FINITE MEMORY DETERMINACY

Transcription

1 p. 1/? FINITE MEMORY DETERMINACY Alexander Rabinovich Department of Computer Science Tel-Aviv University

2 p. 2/? Plan 1. Finite Memory Strategies. 2. Finite Memory Determinacy of Muller games. 3. Latest Appearance Record. 4. Reduction of Muller Games to parity games.

3 p. 3/? Finite Memory Strategies G(V 1, V 2, E) - arena or game graph. A strategy for I is a function f I : (V 1 V 2 ) V 1 V 2. A strategy is memoryless if it depends only on the last node - does not remember the past (history). A memoryless strategy for I corresponds - to a function σ I : V 1 V 2 - does not remember anything about the past. Finite memory strategy remembers a finite amount of information about the past.

4 p. 4/? Finite Memory Strategies Finite memory strategy remembers a finite amount of information about the past. Def. Let M be a finite set. A strategy σ 1 for I with the memory M consists of 1. next move function next 1 : M V 1 V the memory update function: update : M V M 3. m init : V M the initial state of the memory. Def. Let x be a play x 1 x 2... x i... and let m 1 m i m 2... be the sequence of memory states computed by m i = update(m i 1, x i ). The play x is consistent with σ 1 if x 2i = next 1 (m 2i 1, x 2i 1 ) Note. If the game arena is finite then a finite memory strategy can be implemented by a finite automata.

5 p. 5/? Example V 1 V 2 V 3 Winning conditions for : Inf(ρ) = {v 1, v 2, v 3 } Can win? A winning finite memory strategy for : M = {1, 2, 3} - remember the previous state. update(, v i ) = i; move 1 (v 2, i) = 4 i.

6 p. 6/? Muller Winning Conditions G(V 1, V 2, E) - game graph. Nodes are colored by a finite set Q (states). col : V 1 V 2 Q. A play x = x 1, x 2... induces an ω-sequence c x of over Q: c x = col(x 1 )col(x 2 )col(x 3 )... Inf(x) - the set of states that appear infinitely often in c x. Muller winning conditions: A family F = {F 1,..., F k } of subsets of Q. Each subset F i is called a macrostate. Player I wins the play x iff Inf(x) F. Otherwise Player II wins the play. Strategies, winning strategies, memoryless strategies... are defined as usual.

7 p. 7/? Finite Memory Determinacy of Muller Games Theorem. In every Muller game one of the players has a Finite memory winning strategy. Moreover, there is a partition (W 1, W 2 ) of the nodes of G and finite memory memory strategies σ 1 and σ 2 such that 1. σ 1 is a winning strategy of Player I from W σ 2 is a winning strategy of Player II from W The memory size is O(n!), where n is the number of states (colors). The proof will reduce a Muller game over G to a parity game over more complex graph G. A major role in the proof plays the Latest Appearance Record - a data structure introduced by McNaughton.

8 p. 8/? Latest Appearance Record Let s be a string over a finite alphabet Σ. The latest appearance records of sσ remembers the order of the last occurrences of each letter in s and the last letter σ. lar(sσ) = abc, σ iff the last occurrence of a in s precedes the last occurrence of b precedes the last occurrence of c. s =... a... b...c The recent letters set of sa the set of letters that occurs between the last two occurrences of σ (including σ). string LAR RLS a ǫ, a ac a, c acc ac, c {c} accb ac, b accba acb, a {a, b, c} accbab cba, b {a, b} shorthand for LAR a ac ac acb cba cab

9 p. 9/? Properties of the Latest Appearance Record Lemma. 1. lar(s) contains each letter at most once. 2. lar(sa) is computable from lar(s). (remove a from lar(s) and then append a to the right.) 3. rls(sa) is computable from lar(s). Lemma. Let u be an a 1 a 2... a i.... Let s i = a 1... a i be the i-th prefix of u. Then inf(u) = F = {q 1,... q k } iff 1. there is j such that rls(s i ) F for all i j and 2. infinitely often the cardinality of rls(s i ) = k and

10 p. 10/? Reduction of Muller Games to Parity Games Muller game: G(V 1, V 2, E, col : V Q) with Muller winning conditions: F = {F 1,..., F k }. Construct a parity game G Nodes: V 1 LAR(Q) V 2 LAR(Q). Edges u, l u, l iff u G u and l = lar(lq) where q is the label of u. Coloring Col (u, l) = { 2 rls(l) + 1 if rls(l) F 2 rls(l) otherwise A play ρ in G induces a play ρ in G. Moreover I wins ρ in G if he wins ρ in G.

11 p. 11/? Reduction of Muller Games to Parity Games A play ρ in G from u, q is induced by a play ρ in G. Moreover I wins ρ in G if he wins ρ in G. A memoryless strategy f in G defines a strategy f with the memory LAR(Q) in G. Moreover, f is winning iff f is winning. Hence, the memoryless determinacy of parity games implies Theorem. In every Muller game one of the players has a Finite memory winning strategy. Moreover, there is a partition (W 1, W 2 ) of the nodes of G and finite memory memory strategies f 1 and f 2 such that 1. f 1 is a winning strategy of Player I from W f 2 is a winning strategy of Player II from W 2.

12 p. 12/? Decidability of Muller Games Input A parity game over a finite graph, a vertex v. Task Find the winner of the game and his winning strategy Theorem The Problem is computable.

13 p. 13/? Game version of Church s Problem Consider a bit by bit transformation of bit streams...b t...b 2 b 1 F...a t...a 2 a 1 Question. Given a logical specification ϕ(x, Y) of the input-output relation. 1. Does the output player has a strategy to satisfy ϕ? 2. If Yes, compute such a strategy. Solution of Church s Problem Reduce to ( a finite state) Muller Game. From the solution of Muller Game find the solution of the Church Problem.

14 p. 14/? From Logic to Muller Games Each formula ϕ(x, Y) of MLO can be transformed into a Muller game over a finite graph with designated node v 0 such that 1. Player I has a winning strategy for ϕ iff Player I has a winning strategy in the corresponding Muller game. 2. A finite-state winning strategy for Player I in the Muller game from v 0 allows to construct a finite state winning strategy for Player I to satisfy ϕ. Proof. (1) Transform formula to an equivalent deterministic Muller automata. (2) Transform the automaton to Muller game. q1 q1 (0,0) 0 (0,1) q2 0 1 q2 q (1,0) q3 q 1 0 q3 (1,1) 1 q4 q4

15 p. 15/? Game version of the BL Theorem Theorem. Let ϕ(x, Y) be an MLO formula. One of the player has a finite state winning strategy in the game with winning condition ϕ. Moreover, it is decidable who wins and his finite state winning strategy is computable from ϕ. Corollary. The Church Problem is decidable.