On the computation of quotients and factors of regular languages

Abstract

Quotients and factors are important notions in the design of various computational procedures for regular languages and for the analysis of their logical properties. We propose a new representation of regular languages, by linear systems of language equations, which is suitable for the following computations: language reversal, left quotients and factors, right quotients and factors, and factor matrices. We present algorithms for the computation of all these notions, and indicate an application of the factor matrix to the computation of solutions of a particular language reconstruction problem. The advantage of these algorithms is that they all operate only on linear systems of language equations, while the design of the same algorithms for other representations often require translation to other representations.

Keywords

regular language / language factorization / language quotient

References

[1] Brzozowski J A. Derivatives of regular expressions. Journal of the Association for Computing Machinery, 1964, 11(4): 481–494

[2] Antimirov V M. Partial derivatives of regular expressions and finite automaton　constructions. Theoretical Computer Science, 1996, 155: 291–319

[3] Conway J H. Regular Algebra and Finite Machines. Mathematics series. Chapman and Hall, 1971

[4] Suzuki T, Okui S. Product derivatives of regular expressions. ISPJ Online Transactions, 2008, 1: 53–65

[5] Hopcroft J E, Ullman J D. Introduction to Automata Theory, Languages, and Computation. Series in Computer Science. Addison-Wesley Publishing Company, Inc., 1979

[6] Kozen D C. Automata and Computability. Undergraduate Texts in Computer Science. Springer-Verlag, New York, Inc., 1997

[7] Berstel J. Transductions and Context-Free Languages. B.G. Teubner Stuttgart, 1979

[8] Mateescu A, Salomaa A, Yu S. On the decomposition of finite languages. In: Rozenberg G, Thomas W, eds. Developments in Language Theory: Foundations, Applications, Perspectives. World Scientific, 2000, 22–31

[9] Mateescu A, Salomaa A, Yu S. Factorizations of languages and commutativity conditions. Acta Cybernetica, 2002, 15(3): 339–351

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg