Hashing Modulo Context-Sensitive $\alpha$-Equivalence

TL;DR

This paper introduces a new context-sensitive notion of $oldsymbol{ extalpha}$-equivalence for $oldsymbol{ extlambda}$-terms, along with an efficient hashing scheme that improves subterm identification and sharing, with applications in large-scale mathematical knowledge graphs.

Contribution

It formalizes context-sensitive $oldsymbol{ extalpha}$-equivalence, proves its equivalence to bisimulation, and develops an $O(n extlog n)$ hashing algorithm that outperforms previous methods.

Findings

The hashing scheme correctly identifies $oldsymbol{ extlambda}$-terms modulo context-sensitive $oldsymbol{ extalpha}$-equivalence.

The algorithm operates in $O(n extlog n)$ time, improving upon previous bounds.

Application to Coq proof assistant data demonstrates practical utility in large-scale knowledge graph construction.

Abstract

The notion of $\alpha$-equivalence between $\lambda$-terms is commonly used to identify terms that are considered equal. However, due to the primitive treatment of free variables, this notion falls short when comparing subterms occurring within a larger context. Depending on the usage of the Barendregt convention (choosing different variable names for all involved binders), it will equate either too few or too many subterms. We introduce a formal notion of context-sensitive $\alpha$-equivalence, where two open terms can be compared within a context that resolves their free variables. We show that this equivalence coincides exactly with the notion of bisimulation equivalence. Furthermore, we present an efficient $O(n\log n)$ runtime hashing scheme that identifies $\lambda$-terms modulo context-sensitive $\alpha$-equivalence, generalizing over traditional bisimulation partitioning algorithms and improving upon a previously established $O(n\log^2 n)$ bound for a hashing modulo ordinary $\alpha$-equivalence by Maziarz et al. Hashing $\lambda$-terms is useful in many applications that require common subterm elimination and structure sharing. We have employed the algorithm to obtain a large-scale, densely packed, interconnected graph of mathematical knowledge from the Coq proof assistant for machine learning purposes.