Partial Rewriting and Value Interpretation of Logically Constrained Terms (Full Version)

Logically constrained term rewrite systems (LCTRSs) are a rewriting formalism that naturally supports built-in data structures, including integers and bit-vectors. The recent framework of existentially constrained terms and most general constrained rewriting on them (Takahata et al., 2025) has many advantages over the original approach of rewriting constrained terms. In this paper, we introduce partial constrained rewriting, a variant of rewriting existentially constrained terms whose underlying idea has already appeared implicitly in previous analyses and applications of LCTRSs. We examine the differences between these two notions of constrained rewriting. First, we establish a direct correspondence between them, leveraging subsumption and equivalence of constrained terms where appropriate. Then we give characterizations of each of them, using the interpretation of existentially constrained terms by instantiation. We further introduce the novel notion of value interpretation, that highlights subtle differences between partial and most general rewriting.

💡 Research Summary

The paper investigates two distinct rewriting mechanisms for logically constrained term rewrite systems (LCTRSs) that operate on existentially constrained terms. LCTRSs extend classical term rewriting by allowing rewrite rules to carry logical constraints interpreted over a background theory (e.g., integers, bit‑vectors). While the original notion of constrained rewriting was defined modulo equivalence of constrained terms and proved difficult to implement, recent work introduced “most general constrained rewriting” (MGCR) which extracts the most general part of a rewrite step and enjoys uniqueness of reducts for left‑linear rules.

Building on this, the authors propose a new variant called “partial constrained rewriting” (PCR). The essential difference is that MGCR applies a rule to all instantiations of a constrained term that satisfy the constraint, whereas PCR applies the rule only to some instantiations, i.e., those for which the constraint can be satisfied together with the specific substitution required by the rule. An illustrative example uses a simple summation function over integers. MGCR reduces the term `sum(x)