### Clausal Logic and Logic Programming in Algebraic Domains

William C Rounds, Guo-Qiang Zhang
2001 Information and Computation
We introduce a domain-theoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolutionrule for inferencein such a clausal logic; we introduce a natural declarative semantics and a xed-point semantics for disjunctive logic programs, and prove their equivalence; nally, we apply our results to give both a syntax and semantics for default logic in
more » ... any coherent algebraic dcpo. Key words and phrases. Domain theory and applications, logic programming, logics in articial intelligence. Research supported by NSF grant IRI-9509067. 1 De nition 2.1. A poset is a pair (D; v), where D is a nonempty set and v is a re exive, antisymmetric, and transitive relation on D. If D has a least element we write this as ?. When x v y we sometimes say \x subsumes y". De nition 2.2. A subset X of a poset D is directed i for any x; y in X there is a z in X with x v z and y v z. A typical example of a directed set is a totally ordered subset of D. Chains and directed sets provide us with an abstract notion of \approximatingsequence", where \approximation" is in the sense of learning more and more speci c information. We typically require approximating sequences to \converge" to a \limit", which is the least upper bound of the directed set or chain. This is captured by De nition 2.3. A directed complete partial order (dcpo) is a poset (D; v) with a bottom element ? and least upper bounds of directed sets. We read \x v y" as \x subsumes y".