On the decidability of homomorphism equivalence for languages

AbstractWe consider decision problems of the following type. Given a language L and two homomorphisms h1 and h2, one has to determine to what extent h1 and h2 agree on L. For instance, we say that h1 and h2 are equivalent on L if h1(ω) = h2(ω) holds for each ω ε L. In our main theorem we present an algorithm for deciding whether two given homomorphisms are equivalent on a given context-free language. This result also gives an algorithm for deciding whether the translations defined by two deterministic gsm mappings agree on a given context-free language

Systems of equations over a free monoid and Ehrenfeucht's conjecture

AbstractEhrenfeucht's conjecture states that every language L has a finite subset F such that, for any pair (g, h) of morphisms, g and h agree on every word of L if and only if they agree on every word of F. We show that it holds if and only if every infinite system of equations (with a finite number of unknowns) over a free monoid has an equivalent finite subsystem. It is shown that this holds true for rational (regular) systems of equations.The equivalence and inclusion problems for finite and rational systems of equations are shown to be decidable and, consequently, the validity of Ehrenfeucht's conjecture implies the decidability of the HDOL and DTOL sequence equivalence problems. The simplicity degree of a language is introduced and used to argue in support of Ehrenfeucht's conjecture

Equality languages and fixed point languages

This paper considers equality languages and fixed-point languages of homomorphisms and deterministic gsm mappings. It provides some basic properties of these classes of languages. We introduce a new subclass of dgsm mappings, the so-called symmetric dgsm mappings. We prove that (unlike for arbitrary dgsm mappings) their fixed-point languages are regular but not effectively obtainable. This result has various consequences. In particular we strengthen a result from Ehrenfeucht, A., and Rozenberg, G. [(1978), Theor. Comp. Sci. 7, 169–184] by pointing out a class of homomorphisms which includes elementary homomorphisms but still has regular equality languages. Also we show that the result from Herman, G. T., and Walker, A. [(1976), Theor. Comp. Sci. 2, 115–130] that fixed-point languages of DIL mappings are regular, is not effective

Checking Sets, Test Sets, Rich Languages and Commutatively Closed Languages

AbstractThe problem of homomorphism equivalence is to decide for some language L over some finite alphabet Σ and two homomorphisms f and g whether or not f (x) = g(x) for all x in L. It has been conjectured that each L can be represented by some finite subset F such that for all pairs of homomorphisms f and g: f (x) = g(x) for all x in F implies f (x) = g(x) for all x in L. This conjecture is proved for the families of rich and commutatively closed languages. Lower and upper bounds are derived for the sizes of these finite subsets and examples of language families are given for which there are effective constructions of these subsets

On test sets for checking morphism equivalence on languages with fair distribution of letters

AbstractA test set for a language L is a finite subset T of L with the property that each pair of morphisms that agrees on T also agrees on L. Some results concerning test sets for languages with fair distribution of letters are presented. The first result is that every D0L language with fair distribution of letters has a test set. The second result shows that every language L with fair distribution has a test set relative to morphisms g, h which have bounded balance on L. These results are generalizations of results of Culik II and Karhumäki (1983)