Determinacy strength of infinite games in ω -languages recognized by variations of pushdown automata Wenjuan Li jointly with Prof. Kazuyuki Tanaka Mathematical Institute, Tohoku University Sep. 16, 2016 Workshop on Mathematical Logic and its Applications JSPS Core-to-Core Program Kyoto University, Kyoto W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 1 / 39
Infinite games Gale-Stewart game G ( X ), where X ( ⊆ A ω ) is a winning set for player I Determinacy With the usual convention, C -Det denotes that “A Gale-Stewart game G ( X ) is determined (one of the two players has a winning strategy), if X is contained in the class C ”. ω -languages accepted by automata L ( M ), where M is some kind of automata Question If the winning sets are effectively given, i.e., winning sets are accepted by some kind of automata, how is the determinacy strength of such games? W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 2 / 39
Outline Introduction 1 Pushdown automata, visibly pushdown automata, etc. Determinacy strength and ω -languages 2 2DVPL ω , r - PDL ω , PDL ω , etc. Ongoing and future works 3 W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 3 / 39
Pushdown automata on infinite words ( ω -PDA) Infinite … … a a input tape 1 i top p Finite control Stack A run on a 1 ... a n ... is an infinite sequence of configurations: a 1 or ε a n +1 or ε a n or ε ( q in , ⊥ ) − − − − → ( q 1 , γ 1 ) ... − − − − → ( q s , γ s ) − − − − − → ... An infinite word a 1 ... a n ... ∈ A ω is accepted by a B¨ uchi pushdown automaton if there exists a run visiting a state in F infinitely many times. W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 4 / 39
Visibly Pushdown Automata ◮ For (1-stack) visibly pushdown automata (2VPA), the alphabet A is partitioned into Push , Pop , Int . The transitions are as follows. a b q … … If a ∈ Pop a c a c q p If a ∈ Push b a b If a ∈ Int a q b ◮ For 2-stack visibly pushdown automata (2VPA), the alphabet A is partitioned into Push 1 , Pop 1 , Push 2 , Pop 2 , Int . Example n | n ∈ N } , is recognized Given A = ( { a } , { a } , { b } , { b } , ∅ ), the language { ( ab ) n a n b by a deterministic 2-stack visibly pushdown automaton (2DVPA). W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 5 / 39
Outline Introduction 1 Pushdown automata, visibly pushdown automata, etc. Determinacy strength and ω -languages 2 2DVPL ω , r - PDL ω , PDL ω , etc. Ongoing and future works 3 W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 6 / 39
Determinacy strength of infinite games in deterministic 2-stack visibly ω -languages W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 7 / 39
Recall undecidability results of games in some ω -languages REG ω : ω -regular lang. (FA) CFL ω : context free ω -lang. (PDA) DCFL ω : deterministic CFL ω (DPDA) VPL ω : visibly pushd. ω -lang. (VPA) Effectively determined BTM ω : ω -lang. by B¨ uchi Turing machine Note that these languages are defined with B¨ uchi or Muller condition. W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 8 / 39
Question How about other acceptance conditions of lower levels? In this talk we concentrates on determinacy strength of infinite games specified by nondeterministic pushdown automata and variants of it with various acceptance conditions, e.g., safety, reachability, co-B¨ uchi conditions. W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 9 / 39
Acceptance conditions of infinite words Safety (or Π 1 ) accepance condition L ( M ) = { α ∈ Σ ω | there is a run r = ( q i , γ i ) i ≥ 1 of M on α such that ∀ i , q i ∈ F } . Reachability (or Σ 1 ) acceptance condition L ( M ) = { α ∈ Σ ω | there is a run r = ( q i , γ i ) i ≥ 1 of M on α such that ∃ i , q i ∈ F } . Let Inf( r ) be the set of states that are visited infinite many times during the run r . Co-B¨ uchi (or Σ 2 ) acceptance condition L ( M ) = { α ∈ Σ ω | there is a run r = ( q i , γ i ) i ≥ 1 of M on α such that Inf( r ) ⊆ F } . W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 10 / 39
Acceptance conditions of infinite words (continued) ( Σ 1 ∧ Π 1 ) acceptance condition There exist F r , F s ⊂ Q , L ( M ) = { α ∈ Σ ω | there is a run r = ( q i , γ i ) i ≥ 1 of M on α such that ∃ i , q i ∈ F r ∧ ∀ i , q i ∈ F s } . ( Σ 1 ∨ Π 1 ) acceptance condition There exist F r , F s ⊂ Q , L ( M ) = { α ∈ Σ ω | there is a run r = ( q i , γ i ) i ≥ 1 of M on α such that ∃ i , q i ∈ F r ∨ ∀ i , q i ∈ F s } . W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 11 / 39
Acceptance conditions of infinite words (continued) ∆ 2 acceptance condition There exist F b , F c ⊂ Q , L ( M ) = { α ∈ Σ ω | there is a run r of M on α such that Inf( r ) ∩ F b � = ∅} = { α ∈ Σ ω | there is a run r of M on α such that Inf( r ) ⊂ F c } . W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 12 / 39
Various acceptance conditions of ω -2DVPA ◮ We denote the ω -languages accepted by ω -2DVPA with different acceptance conditions as follows. ◮ ω -languages accepted by deterministic Turing machines with safety (resp., reachabiliy, co-B¨ uchi, B¨ uchi) condition is the collection of all arithmetical Π 0 1 -sets (respectively, Σ 0 1 -sets, Σ 0 2 -sets, Π 0 2 -sets). Acceptance conditions Subclass of 2DVPL ω ⊆ Σ 0 Reachability 2DVPL ω ( Σ 1 ) 1 ⊆ Π 0 Safety 2DVPL ω ( Π 1 ) 1 ⊆ Σ 0 Co-B¨ uchi 2DVPL ω ( Σ 2 ) 2 ⊆ Π 0 B¨ uchi 2DVPL ω ( Π 2 ) 2 Similarly, by 2DVPL ω ( C ) we denote the ω -languages accepted by deterministic 2-stack visibly pushdown automata with an acceptance condition C . W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 13 / 39
Theorem There exists an infinite game in 2 DVPL ω ( Σ 1 ∧ Π 1 ) with only Σ 0 1 -hard winning strategies. Proof. ◮ Let R be a universal 2-counter automaton. ◮ We construct a game G R such that the halting problem of R is computable in any winning strategies of player II, while player I has no winning strategy, and moreover the winning set for player II is accepted by a deterministic 2-stack visibly pushdown automaton with a Σ 1 ∨ Π 1 acceptance condition. W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 14 / 39
Recall 2-counter automata � A 2-counter automaton can be seen as a restricted 2-stack pushdown automaton with just one symbol for each stack: the number of the symbols in a stack is expresses as a nonnegative integer in a counter. � The input is a natural number m which is initially store in one of the counter. � By the current state and the tests results on whether each counter is zero or not, the automaton goes to next state and do operations on the two counters by increasing the counter(s) by 1, or decreasing the counter(s) by 1 if the counter is not zero. � It is known that a (deterministic) 2-counter automaton, is equivalent to a Turing machine. Thus the halting problem for a certain (universal deterministic) 2-counter automaton is Σ 0 1 -complete. W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 15 / 39
Recall 2-counter automata (continued) • A configuration ( q , m , n ) of a 2-counter automaton R is coded as qa m b n , where q ∈ Q , and m , n are non-negative integers in the two counters. • A run for a natural number m on R : q in a m 0 b n 0 �→ R q 1 a m 1 b n 1 �→ R q 2 a m 2 b n 2 �→ R · · · , where q in is the initial state, and m 0 = m , n 0 = 0. • A run is halting if it reaches a halting configuration. • A natural number m ∈ L ( R ) iff there exists a run on m such that q in a m b n 0 �→ R q 1 a m 1 b n 1 �→ R · · · �→ R q s a m s b n s , where n 0 = 0 and q s is a halting state. W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 16 / 39
Back to the proof: construct a game G R Let R be a universal 2-counter automaton. I wins II wins II wins W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 17 / 39
Construct a game G R I wins II wins II wins II wins I wins I wins W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 18 / 39
If player II says “no”, how does she challenge? Player II wants to makes sure (1) the sequence of configurations provided by player I is a sequence of the form qa m b n and connected by ⊲ , (2) it starts with the initial configuration, (3) any two consecutive configurations constitute a valid transition of R , and (4) the sequence of configurations is ended with a halting configuration. The conditions (1), (2) and (4) are easy to check with Σ 1 conditions (i.e., player I lose with Π 1 ). In the following we explain how player II challenges if she thinks player I cheated by disobeying the above rule (3). Check! m n m n m ( )? q a b 0 0 q a b q a 1 b I i i i m L R 0 1 0 i i Challenge II Yes/No min{ , } ( $) ( $) * m n m m cb a 1 q b a 1 q i i i i with a witness: Such a play can be checked by a deterministic 2-stack visibly pushdown automaton. W. Li (Tohohu University) Determ. streng. of infin. games in ω -lang. 19 / 39
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