Nominal Unification and Matching of Higher Order Expressions with Recursive Let

TL;DR

This paper presents a sound and complete algorithm for nominal unification of higher-order expressions with recursive let constructs, analyzing its complexity and exploring special cases like letrec-matching.

Contribution

It introduces a polynomial-time nominal unification algorithm for higher-order expressions with recursive let and atom-variables, and analyzes complexity of related matching problems.

Findings

Algorithm runs in nondeterministic polynomial time.

Specialized algorithms for letrec-matching and environment variables are in NP.

Trade-offs between exponential growth and nondeterminism are identified.

Abstract

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. We also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time. In addition we prove that there is a guessing strategy for nominal unification with letrec and atom-variable that is a trade-off between exponential growth and non-determinism. Nominal matching with variables representing partial letrec-environments is also shown to be in NP.