Decidability of the purely existential fragment of the theory of term algebras

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Abstract
This paper is concerned with the question of the decidability and the complexity of the decision problem for certain fragments of the theory of free term algebras . The existential fragment of the theory of term algebras is shown to be decidable through the presentation of a nondeterministic algorithm, which, given a quantifier-free formula P , constructs a solution for P if it has one and indicates failure if there are no solutions. It is shown that the decision problem is in NP by proving that, if a quantifier-free formula P has a solution, then there is one that can be represented as a dag in space at most cubic in the length of P . The decision problem is shown to be complete for NP by reducing 3-SAT to that problem. Thus it is established that the existential fragment of the theory of pure list structures in the language of NIL, CONS, CAR, CDR, =, ≤ (subexpression) is NP-complete. It is further shown that even a slightly more expressive fragment of the theory of term algebras, the one that allows bounded universal quantifiers, is undecidable.

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