# Weak Minimization of DFA — An Algorithm and Applications

## Abstract

DFA minimization is a central problem in algorithm design and is based on the notion of DFA equivalence: Two DFA’s are equivalent if and only if they accept the same set of strings. In this paper, we propose a new notion of DFA equivalence (that we call weak-equivalence):We say that two DFA’s are weakly equivalent if they both accept the same number of strings of length *k* for every *k*. The motivation for this problem is as follows. A large number of counting problems can be solved by encoding the combinatorial objects we want to count as strings over a finite alphabet. If the collection of encoded strings is accepted by a DFA, then standard algorithms from computational linear algebra can be used to solve the counting problem efficiently. When applying this approach to largwe-scale applications, the bottleneck is the space complexity since the computation involves a matrix of order *k × k* if *k* is the size of the underlying DFA *M*. This leads to the natural question: Is there a smaller DFA that is weakly equivalent to *M*? We present an algorithm of time complexity *O*(*k* ^{2}) to find a compact DFA equivalent to a given DFA. We illustrate, in the case of tiling problem, that our algorithm reduces a (strongly minimal) DFA by a factor close to 2.

## Keywords

Transfer Matrix Simple Path Input Symbol Grid Graph Counting Problem## Preview

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## References

- [Cop]D. Coppersmith and S. Winograd:
*Matrix Multiplication via Arithmetic Progressions*. Journal of Symbolic Computation 9(3): 251–280 (1990).zbMATHMathSciNetCrossRefGoogle Scholar - [Gar]M. Garey and D. Johnson,
*Computers and Intractability-A Guide to the Theory of NP-completeness*, W.H. Freeman & Sons (1979).Google Scholar - [Hop]J. Hopcroft and J. Ullman,
*Introduction to Automata, Languages and Theory of Computation*, Addison-Wesley, Inc. (1979).Google Scholar - [Jia]T. Jiang and B. Ravikumar, Minimal NFA problems are hard,
*SIAM Journal on Computing*Vol. 22, No. 6, 1117–1141, (1993).zbMATHCrossRefMathSciNetGoogle Scholar - [Kla]D. Klarner and J. Pollack, Domino tilings with rectangles of fixed width,
*Discrete Mathematics*32 (1980), 45–52.zbMATHCrossRefMathSciNetGoogle Scholar - [Liu]C. L. Liu,
*Introduction to Combinatorial Mathematics*, McGraw Hill, New York, NY, 1968.zbMATHGoogle Scholar - [Mad]N. Madras and G. Slade,
*The Self-avoiding walk*, Birkhauser, Boston, MA, 1993.zbMATHGoogle Scholar - [Ogi]M. Ogihara and S. Toda, The complexity of computing the number of selfavoiding walks,
*Mathematical Foundations of Computer Science*2001, Editors: J. Sgall et al., Springer-Verlag Lecture Notes in Computer Science, Vol. 2136, 585–597.CrossRefGoogle Scholar - [Pac]L. Pachter, Combinatorial Approaches and Conjectures for 2-Divisibility Problems Concerning Domino Tilings of Polyominoes,
*Electronic Journal of Combinatorics*4 (1997), #R29.Google Scholar - [Pon]A. Ponitz and P. Tittmann, Improved Upper Bounds for Self-Avoiding Walks in Zd,
*Electronic Journal of Combinatorics*7 (2000), # R 21.Google Scholar - [Pro]J. Propp, A reciprocity theorem for domino tilings,
*Electronic Journal of Combinatorics*8 (2001), # R 18.Google Scholar - [Sta]R. Stanley, On dimer coverings of rectangles of fixed width,
*Discrete Applied Mathematics*12 (1985), 81–87.zbMATHCrossRefMathSciNetGoogle Scholar - [Ste]R. Stearns and H. Hunt, On the quivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata,
*SIAM Journal on Computing*14 (1985), 598–611.zbMATHCrossRefMathSciNetGoogle Scholar - [Val]L. Valiant, The complexity of enumeration and reliability problems,
*SIAM Journal on Computing*, 8(3): 410–421, 1979.zbMATHCrossRefMathSciNetGoogle Scholar - [Wil]H. Wilf, The problem of kings,
*Electronic Journal of Combinatorics*. 2 (1995), #R3.Google Scholar