Structural subtyping of non-recursive types is decidable

Abstract

We show that the first-order theory of structural subtyping of non-recursive types is decidable, as a consequence of a more general result on the decidability of term powers of decidable theories. Let /spl Sigma/ be a language consisting of function symbol and let /spl Cscr/; (with a finite or infinite domain C) be an L-structure where L is a language consisting of relation symbols. We introduce the notion of /spl Sigma/-term-power of the structure /spl Cscr/; denoted /spl Pscr/;/sub /spl Sigma//(/spl Cscr/;). The domain of /spl Pscr/;/sub /spl Sigma//(/spl Cscr/;) is the set of /spl Sigma/-terms over the set C. /spl Pscr/;/sub /spl Sigma//(/spl Cscr/;) has one term algebra operation for each f /spl isin/ /spl Sigma/, and one relation for each r /spl isin/ L defined by lifting operations of /spl Cscr/; to terms over C. We extend quantifier for term algebras and apply the Feferman-Vaught technique for quantifier elimination in products to obtain the following result. Let K be a family of L-structures and K/sub P/ the family of their /spl Sigma/-term-powers. Then the validity of any closed formula F on K/sub P/ can be effectively reduced to the validity of some closed formula q(F) on K. Our result implies the decidability of the first-order theory of structural subtyping of non-recursive types with covariant constructors, and the construction generalizes to contravariant constructors as well.