Abstract
We present a minimization algorithm for finite state automata that finds and merges bisimulation-equivalent states, identified through partition aggregation. We show the algorithm to be correct and run in time \( O \left( n^2 d^2 \left| \Sigma \right| \right) \), where n is the number of states of the input automaton \(M\), d is the maximal outdegree in the transition graph for any combination of state and input symbol, and \(\left| \Sigma \right| \) is the size of the input alphabet. The algorithm is slower than those based on partition refinement, but has the advantage that intermediate solutions are also language equivalent to \(M\). As a result, the algorithm can be interrupted or put on hold as needed, and the derived automaton is still useful. Furthermore, the algorithm essentially searches for the maximal model of a characteristic formula for \(M\), so many of the optimisation techniques used to gain efficiency in SAT solvers are likely to apply.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Detailed proofs are provided in the technical report available for download at https://www8.cs.umu.se/research/uminf/index.cgi?year=2016&number=18.
References
Abdulla, P.A., Holík, L., Kaati, L., Vojnar, T.: A uniform (bi-)simulation-based framework for reducing tree automata. Electron. Notes Theor. Comput. Sci. 251, 27–48 (2009)
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The design and analysis of computer algorithms. Addison-Wesley, Reading (1974)
Almeida, M., Moreira, N., Reis, R.: Incremental DFA minimisation. RAIRO - Theor. Inform. Appl. 48(2), 173–186 (2014)
Björklund, J., Maletti, A., May, J.: Backward and forward bisimulation minimization of tree automata. Theor. Comput. Sci. 410(37), 3539–3552 (2009)
Björklund, J., Maletti, A., Vogler, H.: Bisimulation minimisation of weighted automata on unranked trees. Fundamenta Informatica 92(1–2), 103–130 (2009)
Buchholz, P.: Bisimulation relations for weighted automata. Theor. Comput. Sci. 393(13), 109–123 (2008)
Cleophas, L., Kourie, D.G., Strauss, T., Watson, B.W.: On minimizing deterministic tree automata. In: Holub, J., Ž\(\check{\text{d}}\)árek, J. (eds.) Prague Stringology Conference, Prague, Czech Republic, pp. 173–182 (2009)
Daciuk, J.: Optimization of Automata. Gdańsk University of Technology Publishing House, Gdańsk (2014)
Gramlich, G., Schnitger, G.: Minimizing NFA’s and regular expressions. J. Comput. Syst. Sci. 73(6), 908–923 (2007)
Hopcroft, J.E.: An \(n\) log \(n\) algorithm for minimizing the states in a finite automaton. In: Kohavi, Z. (ed.) The Theory of Machines and Computations, pp. 189–196. Academic Press (1971)
Hopcroft, J.E., Ullman, J.D.: Set merging algorithms. SIAM J. Comput. 2(4), 294–303 (1973)
Huffman, D.A.: The synthesis of sequential switching circuits. J. Franklin Inst. 257, 161–190, 275–303 (1954)
Maletti, A.: Minimizing weighted tree grammars using simulation. In: Yli-Jyrä, A., Kornai, A., Sakarovitch, J., Watson, B. (eds.) FSMNLP 2009. LNCS (LNAI), vol. 6062, pp. 56–68. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14684-8_7
Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: 13th Annual IEEE Symposium on Switching and Automata Theory, pp. 125–129 (1972)
Moore, E.F.: Gedanken-experiments on sequential machines. Automata Stud. 34, 129–153 (1956). Princeton University Press, Princeton, New Jersey
Nerode, A.: Linear automaton transformations. Proc. Am. Math. Soc. 9(4), 541–544 (1958)
Paige, R., Tarjan, R.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)
Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22(2), 215–225 (1975)
ten Eikelder, H.: Some algorithms to decide the equivalence of recursive types. Technical report 93/32, Department of Mathematics and Computer Science, Technische Universiteit Eindhoven (1991)
Watson, B.W.: Taxonomies and toolkits of regular language algorithms. Ph.D. thesis, Department of Mathematics and Computer Science, TU Eindhoven (1995)
Watson, B.W., Daciuk, J.: An efficient incremental DFA minimization algorithm. Nat. Lang. Eng. 9(1), 49–64 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Björklund, J., Cleophas, L. (2017). Minimization of Finite State Automata Through Partition Aggregation. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2017. Lecture Notes in Computer Science(), vol 10168. Springer, Cham. https://doi.org/10.1007/978-3-319-53733-7_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-53733-7_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53732-0
Online ISBN: 978-3-319-53733-7
eBook Packages: Computer ScienceComputer Science (R0)