# Rational transductions and complexity of counting problems

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## Abstract

This work presents an algebraic method based on rational transductions to study the sequential and parallel complexity of counting problems for regular and context-free languages. This approach allows to obtain old and new results on the complexity of ranking and unranking as well as on other problems concerning the number of prefixes, suffixes, subwords and factors of a word which belong to a fixed language. Other results concern a suboptimal compression of finitely ambiguous c.f. languages, the complexity of the value problem for rational and algebraic formal series in noncommuting variables and a characterization of regular and *Z*-algebraic languages by means of rank functions.

## Keywords

Formal Series Regular Language Regulate Representation Counting Problem Constant Space
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## References

- [1]A.V. Aho, J.E. Hopcroft and J.D. Ullman, The design and analysis of computer algorithms, Addison-Wesley, Reading Mass. 1974.zbMATHGoogle Scholar
- [2]C. Alvarez and B. Jenner, “A very hard log space counting class”, Proceedings 5th Conference on Structure in Complexity Theory (1990), 154–168.Google Scholar
- [3]J. Berstel, Transductions and context-free languages, Teubner, Stuttgart, 1979.zbMATHGoogle Scholar
- [4]A. Bertoni, D. Bruschi and M. Goldwurm, “Ranking and formal power series”,
*Theoretical Computer Science*79 (1991) 25–35.zbMATHMathSciNetCrossRefGoogle Scholar - [5]A. Bertoni, M. Goldwurm and N. Sabadini, “The complexity of computing the number of strings of given length in context-free languages”,
*Theoretical Computer Science*86 (1991), 325–342.zbMATHMathSciNetCrossRefGoogle Scholar - [6]M. Bogni, Algoritmi per il problema del valore di serie formali a variabili non commutative, Degree Thesis in Mathematics, Dip. Scienze dell'Informazione, Università di Milano, June 1991.Google Scholar
- [7]S.A. Cook, “A taxonomy of problems with fast parallel algorithms,”
*Inform. and Control*64 (1985), 2–22.zbMATHMathSciNetCrossRefGoogle Scholar - [8]J. Earley, “An efficient context-free parsing algorithm”,
*Communications of the ACM*13 n.2 (1970),94–102.zbMATHCrossRefGoogle Scholar - [9]P. Flajolet, “Analytic models and ambiguity of context-free languages”,
*Theoretical Computer Science*49 (1987), 283–309.zbMATHMathSciNetCrossRefGoogle Scholar - [10]A.V. Goldberg and M. Sipser, “Compression and ranking”, Proceedings 17th ACM Symp. on Theory of Comput. (1985), 59–68.Google Scholar
- [11]M. Harrison, Introduction to formal language theory, Addison-Wesley, Reading Mass., 1978.zbMATHGoogle Scholar
- [12]D.T. Huynh, “The complexity of ranking simple languages”,
*Math. Systems Theory*23 (1990), 1–20.zbMATHMathSciNetCrossRefGoogle Scholar - [13]D.T. Huynh, “Effective entropies and data compression”,
*Information and Computation*90 (1991),67–85.zbMATHMathSciNetCrossRefGoogle Scholar - [14]A. Salomaa and M. Soittola, Automata theoretic aspects of formal power series, Springer Verlag, Berlin 1978.zbMATHGoogle Scholar
- [15]D.F. Stanat, “A homomorphism theorem for weighted context-free grammar”,
*J. Comput. System Sci.*6 (1972), 217–232.zbMATHMathSciNetGoogle Scholar - [16]L.G. Valiant, “The complexity of enumeration and reliability problems”,
*SIAM J. Comput.*8 (1979), 410–420.zbMATHMathSciNetCrossRefGoogle Scholar - [17]V. Vinaj, “Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits”, Proc. 6th Conference on Structure in Complexity Theory (1991), 270–284.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1992