Introduction to Neighborhood Semantics for. Modal Logic. ILLC, University of Amsterdam. January 7, 2007

Transcription

1 Introduction to Neighborhood Semantics for Modal Logic Eric Pacuit January 7, 2007 ILLC, University of Amsterdam staff.science.uva.nl/ epacuit

2 Introduction 1. Motivation 2. Neighborhood Semantics for Modal Logic 3. Brief Survey of Results 4. Bisimulations

3 Game Forcing Operator Let G be an extensive game and s a node in G. We say an agent i can force a formula φ at s (written s = iφ) provided 1. i can move at s 2. there is a strategy for i such that for all strategies chosen by the other players, φ will become true.

4 Game Forcing Operator A L R B B l r l r q q p p p

5 Game Forcing Operator A L R B B l r l r p q q p p root = Ap

6 q Game Forcing Operator A L R B B l r l r q p p p root = Aq

7 Game Forcing Operator A L R B B l r l r q q p p p root = A(p q)

8 Review: (Propositional) Modal Logic Basic Modal Language: φ := p φ φ φ φ, where p Φ0. Relational Semantics: A Kripke model is a tuple W, R, V where R W W and V : Φ0 2 W.

9 Review: (Propositional) Modal Logic Basic Modal Language: φ := p φ φ φ φ, where p Φ0. Relational Semantics: A Kripke model is a tuple W, R, V where R W W and V : Φ0 2 W. Let R(w) = {v wrv} Truth at a state: Let M = W, R, V be a model with w W, M, w = φ i R(w) (φ) M where (φ) M is the truth set of φ.

10 Review: (Propositional) Modal Logic Given a relational frame W, R

11 Review: (Propositional) Modal Logic Given a relational frame W, R The set of necessary propositions: Nw = {X R(w) X}

12 Review: (Propositional) Modal Logic Given a relational frame W, R The set of necessary propositions: Nw = {X R(w) X} Fact: Nw is closed under intersections, supersets and contains the unit and contains a minimal element: w, Nw Nw

13 Other Examples Fix t [0, 1]: Intended Interpretation of φ: φ is assigned (subjective) probability > t Fact: φ ψ (φ ψ) is not valid under this interpretation.

14 Other Examples Fix t [0, 1]: Intended Interpretation of φ: φ is assigned (subjective) probability > t Fact: φ ψ (φ ψ) is not valid under this interpretation. Other examples: Concurrent Propositional Dynamic Logic, Parikh's Game Logic, or Pauly's Coalition Logic, Alternating-time Temporal Logic Extensive literature on the logical omniscience problem, Deontic Logics

15 Neighborhood Semantics for Propositional Modal Logic A Neighborhood Frame is a tuple W, N where N : W 2 2W A Neighborhood Model is a tuple W, N, V where V : Φ0 2 W Truth in a model is dened as follows M, w = p i w V (p) M, w = φ i M, w = φ M, w = φ ψ i M, w = φ and M, w = ψ M, w = φ i (φ) M N(w)

16 Some History Neighborhood Models were rst discussed in (Scott 1970, Montague 1970) perhaps (McKinsey and Tarski 1944) should be cited? See (Segerberg 1971) and (Chellas 1980) for discussions of neighborhood semantics for propositional modal logics.

17 Non-normal Modal Logics E φ φ RE φ ψ φ ψ M (φ ψ) ( φ ψ) C ( φ ψ) (φ ψ) N K (φ ψ) ( φ ψ)

18 Non-normal Modal Logics E φ φ RE φ ψ φ ψ M (φ ψ) ( φ ψ) C ( φ ψ) (φ ψ) N K (φ ψ) ( φ ψ) Classical Modal Logics: E = E + RE + P C, EM = E + M, EC = E + C, etc.

19 Non-normal Modal Logics E φ φ RE φ ψ φ ψ M (φ ψ) ( φ ψ) C ( φ ψ) (φ ψ) N K (φ ψ) ( φ ψ) Classical Modal Logics: E = E + RE + P C, EM = E + M, EC = E + C, etc. Fact: K = EMCN

20 Constraints on neighborhood frames Monotonic or Supplemented: If X Y N(w), then X N(w) and Y N(w) Closed under nite intersections: If X N(w) and Y N(w), then X Y N(w) Contains the unit: W N(w) Augmented: Supplemented plus for each w W, N(w) N(w)

21 Denability Results 1. F = (φ ψ) φ ψ i F is closed under supersets (monotonic frames). 2. F = φ ψ (φ ψ) i F is closed under nite intersections. 3. F = i F contains the unit 4. F = EMCN i F is a lter 5. F = φ φ i for each w W, w N(w) 6. And so on...

22 Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under nite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are lters K is sound and strongly complete with respect to the class of all augmented neighborhood frames

23 Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under nite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are lters K is sound and strongly complete with respect to the class of all augmented neighborhood frames

24 Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under nite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are lters K is sound and strongly complete with respect to the class of all augmented neighborhood frames

25 Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under nite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are lters K is sound and strongly complete with respect to the class of all augmented neighborhood frames

26 Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under nite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are lters K is sound and strongly complete with respect to the class of all augmented neighborhood frames

27 Completeness Results E is sound and strongly complete with respect to the class of all neighborhood frames EM is sound and strongly complete with respect to the class of all monotonic neighborhood frames EC is sound and strongly complete with respect to the class of all neighborhood frames that are closed under nite intersections EN is sound and strongly complete with respect to the class of all neighborhood frames that contain the unit K is sound and strongly complete with respect to the class of all neighborhood frames that are lters K is sound and strongly complete with respect to the class of all augmented neighborhood frames

28 Questions What is the precise connection between neighborhood semantics for modal logic and relational semantics for modal logic? What is the expressive power of the basic modal language over neighborhood frames?

29 Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V, and vice versa (see (Chellas, 1980) for more information). There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 200?).

30 Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V, and vice versa (see (Chellas, 1980) for more information). There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 200?).

31 Some Results For each Kripke model W, R, V, there is an pointwise equivalent augmented neighborhood model W, N, V, and vice versa (see (Chellas, 1980) for more information). There are logics which are complete with respect to a class of neighborhood frames but not complete with respect to relational frames (Gabbay 1975, Gerson 1975, Gerson 1976). There are logics incomplete with respect to neighborhood frames (Martin Gerson, 1975; Litak, 200?).

32 Kracht-Wolter Translation Given a neighborhood model M = W, ν, V, dene a Kripke model M = V, R, R, Rν, P t, V as follows: V = W 2 W R = {(v, w) w W, v 2 W, v w} R = {(v, w) w W, v 2 W, v w} Rν = {(w, v) w W, v 2 W, v ν(w)} P t = W

33 Kracht-Wolter Translation Let L be the language φ := p φ φ ψ [ ]φ [ ]φ [ν]φ Pt where p At and Pt is a unary modal operator. Dene ST : LNML L as follows ST (p) = p ST ( φ) = ST (φ) ST (φ ψ) = ST (φ) ST (φ) ST ( φ) = ν ([ ]ST (φ) [ ] ST (φ))

34 Kracht-Wolter Translation Theorem For each neighborhood model M = W, ν, V and each formuala φ LNML, for any w W, M, w = φ i M, w = Pt ST (φ) (The translation is simpler if monotonicity is assumed)

35 Expressive Power of Modal Logic (w.r.t. Relational Frames) Van Benthem Characterization Theorem On the class of Kripke Structures, Modal Logic is the bisimulation invariant fragment of rst-order logic.

36 Bisimulations for Neighborhood Structures First Attempt: Let M = W, N, V and M = W, N, V be two neighborhood structures and s W and t W. A non-empty relation Z W W is a bisimulation between M and M if (prop) If wzw then w and w satisfy the same formulas (back) If wzw and X N(w) then there is a X W such that X N (w ) and x X x X such that xzx (forth) If wzw and X N (w ) then there is a X W such that X N(w) and x X x X such that xzx

37 Bisimulations for Neighborhood Structures First Attempt: Let M = W, N, V and M = W, N, V be two neighborhood structures and s W and t W. A non-empty relation Z W W is a bisimulation between M and M if (prop) If wzw then w and w satisfy the same formulas (back) If wzw and X N(w) then there is a X W such that X N (w ) and x X x X such that xzx (forth) If wzw and X N (w ) then there is a X W such that X N(w) and x X x X such that xzx

38 Bisimulations for Neighborhood Structures First Attempt: Let M = W, N, V and M = W, N, V be two neighborhood structures and s W and t W. A non-empty relation Z W W is a bisimulation between M and M if (prop) If wzw then w and w satisfy the same formulas (back) If wzw and X N(w) then there is a X W such that X N (w ) and x X x X such that xzx (forth) If wzw and X N (w ) then there is a X W such that X N(w) and x X x X such that xzx Only works for monotonic modal logics

39 Bounded Morphism Let M1 = W1, N1, V1 and M2 = W2, N2, V2 be two neighborhood models. A bounded morphism from M1 to M2 is a map f : W1 W2 such that for all X W2 and w W1, f 1 [X] N1(w) i X N2(f(w)) and for all p, w V1(p) i f(w) V2(p)

40 Bounded Morphism Let M1 = W1, N1, V1 and M2 = W2, N2, V2 be two neighborhood models. A bounded morphism from M1 to M2 is a map f : W1 W2 such that for all X W2 and w W1, f 1 [X] N1(w) i X N2(f(w)) and for all p, w V1(p) i f(w) V2(p) Lemma Let M1 = W1, N1, V1 and M2 = W2, N2, V2 be two neighborhood models and f : W1 W2 a bounded morphism. Then for each modal formula φ LNML and state w W1, M1, w = φ i M2, f(w) = φ

41 Behavorial Equivalence Two model-state pairs M1, w1 and M2, w2 are behavorially equivalent provided there is a neighborhood model N = W, N, V such that there are boudned morphisms from f from M1 to N and g from M2 to N and f(w1) = g(w2).

42 Two-Sorted First-Order Language for Neighborhood Structures View M = V, R, R, Rν, P t, V as a 2-sorted rst-order structure. Let L 2 be a two-sorted rst-order language (point variables and set variables) Fact: First-order structures that are generated by neighborhood structures (i.e., of the form M for some neighborhood structure M) can be axiomatized (in L 2 ).

43 Standard Translation We map formulas of the basic modal language to L 2 : stx(p) = P x stx( φ) = stx(φ) stx(φ ψ) = stx(φ) stx(ψ). stx( φ) = u(xrνu ( z(ur z stz(φ)) z (ur z stz (φ)) STx(φ) = W x stx(φ)

44 Characterization Theorem for Classical Modal Logic Theorem (Pauly) On the class of neighborhood models, monotonic modal logic is the monotonic bisimulation invariant fragment of L 2

45 Characterization Theorem for Classical Modal Logic Theorem (Pauly) On the class of neighborhood models, monotonic modal logic is the monotonic bisimulation invariant fragment of L 2 Theorem On the class of neighbourhood models, modal logic is the behavioural equivalence-invariant fragment of L 2. Joint work with Helle Hvid Hansen and Clemens Kupke

46 Conclusions Model theory of modal logic with respect to neighborhood structures Monotonic Modal Logics have been studied (eg. Hansen, 2003) Large literature on the topological interpretation of modal logic Modal Language for Topology: Expressivity and Denability, B. ten Cate, D. Gabelaia and V. Sustretov, 2006.

47 Thank you.