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We consider expressions built up from binary relation names using theoperators union, composition, and set difference. We show that it isundecidable to test whether a given such expression $e$ is finitelysatisfiable, i.e., whether there exist finite binary relations that can besubstituted for the relation names so that $e$ evaluates to a nonempty result.This result already holds in restriction to expressions that mention just asingle relation name, and where the difference operator can be nested at mostonce.

Undecidability of satisfiability in the algebra of finite binary relations with union, composition, and difference