Abstract
Motivated by proof checking, we consider the problem of efficiently establishing equivalence of propositional formulas by relaxing the completeness requirements while still providing certain guarantees. We present a quasilinear time algorithm to decide the word problem on a natural algebraic structures we call orthocomplemented bisemilattices, a subtheory of Boolean algebra. The starting point for our procedure is a variation of Aho, Hopcroft, Ullman algorithm for isomorphism of trees, which we generalize to directed acyclic graphs. We combine this algorithm with a term rewriting system we introduce to decide equivalence of terms. We prove that our rewriting system is terminating and confluent, implying the existence of a normal form. We then show that our algorithm computes this normal form in log linear (and thus sub-quadratic) time. We provide pseudocode and a minimal working implementation in Scala.
We acknowledge the financial support of the Swiss National Science Foundation project 200021_197288 “A Foundational Verifier”.
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Guilloud, S., Kunčak, V. (2022). Equivalence Checking for Orthocomplemented Bisemilattices in Log-Linear Time. In: Fisman, D., Rosu, G. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2022. Lecture Notes in Computer Science, vol 13244. Springer, Cham. https://doi.org/10.1007/978-3-030-99527-0_11
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