Asymptotics of minimal deterministic finite automata recognizing a - PowerPoint PPT Presentation

Compacted Binary Trees Asymptotics of minimal deterministic finite automata recognizing a finite binary language AofA 2020 Andrew Elvey Price, Wenjie Fang, and Michael Wallner Institut Denis-Poisson, Universit´ e de Tours, France September, 2020 Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 1 / 22

Compacted Binary Trees | What is a DFA? What is a DFA? Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 2 / 22

Compacted Binary Trees | What is a DFA? Deterministic finite automata (DFA) DFA on alphabet { a , b } Graph with two outgoing edges from each node (state), labelled a and b An initial state q 0 A set F of final states (coloured green). b q 3 q 1 a a a a b b b a b q 2 q 4 q 0 Figure: A DFA. Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 3 / 22

Compacted Binary Trees | What is a DFA? Deterministic finite automata (DFA) Properties DFA on alphabet { a , b } Language: the set of accepted Graph with words two outgoing edges from each Minimal: no DFA with fewer node (state), labelled a and b states accepts the same language An initial state q 0 Acyclic: no cycles (except loops A set F of final states (coloured at unique sink) green). b q 3 q 1 a a a a b b b a b q 2 q 4 q 0 Figure: A DFA. This is the minimal DFA recognising the language { aa , aab , ab , b , bb } . Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 3 / 22

Compacted Binary Trees | What is a DFA? Counting minimal acyclic DFAs This work: Asymptotics of the numbers m n of minimal, acyclic DFAs on a binary alphabet with n + 1 nodes. Studied by Domaratzki, Kisman, Shallit and Liskovets between 2002 and 2006 Best bounds were out by an exponential factor We gave upper and lower bounds differing by a Θ( n 1 / 4 ) factor, by relating the DFAs to compacted trees. b q 3 q 1 a a a a b b b a b q 2 q 4 q 0 Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 4 / 22

Compacted Binary Trees | Main result Main result Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 5 / 22

Compacted Binary Trees | Main result Main result – A stretched exponential appears Theorem The number m n of minimal DFAs recognising a finite binary for n → ∞ � n ! 8 n e 3 a 1 n 1 / 3 n 7 / 8 � m n = Θ , where a 1 ≈ − 2 . 3381 is the largest root of the Airy function � ∞ � � t 3 Ai( x ) = 1 cos 3 + xt dt. π 0 Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 6 / 22

Compacted Binary Trees | Main result Main result – A stretched exponential appears Theorem The number m n of minimal DFAs recognising a finite binary for n → ∞ � n ! 8 n e 3 a 1 n 1 / 3 n 7 / 8 � m n = Θ , where a 1 ≈ − 2 . 3381 is the largest root of the Airy function � ∞ � � t 3 Ai( x ) = 1 cos 3 + xt dt. π 0 Conjecture Experimentally we find m n ∼ γ n !8 n e 3 a 1 n 1 / 3 n 7 / 8 , where γ ≈ 76 . 438160702 . Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 6 / 22

Compacted Binary Trees | Main result What is the Airy function? Properties � ∞ � � t 3 Ai( x ) = 1 cos 3 + xt dt π 0 Largest root a 1 ≈ − 2 . 3381 lim x →∞ Ai( x ) = 0 Also defined by Ai ′′ ( x ) = x Ai( x ) [Banderier, Flajolet, Schaeffer, Soria 2001]: Random Maps [Flajolet, Louchard 2001]: Brownian excursion area Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 7 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 8 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths a b q 0 b a b a b b a b a a a, b a b Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths a b q 0 b a b a b b a b a a a, b a b Highlight spanning tree given by depth first search (ignoring the sink) I.e., Black path to each vertex is first in lexicographic order Colour other edges red Draw as a binary tree with a edges pointing left and b edges pointing right Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths a b q 0 b a b a b b a b a a a, b a b Highlight spanning tree given by depth first search (ignoring the sink) I.e., Black path to each vertex is first in lexicographic order Colour other edges red Draw as a binary tree with a edges pointing left and b edges pointing right Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths q 0 a b a a b a b a a b a, b b b a b Highlight spanning tree given by depth first search (ignoring the sink) I.e., Black path to each vertex is first in lexicographic order Colour other edges red Draw as a binary tree with a edges pointing left and b edges pointing right Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths 8 7 5 6 4 3 1 2 Label nodes in post-order. By construction red edges point from a larger number to a smaller number → Label pointers Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths 8 7 5 6 6 4 3 3 3 1 5 1 2 1 1 Label nodes in post-order. By construction red edges point from a larger number to a smaller number → Label pointers Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22

Compacted Binary Trees | Bijection to decorated paths Bijection to decorated paths (7 , 7) 8 8 7 7 6 5 6 5 6 4 4 3 3 3 3 1 5 1 2 (0 , 0) 2 ( − 1 , 0) 1 1 1 3 a 2 a 2 b 4 a 4 b 6 a 6 b 7 b When the tree traversal... goes up: add up step with colour matching the corresponding node. passes a pointer: add horizontal step mark box corresponding to pointer label Elvey Price, Fang, Wallner | Bordeaux, Paris | 29.9.2020 9 / 22