Higher-order unification has been shown to be undecidable. Miller discovered the pattern fragment and subsequently showed that higher-order pattern unification is decidable and has most general unifiers. We extend the algorithm to higher-order rational terms (a.k.a. regular Böhm trees, a form of cyclic λ-terms) and show that pattern unification on higher-order rational terms is decidable and has most general unifiers. We prove the soundness and completeness of the algorithm.

Logical frameworks provide natural and direct ways of specifying and reasoning within deductive systems. The logical framework LF and subsequent developments focus on finitary proof systems, making the formalization of circular proof systems in such logical frameworks a cumbersome and awkward task. To address this issue, we propose CoLF, a conservative extension of LF with higher-order rational terms and mixed inductive and coinductive definitions. In this framework, two terms are equal if they unfold to the same infinite regular Böhm tree. Both term equality and type checking are decidable in CoLF. We illustrate the elegance and expressive power of the framework with several small case studies.