Ramified higher-order unification

Abstract

While uni cation in the simple theory of types a.k.a. higher-order logic is undecidable, we s h o w t h a t u nication in the pure rami ed theory of types with integer levels is decidable. But t h e pure rami ed theory of types cannot express even the simplest formulas of logic. The impure rami ed type theory has an undecidable uni cation problem even at order 2. However, the d ecidability result for the pure subsystem indicates that u ni cation should fail to t erminate less often than general higher-order uni cation. We present applications to t w o expressive s u bsystems of secondorder Peano arithmetic, ACA 0 and 1 k -CA 0 .