State complexity of complementing unambiguous fjnite automata - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . State complexity of complementing unambiguous fjnite automata Michael Raskin, raskin@mccme.ru LaBRI, Université de Bordeaux July 10, 2018 Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 15

. . . . . . . . . . . . . . . Non-determinism in automata The basic classes: deterministic and non-deterministic fjnite automata The set of languages is the same State complexity (number of states required) difgers Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 15

. Automata languages . . . . . . . . . . Non-determinism in automata: state complexity Exponentially more succinct . Intersection, union Quadratic state complexity Complement No extra cost Exponential state complexity Reversing direction (left to right/right to left) Exponential state complexity No extra cost Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . 3 / 15 . . . . . . . . . . . . . L ( DFA ) = L ( NFA )

. Automata languages . . . . . . . . . . Non-determinism in automata: state complexity Exponentially more succinct . Intersection, union Quadratic state complexity Complement No extra cost Exponential state complexity Reversing direction (left to right/right to left) Exponential state complexity No extra cost Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . 3 / 15 . . . . . . . . . . . . . L ( DFA ) = L ( NFA )

. Automata languages . . . . . . . . . . Non-determinism in automata: state complexity Exponentially more succinct . Intersection, union Quadratic state complexity Complement No extra cost Exponential state complexity Reversing direction (left to right/right to left) Exponential state complexity No extra cost Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . 3 / 15 . . . . . . . . . . . . . L ( DFA ) = L ( NFA )

. . . . . . . . . . . . . . . Codeterministic automata coDFA: DFA reading the word right-to-left Union/intersection between DFA and coDFA — exponential state complexity Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 15 (if we want to stay in DFA ∪ coDFA)

. UFA . . . . . . . . Unambiguous automata UFA: NFA with at most one accepting run for each word DFA NFA . more succinct even more succinct Intersection, union Complement No extra cost ? Reversing order No extra cost No extra cost Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 15 ( · ) 2 ( · ) 2 , ? ( · ) 2 exp ( · ) exp ( · )

. . . . . . . . . . . . . . . Complementing UFA Known to be at least quadratic Lower bound holds for unary case Conjectured to be polynomial Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 15

. . . . . . . . . . . . Main result . State complexity of complementing UFA: superpolynomial lower bound Even in unary case Even if complement is general NFA Even if language is also easy for DFA with multiple passes A lower bound for complementing a unary UFA must be weak: (Dębski, 2017) Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 15 Very weakly superpolynomial lower bound: n Ω( log log log n ) upper bound n O ( log n )

. . . . . . . . . . . . Main result . State complexity of complementing UFA: superpolynomial lower bound Even in unary case Even if complement is general NFA Even if language is also easy for DFA with multiple passes A lower bound for complementing a unary UFA must be weak: (Dębski, 2017) Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 15 Very weakly superpolynomial lower bound: n Ω( log log log n ) upper bound n O ( log n )

. . . . . . . . . . . . Main result . State complexity of complementing UFA: superpolynomial lower bound Even in unary case Even if complement is general NFA Even if language is also easy for DFA with multiple passes A lower bound for complementing a unary UFA must be weak: (Dębski, 2017) Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 15 Very weakly superpolynomial lower bound: n Ω( log log log n ) upper bound n O ( log n )

. . . . . . . . . . . . Main result . State complexity of complementing UFA: superpolynomial lower bound Even in unary case Even if complement is general NFA Even if language is also easy for DFA with multiple passes A lower bound for complementing a unary UFA must be weak: (Dębski, 2017) Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 15 Very weakly superpolynomial lower bound: n Ω( log log log n ) upper bound n O ( log n )

. . . . . . . . . . . . . . . . Why? Direct construction Simple Chrobak normal form: Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . 8 / 15 unary NFA := collection of cycles C 1 , . . . , C n Input word ≡ length ≡ remainder modulo lcm ( | C 1 | , . . . , | C n | )

. . . . . . . . . . . . . . . Why? Tournaments! Square-free cycle lengths Unambiguity: Tournament of yielding between cycles Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 15 Input word ≡ remainder modulo lcm of cycle lengths Remainder 0 not in language; separation instead of complement Remainder 0 modulo gcd ( | C i | , | C j | ) rules out acceptance by C i or by C j C i yields to C j : remainder 0 modulo gcd ( | C i | , | C j | ) rules out acceptance by C i

. . . . . . . . . . . . . . . Why? Tournament properties Good case: Every small set of cycles yields to some other cycle Random tournament: good case Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 15 Bad case: Everyone yields to C 1 , remainder 0 modulo C 1 separates

. . . . . . . . . . . . . . . Why? Tournament: Yielding between cycles Random tournament is good Technical details: tournament of yielding can be controlled Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 11 / 15 Input ≡ remainder

. . . . . . . . . . . . . . . . Lower bound for construction A proof of non-inclusion proves that every cycle yields No small dominating set: many independent edges among the chosen Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . 12 / 15 Separation ≈ proof of non-inclusion ≈ bad remainder for some modulo

. . . . . . . . . . . . . . Lower bound for construction A proof of non-inclusion proves that every cycle yields No small dominating set: many independent edges among the chosen Choice of accepting states: gcd , corresponding to a chosen edge, divides separating modulo Careful assignment of prime factors superpolynomial size Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 13 / 15 A lot of difgerent gcd ’s divide the length of a cycle ⇒

. . . . . . . . . . . . . . . . Future directions Non-unary case: is the state complexity exponential? how do automata behave? Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . 14 / 15 Hypothesis: at least 2 n Θ (1) Unary case: is the state complexity n Θ( log n ) ? DFA × coDFA : studied as transducers (bimachines) —

. . . . . . . . . . . . . . . . Thanks for your attention. Questions? Michael Raskin, raskin@mccme.ru (LaBRI) State complexity of complementing UFA July 10, 2018 . . . . . . . . . . . . . . . . . . . . . . . . 15 / 15