NFA reduction algorithms by means of regular inequalities

We present different techniques for reducing the number of states and transitions in nondeterministic automata. These techniques are based on the two preorders over the set of states, related to the inclusion of left and right languages. Since their exact computation is NP-hard, we focus on polynomial approximations which enable a reduction of the NFA all the same. Our main algorithm relies on a first approximation, which can be easily implemented by means of matrix products with an O(mn3) time complexity, and optimized to an O(mn) time complexity, where m is the number of transitions and n is the number of states. This first algorithm appears to be more efficient than the known techniques based on equivalence relations as described by Lucian Ilie and Sheng Yu. Afterwards, we briefly describe some more accurate approximations and the exact (but exponential) calculation of these preorders by means of determinization.

[1]  P. Malik.  On equivalence. , 2003, The Canadian journal of cardiology.

[2]  Valentin M. Antimirov.  Rewriting Regular Inequalities (Extended Abstract) , 1995, FCT.

[3]  J. Brzozowski.  Canonical regular expressions and minimal state graphs for definite events , 1962 .

[4]  Udi Manber,et al.  Fast text searching: allowing errors , 1992, CACM.

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

[6]  Tao Jiang,et al.  Minimal NFA Problems are Hard , 1991, SIAM J. Comput..

[7]  Bruce W. Watson,et al.  An efficient incremental DFA minimization algorithm , 2003, Natural Language Engineering.

[8]  Jérôme Amilhastre,et al.  FA Minimisation Heuristics for a Class of Finite Languages , 1999, WIA.

[9]  Joseph JáJá,et al.  An Introduction to Parallel Algorithms , 1992 .

[10]  Jean-Marc Champarnaud,et al.  Compact and fast algorithms for safe regular expression search , 2004, Int. J. Comput. Math..

[11]  Jan Daciuk,et al.  Incremental Construction of Minimal Acyclic Finite State Automata and Transducers , 1998 .

[12]  Jean-Marc Champarnaud,et al.  Compact and Fast Algorithms for Regular Expression Search , 2004 .

[13]  Jean-Marc Champarnaud,et al.  Theoretical study and implementation of the canonical automaton , 2002, Fundam. Informaticae.

[14]  Lucian Ilie,et al.  Algorithms for Computing Small NFAs , 2002, MFCS.

[15]  T. G. Szymanski,et al.  On the Equivalence, Containment, and Covering Problems for the Regular and Context-Free Languages , 1976, J. Comput. Syst. Sci..

[16]  Tsunehiko Kameda,et al.  On the State Minimization of Nondeterministic Finite Automata , 1970, IEEE Transactions on Computers.