On Existential Theories of List

We discuss the existential fragments of two theories of concatenation. These theories describe concatenation of possibly nested lists in the algebra of nite trees with lists and in the algebra of rational trees with lists. Syntax and the choice of models are motivated by the treatment of lists in PROLOG III. In a recent prototype of this language, Colmerauer has integrated a built-in concatenation of lists, and the constraint-solver checks satissability of equations and disequations over concatenated lists. But, for eeciency reasons satissability is only tested in a rather approximative way 1. The question arises if satissability is decidable. Our main results are the following. For the algebra of nite trees with lists, the exi-stential fragment of the theory is decidable. For the algebra of rational trees with lists, the positive existential fragment of the theory is decidable. Problems in the existential fragment may be traced back to a diicult question about solvability of word equations with length constraints for variables.