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

  • Authors:

  • K. N. Venkataraman

  • Affiliations:

  • Tufts Univ., Medford, MA

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1987

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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.