Hashing Modulo Context-Sensitive $α$-Equivalence

The notion of $α$-equivalence between $λ$-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 $α$-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 $λ$-terms modulo context-sensitive $α$-equivalence, generalizing over traditional bisimulation partitioning algorithms and improving upon a previously established $O(n\log^2 n)$ bound for a hashing modulo ordinary $α$-equivalence by Maziarz et al. Hashing $λ$-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.