A logic with temporally accessible iteration

A Logic with T emporally Accessible Iteration Alexei Lisitsa 1 Department of Computer Science, the Univ ersity of Liv erpool, Ashton Building,Ashton Street, Liv erpool, L69 7ZF , U.K. alexei@csc.li v.ac.uk Abstract. Deficiency in exp ressiv e power of the first-order logic has led to deve l- oping its numero us e xtensions by fixed point ope rators, such as Least Fixed -Point (LFP), inflationary fi xed-po int (IFP) , partial fixed-point (PFP), etc. T hese logics hav e been e xtensiv ely studied in finite mo del theory , database theory , descrip- tiv e complex ity . In this paper we introduce unifying frame work, the logic with iteration operator , in which iteration steps may be accessed by temporal logic formulae. W e sho w that proposed logic FO+T AI subsumes all mentioned fixed point extensions as well as many other fixed point logics as natural fragments. On the o ther hand we sh ow that o ver fi nite structures F O+T AI is no more e xpres- siv e than F O+PFP . F urther we show that adding t he same machinery to the logic of monotone inductions (FO+LFP ) do es not increase its expressi ve po wer either . 1 Intr oduction Probably on e of the earliest proposals to extend logic with inductive constructs can be found in the W ittgenstein’ s famous T ractatus Logico-Philosop hicus [20] 4.127 3 If we want to expr ess in con ceptual notation the general pr oposition ‘b is a successor o f a’, then we r eq uir e an e xpres sion for th e general term of the series of forms aRb ( ∃ x ); aRx.X Rb ( ∃ x, y ) : aRx.xRy .y Rb . . . Implicitly Witt genstein admitted in sufficient expr essi ve power of the (first-ord er) predicate logic and p ropo sed an extension which in moder n terms we can call first- order logic augmented with the transitiv e closure opera tor (FO+TC). T ransitive closure is a particular case of more general inductive operators, which have extensiv ely studied in recursion theory and its generaliza tions [17,16,1]. Special role log ics with inductive operato rs play in f oundatio ns of computer sci- ence. Logic langu ages with fixed point construc ts serve theoretical models of query languag es in database theory and, when con sidered over lin early ord ered finite struc- tures, are used to characterize com putation al complexity classes within descriptive com - plexity th eory[1 9 ,12,13]. The relation ships between fixed p oint logics and comp lexity have many interesting aspects - the logics reflect faithfully computatio ns over structures and this led to formu lation o f a n ew no tion of r elational co mplexity [ 3]. On th e other hand, tantalizing open problems in comp utational comp lexity ca n be formulated in log- ical terms, f or example PTIME = PSP A CE if and o nly if log ics with least fixed poin t and partial fixed p oints have the same expressive p ower over classes of fin ite models [2]. In o ther direction, moda l logic with fixed points, µ -c alculus, is one of the un ifying formalisms used in th e re search on model ch ecking and verification [8]. No t nec essar- ily monoton e inductive defin itions also appea r in the research o n seman tics o f lo gic progr amming[ 10 ], in for malization of reasoning [7] and in the re vision theory [15 ]. In this paper we propo se a s imple mechanism allo wing to ”internalize” various vari- ants of the in ductive definitions within a single logic. Sem antics of fixed-point operato rs is u sually de fined by using a n iteratio n, mor e prec isely in term s o f ”to what itera tion conv erge”. W e suggest to look on the iteratio n process itself and aug ment th e logic with an access to th e iteratio n stages via tempo ral form ulae. As a resu lt we get a logic FO+T AI (temporally accessible iteration) which naturally subsumes many (virtually all deterministic variants of ) inducti ve logics, including logics with least fixed point, infla- tionary fi xed point, variants of partial fixed points, as well as logics with anti-monoto ne and non- monoto ne inductions. W e present th e seman tics of FO+T AI fo r fin ite structu res on ly . Th e case of in finite structures requir es considering tr ansfi nite iterations and temporal access to the iteratio n stages would need a v ariant of temporal lo gic over o rdinals (e.g. [5]). Th is case requires further in vestigations and will be treated elsewhere. W e show b y tr anslations that over fin ite structures FO+T AI is not less expre ssi ve than all men tioned in ductive logics an d at the same tim e it is no m ore expressive than FO+PFP . Further, w e show that add ing the same machiner y to the log ic of m onoto ne induction s (FO+LFP) does not increase its expressiv e power either . The pap er is o rganized as fo llows. In th e next section we introd uce classical fixed- point logics and first-order temporal lo gics. Based on that in the Section 3 we define the logic FO+T AI. In Section 4 we demonstra te how to define in FO+T AI classical inductive constructs. In Section 5 it is shown tha FO+T AI sub susmes the lo gic of non- monoto ne in duction FO+ID. In Sectio n 6 we con sider expressi ve power FO+ T AI and its monoton e frag ment. Section 7 concludes the paper . 2 Pr eliminaries 2.1 Fixed point extensions of first-order logic W e start with the sho rt re view of ind uctive definability , which will set up a context in which lo gics with tem porally a ccessible iteration n aturally appear . In th is p aper we will mainly deal with definability over (classes of) finite structures, so unless otherwise stated all structures are assumed to be finite. Let ϕ ( R, ¯ x ) is a first-order fo rmula, wh ere R is a re lation symbol of some arity n and ¯ x is a tuple of in dividual variables of the length n (the same as the arity of R ). Consider a structure M with the dom ain M , interpreting all sym bols in ϕ except R an d ¯ x . Then one can consider a map Φ ϕ : 2 M k → 2 M k , i.e mapping k -a ry relations over M to k -ary relation s o ver M defined by ϕ ( R, ¯ x ) as follows: Φ ϕ ( P ) = { ¯ a | ( M , | = ϕ ( P, ¯ a ) } V arious fixed-po int constructions may then be defined. If operator Φ ϕ is m onoton e then by classical Knaster-T arski theor em [18] it h as a least fixed-point, that is the least relation R , such that R ( ¯ x ) ↔ ϕ ( R, ¯ x ) h olds. This least fixed-po int R ∞ can be obtaned as a limit of the following it eration : – R 0 = ∅ – R i +1 = Φ ( R i ) Over finite stru ctures this iteration stabilizes on some finite step n ≥ 0 : R n +1 = R n . Simple syntactical p roperty o f ϕ ( R, ¯ x ) which gua rantees mo noton icity of Φ ϕ is that this formu la is positive in R . Inflatio nary fixed poin t of a no t n ecessary monoto ne oper ator Φ is defined as the limit of the following iteration: – R 0 = ∅ – R i +1 = Φ ( R i ) ∪ R i The inflationary fixed point exists for an arbitrary oper ator and over finite structures the above iteration reaches it at some finite step. P a rtial fixed point of an operator Φ defined by an a rbitrary for mula ϕ ( R, ¯ x ) is de- fined as follows. Consider the iteration : – R 0 = ∅ – R i +1 = Φ ( R i ) Partial fixed point of Φ is a fixed point (limit) of t he iteration (if it exists) and empty set otherwise. Aiming to re solve difficulties in the definition of semantics o f partial fixed po int operator over infinite structures in [1 4] a n alternative general semantics f or such an operator has been proposed . W e will discuss it later in 4.4. Let I ND is one o f the above fixed poin t oper ators (LFP , IFP , PFP or PFPgen) th en the syntax of logic FO+IND extends the standard syntax of first-or der lo gic with the follow- ing c onstruct. L et ϕ ( R, ¯ x ) be a formula with free indi vidu al variables ¯ x = x 1 , . . . , x k and free predicate variable R . For the c ase IND ≡ LFP we add itionally requir e that ϕ ( R, ¯ x ) is p ositiv e in R . Then ρ := [ I N D R, ¯ x ϕ ] ¯ t is also for mula. Free variables of ρ are free v ariables occu rring in ϕ and t other than ¯ x . Semantics of such formula ρ is read then as f ollows: an interpretation of tuple o f terms ¯ t belon g to the relation which is a fixed p oint o f the operator Φ ϕ of the correspond ing type IND (i. e. least, inflationary , partial, or g eneralized partial fixed poin t, for IND ≡ LFP , IFP , PFP , genPFP , r espec- ti vely .) Usually the above lo gics defined in a way allowing also simultane ous inductive definitions, i.e the formu lae of the form [ I N D R i : S ] ¯ t where S :=        R 1 ( ¯ x 1 ) ← ϕ 1 ( R 1 , . . . , R k , ¯ x 1 ) . . . R k ( ¯ x k ) ← ϕ 1 ( R 1 , . . . , R k , ¯ x k ) is a system of for mulae. Consider a stru cture M with the domain M , interpr eting a ll symbols in ϕ i except R j and ¯ x . The n ϕ i defines a map ping Φ ϕ i : 2 M r 1 × . . . 2 M r k → 2 M r i , where all r j are arities of R j , as follows: Φ ( P 1 , . . . , P k ) = { ¯ a | ( M | = ϕ ( P 1 , . . . , P k , ¯ a ) } . Definitions of all men tioned fixed p oints n aturally g eneralize to the c ase of simultane- ous iteration R 0 i = ∅ R j +1 i = Φ ϕ i ( R j 1 , . . . R j k ) . The fo rmula [ I N D R i : S ] ¯ t is tru e f or a tuple of terms ¯ t if its interpretation belong s to i -th componen t R ∞ i of the correspond ing simultaneous fixed point. For all mentio ned logics, simultaneous in duction can be elim inated and equivalent formulae with simple induction can be prod uced [9,14]. 2.2 First-order temporal logic The lang uage T L of first order temp oral logic over the n atural nu mbers is con structed in the stan dard way from a classical (non -tempo ral) first order lang uage L and a set of futu re-time temporal o perators ‘ ♦ ’ ( some time ), ‘ ’ ( a lways ), ‘ ❣ ’ ( in the next mo- ment ), ‘ U ’( until ). Formulae in T L ar e interpr eted in fir st-order temporal s tructures of the form M = h D , I i , where D is a n on-emp ty set, the d omain of M , and I is a fu nction associating with every moment of time n ∈ N an interpretation of predicate, fu nction and constant symbols of L ov er D . First-ord er (nontempo ral) structures correspon ding to each point of time will be denoted M n = h D , I ( n ) i . Intuitively , th e interp retations of T L -formulae ar e sequ ences of first-orde r struc- tures, or states of M , such as M 0 , M 1 , . . . , M n . . . . An assignment in D is a fu nction a from the set L v of individual variables of L to D . If P i s a pred icate symbol then P I ( n ) (or simply P n if I is understood ) is the interpretatio n of P in the state M n . W e requir e that (in dividual) v ariables an d co nstants of T L are rigid , th at is nei- ther assignmen ts nor in terpretation s of con stants depend o n th e state in wh ich they are ev aluated. The sa tisfaction relation M n | = a ϕ (or simp ly n | = a ϕ , if M is u nderstood ) in the stru cture M for the assignment a is d efined ind uctively in th e usual way und er the following semantics of temporal operators: n | = a ❣ ϕ if f n + 1 | = a ϕ n | = a ♦ ϕ iff there is m ≥ n s uch that m | = a ϕ n | = a ϕ iff m | = a ϕ for all m ≥ n n | = a ϕ U ψ iff there is m ≥ n such that m | = a ψ and k | = a ϕ for every n ≤ k < m Let M be a tempo ral struc ture and ψ ( ¯ x ) be a temporal formula with ¯ x on ly free variables a nd | ¯ x | = k . Then ψ ( ¯ x ) defines a k -ary relation P on M 0 as follows: P (¯ a ) ↔ M 0 | = a ψ ( ¯ x ) where a : ¯ x 7→ ¯ a . 3 Logic with temporally accessible iteration In all variants o f in ductive log ics we have discussed in the previous section, th e semantics o f fixed-p oint con struction ca n b e defin ed in terms of iteration of operators, associated with some formulae . In this section we described a log ic which gen eralize and subsu me all these log ics. The id ea is simple: in stead of d efining a par ticular fixed- point construct we allow arbitrar y itera tions of operato rs defin ed by fo rmulae. These iterations wh en ev aluated over a struc ture give r ise to the seq uences of r elations over that structure. The n we allow first-order tempora l lo gic machin ery to access these se- quences of relations (tem poral structure s) and define new r elations in terms of th ese sequences. The syntax of F O + T AI ( first-or der logic with temporally a ccessible iterations ) extends the standard syntax of first-order logic with the following c onstruct. Let ϕ ( R, ¯ x ) be a form ula with free in dividual variables ¯ x = x 1 , . . . , x k and f ree predicate variable R of arity k . Let ψ ( ¯ z ) be a first-ord er tempo ral formula ( T L -for mula) with free indi vidua l variables ¯ z = z 1 , . . . , z m . Then τ := [ ψ ( ¯ z )][ I R, ¯ x ϕ ] ¯ t is also for mula, whe re ¯ t is a tuple of terms o f the same length as ¯ z . The free variables of τ ar e the fre e variables occurrin g in ¯ t and the free variables of ψ and ϕ other than ¯ z and ¯ x , respectively . The semantics of this construct is defined as follows. Let M be the struc ture with th e do main M and interpretations of all p redicate an d function al symbols in M , which will denote by P M and f M . Let a be assignment pro- viding an interpretation of free variables of ϕ and ψ im M . Consider the iteration R 0 = ∅ and R i +1 = Φ ϕ ( R i ) . It gives rise to th e tempora l structur e M = M 0 , . . . , M i , . . . , where e very M i is a structur e M extended by an interp retation o f R b y R i . In partic- ular M 0 is M aug mented with empty interpre tation o f R . L et P is an m -ary relatio n defined by ψ ( ¯ z ) on M 0 (i.e on M). Then f or any tuple ¯ a ∈ M m , M | = [ ψ ( ¯ z )][ I R, ¯ x ϕ ]¯ a iff ¯ a ∈ P . As i n other fixed point logics, we also allo w simultaneo us iteration formu lae, i.e. the formulae of the form τ := [ ψ ( ¯ z )][ I : S ] ¯ t where S :=        R 1 ( ¯ x 1 ) ← ϕ 1 ( R 1 , . . . , R k , ¯ x 1 ) . . . R k ( ¯ x k ) ← ϕ 1 ( R 1 , . . . , R k , ¯ x k ) is a system of formula e. Simultaneo us iteration R 0 i = ∅ R j +1 i = Φ ϕ i ( R j 1 , . . . R j k ) induces a tempo ral structure M = M 0 , . . . , M i , . . . , wher e e very M j is a structu re M e xtend ed by interpretation of R i by R j i . Let P is an m -ar y relation defined by ψ ( ¯ z ) o n M 0 (i.e on M). Then for any tup le ¯ a ∈ M m , M | = [ ψ ( ¯ z )][ I : S ]¯ a if f ¯ a ∈ P . Proposition 1. FO+T AI with simultaneous iteration has the s ame expr essive p ower as FO+T AI with sing ular iter ation . Proof (hin t). The p roof pro ceed by standard argum ent based o n faithf ul mo delling of s imultan eous it eration by a single iteration of higher -dim ensional jo int oper ator . F ull details of such modelling (for LFP , IFP , PFP) can b e found in [9]. 4 FO+T AI vs other fixed point logics In this section we show that FO+T AI subsumes m any fixed po int logics. W e start with classical fixed point constructs. 4.1 Least Fixed P oint T ranslation of LFP constru ct in FO+T AI follo ws l iterally a description of the least fi xed point as a limit - least fixed po int co nsists of pr ecisely tho se tu ples which eventua lly appear in approx imations: LFP: [ LF P R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ ♦ R ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t Here we assume of course that R is positiv e in ϕ ( R, ¯ x ) . 4.2 Inflationary Fixed Point Similarly to th e case o f LFP we h ave for Inflation ary Fixed Point the following defini- tion: IFP: [ I F P R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ ♦ R ( ¯ z )][ I R, ¯ x ( R ( ¯ x ) ∨ ϕ ( R , ¯ x ))] ¯ t 4.3 Partial Fixed P oint The following definition PFP: [ P F P R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ ♦ ( R ( ¯ z ) ∧ ∀ ¯ v ( R ( ¯ v ) ⇔ ❣ R ( ¯ v )))][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t says that Partial Fixed Poin t con sists of the tu ples satisfyin g two co nditions: 1 ) a tu ple should ap pear at some stage i of iteratio ns, and f urtherm ore 2) approx imations at the stages i and i + 1 should be the same. 4.4 General PFP In [14] a n alternative semantics for PFP has been defin ed und er the n ame general PFP . Unlike the standard PF P general PFP generalizes easily to infinite struc tures and ha ving the same expressi ve power as standar d PFP over finite stru ctures p rovides sometimes with more c oncise a nd natural eq uiv alent f ormulae. As we mention ed in the Introduc- tion, in th is p aper we consider o nly finite stru ctures sem antics an d fo r th is case defini- tion o f g eneral PFP is as follows. Let Φ is an o perator defin ed b y an arbitrary f ormula ϕ ( R, ¯ x ) . Consider the iteration: – R 0 = ∅ – R i +1 = Φ ( R i ) Then general partial fixed poin t of Φ is defined [14] as a set of tuples which occur in every stage of the fi rst cycle in th e sequ ence of stages. . As n oticed in [ 14], in genera l, this definition is not equiv alent to say ing that the fixed p oint c onsists of those tuples which occur at all stages starting fr om so me stage . Non-equ iv a lence of two definitio ns can be establishe d if tr ansfinite iteration is allowed. Since w e con sider the iteration over finite structures on ly , a cycle, that is a sequen ce R i , . . . , R j with R i = R j , will necessarily appear at some finite stages i an d j . Based in that, for th e case of finite structures we have the following equivalent definition of PFPgen in terms of FO+T AI: PFPgen: [ P F P g en R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ ♦ R ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t The definition s ays that general PFP consists o f those tuples which occur at all finite stages starting from some stage of iteration. 4.5 Anti-monotone induction Let Φ ϕ be an operato r associated with a formu la ϕ ( P, ¯ x ) . It m ay tu rn out that this o p- erator is an ti-monoto ne , that is P ⊆ P ′ = > Φ ϕ ( P ′ ) ⊆ Φ ϕ ( P ) . Syntactical con dition which entails anti-monoto nicity is that the pred icate variable P has only negative oc - currenc es in ϕ ( P, ¯ x ) . As befo re consid er the iteration R 0 = ∅ , R i +1 = Φ ( R i ) . An in teresting analo gue of classical Knaster -T arski r esult ho lds [2 1,10]: the above iteration of anti-mon otone operator converges to a pair of oscillating points P and Q that is Q = Φ ( P ) and P = Φ ( Q ) . What is more, one of the o scillating poin ts is a least fi xed point µ and another is the greatest fixed point ν of the mon otone ope rator Φ 2 (where Φ 2 ( X ) = Φ ( Φ ( X ))) .) One may e xtend then the first-order logic with suitable oscillating points constructs [ OP µ R, ¯ x ϕ ( R, ¯ x )] ¯ t and [ O P ν R, ¯ x ϕ ( R, ¯ x )] ¯ t for ϕ ( R, ¯ x ) negativ e in R , with obvious seman - tics. B ecause of the definability of oscillating points as t he fixed po ints of Φ 2 , first order logic extended with these constructs is no more expressi ve than FO+LFP and therefo re than FO+T AI. Wh at is interesting h ere is that FO+T AI allo ws to define oscillating poin ts directly , no t ref erring to LFP construct. Her e it goes. For the greater o f two oscillating points we hav e [ OP ν R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ ψ ν ( R )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t where ψ ν ( R ) is the temporal form ula ♦ ( R ( ¯ z ) ∧ ∀ ¯ y ( R ( ¯ y ) ↔ ❣ ❣ R ( ¯ y )) ∧ ( ∃ ¯ y ( R ( ¯ y ) ∧ ❣ ¬ R ( ¯ y )) ∨ ∀ ¯ y ( R ( ¯ y ) ↔ ❣ R ( ¯ y )))) Similarly , [ OP µ R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ ψ µ ( R )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t where ψ µ ( R ) is the temporal formula ♦ ( R ( ¯ z ) ∧ ∀ ¯ y ( R ( ¯ y ) ↔ ❣ ❣ R ( ¯ y )) ∧ ( ∃ ¯ y ( ¬ R ( ¯ y ) ∧ ❣ R ( ¯ y )) ∨ ∀ ¯ y ( R ( ¯ y ) ↔ ❣ R ( ¯ y )))) 4.6 Some variations In the above FO+T AI definition f or LFP it is assumed that ϕ ( R , ¯ x ) is po siti ve in R . I f we consider the same right-hand side definitio n [ ♦ R ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t for not n ecessarily positive (and m onoto ne) ϕ ( R, ¯ x ) than we get defin ition of an o perator which does no t have direct analogue in standard fixed-point logic s and may be considered as a variation of PFP , which we denote by P F P ∪ . Similarly , one can define: [ P F P ∩ R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ R ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t It has tu rned o ut tho ugh th at both P F P ∪ and P F P ∩ are ea sily definable b y simu lta- neous partial fixed-points, for details see Theorem 1. If in defin ition o f PFPgen we swap temp oral operators we get a defin ition o f what can be called Recurrent Fixed Point (RFP) 1 : RFP: [ RF P R, ¯ x ϕ ( R, ¯ x )] ¯ t ⇔ [ ♦ R ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t Again it is not difficult to demonstrate that RF P is definable in terms of either PFP or PFPgen. 5 ID-logic of non-monotone induction In [6] a log ic o f no n-mon otone definitions h as been intr oduced . Motiv ated by well found ed semantics for logic program ming, ID-logics formalises non-monoto ne, in gen- eral, indu ctiv e d efinitions of th e form P ← ϕ ( P ) where pr edicate variable P may have both positi ve and negati ve occurrences in ϕ ( P ) . It s ubsum es and generalizes both monoto ne a nd anti-m onoton e inductio ns. Th e main point in the d efinition of ID-logic is a semantics given to no n-mo notone in ductive d efinition which we present h ere in 1 Notice, than in general, and similarly to P F P g en , neither of P F P ∪ , P F P ∩ , RF P define fixed points of any natural operators. But we follo w [14] and preserve the name “fixed points” and FP in abbre viations. a operator fo rm 2 . Similarly to already discussed fixed p oint exten sions, the syntax of ID-logic (this version we call FO+ID) extend s the standard syntax of first-order log ic with the f ollowing constru ct. Let ϕ ( P, ¯ x ) b e a fo rmula with fr ee ind ividual variables ¯ x = x 1 , . . . , x k and fre e pr edicate variable P . Then ρ := [ I D P, ¯ x ( P ( ¯ x ) ← ϕ ( P, ¯ x ))] ¯ t is also formula. No w we e xplain semantics of this construct in terms of FO+T AI , show- ing thereby that FO+ID is also subsume d by FO+T AI. Since ϕ ( P, ¯ x ) may have bo th negativ e a nd positive occu rrences of P the itera tion of the operator Ψ ϕ applied to the empty interpretation of P will not necessary con verge to a fixed point. In the semantics adopted in FO+ ID, the extension o f defined pr edicate is o btained as a co mmon limit of iterativ ely computed lower and upper bounds (if it exists). Introduce tw o new auxiliary predicate variables P l and P u , with th e inten ded meanin g to be lower and negated up- per approxim ations fo r the defined pred icate. Further, denote by ϕ ( P l ) , respectively , by ϕ ( ¬ P u ) th e result of replace ment of all negativ e occu rrences o f P in ϕ ( P , ¯ x ) with P l , resp. with ¬ P u . All positive occurrence s o f P remain s unaffected in both cases. Consider then the following definitio n of the step of simultaneous iteration: S :=        P u ( ¯ y ) ← ¬ [ LF P P, ¯ x ( ϕ ( P l ))] ¯ y . . . P l ( ¯ y ) ← [ LF P P, ¯ x ( ϕ ( ¬ P u ))] ¯ y Since both ϕ ( P l ) and ϕ ( ¬ P u ) are positi ve in P the least fixed point oper ators in the right hand sides of the definitions are well-defined. Starting w ith P 0 l = P 0 u = ∅ and iterating this defin ition o ne g ets a sequen ces of lower and negated up per app roximation s P i l and P i u . If th e lower and upper app roxi- mations converge to the same limit, i.e. P ∞ l = ¬ P ∞ u then by definitio n [6] this limit is taken as the predicate d efined by the above I D-construct. Summing up , the FO+ID formu la ρ shown above is equiv alent to the follo wing formula of FO+T AI: [ ♦ ( P L ( ¯ x ) ∧ ∀ ¯ y ( P l ( ¯ y ) ↔ ¬ P u ( ¯ y ))][ I : S ∗ ] ¯ t where S ∗ is obtain ed of the above S by tran slation of the right han d side parts of S into FO+T AI. 6 Expr essive power W e hav e seen in previous sections that FO+T AI is very expressi ve logic an d subsum es many other fixed-p oint logics, inclu ding most expressive (among mentioned ) FO+PFP (and FO+PFPgen). The n atural question is whether FO+T AI is more expressiv e th an FO+PFP? I n this section we answer this question negatively and sh ow that for any formu la in FO+T AI one can effecti vely prod uce an equivalent (over finite stru ctures) FO+PFP formula. 2 In [6] inducti ve definitions of I D-logic are presented not by operators, but by special f or- mulae called definitions . The difference is purely syntactical and insignificant for our discussion here. Theorem 1. F or every formula τ := [ ψ ][ I R, ¯ x ϕ ] ¯ t of FO+T AI there is an equivalen t formula τ ∗ of FO + PFP Proof The formula τ ∗ equiv alent to a τ is build by induction on the construction of τ . Corr ectness of the proposed translation τ 7→ τ ∗ is established by induction along the construction . Correctness of the base case and induction steps follows by routine check of definitions. If τ := [ ψ ( ¯ z )][ I R, ¯ x ϕ ] ¯ ( t ) with [ ψ ] non-temp oral formula then [ τ ] ∗ := ψ ( ¯ z ) | R ←∅ , ¯ z ← ¯ t where R ← ∅ mean s substitute all o ccurren ces of R in ψ with ∃ x 6 = x and ¯ z ← ¯ t substitute ¯ t into ¯ z . The cases of bo olean conn ectiv es an d quan tifiers in the head and bo dy of the fo r- mula are straightfor ward. – ([ ψ 1 ∧ ψ 2 ][ I R, ¯ x ϕ ] ¯ t ) ∗ = ([ ψ 1 ][ I R, ¯ x ϕ ] ¯ t ) ∗ ∧ ([ ψ 2 ][ I R, ¯ x ϕ ] ¯ t ) ∗ – ([ ψ 1 ∨ ψ 2 ][ I R, ¯ x ϕ ] ¯ t ) ∗ = ([ ψ 1 ][ I R, ¯ x ϕ ] ¯ t ) ∗ ∨ ([ ψ 2 ][ I R, ¯ x ϕ ] ¯ t ) ∗ – ([ ¬ ψ ][ I R, ¯ x ϕ ] ¯ t ) ∗ = ¬ ([ ψ ][ I R, ¯ x ϕ ] ¯ t ) ∗ – ([ ∃ y .ψ ][ I R, ¯ x ϕ ] ¯ t ) ∗ = ∃ y . ([ ψ ][ I R, ¯ x ϕ ] ¯ t ) ∗ – ([ ∀ y .ψ ][ I R, ¯ x ϕ ] ¯ t ) ∗ = ∀ y . ([ ψ ][ I R, ¯ x ϕ ] ¯ t ) ∗ – ( ϕ ∧ ψ ) ∗ = ϕ ∗ ∧ ψ ∗ – ( ¬ ϕ ) ∗ = ¬ ϕ ∗ – ( ∀ y ϕ ) ∗ = ∀ y ϕ ∗ – If τ = [ ♦ ψ ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t then τ ∗ := [ P F P Q , ¯ v : S ] ¯ t where S :=        R ( ¯ x ) ← ( ϕ ( R, ¯ x )) ∗ . . . Q ( ¯ v ) ← Q ( ¯ v ) ∨ ([ ψ ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ v ) ∗ – The case of -modality as the main con nective in the head of iteration is red uced to the case of ♦ modality: ([ ψ ][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t ) ∗ = ([ ¬ ♦ ¬ ψ ][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t ) ∗ – If τ = [ ❣ ψ ( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t then τ ∗ := [ P F P Q, ¯ x : S ] ¯ t where S :=        R ( ¯ x ) ← ( ϕ ( R, ¯ x )) ∗ . . . Q ( ¯ x ) ← [( ϕ ( R, ¯ x ) ∗ ] 2 – If τ = [( ψ 1 U ψ 2 )( ¯ z )][ I R, ¯ x ϕ ( R, ¯ x )] ¯ t then τ ∗ := [ P F P Q ¯ x : S ] ¯ t where S :=      R ( ¯ x ) ← ( ϕ ( R, ¯ x )) ∗ P ( ¯ z ) ← P ( ¯ z ) ∨ ¬ ψ 1 ( ¯ z ) Q ( ¯ z ) ← Q ( ¯ z ) ∨ ( ¬ P ( ¯ z ∧ ψ 2 ( ¯ z ) 6.1 T emporally accessible monot one induction What h appens if we ap ply tem poral logic based access to the iteration steps of monotone induction ? Will the resulting logic be more expressiv e th an the logic of the m onoton e induction ? Negativ e answer is gi ven by the following theorem. Theorem 2. F or every formula τ := [ ψ ][ I R, ¯ x ϕ ] ¯ t of FO+T AI with ϕ positive in R there is an equivalen t formula ( τ ) ∗ of FO + LFP Proof The translation her e u ses the stage comparison theo r em of Moschovakis [16]. W ith any mon otone ma p Φ ϕ of arity k de fined by a p ositi ve in R fo rmula and a s tructu re with finite domain M on can associate a r ank fu nction | | Φ : M k → N ∪ {∞} which when app lied to any tuple of elements ¯ a ∈ M k yeilds the least numbe r n such that ¯ a ∈ Φ n ( ∅ ) if such n e xists and ∞ otherwise, i.e. when ¯ a 6∈ Φ ∞ Stage comparison relation ≤ Φ defined as ¯ a ≤ Φ ¯ b ⇔ ¯ a, ¯ b ∈ Φ φ ( ∅ ) and | ¯ a |≤| ¯ b | . Theorem 3. F or any LF P ϕ operator associated with a first-or der fo rmula ϕ ( P , ¯ x ) positive i n P the stage compa rison r ela tion ≤ ϕ is defin able in FO+LFP uniformly over all finite structur es. The stage comp arison relation can b e used then to simu late time in m odelling tem- poral access to the iteration step s with in FO+L FP . As a bove, the translation is defined by indu ction on formula structure. W e present here only translatio n of [ ψ ( ¯ z )][ I R, ¯ x ϕ ] ¯ t where ψ ( ¯ z ) is a tempo ral formula and ϕ is in FO+LFP . For a formu la [ ψ ( ¯ z )][ I R, ¯ x ϕ ] ¯ t , define tran slation o f its tempor al hea der ψ ( ¯ z ) in the context of iteration [ I R, ¯ x ϕ ] , to a form ula in FO+LFP . Translation is ind exed b y either a constant s ( from s tart ) or a tup le of v ariables of the same length as the ar ity of the predicate used in iteration definition, i.e. of R : – [ P ( ¯ x )] s := P ( ¯ x ) (for any predicate P ). – [ R ( ¯ x )] ¯ u := ¯ x ≤ ϕ ¯ u ∧ R ( ¯ x ) (for the iteration predicate R ) – [ P ( ¯ x )] ¯ u := P ( ¯ x ) (for any predicate P other than iteration predicate) – [ ρ ∧ τ ] s := [ ρ ] s ∧ [ τ ] s – [ ρ ∧ τ ] ¯ u := [ ρ ] ¯ u ∧ [ τ ] ¯ u – [ ¬ ρ ] s := ¬ [ ρ ] s – [ ¬ ρ ] ¯ u := ¬ [ ρ ] ¯ u – [ ∀ xρ ] s := ∀ x [ ρ ] s – [ ∀ xρ ] ¯ u := ∀ x [ ρ ] ¯ u – [ ❣ τ ] s := ∃ ¯ u ( ϕ ( ¯ u ) ∧ [ τ ] ¯ u ) – [ ❣ τ ] ¯ u := ∃ ¯ u ′ ( next ϕ ( ¯ u, ¯ u ′ ) ∧ [ τ ] ¯ u ′ ) – [ ♦ τ ] s := ∃ ¯ u [ LF P R, ¯ x ϕ ] ¯ u ∧ [ τ ] ¯ u – [ ♦ τ ] ¯ u := ∃ ( ¯ u ′ ( ¯ u ≤ ϕ ¯ u ′ ) ∧ [ τ ] ¯ u ′ ) – [ ρ U τ ] s := ∃ ¯ u ([ LF P R, ¯ x ϕ ] ¯ u ∧ [ τ ] ¯ u ) ∧ ∀ ¯ u ′ ( ¯ u ′ < ϕ ¯ u ) → [ ρ ] ¯ u ′ – [ ρ U τ ] ¯ u := ∃ ¯ u ′ ( ¯ u ≤ ϕ ¯ u ′ ∧ [ τ ] ¯ u ′ ) ∧ ∀ ¯ u ′′ ( ¯ u ≤ ϕ ¯ u ′′ < ϕ ¯ u ′ ) → [ ρ ] ¯ u ′′ Now to get a formula in FO+LFP equiv alent to [ ψ ( ¯ z )][ I R, ¯ x ϕ ] ¯ t we take translatio n [ ψ ( ¯ z )] s in the context of [ I R, ¯ x ϕ ] ¯ t . 7 Concluding re marks W e proposed in this paper the logic with temporally accessible iteration which p rovides the simple unify ing framework for studying logics with inductiv e fixed point operato rs. 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