Publisher Summary This chapter discusses the recent results and open problems on homomorphisms on free monoids. Most of the results are decidability results. They are motivated or directly constitute problems in L-systems theory. All of the results are the basic problems about free monoids and as such are not only of purely mathematical interest but also are of fundamental interest for theoretical computer science. The chapter discusses the iterations of one or more homomorphisms—daily oral language (DOL), HDOL, and DTOL systems—and some generalizations thereof. The problem to test whether two homomorphisms agree on a given language from family L is called the homomorphic equivalence problem for L. The chapter discusses whether two given homomorphisms agree string by string on a given language and its applications to transducers. A deterministic generalized sequential mapping (dgsm) is a mapping defined by deterministic generalized sequential machine with accepting states. The decidability of homomorphic equivalence is open for all the families of languages between DOL and indexed.

Homomorphisms: decidability, equality and test sets

On Recent Trends in Formal Language Theory

A representation of recursively enumerable languages by two homomorphisms and a quotient

2DST mappings on languages and related problems

Based on the notions of locality and recognizability for n-dimensional languages n-dimensionally colourable 1-dimensional languages are introduced. It is shown: A language L is in NP if and only if L is n-dimensionally colourable for some n. An analogous characterization in terms of deterministic n-dimensional colourability is obtained for P. The addition of one unbounded dimension for colouring leads to a characterization of PSPACE.

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Formal language characterizations of P, NP, and Pspace

Recently, it has been shown that for each recursively enumerable language there exists an erasing homomorphism h 0 and homomorphisms h 1,h 2 such that L= h 0(e(h 1,h 2)) where (e(h 1,h 2)) is the set of minimal words on which h 1 and h 2 agree. Here we show that by restrictions on the erasing h 0 we obtain most time-complexity language classes, and by restrictions on the pair (h 1 h 2) we characterize all space complexity language classes.

K. Cullik

Papers

A homomorphic characterization of time and space complexity classes of languages