Survey on IF Modal Logic

Transcription

1 Survey on IF Modal Logic Tero Tulenheimo Laboratoire Savoirs, Textes, Langage CNRS Université Lille 3 France Seminario de Lógica y Lenguaje Universidad de Sevilla

2 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

3 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

4 Two notions of scope Consider the formulas 1 x y Rxy 2 x(px Qx) In (1), y is in the syntactic scope of x. In (2), is in the syntactic scope of x. Further, y is also in the semantic scope of x in (1): the semantic value for y can be chosen depending on the semantic value corresponding to x. Similarly, is in (2) in the semantic scope of x.

5 Syntactic scope: relation determined by the syntactic tree (subformula structure) of a formula. Semantic scope: O 2 is in the semantic scope of O 1 in φ if the semantic value corresponding to O 2 is allowed to depend on the semantic value corresponding to O 1. If we are interested in truth and if O 2 has existential force, this means that a value witnessing O 2 is a function of a value corresponding to O 1. In FO, the two notions of scope coincide. In particular, whenever an operator is in the syntactic scope of another operator in an FO formula, it thereby is also in its semantic scope.

6 In IF logic, the two notions are dissociated. Motivation: in a first-order language the only way to express dependencies between variables is via formal dependencies between quantifiers: via semantic scopes. By identifying syntactic and semantic scope, many such dependence relations remain inexpressible in FO. The slash notation (/) used to indicate semantic scopes. 1 x( y/ x)rxy [truth-equivalent to y xrxy] 2 x(px ( / x) Qx) [ " ( xpx xqx)] 3 x y z( v/ x, y)rxzyv [ " x y z v Rxzyv]

7 Writing ( x/q 1 x 1,..., Q n x n ) indicates that x occurring in a formula in the syntactic scope of the quantifiers Q 1 x 1,..., Q n x n is nevertheless not in their semantic scope. This is semantically interpreted as follows: x must be witnessed uniformly w.r.t. the semantic values interpreting the syntactically preceding quantifiers Q 1 x 1,..., Q n x n. Only slashes on operators with existential force matter for truth; similarly for operators with universal force and falsity. In the literature two conceptualizations can be found: IF first order logic proper and slash logic (Hodges 2007).

8 What results if these ideas are studied in connection with modal logic? First studied in a concurrency-theoretical setting by Bradfield (2000), Bradfield & Fröschle (2002). Semantics relative to standard modal structures in T (2003, 2004).

9 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

10 Choosing the syntax Consider basic modal logic (or ML) in negation normal form. Formulas obtained form literals by,,,. We restrict our attention to constraints on finding a witness when interested in truth: slashed diamonds. We wish to allow diamonds to be independent of a selection of syntactically preceding boxes and diamonds. Further options could be explored (but will not in this talk).

11 Choosing the syntax (cont.) How to indicate independence? Note that in modal syntax there is nothing corresponding to variables. We opt for using relative de Bruijn indices. E.g., ( /1)p and (p ( /1, 2)q). The syntax of IFML is, then, given as follows: 1 If ψ ML and ψ results from replacing in ψ all tokens of by the symbol ( / ), then ψ is a formula. 2 If ψ is a formula, and i 1,..., i k are (numerals standing for) positive integers, and α is a token of ( / ) that lies in ψ in the syntactic scope of at least max{i 1,..., i k } modal operators, the result of replacing α in ψ by the symbol ( also a formula. /i 1,..., i k ) is

12 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

13 Models Models M structures of the form (M, R, V ), where M, R M M and V is a function of type prop Pow(M). Semantics relative to pointed models (M, w).

14 Example (1) On what condition should we have M, w = ( On the condition that M, w = p and the witness of can be chosen independently of the state interpreting. c c 1 c 2 c 1 c 2 3 p c 3 p p b b 1 b 1 b 2 2 M a a /1)p? False in M at a True in N at a N

15 Game-theoretical semantics Semantics in terms of two-player games. We may call the players A and E. We introduce such a game for every triple (φ, M, w), where φ is a formula of IFML and (M, w) is a poited model, and denote it by G(φ, M, w). A will make a choice corresponding to the operators and (operators with universal force) 1 E will make a choice corresponding to the operators and (operators with existential force) 1 Recall that the negation sign may only appear in front of an atom.

16 Example (1) (cont., ML) How is the game for the basic modal formula p played relative to the pointed model (M, w)? c 1 c 2 c 3 p p b 1 b 2 M a A play (a, x, y) consists of an accessible state x chosen by A corresponding to, followed by an accessible state y chosen by E corresponding to. E wins the play if y makes p true, otherwise A wins. There are 4 possible plays: (a, b 1, c 1 ) and (a, b 1, c 2 ) and (a, b 2, c 2 ) and (a, b 2, c 3 ). So there is a winning strategy for E: choose c 1 if A chose b 1, and choose c 3 if A chose b 2.

17 Example (1) (cont., IFML) What about the game for the IF modal formula ( played relative to (M, w)? c 1 c 2 c 3 p p b 1 b 2 M a /1)p The set of possible plays the same as in the ML case above. Also the winning condition of plays are the same. The difference is that here a restriction is imposed on available winning strategies: A strategy may only be winning if it is uniform in the sense of assigning the same state to ( /1) regardless of the state chosen for. No such w.s. exists.

18 Game rules Game rules for semantic games G(φ, M, w) can be recursively defined as follows: If φ = p, the play has come to an end. E wins if w V (p), otherwise A wins. If φ = p, the play has come to an end. E wins if w / V (p), otherwise A wins. If φ = (ψ χ), player E chooses a disjunct θ {ψ, χ} [LEFT or RIGHT] and the play continues as G(θ, M, w). If φ = (ψ χ), player A chooses a conjunct θ {ψ, χ} [LEFT or RIGHT] and the play continues as G(θ, M, w).

19 Game rules (cont.) If φ = ( /i 1,..., i k )ψ, player E chooses a state v M. If (w, v) / R, the play has come to an end and E loses. Otherwise the play continues as G(ψ, M, v). If φ = ψ, player A chooses a state v M. If (w, v) / R, the play has come to an end and A loses. Otherwise the play continues as G(ψ, M, v). Note: The game rule for slashed diamonds ( /i 1,..., i k ) makes no use of the independence indication /i 1,..., i k. The independence indications have a role elsewhere: they regulate winning strategies available to E.

20 A strategy σ for E in game G(φ, M, w) is a tuple (σ 1,..., σ n ) of strategy functions, one strategy function for every token of an operator with existential force in φ: disjunctions, diamonds. The strategy σ is uniform if for every diamond token ( /i 1,..., i k ) in φ, the following holds: for any two plays which (a) are both admitted by σ, (b) at both of which a move corresponding to this diamond token must be made, and (c) differ at most in the choices made for the preceding modal operators identified by i 1,..., i k, this strategy yields the same choice.

21 Truth, falsity, non-determinacy The notions of truth and falsity are defined as follows: φ is true in M at w: there is a uniform winning strategy for E in G(φ, M, w). φ is false in M at w: there is a winning strategy for A in G(φ, M, w). We will see that there are pointed models (M, w) and formulas φ such that φ is in the above sense neither true nor false in M at w. We then say that φ is non-determined in M at w.

22 Example (2) Consider evaluating ( x 1 / )( q x 2 x 3 /1, 2)q at w in M: u 1 u 2 u 3 u 4 v 1 First, (σ 1, σ 2 ) is a w.s. for E in the relevant game: w v 2 σ 1 (v 1 ) = u 2 and σ 1 (v 2 ) = u 3. σ 2 (v 1, u 2 ) = x 2 and σ 1 (v 2, u 3 ) = x 2. Second, this strategy is indeed uniform: of the possible plays at which a choice for ( /1, 2) is made, it admits only (w, v 1, u 2 ) and (w, v 2, u 3 ). It maps them both to x 2.

23 Example (3) Let us return to the formula and the pointed model of Example (1). We may note that ( /1)p is non-determined in M at w. c 1 c 2 c 3 p p b 1 b 2 M It was already seen that ( a /1)p is not true in M at w. It is not false either. There are two possible strategies for A, namely choosing b 1 or choosing b 2. In the former case E may choose c 1 and win; in the latter case she may choose c 3 and win.

24 Example (4) Consider the model M: q u 1 u 2 q v 1 v 2 v 4 v 3 M Obviously M, w = ( However, M, w = (( w /1)q. /1)q ( /1)q). Here is a winning strategy (σ 1, σ 2 ) for E in the game corresponding to the latter formula: σ 1 (w, v 1 ) = LEFT = σ 1 (w, v 2 ) and σ 1 (w, v 3 ) = RIGHT = σ 1 (w, v 4 ) σ 2 (w, v 1,LEFT) = u 1 = σ 2 (w, v 2,LEFT) and σ 2 (w, v 3,RIGHT) = u 2 = σ 2 (w, v 4,RIGHT).

25 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

26 Expressive power Suppose the semantics of L and L are defined relative to pointed models. Logic L translatable into logic L (in symbols L L ) if for every φ L there is ψ φ L such that for all M and w: M, w = φ iff M, w = ψ φ. L is more expressive than L (in symbols L < L ) if L L but L L.

27 IFML compared with ML

28 ML < IFML Fact (1) ML < IFML. Proof. Let us go back to the structures of Example (1): c c 1 c 2 1 c 2 c 3 p p p b b 1 b 1 b 2 2 M a 1 ML IFML. If φ ML, it is translated by the result φ of replacing every by ( / ). (The standard semantics of ML coincides with the GTS of its translation by AC.) 2 M, a = ( /1)p but N, a = ( a /1)p. c 3 3 (M, a) and (N, a ) are bisimilar and so ML-equivalent. N

29 ...with Σ 1 1

30 IFML < Σ 1 1 Fact (2) IFML < Σ 1 1. Proof. 1 It is straightforward to translate IFML to IF first-order logic. Since the latter is equiexpressive with Σ 1 1, we have IFML Σ Let M and N be models with domains {1} resp. {1, 2} such that R M = =R N, Q M = and Q N = {2}. Hence M, 1 = ( Qx yqy) but N, 1 = ( Qx yqy). 3 Clearly IFML is invariant under generated submodels, so (M, 1) and (N, 1) are IFML-equivalent. 4 By (2) and (3), FO IFML and so a fortiori Σ 1 1 IFML.

31 ...with FO

32 IFML FO It is well known that ML FO. By contrast: Theorem (1) (T & Sevenster 2007) IFML FO. Idea of the proof: We discern a certain IFML formula χ, define for every n < ω a pointed model (M n, a), and show: 1 M n, a = χ iff n is even. 2 (M 2 n, a) and (M 2 n +1, a) are elementarily equivalent up to quantifier rank n + 1. Hence χ has no FO translation. For if it had one, say φ χ, let r be its quantifier rank. Then (M 2 r, a) and (M 2 r +1, a) are (r + 1) equivalent and in particular not distinguished by φ χ. Yet they are distinguished by χ.

33 IFML FO Let every (M n, a) consist of 4 layers, each layer with n points. As to how the points are related, here is the case of M 5 : e 1 d 5 e 5 d 4 e 4 c 4 d 3 c 5 b 5 b 4 b 1 a c 3 b 3 c 1 b 2 c 2 d 2 e 3 d 1 e 2

34 IFML FO Let χ be the following formula: ( ( /1)( /1, 3) ( e 1 d 5 /1)( /1, 3) ) e 5 d 4 e 4 c 4 d 3 c 5 b 5 b 4 b 1 a c 3 b 3 c 1 b 2 c 2 d 2 e 3 d 1 e 2

35 The fragment IFML 0 compared with FO, FO n and GF

36 IFML 0 Let IFML 0 be the very simple fragment of IFML obtained by only allowing formulas whose all diamonds are of the form ( / ) or ( /1) where 1 identifies a preceding box token. E.g., ( /1)p and (p (( are formulas, while are not. ( / )( / )( /1)p and ( /1)q ( / )( /1)r)) /1, 2)q IFML 0 is a notational variant of the logic L SD of T & Sevenster (AiML 2006).

37 IFML 0 < FO Fact (3) IFML 0 IFML. To see that IFML 0 FO, consider examples. Example (5) ( /1)p is translated by x(rx 0 x ( y/x)(rxy Py)) which is truth-equivalent to y x(rx 0 x (Rxy Py)). ( ( /1)p ( /1)p ) is translated by x(rx 0 x ( ( y/x)(rxy Py) ( z/x)(rxz Pz) ) which is truth-equivalent to y z x(rx 0 x ( (Rxy Py) (Rxz Pz) ).

38 IFML 0 < FO Lemma IFML 0 FO. Proof. If φ IFML 0, let φ be its translation into IF first-order logic, in which no two quantifiers carry the same variable. For any existential quantifier ( x/y) occurring in φ, the variable y is bound by a preceding universal quantifier y (such that between the two quantifiers there is no further quantifier). Erase each such quantifier ( x/y) in φ, and prefix the preceding universal quantifier y by x. If φ is the result of all these modifications, it is a first-order formula truth-equivalent to φ and therefore a first-order translation of φ.

39 IFML 0 < FO By the above Lemma and the fact that FO IFML (0), we have: Theorem (2) IFML 0 < FO. By Theorems (1) and (2), IFML IFML 0. So by the fact that IFML 0 IFML, we have: Corollary IFML 0 < IFML.

40 IFML 0 FO n Write FO n for the n-variable fragment of FO. It is well known that ML FO 2. Recall that we just saw that IFML 0 FO. However: Theorem (3) For all n < ω, IFML 0 FO n. Proof. Let n 2 be arbitrary and consider the formula (( } /1)... ( {{ /1) ). } n 1 times It is translated into FO n+1 by the formula z 1... z n 1 y(rx 0 y (Ryz 1... Ryz n 1 )) which can be shown not to be translatable into FO n.

41 IFML 0 GF Guarded fragment (GF) of FO: Atomic formulas (including identities). Closed under and. If ψ is a formula and G is an atomic formula with Free(ψ) Free(G), then x 1... x n (G ψ) is a formula. It is well known that ML GF. More generally, Andréka, van Benthem and Németi (1998) suggested that the distinguishing feature of the modal fragments of FO and notably their good computational properties is that quantifiers appear guarded in them. Postponing comments on the computational properties of IFML 0, we show:

42 IFML 0 GF Theorem (4) IFML 0 GF. Proof. The pointed models (M, a) and (M, a ) can be shown to be GF-equivalent. Yet the IFML 0 formula ( in M at a. c c 1 a c 2 b b 1 b 2 M a M /1) is true in M at a but not true

43 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

44 SAT and VAL Fact (4) For IFML 0, the satisfiability and validity problems are not each other s duals. First we may note that it is not the case that an arbitrary formula φ is valid iff φ is satisfiable. Suppose χ is non-determinate in M at w. [Cf. Ex. (3).] Then also χ and (χ χ) are non-determinate in M at w. Hence, while (χ χ) indeed is not satisfiable, (χ χ) is not valid. More generally, it can be shown that IFML 0 is not closed under contradictory negation: it is not the case that for every φ IFML 0 there is a formula neg(φ) IFML 0 such that for all N and v: N, v = neg(φ) iff N, v = φ. So, the relevant notion of duality is not syntactically expressible in IFML 0.

45 SAT, VAL and model-checking We recall that ML-SAT and ML-VAL are duals to each other and PSPACE-complete. We further recall that the (combined) model-checking complexity for ML is complete for PTIME. Theorem (5) (T & Sevenster 2006) (a) IFML 0 has strong finite model property; (b) IFML 0 -SAT is PSPACE-complete. Theorem (6) (ibid) IFML 0 -VAL is decidable in PSPACE-complete. Theorem (7) (ibid) The (combined) model-checking complexity for IFML 0 is complete for NPTIME.

46 Coming back to GF... By now we have seen that: 1 IFML 0 enjoys good computational properties. 2 It appears to deviate from ML sufficiently little to count as a modal logic. 3 It is translatable into FO. 4 It is not translatable into GF. [Thm. (4)] But if so, then the key factor explaining why modal logics have good computational properties is not that they are fragments of FO in which quantifiers appear guarded pace Andréka, van Benthem and Németi.

47 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

48 Positionality ML is positional in the following sense: 2 in order to evaluate φ in M at w, all that needs to be known are the state w and the truth-values of the immediate subformulas of φ at the successor states of w. Or in game-theoretic terms: if there is a w.s. for a player in game G(φ, M, w), there is a winning strategy for this player all of whose strategy functions are one-argument functions, taking as argument the state that has been introduced most recently. 2 Given the compositional semantics of ML, it suffices to note that ML is invariant under generated submodels.

49 Fact (5) IFML 0 and therefore IFML is not positional. Proof. Think of evaluating the formula ( d 1 d 2 c 1 c 2 /1) in M at a: b 1 b 2 a The formula is true; here is a non-positional w.s. σ for E: σ(a, b 1, c 1 ) = d 1 = σ(a, b 1, c 2 ) σ(a, b 2, c 2 ) = d 2 = σ(a, b 2, c 3 ) Let, then, τ be a positional strategy for E. By positionality its arguments are c 1, c 2, c 3. By the independence requirement it must be uniform. So actually τ must map all states to d 1 or all states to d 2. Hence it cannot be winning. c 3

50 Quasi-positionality There is, however, a generalization of the notion of positionality that IFML 0 enjoys. A strategy function for a diamond token ( /1) is quasi-positional if it is a one-argument function, taking as argument the most recent of those states that precede the box token referred to by the independence indication /1. A strategy function for a disjunction token or a diamond token ( / ) is quasi-positional simply if its value depends only on the most recent state introduced in the play. A strategy is quasi-positional if all its strategy function are. A logic is quasi-positional if the truth of any formula in a model at a state implies that there is a quasi-positional w.s. for E in the corresponding game.

51 Fact (6) IFML 0 is quasi-positional. Proof. The formulas of IFML 0 can be generated as the formulas α of the following grammar: α ::= p p (α α) (α α) β ::= ( α α β /1)α (α β) (β α) (β β) (α β) (β α) (β β), where p prop. Clearly in order to evaluate a formula β at w, it suffices to know w, its successors, and the successors of those successors, as well as the truth-values of the terms of the Boolean combination β at those states. This guarantees that the strategy functions for the ( /1) can be chosen to be quasi-positional and those for the disjunction symbols in the formulas β positional.

52 Failure of quasi-positionality for IFML Example (6) Consider the pointed model (M, a): e 1 e 2 1 The formula ( d 1 c 1 / )( d 2 b 1 c 2 a d 3 b 2 d 4 c 3 /1, 2) is true in M at a. 2 However, there is no w.s. for E whose strategy function for ( / ) is quasi-positional. Fact (7) IFML is not quasi-positional.

53 Outline 1 Syntactic vs. semantic scope 2 Syntax 3 Semantics 4 Expressivity 5 Complexity 6 Positionality 7 Comparison with HyLo

54 Like (versions of) IF modal logic, also (versions of) hybrid logic extend ML. For a systematic comparison, see T: Hybrid logic meets IF modal logic, JoLLI 18, Let us here take some linguistic examples. We use the following notation: H for always in the past F for sometimes in the t for at time t.

55 Consider the English sentence (+) John believed that Harry would leave uttered on a certain day at 8 o clock (= t 0 ). Here would is a morphological form of will in a SOT structure. Suppose believed is here construed as believed for the preceding 24 hours.

56 Think of readings of (+): (+) John believed that Harry would leave (1) H John believes that F Harry leaves. (2) H John believes t0 F Harry leaves. [indexical reading] Is (+) falsified in this scenario: at 6.15 John believes that Harry will leave at 9.30, while at 7.45 he believes that Harry s departure will take place at 8.30? Not under either of the readings (1) or (2).

57 (+) John believed that Harry would leave Suppose then that someone asks, upon hearing the utterance of (+) at t 0 : (Q) When did John believe that Harry would leave? If the question makes sense to the utterer, the intended reading of (+) is neither (1) nor (2). Instead, there is an existential presupposition according to which there is a time at which Harry will leave according to what John believed: (3) H John believes t0 (F/H) Harry leaves. [ uniform reading ]

58 Hybrid logic: temporal (interpretive) dependencies. IF modal logic: functional dependencies. Both types of dependence are present in modal semantics and, e.g., in natural language temporal discourse. The two types of independence are conceptually unrelated.

59 Consider whether the witness of F is functionally resp. interpretively dependent of H: (1) H John believes that F Harry leaves. (2) H John believes t0 F Harry leaves. (3) H John believes t0 (F/H) Harry leaves. (4) H John believes that (F/H) Harry leaves. (yes,yes) (yes,no) (no,no) (no,yes) Moral: The expressive resources that go together with the two logics (HyLo, IFML) are rather different.

60 References H. Andréka, J. van Benthem, and I. Németi (1998). Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27: Bradfield, J. (2000). Independence: logics and concurrency. In Lecture Notes in Computer Science, (Vol. 1862, pp ). London: Springer. Bradfield, J., & Fröschle, S. (2002). Independence-friendly modal logic and true concurrency. Nordic Journal of Computing, 9,

61 Hodges, W. (2007). Logics of Imperfect Information: Why Sets of Assignments? In J. van Benthem, D. Gabbay, & B. Löwe (Eds.), Interactive Logic, (Texts in Logic and Games Vol. 1, pp ). Amsterdam: Amsterdam University Press. Hyttinen, T., & Tulenheimo, T. (2005a). Decidability of IF modal logic of perfect recall. In R. Schmidt, I. Pratt-Hartmann, M. Reynolds, & H. Wansing (Eds.), Advances in Modal Logic, (Vol. 5, pp ). London: KCL Publications. Hyttinen, T., & Tulenheimo, T. (2005b). Decidability and undecidability results for some IF modal logics, unpublished manuscript. Sevenster, M. (2006). Branches of Imperfect Information. Ph.D. thesis, University of Amsterdam.

62 Tulenheimo, T. (2003). On IF modal logic and its expressive power. In P. Balbiani, N.-Y. Suzuki, F. Wolter, & M. Zakharyaschev (Eds.), Advances in Modal Logic, (Vol. 4, pp ). London: KCL Publications. Tulenheimo, T. (2004). Independence-Friendly Modal Logic: Studies in its Expressive Power and Theoretical Relevance. Ph.D. thesis, University of Helsinki. Tulenheimo, T. (2009). Hybrid Logic Meets IF Modal Logic, Journal of Logic, Language and Information 18(4): Tulenheimo, T., & Rebuschi, M. (2009). Equivalence criteria for compositional IF modal logics, Electronic Notes in Theoretical Computer Science 231:

63 Tulenheimo, T., & Sevenster, M. (2006). On modal logic, IF logic and IF modal logic. In G. Governatori, I. Hodkinson, & Y. Venema (Eds.), Advances in Modal Logic, (Vol. 6, pp ). London: College Publications. Tulenheimo, T., & Sevenster, M. (2007). Approaches to independence friendly modal logic. In J. van Benthem, D. Gabbay, & B. Löwe (Eds.), Interactive Logic, (Texts in Logic and Games Vol. 1, pp ). Amsterdam: Amsterdam University Press. Väänänen, J. (2007). Dependence Logic: A New Approach to Independence Friendly Logic. Cambridge: Cambridge University Press.