Topologies on pseudoinnite paths

Transkriptio

1 Topologies on pseudoinnite paths Andrey Kudinov Institute for Information Transmission Problems, Moscow National Research University Higher School of Economics, Moscow Moscow Institute of Physics and Technology June 14, 216

2 PLAN 1. Language 2. Topological semantics 3. Products of logics 4. Neighborhood frames 5. Without seriality 6. Dense topological spaces 7. Logic S5 8. Ideas of the proof

3 Language and logics φ ::= p φ φ φ iφ, i = 1, 2., and i are expressible in the usual way. Normal modal logic. K n denotes the minimal normal modal logic with n modalities and K = K 1. L 1 and L 2 two modal logics with one modality then the fusion of these logics is dened as L 1 L 2 = K 2 + L 1 + L 2 ; where L i is the set of all formulas from L i where in all formulas is replaced by i.

4 Topological semantics We can dene topology on set X by specifying a closure operator C : 2 X 2 X, satisfying the Kuratowski axioms: 1. C( ) =, 2. A C(A), for A X, p p 3. C(A B) = C(A) C(B), for A, B X, (p q) p q 4. C ( C(A) ) C(A). A X, p p Logic S4 In topological semantics for modal logic closure operator correspond to. Topological model (X, θ), where X = (X, C) topological space: p θ(p) X θ(φ ψ) = θ(φ) θ(ψ) θ( φ) = X \ θ(φ) θ( φ) = C(θ(φ)). X = φ θ(θ(φ) = X), Log(X) = {φ X = φ}.

5 Topological semantics We can dene topology on set X by specifying a closure operator C : 2 X 2 X, satisfying the Kuratowski axioms: 1. C( ) =, 2. A C(A), for A X, p p 3. C(A B) = C(A) C(B), for A, B X, (p q) p q 4. C ( C(A) ) C(A). A X, p p Logic S4 In topological semantics for modal logic closure operator correspond to. Topological model (X, θ), where X = (X, C) topological space: p θ(p) X θ(φ ψ) = θ(φ) θ(ψ) θ( φ) = X \ θ(φ) θ( φ) = C(θ(φ)). X = φ θ(θ(φ) = X), Log(X) = {φ X = φ}.

6 Alexandro topology On a transitive reexive Kripke frame F = (W, R) we can dene topology, i.e. closure operator: C F (A) = R 1 (A). It will be an Alexandro topology (any intersection of open sets is open, all points have minimal neighborhood). Lemma F = φ (W, C F ) = φ. We dene T op(f ) = (W, C F ) Completeness of S4 w.r.t. all Alexandro spaces. Many topologies are not Alexandro: R n, Cantor space, Q or any metric space.

7 Alexandro topology On a transitive reexive Kripke frame F = (W, R) we can dene topology, i.e. closure operator: C F (A) = R 1 (A). It will be an Alexandro topology (any intersection of open sets is open, all points have minimal neighborhood). Lemma F = φ (W, C F ) = φ. We dene T op(f ) = (W, C F ) Completeness of S4 w.r.t. all Alexandro spaces. Many topologies are not Alexandro: R n, Cantor space, Q or any metric space.

8 Alexandro topology On a transitive reexive Kripke frame F = (W, R) we can dene topology, i.e. closure operator: C F (A) = R 1 (A). It will be an Alexandro topology (any intersection of open sets is open, all points have minimal neighborhood). Lemma F = φ (W, C F ) = φ. We dene T op(f ) = (W, C F ) Completeness of S4 w.r.t. all Alexandro spaces. Many topologies are not Alexandro: R n, Cantor space, Q or any metric space.

9 Derivational semantics We can dene topological space using derivative operator d : 2 X 2 X, where d(a) is the set of all limit points of A. In a similar way we dene derivational semantics: θ( φ) = d(θ(φ)) The logic of all topological space is wk4 = K + p p p (Esakia'1981). The logic of Q, Cantor space (or any dense-in-itself zero-dimensional metric space) is D4 = K + p p + (Shehtman'199).

10 The product of Kripke frames For two frames F 1 = (W 1, R 1) and F 2 = (W 2, R 2) F 1 F 2 = (W 1 W 2, R 1, R 2), where (a 1, a 2)R 1(b 1, b 2) a 1R 1b 1 & a 2 = b 2 (a 1, a 2)R 2(b 1, b 2) a 1 = b 1 & a 2R 2b 2 For two logics L 1 and L 2 L 1 L 2 = Log({F 1 F 2 F 1 = L 1 & F 2 = L 2}) (Shehtman, 1978) For two classes of frames F 1 and F 2 Log({F 1 F 2 F 1 F 1 & F 2 F 2}) Log(F 1) Log(F 2) p 1 2p + 1 2p 2 1p. K K = K K + 1 2p 1 2p + 1 2p 2 1p S4 S4 = S4 S p 1 2p + 1 2p 2 1p.

11 The product of topological spaces (van Benthem et al, 25) For two topological space X 1 = (X 1, τ 1) and X 2 = (X 2, τ 2) X 1 X 2 = (X 1 X 2, τ 1, τ 2 ), where τ 1 has base {U 1 x 2 U 1 τ 1 & x 2 X 2} τ 2 has base {x 1 U 2 x 1 X 1 & U 2 τ 2}

12 The product of topological spaces (van Benthem et al, 25) For two topological space X 1 = (X 1, τ 1) and X 2 = (X 2, τ 2) X 1 X 2 = (X 1 X 2, τ 1, τ 2 ), where τ 1 has base {U 1 x 2 U 1 τ 1 & x 2 X 2} y τ 2 has base {x 1 U 2 x 1 X 1 & U 2 τ 2} x

13 The product of topological spaces (van Benthem et al, 25) For two topological space X 1 = (X 1, τ 1) and X 2 = (X 2, τ 2) X 1 X 2 = (X 1 X 2, τ 1, τ 2 ), where τ 1 has base {U 1 x 2 U 1 τ 1 & x 2 X 2} For two logics L 1 and L 2 τ 2 has base {x 1 U 2 x 1 X 1 & U 2 τ 2} L 1 t L 2 = Log({X 1 X 2 X 1 = L 1 & X 2 = L 2} S4 t S4 = Log(Q Q) = S4 S4 (van Benthem et al, 25) Log(R R) S4 S4 (Kremer, 21?) Log(Cantor Cantor) S4 S4 d-logic of product of topological spaces was considered by L. Uridia (211). Log d (Q Q) = D4 D4 Generalization to neighborhood frames was done by K. Sano (211).

14 Known results Theorem (212) Let L 1 and L 2 be from the set {D, T, D4, S4} then L 1 n L 2 = L 1 L 2. Not straightforward but still a Corollary In derivational semantics 1. D4 d D4 = D4 D4. 2. [Uridia'211] Log d (Q Q) = D4 D4

15 Topological semantics based on closure operator or derivative operator can be generalized in the neighborhood semantics. We can consider neighborhood function τ : X 2 2X. For x X τ(x) is a set of neighborhoods of x. It connected with C is the following way: A τ(x) x I(A), where I(A) = X \ C(X \ A). And for derivational semantics A τ(x) x d(a), where d(a) = X \ d(x \ A).

16 Neighborhood frames A (normal) neighborhood frame (or an n-frame) is a pair X = (X, τ), where X ; τ : X 2 2X, such that τ(x) is a lter on X; τ neighborhood function of X, τ(x) neighborhoods of x. Filter on X: nonempty F 2 X such that 1) U F & U V V F 2) U, V F U V F (lter base) The neighborhood model (n-model) is a pair (X, V ), where X = (X, τ) is a n-frame and V : P V 2 X is a valuation. Similar: neighborhood 2-frame (n-2-frame) is (X, τ 1, τ 2) such that τ i is a neighborhood function on X for each i. Validity in model: M, x = iψ V τ i(x) y V (M, y = ψ). M = ϕ X = ϕ X = L Log(C) = {ϕ X = ϕ for some X C} nv (L) = {X X is an n-frame and X = L}

17 Connection with Kripke frames Denition Let F = (W, R) be a Kripke frame. We dene neighborhood frame N (F ) = (W, τ) as follows. For any w W τ(w) = {U R(w) U W }. Lemma Let F = (W, R) be a Kripke frame. Then Log(N (F )) = Log(F ).

18 Bounded morphism for n-frames Denition Let X = (X, τ 1,...) and Y = (Y, σ 1,...) be n-frames. Then function f : X Y is a bounded morphism if 1. f is surjective; 2. for any x X and U τ i(x) f(u) σ i(f(x)); 3. for any x X and V σ i(f(x)) there exists U τ i(x), such that f(u) V. In notation f : X Y. Lemma If f : X Y then Log(Y) Log(X).

19 Not always fusion It is not the case for logic K! Lemma For any two n-frames X 1 and X 2 X 1 X 2 = And even more, for any closed 1-free formula φ and any closed 2-free formula ψ X 1 X 2 = φ 1φ, X 1 X 2 = ψ 2ψ. Proof. X 1 X 2, (x, y) = 1 τ 1(x, y) τ 1(x) y X 2 ( τ 1(x, y )) y X 2 (X 1 X 2, (x, y ) = 1 ) = X 1 X 2, (x, y) = 2 1. Hence, X 1 X 2 =

20 Not always fusion It is not the case for logic K! Lemma For any two n-frames X 1 and X 2 X 1 X 2 = And even more, for any closed 1-free formula φ and any closed 2-free formula ψ X 1 X 2 = φ 1φ, X 1 X 2 = ψ 2ψ. Proof. Since ψ does not contain neither 2, nor variables, its value does not depend on the second coordinate. Let F = X 1 X 2. So F, (x, y) = ψ, then y (F, (x, y ) = ψ), hence, F, (x, y) = 2ψ.

21 Not always fusion Lemma For any two n-frames X 1 and X 2 X 1 X 2 = And even more, for any closed 1-free formula φ and any closed 2-free formula ψ X 1 X 2 = φ 1φ, X 1 X 2 = ψ 2ψ. Denition For two unimodal logics L 1 and L 2, we dene L 1, L 2 = L 1 L 2 +, where = {φ 2φ φ is closed and 2-free} {ψ 1ψ ψ is closed and 1-free}. Lemma For any two normal modal logics L 1 and L 2 L 1, L 2 L 1 n L 2. Note that if L 1 L 2 then L 1 L 2 =.

22 Cantor space and innite paths Standart construction: Cantor space as the set on innite paths on innite binary tree T 2. The base of topology is the sets of the following type: where α and β are two innite paths: U m(α) = {β a 1 = b 1,... a m = b m}. α = a 1a 2a 3..., β = b 1b 2b In order to proof completeness of S4 w.r.t. Cantor space we need to construct p-morphism from Cantor space to arbitrary nite S4-frame. [Mints, 1998]

23 Cantor space and innite paths Standart construction: Cantor space as the set on innite paths on innite binary tree T 2. The base of topology is the sets of the following type: where α and β are two innite paths: U m(α) = {β a 1 = b 1,... a m = b m}. α = a 1a 2a 3..., β = b 1b 2b In order to proof completeness of S4 w.r.t. Cantor space we need to construct p-morphism from Cantor space to arbitrary nite S4-frame. [Mints, 1998]

24 Cantor space and innite paths Standart construction: Cantor space as the set on innite paths on innite binary tree T 2. The base of topology is the sets of the following type: where α and β are two innite paths: U m(α) = {β a 1 = b 1,... a m = b m}. α = a 1a 2a 3..., β = b 1b 2b In order to proof completeness of S4 w.r.t. Cantor space we need to construct p-morphism from Cantor space to arbitrary nite S4-frame. [Mints, 1998]

25 Constructing dense topologies from Kripke frames We need to construct a dense topological space based on a Kripke frame. This becomes important in studying of products of topological spaces.

26 Example r q h i j k l m n p d e f g b c a

27 Example r q h i j k l m n p d e f g b c a a bej ω

28 Example r q h i j k l m n p d e f g b c b a a bej ω

29 Example r q h i j k l m n p d e f g e b c b a a bej ω

30 Example r q h i j k l m n p j d e f g e b c b a a bej ω

31 Example r q h i j k l m n p j d e f g e b c b a a bej ω a bej ω

32 Example r q h i j k l m n p j d e f g e b c b a a bej ω a bej ω

33 Example r q h i j k l m n p j d e f g e b c b a a bej ω a bej

34 Example r q h i j k l m n p j d e f g e b c b a α = a bej ω f F (α) = j

35 N ω (F ) Denition Sets U n(α) form a lter base. So we can dene τ(α) the lter with base {U n(α) n N} ; N ω(f ) = (W ω, τ) is a dense n-frame based on F. Frame N ω(f ) is dense in a sense that the intersection of all neighborhoods of a point is empty. So, there are no minimal neighborhoods unlike in T op(f ). Lemma Let F = (W, R) be a Kripke frame with root a, then f F : N ω(f ) N (F ). Corollary For any frame F Log(N ω(f )) Log(N (F )) = Log(F ).

36 N ω (F ) Denition Sets U n(α) form a lter base. So we can dene τ(α) the lter with base {U n(α) n N} ; N ω(f ) = (W ω, τ) is a dense n-frame based on F. Frame N ω(f ) is dense in a sense that the intersection of all neighborhoods of a point is empty. So, there are no minimal neighborhoods unlike in T op(f ). Lemma Let F = (W, R) be a Kripke frame with root a, then f F : N ω(f ) N (F ). Corollary For any frame F Log(N ω(f )) Log(N (F )) = Log(F ).

37 N ω (F ) Denition Sets U n(α) form a lter base. So we can dene τ(α) the lter with base {U n(α) n N} ; N ω(f ) = (W ω, τ) is a dense n-frame based on F. Frame N ω(f ) is dense in a sense that the intersection of all neighborhoods of a point is empty. So, there are no minimal neighborhoods unlike in T op(f ). Lemma Let F = (W, R) be a Kripke frame with root a, then f F : N ω(f ) N (F ). Corollary For any frame F Log(N ω(f )) Log(N (F )) = Log(F ).

38 N ω (F ) Denition Sets U n(α) form a lter base. So we can dene τ(α) the lter with base {U n(α) n N} ; N ω(f ) = (W ω, τ) is a dense n-frame based on F. Frame N ω(f ) is dense in a sense that the intersection of all neighborhoods of a point is empty. So, there are no minimal neighborhoods unlike in T op(f ). Lemma Let F = (W, R) be a Kripke frame with root a, then f F : N ω(f ) N (F ). Corollary For any frame F Log(N ω(f )) Log(N (F )) = Log(F ).

39 Counterexample It possible that Log(N ω(f )) Log(F ). Consider: Obviously G = p p. Lemma N ω(g) p p G = ({1}, S), 1 n S1 m m = n + 1. Proof. Consider valuation θ(p) = { 2n 1 ω n N }. Then in any neighbourhood of point ω there are points where p is true and there are points where p is false. Hence, N ω(g) = p p.

40 Completeness results Theorem (214) K n K = K, K. Theorem If logics L 1 and L 2 are axiomatizable by closed formulas and by axioms like k p p then L 1 n L 2 = L 1, L 2. Corollary K4 d K4 = K4, K4.

41 Logic S5 We put 1 = {φ 2φ φ is closed and 2-free}, com 12 = 1 2p 2 1p, com 21 = 2 1p 1 2p, chr = 1 2p 2 1p. Theorem If logic L is axiomatizable by closed formulas and by axioms like k p p then L n S5 = L S com 12 + chr.

42 How to prove PLAN We have two loogic L 1 and L 2 Canonicity of the logic L 1, L 2. Construct F 1 = L 1 and F 2 = L 2 and F 1, F 2 F L1,L 2. Construct N ω Γ 1 (F 1) N ω Γ 2 (F 2) N ( F 1, F 2 Γ 1 Γ 2 ). Check that N ω Γ 1 (F 1) = L 1 and N ω Γ 2 (F 2) = L 2

43 a 1b a 2b a1 a2 aa ab b1 b2 ba bb 1 2 a b

44 1b a 1b a 2b a1 a2 aa ab b1 b2 ba bb a b Λ (α, β) α = 1... β = b...

45 a 1b a 2b a1 a2 aa ab b1 b2 ba bb 1 2 a b (α, β) α = 1... β = b...

46 a 1b a 2b a1 a2 aa ab b1 b2 ba bb 1 2 a b 1 (α, β) α = 1... β = b...

47 a 1b a 2b a1 a2 aa ab b1 b2 ba bb b 1 2 a b 1 (α, β) α = 1... β = b...

48 a 1b 21 1b 22 2a 2b a1 a2 aa ab b1 b2 ba bb b 1 2 a b 1 (α, β) α = 1... β = b...

49 How to prove for S5 PLAN We have two loogic L and S5 Canonicity of the logic L, S5]. Construct F 1 = L and F 2 = (R, R 2 ) and F 1, F 2 F L,S5]. ( Construct N Γ ω (F 1) N ω(f 2) N F 1, F 2] Γ). Check that N Γ ω (F 1) = L

50 We dene C 12 = {ab ba a W 1, b W 2} We also dene three Kripke frames: F 1, F 2 = (F 1 F 2, R < 1, R < 2 ) F 1, F 2] = (F 1 F 2, R < 1, R 2) ar < 1 b u W 1( b = au) ar < 2 b v W 2( b = av) ar 2 b b ( ar < 2 b & b == C12 b) Lemma For F 1 and F 2 dened above F 1, F 2] = com 12, chr 1.

51 THANK YOU!