Modal logics and their semantics

Transcription

1 Modal logics and their semantics Joshua Sack Department of Mathematics and Statistics, California State University Long Beach California State University Dominguez Hills Feb 22, 2012

2 Relational structures Definition A finite relational structure is a pair (S,R), where S is a finite set R is a (binary) relation over S. related notions A relational structure is also known as a directed graph (graph theory) Kripke frame (modal logic) network (network theory) finite labelled transition system (computer science) automaton (computer science)

3 Logics that describe variations of graphs Two basic logics are First order logic first order logic describes a graph using quantifiers and relations. can provide a complete description of a finite graph the truths of the logic are generally undecidable (the answer to the question is φ true is some model/graph? cannot necessarily be answered in a finite number of time-steps). Modal logic modal logic describes properties of vertices in a graph using locally defined quantifiers can only express bisimulation equivalence classes of a graph (highly relevant in modeling computation) the logic is generally decidable

4 What is modal logic? Modal logic formulas ϕ are given by ϕ ::= p ϕ ϕ 1 ϕ 2 ϕ ϕ where p is an atomic proposition coming from a fixed set Φ is a formula representing falsity is negation is conjunction is necessity is possibility (disjunction) and (only if) are derived from and Example p p (related to transitivity)

5 Pointed models Whereas first-order logic describes properties of a model, modal logic describes properties of a pointed model Definition (Pointed Kripke model) A pointed Kripke model over a set Φ is a tuple ((S,R,V),s), where (S,R) is a directed graph (or Kripke frame ) V : Φ P(S) is a function assigning atomic propositions p Φ to subsets S. s S is the point The tuple (S,R,V) is called a Kripke model, and is a kind of directed vertex-labelled graph.

6 Semantics Truth is defined by a relation = between pointed models and formulas as follows: Remark M,s = p iff s V(p) M,s = (where = is the complement of =) M,s = ϕ iff M,s = ϕ M,s = ϕ ψ iff M,s = ϕ and M,s = ψ M,s = ϕ iff M,t = ϕ for all t {t srt} M,s = ϕ iff M,t = ϕ for some t {t srt}. Note: if {t srt} =, then M,s =, while M,s =

7 Example S = {s,t}, Φ = {p} s,p t, p t = s = p s = p s = p s = s =

8 Relationship to 1st order logic: Standard translation Definition (Standard translation) Let x be a first-order variable. The standard translation ST x mapping modal to first order formulas is defined as follows ST x (p) = Px (a predicate P for each proposition letter p.) ST x ( ) = x x (a symbol for inequality in 1st order lang.) ST x ( ϕ) = ST x (ϕ) ST x (ϕ ψ) = ST x (ϕ) ST x ψ ST x ( ϕ) = y(rxy ST y ϕ) ST x ( ϕ) = y(rxy ST y ϕ) Proposition For all M,s, and modal formulas ϕ, M,s = ϕ iff M = ST x (ϕ)[s].

9 Expressivity and Invariants s,p t,p For any modal formula, ϕ s = ϕ if and only if t = ϕ

10 Definition Bisimulation A bisimulation between models M = (S M,R M,V M ) and N = (S N,R N,V N ) is a relation Z S M S N obeying the constraint that if szt, then s V M (p) if and only if t V N (p) if sr M s then there exists t, such that tr N t and s Zt if tr N t then there exists s, such that sr M s and s Zt s Z t R R s t Z s Z t R R s t Z

11 Bisimulation example M s,p t,p A bisimulation between M and itself: the total Cartesian product Z = {(s,s),(s,t),(t,s),(t,t)}. N s,p t,p Largest bisimulation between N and itself: Z = {(s,s),(t,t)}. Largest bisimulation between M and N is Z =.

12 Relationship to 1st order logic: Bisimulation Let L be a first-order language with inequality unary predicates P from some set Φ, and a relation symbol R. Theorem (Van Benthem Characterization Theorem) A first-order formula α in L is invariant under bisimulation if and only if α = ST x (ϕ) for some modal formula ϕ. in other words, modal logic is the bisimulation invariant fragment of 1st order logic.

13 Definition (frame validity) Frame semantics A formula ϕ is valid in a pointed frame ((S,R),s), written (S,R),s = ϕ if and only if for all valuations V, (S,R,V),s = ϕ. A formula ϕ is valid in a frame (S,R), written (S,R) = ϕ if for all s S, (S,R),s = ϕ. Observation Modal logic with frame validity is a fragment monadic second order logic. This is because we quantify over valuations (subsets of the model). F,s = ϕ iff F = P 1... P n ST x (ϕ)[s] F = ϕ iff F = P 1... P n xst x (ϕ)

14 Definition (frame definability) Definability A modal formula ϕ defines a class of frames F if F = {F F = ϕ}. Examples F = p p iff F = xrxx (reflexivity) F = p p iff F = x y z(rxy Ryz Rxz) (transitivity) F = p p iff F = x y z(rxy Rxz Ryz) (Euclidean) But here is a modal formula that defines a class of frames that cannot be defined by a first-order logic formula: ( p p) p characterizes all frames where R is transitive and converse well-founded (there is no infinite path emanating from a point).

15 Goldblatt-Thomason Theorem Theorem A first order definable class of frames F is modally definable if and only if it is closed under each of the following: bounded morphic images generated subframes disjoint unions ultrafilter extensions. Definition (bounded morphic image) A bounded morphism from (S,R) to (S,R ) is a function f : S S, such that both of the following hold: srt implies f(s)r f(t) if f(s)r t, then there exists t, such that srt and t = f(t).

16 Goldblatt-Thomason Theorem Theorem A first order definable class of frames F is modally definable if and only if it is closed under each of the following: bounded morphic images generated subframes disjoint unions ultrafilter extensions. Definition (bounded morphic image) A bounded morphism from (S,R) to (S,R ) is a function f : S S, such that both of the following hold: srt implies f(s)r f(t) if f(s)r t, then there exists t, such that srt and t = f(t).

17 Generated submodel Definition (Generated submodel) (S,R) is a generated subframe of (S,R ) if S S, R R, and (s S and sr t) implies (t S and srt) Example a b e c d

18 Ultrafilter extensions Definition (Ultrafilter) An ultrafilter is a set F of sets in S, such that F and A F and A B implies B F (closed under superset) A,B F implies A B F (closed under finite intersection) A F or A c F (either a set or its complement is in F. Definition (Ultrafilter extension) Given a frame (S,R) its ultrafilter extension is the frame (S,R ), where S is the set of ultrafilters on S FR F if and only if {s s, srs, s X} F whenever X F.

19 Graded modal logic is given by Expressing the out-degree ϕ ::= p ϕ ϕ 1 ϕ 2 n ϕ n ϕ M,s = n ϕ if and only if for all distinct t 1,...,t n {t srt} for some i, M,t i = ϕ. M,s = n ϕ if and only if for some distinct t 1,...,t n {t srt} for all i, M,t i = ϕ. Note that n ϕ is equivalent to n ϕ ( n, n are duals) 1 ϕ is equivalent to ϕ 1 ϕ is equivalent to ϕ The out-degree is n can be expressed as n n+1.

20 Graded modal logic is given by Expressing the out-degree ϕ ::= p ϕ ϕ 1 ϕ 2 n ϕ n ϕ M,s = n ϕ if and only if for all distinct t 1,...,t n {t srt} for some i, M,t i = ϕ. M,s = n ϕ if and only if for some distinct t 1,...,t n {t srt} for all i, M,t i = ϕ. Note that n ϕ is equivalent to n ϕ ( n, n are duals) 1 ϕ is equivalent to ϕ 1 ϕ is equivalent to ϕ The out-degree is n can be expressed as n n+1.

21 Example M s,p t,p The formula 2 is false at s but true at t.

22 Additional property of graded modal logic Satisfiability of graded modal logic is decidable, The complexity of the satisfiability problem is PSPACE-complete (as is ordinary modal logic).

23 Weighted edges Weighted modal logic is given by ϕ ::= p ϕ ϕ 1 ϕ 2 a ϕ a ϕ Replace R with directed weighted edges R q with q Q. For t S, let P s (t) = q where sr q t For T S, let P s (T) = {t T q. sr qt} P s(t). Then M,s = a ϕ if and only if for all T S, such that P s (T) a, for some t T, M,t = ϕ. M,s = a ϕ if and only if for some T S, such that P s (T) a, for all t T, M,t = ϕ. If q = 1 always, then this is graded modal logic. If P s (S) = 1 always, then this is modal probability logic: a ϕ is read the probability of ϕ is at least a

24 Weighted edges Weighted modal logic is given by ϕ ::= p ϕ ϕ 1 ϕ 2 a ϕ a ϕ Replace R with directed weighted edges R q with q Q. For t S, let P s (t) = q where sr q t For T S, let P s (T) = {t T q. sr qt} P s(t). Then M,s = a ϕ if and only if for all T S, such that P s (T) a, for some t T, M,t = ϕ. M,s = a ϕ if and only if for some T S, such that P s (T) a, for all t T, M,t = ϕ. If q = 1 always, then this is graded modal logic. If P s (S) = 1 always, then this is modal probability logic: a ϕ is read the probability of ϕ is at least a

25 Probability logic example 1.6 s,p t, p.4 a ϕ is read the probability of (transitioning to a state where) ϕ is at least a. a ϕ is read the probability of (transitioning to a state where) ϕ is less than a. s =.4 p.4 p s =.6 p s =.4 1 p (higher-order probabilities)

26 Linear combinations in probability logic Additivity conditions are prevalent in probability: some probability logics make linear combinations explicit as terms t Here we can express additivity: ϕ ::= p ϕ ϕ 1 ϕ 2 t a t ::= ap(ϕ) t 1 +t 2 P(ϕ ψ)+p(ϕ ψ) P(ϕ) = 0 where t = 0 is an abbreviation for (t 0) ( t 0).

27 Linear combinations in probability logic Additivity conditions are prevalent in probability: some probability logics make linear combinations explicit as terms t Here we can express additivity: ϕ ::= p ϕ ϕ 1 ϕ 2 t a t ::= ap(ϕ) t 1 +t 2 P(ϕ ψ)+p(ϕ ψ) P(ϕ) = 0 where t = 0 is an abbreviation for (t 0) ( t 0).

28 Probability logic with linear combinations example 1.6 s,p t, p.4 s = P(p).4 P( p).4 s = P(p).6 s = P(P(p) 1).4 (higher-order probabilities) s = P(p) P( p).2 s = P(p)+P( p).2 The last two examples imply that P(p) P( p) =.2.

29 Decidability Satisfiability of modal probability logic is decidable, The complexity of the satisfiability problem is PSPACE-complete (as is ordinary and graded modal logics).

30 Topological semantics Replace the relation with a topology T. Topological model: (S,T,V) M,s = ϕ if and only if for some O T, such that s O for all t O, M,t = ϕ ϕ is true in the interior of the set {t t = ϕ} M,s = ϕ if and only if for all O T, such that s O for some t O, M,t = ϕ. ϕ is true in the closure of the set {t t = ϕ} Remark Note that the quantifier pattern here is the opposite of those used for weighted modal logic (including graded modal logic and modal probability logic).

31 Topological semantics Replace the relation with a topology T. Topological model: (S,T,V) M,s = ϕ if and only if for some O T, such that s O for all t O, M,t = ϕ ϕ is true in the interior of the set {t t = ϕ} M,s = ϕ if and only if for all O T, such that s O for some t O, M,t = ϕ. ϕ is true in the closure of the set {t t = ϕ} Remark Note that the quantifier pattern here is the opposite of those used for weighted modal logic (including graded modal logic and modal probability logic).

32 Neighborhood semantics Let a neighborhood model be M = (S,R,V) where R S P(S). Let N s = {T S srt}. M,s = ϕ if and only if {t S M,t = ϕ} N s. M,s = ϕ if and only if {t S M,t = ϕ} N s. If N s is upward closed (T T and T N s implies T N s ), then M,s = ϕ if and only if for some T N s for all t T, M,t = ϕ. M,s = ϕ if and only if for all T N s for some t T, M,t = ϕ. Remark In the case of topological semantics, N s includes every set with s in its interior.

33 Neighborhood semantics Let a neighborhood model be M = (S,R,V) where R S P(S). Let N s = {T S srt}. M,s = ϕ if and only if {t S M,t = ϕ} N s. M,s = ϕ if and only if {t S M,t = ϕ} N s. If N s is upward closed (T T and T N s implies T N s ), then M,s = ϕ if and only if for some T N s for all t T, M,t = ϕ. M,s = ϕ if and only if for all T N s for some t T, M,t = ϕ. Remark In the case of topological semantics, N s includes every set with s in its interior.

34 Neighborhood semantics Let a neighborhood model be M = (S,R,V) where R S P(S). Let N s = {T S srt}. M,s = ϕ if and only if {t S M,t = ϕ} N s. M,s = ϕ if and only if {t S M,t = ϕ} N s. If N s is upward closed (T T and T N s implies T N s ), then M,s = ϕ if and only if for some T N s for all t T, M,t = ϕ. M,s = ϕ if and only if for all T N s for some t T, M,t = ϕ. Remark In the case of topological semantics, N s includes every set with s in its interior.

35 Generalizations Generalization of weighted modal logic If N s consists of all sets whose weighted edges from s add to at least a, then ϕ in the neighborhood semantics is equivalent to a ϕ in the weighed modal logic semantics. Recall that weighed modal logic generalizes graded modal logic and modal probability logic. Recall furthermore that graded modal logic generalizes standard relational modal logic. Generalization of standard relational modal logic If N s consists of all supersets of the the {t srt}, then ϕ is the same in both the neighborhood semantics and in the standard relational model semantics.

36 Generalizations Generalization of weighted modal logic If N s consists of all sets whose weighted edges from s add to at least a, then ϕ in the neighborhood semantics is equivalent to a ϕ in the weighed modal logic semantics. Recall that weighed modal logic generalizes graded modal logic and modal probability logic. Recall furthermore that graded modal logic generalizes standard relational modal logic. Generalization of standard relational modal logic If N s consists of all supersets of the the {t srt}, then ϕ is the same in both the neighborhood semantics and in the standard relational model semantics.

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