Enumeration and generation with a string automata representation

In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initially-connected deterministic automata (ICDFAs) with n states over an alphabet of k symbols can be used for counting, exact enumeration, sampling and optimal coding, not only the set of ICDFAs but, to some extent, the set of regular languages. An exact generation algorithm can be used to partition the set of ICDFAs in order to parallelize the counting of minimal automata (and thus of regular languages). We present also a uniform random generator for ICDFAs that uses a table of pre-calculated values. Based on the same table it is also possible to obtain an optimal coding for ICDFAs.

[1]  Thomas Paranthoën.  Génération aléatoire et structure des automates à états finis , 2004 .

[2]  Valery A. Liskovets,et al.  Exact enumeration of acyclic deterministic automata , 2006, Discret. Appl. Math..

[3]  D. R. Lick,et al.  Graph theory with applications to algorithms and computer science , 1985 .

[4]  Sartaj Sahni,et al.  Analysis of algorithms , 2000, Random Struct. Algorithms.

[5]  Cyril Nicaud,et al.  Etude du comportement en moyenne des automates finis et des langages rationnels , 2000 .

[6]  Guy Louchard,et al.  Boltzmann Samplers for the Random Generation of Combinatorial Structures , 2004, Combinatorics, Probability and Computing.

[7]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[8]  Michael Domaratzki,et al.  Enumeration of Formal Languages , 2006, Bull. EATCS.

[9]  Robert W. Robinson.  Counting strongly connected finite automata , 1985 .

[10]  Frank Harary,et al.  Graphical enumeration , 1973 .

[11]  M. Lothaire.  Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications) , 2005 .

[12]  Philippe Flajolet,et al.  Analysis of algorithms , 2000, Random Struct. Algorithms.

[13]  Frédérique Bassino,et al.  Enumeration and random generation of accessible automata , 2007, Theor. Comput. Sci..

[14]  Jeffrey Shallit,et al.  On the Number of Distinct Languages Accepted by Finite Automata with n States , 2002, DCFS.

[15]  M. Domaratzki.  Combinatorial Interpretations of a Generalization of the Genocchi Numbers , 2004 .

[16]  Nelma Moreira,et al.  On the density of languages representing finite set partitions. , 2005 .

[17]  John E. Hopcroft,et al.  An n log n algorithm for minimizing states in a finite automaton , 1971 .

[18]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[19]  Frank Harary,et al.  Enumeration of Finite Automata , 1967, Inf. Control..

[20]  Jean-Marc Champarnaud,et al.  Random generation of DFAs , 2005, Theor. Comput. Sci..

[21]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[22]  Nelma Moreira,et al.  On the Representation of Finite Automata , 2005, DCFS.

[23]  Rogério Reis,et al.  Efficient representation of integer sets , 2006 .

[24]  M. Lothaire,et al.  Applied Combinatorics on Words , 2005 .

[25]  Michael A. Harrison,et al.  A Census of Finite Automata , 1965, Canadian Journal of Mathematics.

[26]  Nelma Moreira,et al.  Interactive manipulation of regular objects with FAdo , 2005, ITiCSE '05.

[27]  Donald E. Knuth.  The Art of Computer Programming 2 / Seminumerical Algorithms , 1971 .