On the sequential nature of unification
Journal of Logic Programming
Foundations of logic programming
Foundations of logic programming
New Classes for Parallel Complexity: A Study of Unification and Other Complete Problems for P
IEEE Transactions on Computers
Term matching on parallel computers
14th International Colloquium on Automata, languages and programming
A Machine-Oriented Logic Based on the Resolution Principle
Journal of the ACM (JACM)
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
Assignment Commands with Array References
Journal of the ACM (JACM)
Fast Decision Procedures Based on Congruence Closure
Journal of the ACM (JACM)
Reasoning About Recursively Defined Data Structures
Journal of the ACM (JACM)
Variations on the Common Subexpression Problem
Journal of the ACM (JACM)
An Efficient Unification Algorithm
ACM Transactions on Programming Languages and Systems (TOPLAS)
Complexity of finitely presented algebras
STOC '77 Proceedings of the ninth annual ACM symposium on Theory of computing
Parallelism in random access machines
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
The circuit value problem is log space complete for P
ACM SIGACT News
On simultaneous resource bounds
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Fundamenta Informaticae - Concurrency specification and programming
Fundamenta Informaticae - Concurrency Specification and Programming (CS&P'2002), Part 2
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Congruence closure is a fundamental operation for symbolic computation. Unification closureis defined as its directional dual, i.e., on the same inputs but top-down as opposed to bottom-up. Unifying terms is another fundamental operation for symbolic computation and is commonly computed using unification closure. We clarify the directional duality by reducing unification closure to a special form of congruence closure. This reduction reveals a correspondence between repeated variables in terms to be unified and equalities of monadic ground terms. We then show that: (1) single equality congruence closure on a directed acyclic graph, and (2) acyclic congruence closure of a fixed number of equalities, are in the parallel complexity class NC. The directional dual unification closures in these two cases are known to be log-space complete for PTIME. As a consequence of our reductions we show that if the number of repeated variables in the input terms is fixed, then term unification can be performed in NC; this extends the known parallelizable cases of term unification. Using parallel complexity we also clarify the relationship of unification closure and the testing of deterministic finite automata for equivalence.