Abstract. We prove an existential version of Gaifman's locality theorem andshow how it can be applied algorithmically to evaluate existential first-order sentences in finite structures. 1 Introduction Gaifman's locality theorem [12] states that every first-order sentence is equivalent toa Boolean combination of sentences saying: There exist elements a 1; : : : ; ak that arefar apart from one another, and each ai satisfies some local condition described by afirst-order formula whose quantifiers only range over a fixed-size neighborhood of an

Archives are implemented as an extension of Theorema for representing large bodies of mathematics. They provide various constructs for organizing knowledge bases in a natural way: breaking formulae across cells, grouping them in a hierarchical structure, attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces, addressing the domains of categories and functors as namespaces with variable opera− tions. All constructs are logic–internal in the sense that they have a natural translation to higher–order logic so that �mathematical knowledge management � can be treated by the object logic itself. 1

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