A propositional encoding for first-order clausal entailment over infinitely many constants

Abstract

There is a fundamental trade-off between the expressiveness of the language and the tractability of the reasoning task in knowledge representation. On the one hand it is widely acknowledged that relations and more generally, the expressiveness of first-order logic is extremely useful for capturing concepts required for common-sense reasoning. But at the same time the entailment problem is only semi-decidable.

There have been a wide range of approaches to deal with this trade-off, from restricting the language to propositional logic to limit the expressiveness of the language in terms of the arity of the predicates (as in description logics) or the use of negation (as in Horn logic) to limit reasoning by weakening the entailment relation using non-standard semantics.

In this work, we address a gap in this literature. We show that there is an intuitive fragment of first-order disjunctive knowledge, for which reasoning is decidable and can be reduced to propositional satisfiability. Knowledge bases in this fragment correspond to universally quantified first-order clauses, but without arity restrictions and without restrictions on the appearance of negation. Queries, however, are expected to be ground formulas. We achieve this result by showing how the entailment over infinitely many infinite-sized structures can be reduced to a search over finitely many finite-size structures. The crux of the argument lies in showing that constants not mentioned in the knowledge base and/or query behave identically (in a suitable formal sense). We then go on to also show that there is also an extension to this result for function symbols.