Goal of the Talk Theorem The class of languages recognisable by T -coalgebra automata is closed under taking complements.

Transcription

1 Complementation of Coalgebra Automata Christian Kissig (University of Leicester) joint work with Yde Venema (Universiteit van Amsterdam) 07 Sept 2009 / Universitá degli Studi di Udine / CALCO 2009

2 Goal of the Talk Theorem The class of languages recognisable by T -coalgebra automata is closed under taking complements.

3 Goal of the Talk Theorem The class of languages recognisable by T -coalgebra automata is closed under taking complements. T preserves weak pullbacks T restricts to finite sets T is standard

4 Outline 1. One Step Complementation Lemma Moss Modality Boolean Dual of Moss Modality 2. Game Bisimulation Parity Graph Games Basic Sets and Local Games Powers and Game Normalisation Game Bisimulation 3. Complementation Lemma for Coalgebra Automata Coalgebra Automata Complementation of Trans-alternating Automata Equivalence of Transalternating and Alternating Automata

5 Outline 1. One Step Complementation Lemma Moss Modality Boolean Dual of Moss Modality 2. Game Bisimulation Parity Graph Games Basic Sets and Local Games Powers and Game Normalisation Game Bisimulation 3. Complementation Lemma for Coalgebra Automata Coalgebra Automata Complementation of Trans-alternating Automata Equivalence of Transalternating and Alternating Automata

6 Moss Modality Definition (Moss Modality) α LT Q where α T LQ where L is the functor taking a set Q to the set of bounded lattice terms t ::= q Q t t t t over Q

7 Moss Modality Definition (Moss Modality) α LT Q where α T LQ Let S = S,σ : S T S, s I and s S, then S, s α iff (σ(s),α) T ( ) where L is the functor taking a set Q to the set of bounded lattice terms t ::= q Q t t t t over Q

8 Moss Modality Definition (Moss Modality) α LT Q where α T LQ Let S = S,σ : S T S, s I and s S, then S, s α iff (σ(s),α) T ( ) where L is the functor taking a set Q to the set of bounded lattice terms t ::= q Q t t t t over Q Example For T = P ω, we get α α [α]

9 Positive Coalgebraic Logics Definition (Positive Coalgebraic Logics) LQ is the set of depth-zero formulas

10 Positive Coalgebraic Logics Definition (Positive Coalgebraic Logics) LQ is the set of depth-zero formulas LT ω LQ is the set of depth-one formulas where T ω (X ) := {T Y Y ω X } is the finitary version of T T ω (X ) := { α α T ω X }

11 One-Step Semantics of Positive Coalgebraic Logics Definition (One-Step Semantics) Valuation V : Q P(S)

12 One-Step Semantics of Positive Coalgebraic Logics Definition (One-Step Semantics) Valuation V : Q P(S) s V 0 q iff s V (q) σ V 1 α if (σ, α) T ( V 0 )

13 One-Step Semantics of Positive Coalgebraic Logics Definition (One-Step Semantics) Valuation V : Q P(S) s V 0 q iff s V (q) σ V 1 α if (σ, α) T ( V 0 ) Definition (Boolean Duals of Depth-One Formulas) Depth-Zero Formulas a and b are boolean duals if for all sets S, all valuations V : Q P(S), and all s S s V c 0 a iff s V 0 b where V c (q) := P(S) \ V (q) is the complementary valuation

14 One-Step Semantics of Positive Coalgebraic Logics Definition (One-Step Semantics) Valuation V : Q P(S) s V 0 q iff s V (q) σ V 1 α if (σ, α) T ( V 0 ) Definition (Boolean Duals of Depth-One Formulas) Depth-One Formulas a and b are boolean duals if for all sets S, all valuations V : Q P(S), and all s S where s V c 1 a iff s V 1 b V c (q) := P(S) \ V (q) is the complementary valuation

15 One-Step Complementation Lemma Definition (Boolean Dual of ) Let α Tω Q, define a set D(α) T ω PQ as follows { } D(α) := Φ T ω P ω Base(α) (α, Φ) (T ) where Base(α T ω Q) is the smallest X ω Q such that α T ω X

16 One-Step Complementation Lemma Definition (Boolean Dual of ) Let α Tω Q, define a set D(α) T ω PQ as follows { } D(α) := Φ T ω P ω Base(α) (α, Φ) (T ) where Base(α T ω Q) is the smallest X ω Q such that α T ω X α := { (T } )Φ Φ D(α)

17 One-Step Complementation Lemma Definition (Boolean Dual of ) Let α Tω Q, define a set D(α) T ω PQ as follows { } D(α) := Φ T ω P ω Base(α) (α, Φ) (T ) where Base(α T ω Q) is the smallest X ω Q such that α T ω X α := { (T } )Φ Φ D(α) Example For T = P ω, we get α := { {a} a α} { α, }.

18 One-Step Complementation Lemma Theorem (One-Step Complementation Lemma) For all α T ω Q, α and α are Boolean duals. For all sets S, all valuations V : Q P(S), and all s S, s V c 1 α iff s V 1 α

19 One-Step Dualisation Definition (One-Step Dualisation) δ 0 : LQ LQ δ 0 (q) := q δ 0 ( φ) := δ 0 [φ] δ 0 ( φ) := δ 0 [φ] δ 1 : LT ω LQ LT ω LQ δ 1 ( α) := (T δ 0 )α δ 1 ( φ) := δ 1 [φ] δ 1 ( φ) := δ 1 [φ]

20 One-Step Dualisation Definition (One-Step Dualisation) δ 0 : LQ LQ δ 0 (q) := q δ 0 ( φ) := δ 0 [φ] δ 0 ( φ) := δ 0 [φ] δ 1 : LT ω LQ LT ω LQ δ 1 ( α) := (T δ 0 )α δ 1 ( φ) := δ 1 [φ] δ 1 ( φ) := δ 1 [φ] Corollary For any a LTω LQ, the depth-one formulas a and δ 1 (a) are Boolean duals. For all sets S, all valuations V : Q P(S), and all s S, s V c 1 a iff s V 1 δ 1(a)

21 Outline 1. One Step Complementation Lemma Moss Modality Boolean Dual of Moss Modality 2. Game Bisimulation Parity Graph Games Basic Sets and Local Games Powers and Game Normalisation Game Bisimulation 3. Complementation Lemma for Coalgebra Automata Coalgebra Automata Complementation of Trans-alternating Automata Equivalence of Trans-alternating and Alternating Automata

22 Parity Graph Games Definition (Arena) Arenas of parity graph games are structures G = V 0, V 1, E, v I, Ω: V N sets V = V 0 V 1 of positions an edge relation E V V an initial position v I V a priority function Ω : V N with finite range

23 Parity Graph Games Definition (Arena) Arenas of parity graph games are structures G = V 0, V 1, E, v I, Ω: V N sets V = V 0 V 1 of positions an edge relation E V V an initial position v I V a priority function Ω : V N with finite range Definition (Winning Condition) Player Π {0, 1} ( Σ = 1 Π ) wins a total play p of G iff p finite and last(p) V Π p infinite and largest priority occuring infinitely often has parity Π

24 Basic Sets G = V 0, V 1, E, v I, Ω: V N Definition We call a set B V basic if 1. v I B 2. any total play from v B either ends in a terminal position or it passes through another position in B 3. v B iff Ω(v) > 0

25 Local Games G = V 0, V 1, E, v I, Ω with basic set B V, b B Definition (Local Game Trees) V b := T b = V b 0, V b 1, E b, (b) { β V first(β) =b, i < β.β(i) B = i = 0 or i = β 1 V b Π := {β V b last(β) V Π } for both Π {0, 1} E b (β) := {βv v E(β)} }

26 Powers G = V 0, V 1, E, v I, Ω with basic set B V b B, T b = V b 0, V b 1, E b, (b) Definition (Powers) We define the power P Π (b) B of Π {0, 1} (Σ = 1 Π) as { } If β Leaves(T b ), put P Π (β) := {last(β)} If β Leaves(T b ), put {PΠ (γ) γ E b (β)} P Π (β) := { } γ E b (β) Y γ Y γ P Π (γ), all γ P Π (b) := P Π ((b) ) if β V b Π if β V b Σ

27 Powers G = V 0, V 1, E, v I, Ω, basic set B V Π {0, 1}, Σ = 1 Π Proposition Let W be a subset of B. Then the following are equivalent: 1. W P Π (b); 2. Π has a surviving strategy f in G b such that W is the set of next basic positions in some play consistent with f

28 Powers G = V 0, V 1, E, v I, Ω, basic set B V Π {0, 1}, Σ = 1 Π Proposition Let W be a subset of B. Then the following are equivalent: 1. W P Π (b); 2. Π has a surviving strategy f in G b such that W is the set of next basic positions in some play consistent with f Proposition The following are equivalent 1. P Π (b) 2. P Σ (b) = 3. Π has a local winning strategy in G b

29 Game Bisimulation G = V 0, V 1, E, Ω, basic set B V,Π {0, 1} G = V 0, V 1, E, Ω, basic set B V,Π {0, 1 } Definition (Game Simulation) AΠ, Π -game simulation is a relation Z B B such that for all v V and v V with vzv, Z satisfies the structural conditions (pro) W P G Π (v). W P G Π (v ). w W. w W. wzw, (op) W P G Σ (v ). W P G Σ (v). w W. w W. wzw, and the priority conditions (parity) Ω(v) mod 2 = Π iff Ω (v ) mod 2 = Π, (contraction) for all v, w V and v, w V with vzv and wzw, Ω(v) Ω(w) iff Ω(v ) Ω(w ).

30 Game Bisimulation G = V 0, V 1, E, Ω, basic set B V,Π {0, 1} G = V 0, V 1, E, Ω, basic set B V,Π {0, 1 } Definition (Game Bisimulation) Z B B is a Π,Π -game bisimulation if Z is a Π,Π -game simulation Z is a Π,Π-game simulation

31 Game Bisimulation G = V 0, V 1, E, Ω, basic set B V,Π {0, 1} G = V 0, V 1, E, Ω, basic set B V,Π {0, 1 } Definition (Game Bisimulation) Z B B is a Π,Π -game bisimulation if Z is a Π,Π -game simulation Z is a Π,Π-game simulation Theorem If Z B B Π,Π -game bisimulation between parity graph games G and G, then if vzv then v Win Π (G) v Win Π (G )

32 Outline 1. One Step Complementation Lemma Moss Modality Boolean Dual of Moss Modality 2. Game Bisimulation Parity Graph Games Basic Sets and Local Games Powers and Game Normalisation Game Bisimulation 3. Complementation Lemma for Coalgebra Automata Coalgebra Automata Complementation of Trans-alternating Automata Equivalence of Trans-alternating and Alternating Automata

33 Alternating Automata, Syntax Definition (Alternating Automata in Logical Form) Alternating T -automata are structures A = Q,θ : Q LT Q, q I, Ω consisting of a finite set Q of states a transition function θ : Q LT Q an initial state q I Q a priority function Ω : Q N

34 Alternating Automata, Semantics A = Q,θ : Q LT Q, q I, Ω is an alternating automaton S = S,σ : S T S, s I is a pointed T -coalgebra Definition Acceptance games are parity graph games G(A, S) = V, V, E, (q I, s I ), Ω G Position Sets of Admissible Moves Ω G (q, s) Q S - {(θ(q), s)} Ω(q) ( τ, s) LT Q S {(q, s) q τ} 0 ( τ, s) LT Q S {(q, s) q τ} 0 ( α, s) T Q S {Z Q S (α, σ(s)) T Z} 0 Z Q S Z 0

35 Trans-alternating Automata Alternating automata: Q,θ : Q LT Q, q I, Ω Definition (Trans-alternating Automata) Q,θ : Q LT LQ, q I, Ω Definition (Acceptance Games) similar to the acceptance games of alternating automata

36 Complements of Trans-alternating Automata A = Q,θ : Q LT LQ, q I, Ω is a trans-alternating aut on Definition (Complements of Trans-alternating Automata) Define the complementary automaton A c = Q,θ c : Q LT LQ, q I, Ω c such that θ c (q) := δ 1 (θ(q)) Ω c (q) := Ω(q) + 1, for all q Q. δ 0 : LQ LQ δ 0 (q) := q δ 0 ( φ) := δ 0 [φ] δ 0 ( φ) := δ 0 [φ] δ 1 : LT ω LQ LT ω LQ δ 1 ( α) := (T δ 0 )α δ 1 ( φ) := δ 1 [φ] δ 1 ( φ) := δ 1 [φ]

37 Complements of Trans-alternating Automata Theorem For every trans-alternating automaton A, the automaton A c accepts precisely those pointed T -coalgebras that are rejected by A.

38 Trans-alternating and Alternating Automata Theorem There is an effective translation between 1. Alternating Automata 2. Trans-alternating Automata 1 2 is trivial

39 Trans-alternating and Alternating Automata Alternating automata: Q,θ : Q LT Q, q I, Ω Trans-alternating automata: Q,θ : Q LT LQ, q I, Ω Definition (Semi-Transalternating Automata) where Q,θ : Q LT SQ, q I, Ω S is the functor taking a set Q to the set of bounded meet-semilattice terms t ::= q Q t t over Q Definition (Acception Games) similar to the acceptance games of alternating automata

40 Trans-alternating and Alternating Automata Theorem There is an effective translation between 1. Alternating Automata 2. Trans-alternating Automata 3. Semi-Transalternating Automata We showed

41 Size Matters Theorem For every alternating automaton A with n states there is a complementing alternating automaton A c with 2 n n states.

42 Size Matters Theorem For every alternating automaton A with n states there is a complementing alternating automaton A c with 2 n n states. Theorem If T is such that α LT Q for any α T Q, then for any alternating T -automaton of n states there is a complementing alternating automaton with at most n + c states, for some constant c.

43 Some Conclusions Summary Effective Complementation Procedure for Coalgebra Automata Coinductive Method of Game (Bi)Simulation for (some) Parity Graph Games

44 Some Conclusions Summary Effective Complementation Procedure for Coalgebra Automata Coinductive Method of Game (Bi)Simulation for (some) Parity Graph Games Corollaries (Boolean) Coalgebraic Logic is Negation-free Correspondence between (Second-Order Monadic) Coalgebraic Logic and Coalgebra Automata

45 Some Conclusions Summary Effective Complementation Procedure for Coalgebra Automata Coinductive Method of Game (Bi)Simulation for (some) Parity Graph Games Corollaries (Boolean) Coalgebraic Logic is Negation-free Correspondence between (Second-Order Monadic) Coalgebraic Logic and Coalgebra Automata Open Questions Categorical Nature of the Correspondence Characterisation of Game (Bi)Similarity

46 Conclusions and References References Thank You Moss, Coalgebraic Logic, APAL, 1999 Venema, Automata and Fixed Point Logics: a Coalgebraic Perspective, Information and Computation, 2006 Kupke, Venema, Coalgebraic automata theory: basic results, LMCS, 2008 Kupke, Kurz, Venema, Completeness of Finitary Moss Logic, AiML 2008 Kissig, Decidability of S2S, MSc Thesis, ILLC, UvA, 2007 Kissig, Venema, Complementation of Coalgebra Automata van Benthem, Extensive Games as Process Models, 2002