Compactification and completion as absolute closure

Shostak's Congruence Closure as Completion

Shostak's congruence closure algorithm is demystiied, using the framework of ground completion on (possibly nonterminating, non-reduced) rewrite rules. In particular, the canonical rewriting relation induced by the algorithm on ground terms by a given set of ground equations is precisely constructed. The main idea is to extend the signature of the original input to include new constant symbols ...

Transitive Closure, Answer Sets and Predicate Completion

Transitive Closure, Answer Sets and Predicate Completion

We prove that the usual logic programming definition of transitive closure is correct under the answer set semantics, and investigate under what conditions it is correct under the completion semantics. That definition is allowed here to be combined with an arbitrary set of rules that may contain negation as failure, not merely with a set of facts. This work is motivated by applications to answe...

Lawvere Completion and Separation Via Closure

Bill Lawvere’s 1973 milestone paper “Metric spaces, generalized logic, and closed categories” helped us to detect categorical structures in previously unexpected surroundings. His revolutionary idea was not only to regard individual metric spaces as categories (enriched over the monoidal-closed category given by the non-negative extended real half-line, with arrows provided by ≥ and tensor by +...

Transitive Closure, Answer Sets and Predicate Completion