Conjunctive Queries, Existentially Quantified Systems of Equations and Finite Substitutions
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This report presents an elementary theory of unification for positive conjunctive queries. A positive conjunctive query is a formula constructed from propositional constants, equations and atoms using the conjunction ∧ and the existential quantifier ∃. In particular, empty queries correspond to existentially quantified systems of equations – called E-formulas. We provide an algorithm which transforms any conjunctive query into a solved form. We prove some lattice-theoretic properties of queries. In particular, the quotient set of E-formulas under an equivalence relation forms a complete lattice. Then we present another lattice – a lattice of finite substitutions. We prove that the both lattices are isomorphic. Finally, we introduce the notion of application of substitutions to formulas and clarify its relationship to E-formulas. This theory can be regarded as a basis for alternative presentation of logic programming.
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