. Operations on Unambiguous Finite Automata . Galina Jir´ askov´ a Mathematical Institute, Slovak Academy of Sciences, Koˇ sice, Slovakia ❁ asek, Jr., and Juraj ˇ Joint work with Jozef Jir´ Sebej DLT 2016, Montr´ eal, Qu´ ebec, Canada . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Nondeterministic and Deterministic Finite Automata . . . NFA N = ( Q , Σ , δ, I , F ): Example (An NFA) . . a,b δ ⊆ Q × Σ × Q a,b a a,b computation on w = a 1 a 2 · · · a k q0 q1 q2 q 3 a k a 1 a 2 a 3 − → q 1 − → q 2 − → · · · − → q k q 0 w = aaa q 0 ∈ I a a a accepting if q k ∈ F q 0 − → q 1 → q 2 − → q 3 − (acc.) a a a rejecting if q k / ∈ F → q 0 − − → q 0 → q 0 − q 0 (rej.) . . . . NFA N = ( Q , Σ , δ, I , F ) is a DFA: Example (An incomplete DFA) . . | I | = 1 b if ( q , a , p ) and ( q , a , r ) are in δ , a a,b a,b q0 q1 q2 q 3 then p = r . . NFAs may have multiple initial states DFAs may be incomplete . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Subset Automaton and Reverse of NFA . . . Definition Example (Subset automaton) . . The (incomplete) subset automaton N a,b a,b a of NFA N = ( Q , Σ , δ, I , F ) q 0 q 2 q 1 is the DFA (2 Q \ {∅} , Σ , δ ′ , I , F ′ ) . . . . (N) b a a . a b b q 02 q 0 q 01 q 012 Proposition . b a Every n -state NFA can be simulated a,b a,b q q 2 q by an (2 n − 1)-state incomplete DFA. 12 1 . . . . Example (Reverse of NFA) Definition . . The reverse of an NFA N q4 b b N = ( Q , Σ , δ, I , F ) is the NFA a a,b a,b q0 q1 q2 q 3 N R = ( Q , Σ , δ R , F , I ) , N R q4 b b a a,b a,b where ( p , a , q ) ∈ δ R iff ( q , a , p ) ∈ δ q0 q1 q2 q 3 . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Unambiguous Finite Automata . . . Definition ( N = ( Q , Σ , δ, I , F )) Example (not unambiguous) . . An NFA is unambiguous if it has q2 at most one accepting computation a q0 q1 on every input string. a − two accepting computations on a S ⊆ Q is reachable in N q4 if S = δ ( I , w ) for some w b b a,b a,b S ⊆ Q is co-reachable in N a q0 q1 q2 q 3 if S is reachable in N R − two accepting computations on abb . . . . Proposition Example (unambiguous) . . An NFA is unambiguous iff (in)complete DFA | S ∩ T | ≤ 1 NFA N s.t. N R deterministic for each reachable S NFA in the first slide and each co-reachable T . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Why Unambiguous Finite Automata? . . Motivation and History . fundamental notion in the theory of variable-length codes [Bersten, Perrin, Reutenauer: Codes and Automata] ambiguity in CF languages: ambiguous, unambiguous, and deterministic CF languages are all different ambiguity in finite automata [Schmidt 1978] - lower bound method based on ranks of matrices elaborated in [Leung 2005] UFA-to-DFA conversion: 2 n NFA-to-UFA conversion: 2 n − 1 lower bound method further elaborated in 2002 by Hromkoviˇ c, Seibert, Karhum¨ aki, Klauck & Schnitger . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Why Operations on Unambiguous Finite Automata? . . Motivation for me:-) . conference trip at DLT 2008 (Kyoto): A. Okhotin - ... ”What is the complexity of complementation on UFAs?” operations on unary UFAs investigated by him in 2012 - lower bound n 2 − o (1) for complementation the second problem for which ”give me a large enough alphabet” method didn’t work ... . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Lower Bounds Methods I . . Well known: To prove that a DFA is minimal, show that . - all its states are reachable, and - no two distinct states are equivalent. . . Well known(?): To prove that an NFA is minimal, describe . a fooling set for the accepted language. . . For UFAs: rank of matrices [Schmidt 78, Leung 05]: . Let N be an NFA. Let M N be the matrix in which rows indexed by non-empty reachable sets columns indexed by non-empty co-reachable sets in entry ( S , T ) we have 0/1 if S and T are/are not disjoint. Then every UFA for L ( N ) has at least rank( M N ) states. . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Lower Bounds Methods II . . Lemma (Leung 1998, Lemma 3) . Let M n be the (2 n − 1) × (2 n − 1) matrix with • rows and columns indexed by non-empty subsets of { 1 , 2 , . . . , n } • M n ( S , T ) = 0 / 1 iff S and T are/are not disjoint. Then rank(M n ) = 2 n − 1 . . . Corollary . If each non-empty set is co-reachable in NFA N, then every UFA equivalent to N has ≥ | non-empty reachable | states. . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. The Complexity of Regular Operations on DFAs . . Maslov 1970 . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. A General Formulation of the Problem . . Maslov 1970 . ”We have languages L ( A i ) (1 ≤ i ≤ k ) recognized by automata A i with n i states, respectively, and a k-ary regular operation f . What is the maximal number of states of a minimal automaton recognizing f ( L ( A 1 ) , . . . , L ( A k )) , for the given n i ?” . In this paper: - automata are unambiguous (UFAs) - f : intersection, reversal, shuffle, star and positive closure, left and right quotients, concatenation, complementation, and union . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Intersection on Unambiguous Finite Automata . . Intersection: K ∩ L = { w | w ∈ K and w ∈ L } . . Known results for intersection: . DFA: mn binary [Maslov 1970] NFA: mn binary [Holzer & Kutrib 2003] . . Our result for intersection on UFAs: . . UFA: mn | Σ | ≥ 2 . Proof sketch: . upper bound: given UFAs A and B , construct the direct product automaton A × B ; it is a UFA lower bound: the witnesses in [HK’03] for NFA intersection are deterministic, so UFAs . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
. Shuffle on Unambiguous Finite Automata . . Shuffle: K � L = { u 1 v 1 u 2 v 2 · · · u k v k | u 1 u 2 · · · u k ∈ K and v 1 v 2 · · · v k ∈ L } . . Known results for shuffle: . DFA: ??? 2 mn − 1 in-DFA: 5-letter [Cˆ ampeanu, Salomaa & Yu 2002] NFA: mn binary [G. J. & Masopust, DLT 2010] . . Our result for shuffle on UFAs: . 2 mn − 1 . UFA: | Σ | ≥ 5 . Proof sketch for lower bound: . take the witness incomplete DFAs from [CSY’02] in the mn -state NFA for shuffle - each non-empty set is reachable [CSY’02] - each non-empty set is co-reachable . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a Operations on Unambiguous Finite Automata
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