Deciding the weak definability of B¨ uchi definable tree languages Michael Vanden Boom Department of Computer Science University of Oxford Queen Mary Theory Group Seminar 13 November 2013 Joint work with Thomas Colcombet, Denis Kuperberg, and Christof L¨ oding
Setting Infinite labelled tree: model of possible execution of a system where ◮ branching represents non-determinism in system, or different possibilities when the environment interacts with the system; ◮ label describes behavior of the system. a a b a b b b a a a a a b b b . . .
Setting Let L ( ϕ ) denote the set of infinite trees over some fixed finite alphabet A that satisfy some property ϕ . Question Given some property ϕ , is there a “simpler” ϕ ′ such that L ( ϕ ) = L ( ϕ ′ )?
Setting Let L ( ϕ ) denote the set of infinite trees over some fixed finite alphabet A that satisfy some property ϕ . Question smaller size, Given some property ϕ , restricted set of operations, is there a “simpler” ϕ ′ different specification language, such that L ( ϕ ) = L ( ϕ ′ )? ...
Setting Let L ( ϕ ) denote the set of infinite trees over some fixed finite alphabet A that satisfy some property ϕ . Question smaller size, Given some property ϕ , restricted set of operations, is there a “simpler” ϕ ′ different specification language, such that L ( ϕ ) = L ( ϕ ′ )? ... Goal: analyze/decide questions like this for regular languages L ( ϕ ).
Regular languages of infinite trees Regular Languages monadic second-order logic (MSO) µ -calculus parity automata
Regular languages of infinite trees Regular Languages monadic second-order logic (MSO) µ -calculus parity automata weakly definable weak MSO alternation-free µ -calculus weak automata
Regular languages of infinite trees Regular Languages monadic second-order logic (MSO) µ -calculus parity automata weakly definable weak MSO alternation-free µ -calculus weak automata Weakly definable languages are expressive (subsuming CTL), but still have good computational properties (model-checking can be done in linear time).
Alternating parity automata on infinite trees A = � A , Q , q 0 , δ, Ω � Ω : Q → P δ describes possible moves for a finite set of for Eve and Adam priorities P Acceptance game A × t ◮ Positions in the game are Q × dom( t ). ◮ Eve and Adam select the next position in the play based on δ . ◮ Eve is trying to ensure the play satisfies the parity condition : the maximum priority occurring infinitely often in the play is even.
Alternating parity automata on infinite trees A = � A , Q , q 0 , δ, Ω � Ω : Q → P δ describes possible moves for a finite set of for Eve and Adam priorities P Acceptance game A × t ◮ Positions in the game are Q × dom( t ). ◮ Eve and Adam select the next position in the play based on δ . ◮ Eve is trying to ensure the play satisfies the parity condition : the maximum priority occurring infinitely often in the play is even. L ( A ) := { t : Eve has a winning strategy in A × t }
Example L 1 := { t : there is some a in t with no b below it } . Construct A 1 with Q = { q 0 , q a , q ⊥ } and Ω : q 0 , q ⊥ �→ 1; q a �→ 2. ◮ In state q 0 , Eve selects a path in the tree. If she sees an a , Eve can choose to switch to state q a . ◮ In state q a , Adam selects a path in the tree. If he sees a b , then he can switch to a sink state q ⊥ .
Example L 1 := { t : there is some a in t with no b below it } . Construct A 1 with Q = { q 0 , q a , q ⊥ } and Ω : q 0 , q ⊥ �→ 1; q a �→ 2. ◮ In state q 0 , Eve selects a path in the tree. If she sees an a , Eve can choose to switch to state q a . ◮ In state q a , Adam selects a path in the tree. If he sees a b , then he can switch to a sink state q ⊥ . L 2 := { t : every a in t has a b below it } . Construct A 2 with Q = { q 0 , q b , q ⊤ } and Ω : q 0 , q ⊤ �→ 2; q b �→ 1. ◮ In state q 0 , Adam selects a path in the tree. If he sees an a , Adam can choose to switch to state q b . ◮ In state q b , Eve selects a path in the tree. If she sees a b , then she can switch to a sink state q ⊤ .
Special types of alternating parity automata B¨ uchi automaton parity automaton using only priorities { 1 , 2 } (we call states of priority 2 the accepting states and states of priority 1 the non-accepting states)
Special types of alternating parity automata B¨ uchi automaton parity automaton using only priorities { 1 , 2 } (we call states of priority 2 the accepting states and states of priority 1 the non-accepting states) Nondeterministic B¨ uchi automaton alternating B¨ uchi automaton such that a strategy for Eve in an acceptance game is just a labelling of the input tree with states (called a run )
Special types of alternating parity automata B¨ uchi automaton parity automaton using only priorities { 1 , 2 } (we call states of priority 2 the accepting states and states of priority 1 the non-accepting states) Nondeterministic B¨ uchi automaton alternating B¨ uchi automaton such that a strategy for Eve in an acceptance game is just a labelling of the input tree with states (called a run ) Alternating weak automaton alternating B¨ uchi automaton such that no cycle visits both accepting and non-accepting states
Regular languages of infinite trees Regular Languages monadic second-order logic (MSO) µ -calculus parity automata weakly definable complement is weak MSO B¨ uchi B¨ uchi alternation-free µ -calculus definable definable weak automata Theorem [Rabin ’70, Kupferman+Vardi ’99] A language L is weakly definable iff L and L are B¨ uchi definable.
Weak definability problem Weak definability decision problem I NPUT : parity automaton U O UTPUT : YES if there exists weak automaton W with L ( W ) = L ( U ), NO otherwise
Weak definability problem Weak definability decision problem I NPUT : parity automaton U O UTPUT : YES if there exists weak automaton W with L ( W ) = L ( U ), NO otherwise Theorem [Niwi´ nski+Walukiewicz ’05] The weak definability problem is decidable if L ( U ) is deterministic .
Weak definability problem Weak definability decision problem I NPUT : parity automaton U O UTPUT : YES if there exists weak automaton W with L ( W ) = L ( U ), NO otherwise Theorem [Niwi´ nski+Walukiewicz ’05] The weak definability problem is decidable if L ( U ) is deterministic . Theorem [Facchini+Murlak+Skrzypczak ’13] The weak definability problem is decidable if L ( U ) is a game language .
Weak definability problem Weak definability decision problem I NPUT : parity automaton U O UTPUT : YES if there exists weak automaton W with L ( W ) = L ( U ), NO otherwise Theorem [Niwi´ nski+Walukiewicz ’05] The weak definability problem is decidable if L ( U ) is deterministic . Theorem Theorem [Facchini+Murlak+Skrzypczak ’13] [Colcombet,Kuperberg,L¨ oding,VB ’13] The weak definability problem is The weak definability problem is decidable if L ( U ) is a decidable if U is B¨ uchi . game language .
Cost automata Finite state automaton A + finite set of counters (initialized to 0, values range over N ) + counter operations on transitions (increment i , reset r , no change ε ) Semantics � A � : infinite trees → N ∪ {∞} � A � ( t ) := min { n : ∃ winning strategy for Eve in A × t such that every play has counter values at most n }
Example f ( t ) := min { n : every a has a b at most n nodes below it } . Construct A with Q = { q 0 , q b , q ⊤ } , Ω : q 0 , q ⊤ �→ 2; q b �→ 1, 1 counter. ◮ In state q 0 , Adam selects a path in the tree. The counter operation is ε . If he sees an a , Adam can choose to switch to state q b . ◮ In state q b , Eve selects a path in the tree. If she sees an a , then the counter is incremented. If she sees a b , then she can switch to a sink state q ⊤ .
Cost automata Finite state automaton A + finite set of counters (initialized to 0, values range over N ) + counter operations on transitions (increment i , reset r , no change ε ) Semantics � A � : infinite trees → N ∪ {∞} � A � ( t ) := min { n : ∃ winning strategy for Eve in A × t such that every play has counter values at most n } Boundedness with respect to language K (written � A � ≈ χ K ) � A � ≈ χ K if there is bound n ∈ N such that � A � ( t ) ≤ n if t ∈ K and � A � ( t ) = ∞ if t / ∈ K
Decidability of boundedness for cost automata Decidability of ≈ is known for some types of cost automata. ◮ cost automata over finite words [Colcombet ’09, Bojanczyk+Colcombet ’06] ◮ cost automata over infinite words [Colcombet unpublished] ◮ cost automata over finite trees [Colcombet+L¨ oding ’10]
Decidability of boundedness for cost automata Decidability of ≈ is known for some types of cost automata. ◮ cost automata over finite words [Colcombet ’09, Bojanczyk+Colcombet ’06] ◮ cost automata over infinite words [Colcombet unpublished] ◮ cost automata over finite trees [Colcombet+L¨ oding ’10] ◮ counter-weak automata over infinite trees [Kuperberg+VB ’11]
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