# The equivalence of dgsm replications on Q-rational languages is decidable

## Abstract

The notion of a morphic replication is extended by replacing morphisms with deterministic generalized sequential machines (dgsm). As is well-known, the equivalence problem for gsm mappings is undecidable, whereas for dgsm mappings or morphic replications it is decidable. It is shown that, more generally, two dgsm replications *τ*_{1} and *τ*_{2} are equivalent on a *Q*-rational language *L* if and only if *τ*_{1}(*w*)=*τ*_{2}(*w*) for all *w* in *L* such that the length of *w* does not exceed a certain polynomial bound whose parameters are the numbers of dgsm's in *τ*_{1} and *τ*_{2}, the cardinalities of their state sets, and the size *n* of the *Q*-matrix system accepting *L*. Accordingly, two dgsm's with *k*_{1} and *k*_{2} states, respectively, are equivalent on *L* if and only if they are (2*nk*_{1}+2*nk*_{2}−1)-equivalent on it. A corresponding result is derived for the morphic replication equivalence on linear languages.

The above results provide an algorithm being of exponential time complexity. Polynomial time bounds for the dgsm equivalence and the morphic replication equivalence on *Q*-rational languages are found by means of the result that the emptiness problem for *Q*-rational languages is decidable in polynomial time.

Finally, a modified dgsm replication is defined, and it is shown that both the original and the modified dgsm replication equivalence problems are decidable for any family of languages satisfying certain conditions. One such family consists of languages given in the form *τ*(*L*) where *τ* is a morphic replication and *L* is a *Q*-rational language.

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