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

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)

The decidability of the equivalence problem for DOL-systems

The language and sequence equivalence problem for DOL-systems is shown to be decidable. In an algebraic formulation the sequence equivalence problem for DOL-systems can be stated as follows: Given homomorphisms h1 and h2 on a free monoid Σ* and a word σ from Σ*, is h1n(σ) = h2n(σ) for all n ⩾ 0