Resolutionwith

Resolutionwith is a term used in formal logic and automated reasoning to denote an extension of the classical resolution rule that incorporates an ancillary condition or guide, referred to as a "with" constraint. The operator takes two clauses that are complementary on a literal and produces a resolvent, but only when a specified side-condition is satisfied. This allows constrained or guided resolution.

In typical usage, if C1 ∨ L and C2 ∨ ¬L are two clauses and W is a constraint, then resolutionwith(C1 ∨ L, C2 ∨ ¬L, W) yields the standard resolvent C1 ∨ C2 restricted by W, or it may filter resolvents to those that satisfy W. The exact formalization varies by framework, but the common idea is to control which resolutions are allowed, or to annotate derived clauses with their justification.

Applications include constrained theorem proving, where theory-specific rules impose side-conditions; proof assistants that require annotated derivations; and SAT/SMT solvers that integrate theory constraints into resolution steps. Benefits include reduced search space, better traceability of proofs, and easier integration with background theories.

Variants of the concept appear under different names, such as constrained resolution or annotated resolution, and implementations differ across tools. Resolutionwith is not a universally standardized operator; it is discussed in some research literature and tool documentation as an approach to guide inference in complex reasoning tasks.

See also: Resolution (logic); Constrained resolution; Automated theorem proving; SAT solving; SMT.