A split-based incremental deterministic automata minimization algorithm

The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-014-9588-y. La fecha de publicación corresponde a la versión First OnlineWe here study previous results due to Hopcroft and Almeida et al. to propose an incremental split-based deterministic automata minimization algorithm whose average running-time does not depend on the size of the alphabet. The experimentation carried out shows that our proposal outperforms the algorithms studied whenever the automata have more than a (quite small) number of states and symbols.García Gómez, P.; Vázquez-De-Parga Andrade, M.; Velasco, JA.; López Rodríguez, D. (2014). A split-based incremental deterministic automata minimization algorithm. Theory of Computing Systems. 1-18. doi:10.1007/s00224-014-9588-y118Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley Publishing Company (1979)Watson, B.W., Daciuk, J.: An efficient incremental DFA minimization algorithm. Nat. Lang. Eng. 9(1), 49–64 (2003)Almeida, M., Moreira, N., Reis, R.: Incremental DFA minimisation. In: Domaratzki, M., Salomaa, K. (eds.) CIAA, of Lecture Notes in Computer Science, vol. 6482, pp 39–48. Springer (2010)Hopcroft, J.E.: An n ⋅ log n $n\cdot \log n$ algorithm for minimizing states in a finite automaton. Technical report, Stanford, University, Stanford (1971)Moore, E.F.: Gedanken experiments on sequential machines. In: Shannon, C.E., Mc-Carthy, J. (eds.) Automata Studies. Princeton Universty Press, Princeton (1956)Berstel, J., Boasson, L., Carton, O., Fagnot, I.: Automata: from Mathematics to Applications, chapter Minimization of automata. European Mathematical Society. (arXiv: 1010.5318v3. ) To appear.David, J.: Average complexity of Moore’s and Hopcroft’s algorithms. Theor. Comput. Sci. 417, 50–65 (2012)Almeida, M., Moreira, N., Reis, R.: Aspects of enumeration and generation with a string automata representation. In: Leung, H., Pighizzini, G. (eds.) DCFS, pp 58–69. New Mexico State University, Las Cruces (2006)Gries, D.: Describing an algorithm by Hopcroft. Acta Informatica 2, 97–109 (1973)Aho, A., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley Publishing Company (1974)Blum, N.: A O ( n log n ) $\mathcal {O}(n\log n)$ implementation of the standard method for minimizing n-state finite automata. Inf. Process. Lett. 57, 65–69 (1996)Knuutila, T.: Re-describing an algorithm by Hopcroft. Theor. Comput. Sci. 250, 333–363 (2001)Veanes, M.: Minimization of symbolic automata. Technical report, Microsoft Research, MSR-TR-2013-48 (2013)Lothaire, M.: Applied Combinatorics on Words chap. 1. Cambridge University Press, Cambridge (2005

Aspects of enumeration and generation with a string automata representation

The representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we present a unique string representation for complete initially-connected deterministic automata (ICDFAs) with n states over an alphabet of k symbols. For these strings we give a regular expression and show how they are adequate for exact and random generation, allow an alternative way for enumeration and lead to an upper bound for the number of ICDFAs. The 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. A uniform random generator for ICDFAs is presented that uses a table of pre-calculated values. Based on the same table, an optimal coding for ICDFAs is obtained. Key words: finite automata, initially connected deterministic finite automata, exact enumeration, random generation, minimal automata