x. ρ ρ [ y /x ] ∀ E ∀ x. x = x r e ﬂ = ∀ x. ¬ ( x < x ) irr eﬂ < ∀ x.y .z . ( x < y ⊓ y < z ) ⊐ x < z tr ans < ∀ x.y . x < y ⊔ x = y ⊔ y < x c onn ϕ x = y ϕ [ y /x ] mon x : ⊥ ∅ uf 1 ∅ x : ⊥ uf 2 *In G I (respecti vel y , H I ), y is diffe rent from x and does not occur in any assumption on which y : A depe nds other than the discarded assumption x < y (respect iv ely , y < x ). In ∀ I , the v ariabl e x must not occur in any open assumption on which ρ depend s. Figure 1. The rules of N ( K l ) 3 A labeled natural deduction system for Kl Our labeled ND system N ( K l ) = N ( K l L ) + N ( K l R ) + N ( K l G ) comprises of t hree sub-systems, whose ru les ar e gi ven in Figure 1. The pr opositional a nd tempora l rules o f N ( K l L ) allow u s to de riv e lwffs from othe r lwf fs with the help of r wffs. The rules ⊃ I and ⊃ E ar e just the labeled version o f the standard ([14, 17]) ND rules for implication introduction and elimination, where the no tion of dischar ged/open assumption is also standar d (e.g. the for mula [ x : A ] is discha rged in the rule ⊃ I ). The rule R AA ⊥ is a labe led version o f reductio ad absurdum , where we do not enforce Prawitz’ s side conditio n that A 6 = ⊥ . 2 The tem poral op erators G and H share the structure of the basic introdu ction/elimination rules, with respect to the same accessibility relation < ; this ho lds becau se, for instance, we express x : G A as the m etalev el im plication x < y = ⇒ y : A for an arbitrary y accessible from x (as we did for  in the introdu ction). The r elational rules of N ( K l R ) allow us to derive rwffs from o ther rwffs only . Th e rules R AA ∅ , ⊐ I , and ⊐ E are reductio ad absurdum and implication introd uction and eliminatio n for rwffs, while ∀ I and ∀ E ar e the standard rules for universal quan tiﬁcation, with the usua l p roviso for ∀ I . T here a re also four ax iomatic ru les ( or “ax ioms”, for short) r eﬂ = , irr eﬂ < , tr ans < , and c onn , which express the prope rties of = 3 and < , wher e, for readability , we employed the symbols for disjunction, conjunction , and negation. The general rule s o f N ( K l G ) allow us to deriv e lwffs from rwf fs and vice versa. The rule mon app lies m onoton icity to an lwf f or rwff ϕ , while the rules u f 1 and uf 2 export falsum (and we thus call it a universal fa lsum ) fr om the lab eled sub-system to the relational one, and vice versa. 4 Deﬁnition 5 (Derivations and proofs) A deriv ation of a formula (lwf f or r wff) ϕ fr om a pr oo f conte xt (Γ , ∆) in N ( K l ) is a tr ee formed using the rules in N ( K l ) , ending with ϕ and depending only on a ﬁnite s ubset of Γ ∪ ∆ . W e then write Γ , ∆ ⊢ ϕ . A derivation of ϕ in N ( K l ) d epending on the empty set, ⊢ ϕ , is a proo f of ϕ in N ( K l ) an d we then say that ϕ is a theorem of N ( K l ) . 2 See [19] for a detaile d discussion on RAA ⊥ , which in particular expl ains ho w , in order to maintain the duality of modal operators like  and ♦ , the rule must all ow one to deriv e x : A from a contrad iction ⊥ at a possibly dif ferent world y , and thereby dischar ge the assumption x : A ⊃⊥ . 3 Note that we d o not need further a xioms to express symmetry and t ransiti vity of = , since the former ca n be deri ved by using mon , c onn , and irr eﬂ < , and the latt er by using mon . 4 Note that t he presentat ion of the system co uld be simpliﬁed by int roducing a unique symbol for f alsum (say f ), shared by the l abeled and the rel ational sub-systems. In that case, we would not need the rules uf 1 and uf 2 , while the rules for falsum elimination RAA ⊥ and RAA ∅ could be replace d by the follo wing rule, where with − ϕ we denote the nega tion of a generic formula (label ed or relation al): [ − ϕ ] . . . . f ϕ RAA f Ho wev er , we prefer to maintain a clear separati on between the two sub-systems, as it will allo w us to giv e a simpler presenta tion of normalization . 4 y : A x < y x : F A F I x : F A [ y : A ] [ x < y ] . . . . z : B z : B F E ∗ y : A y < x x : P A P I x : P A [ y : A ][ y < x ] . . . . z : B z : B P E ∗ ρ 1 ρ 1 ⊔ ρ 2 ⊔ I 1 ρ 2 ρ 1 ⊔ ρ 2 ⊔ I 2 ρ 1 ⊔ ρ 2 [ ρ 1 ] . . . . ρ [ ρ 2 ] . . . . ρ ρ ⊔ E ρ [ y /x ] ∃ x. ρ ∃ I ∃ x. ρ [ ρ [ y /x ]] . . . . ρ ′ ρ ′ ∃ E ∗ *In F E (respecti ve ly , P E ), y is differen t from x and z , and does not occur in any assumption on which the upper occurren ce of z : B depends other than y : A or x < y (respecti ve ly , y < x ). In ∃ E , y does not occur in any assumption on which the upper occurrenc e of ρ ′ depends other than ρ [ y /x ] . Figure 2. Some derived rules [ x

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