Formal Correctness of a Quadratic Unification Algorithm

Abstract

We present a case study using ACL2 to verify a nontrivial algorithm that uses efficient data structures. The algorithm receives as input two first-order terms, and it returns a most general unifier of these terms if they are unifiable, failure otherwise. The verified implementation stores terms as directed acyclic graphs by means of a pointer structure. Its time complexity is O(n 2 ) and its space complexity O(n), and it can be executed in ACL2 at a speed comparable to a similar C implementation. We report the main issues encountered to achieve this formally verified implementation.

(Nominal) Unification by Recursive Descent with Triangular Substitutions

“…The other main approach to the presentation and formalisation of unification algorithms is the Martelli-Montanari transformation system, introduced in [1]. Ruiz-Reina et al [14] formalise a quadratic unification algorithm (using term graphs, due to Corbin and Bidoit) in ACL2 in the transformation style. Urban et al [11] formalise nominal unification in Isabelle/HOL in transformation style.…”

Efficient execution in an automated reasoning environment

“…However, non-trivial reflexive definitions arise in practice. The TR describes such a case study, namely a sophisticated implementation of a unification algorithm using term dags [48]. Furthermore, authors Greve and Wilding describe the use of the same approach in an efficient implementation of a path finding algorithm in a graph [23].…”

“…For the sake of clarity, we will not introduce the technical improvements that can make the algorithm quadratic in time. The interested reader may consult [48], where the complete development of a formally verified quadratic unification algorithm is described.…”

“…But it has to be emphasized that internally, the main process is carried out on term dags. A description of the formal proof of these theorems in ACL2 can be found in [48].…”

“…Essentially, the measure constructs the associated unification problem in prefix form and applies a measure that was previously proved suitable for the prefix version of the algorithm. See [48] for details.…”