On the Undecidability of Equivalence Problems for Relational Expressions
We consider two versions of the relational algebra: (a) the attribute relational algebra, based on the natural join and relations with columns corresponding to attributes, and (b) the positional relational algebra, based on the cross product and relation with an order on the columns, and with any column identified by its position in that order. For the attribute relational algebra, we show that both the equivalence and the finite equivalence (i.e., equivalence over finite relations only) of expressions involving just one ternary relation and the operators of projection, selection, join and difference, are undecidable. For the positional relational algebra, we show that both the equivalence and finite equivalence of expressions involving just one binary relation and the operators of projection, selection, cross product, restriction and difference, are undecidable.
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