Abstract
The elementary calculus of binary relations as developed by Tarski in [5] may be thought of as a certain part of the first-order predicate calculus. Though less expressive, its theory (i.e. the set of its valid sentences) was shown to be undecidable by Tarski in [6]. Translated into algebraic logic this means that the equational theory of the class of relation algebras is undecidable. Similarly it can be proved that the same holds for the (sub-) class of proper relation algebras.The idea in Tarski's proof is to describe a pairing function by which any quantifier prefix may be contracted. In this note we apply a different method to treat the case of finite structures. We prove theTheorem. The equational theory of the class of finite proper relation algebras is undecidable.This result was announced in [4]. The main tool is representing the graph of primitive recursive functions via the cardinalities in finite simple models of equations.
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