The theory of formal languages is the oldest and most fundamental area of theoretical computer science. It has served as a basis of formal modeling from the early stages of programming languages to the recent beginnings of DNA computing. This first handbook of formal languages gives a comprehensive up-to-date coverage of all important aspects and subareas of the field. Best specialists of various subareas, altogether 50 in number, are among the authors. The maturity of the field makes it possible to include a historical perspective in many presentations. The individual chapters can be studied independently, both as a text and as a source of reference. The Handbook is an invaluable aid for advanced students and specialists in theoretical computer science and related areas in mathematics, linguistics, and biology.

Handbook of Formal Languages

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

We survey the properties of sets of integers recognizable by automata when they are written in p-ary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in dierent bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of Cobham-Semenov, the original proof being published in Russian.

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LOGIC AND p-RECOGNIZABLE SETS OF INTEGERS

The purpose of Chapter 9 is to develop some classical results on formal languages and automata by an algebraic treatment using semirings, formal power series and matrices. The use of semirings, formal power series and matrices makes the proofs computational in nature and, consequently, more satisfactory from the mathematical point of view than the customary proofs.

Semirings and formal power series: their relevance to formal languages and automata

In this paper we introduce a sharpening of the Parikh map- ping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix prod- uct gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting in- terconnection between mirror images of words and inverses of matrices. Mathematics Subject Classification. 68Q45, 68Q70.

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A sharpening of the Parikh mapping

— We show that it is decidable whether or not a k-recognizable set is recognizable. Consequently, it is decidable whether or not the set defined by a number System is recognizable. Résumé. Nous montrons qu'il est décidable si un ensemble k-reconnaissable est reconnaissabîe. En conséquence, il est décidable si Vensemble défini par un système de numération est reconnaissabîe.

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A decision method for the recognizability of sets defined by number systems

Bases and ambiguity of number systems

We study the freeness problem for matrix semigroups. We show that the freeness problem is decidable for upper-triangular 2x2 matrices with rational entries when the products are restricted to certain bounded languages. We also show that this problem becomes undecidable for sufficiently large matrices.

Juha Honkala

Papers

A decision method for the recognizability of sets defined by number systems

Bases and ambiguity of number systems

The freeness problem over matrix semigroups and bounded languages

On Parikh Slender Languages and Power Series

On the simplification of infinite morphic words