Abstract Church's simple theory of types is a system of higher-order logic in which functions are assumed to be total. We present in this paper a version of Church's system called PF in which functions may be partial. The semantics of PF, which is based on Henkin's general-models semantics, allows terms to be nondenoting but requires formulas to always denote a standard truth value. We prove that PF is complete with respect to its semantics. The reasoning mechanism in PF for partial functions corresponds closely to mathematical practice, and the formulation of PF adheres tightly to the framework of Church's system.

Turner popularized a technique of Wadsworth in which a cyclic graph rewriting rule is used to implement reduction of the fixed point combinator Y . We examine the theoretical foundation of this approach. Previous work has concentrated on proving that graph methods are, in a certain sense, sound and complete implementations of term methods. This work is inapplicable to the cyclic Y rule, which is unsound in this sense since graph normal forms can exist without corresponding term normal forms. We define and prove the correctness of combinator head reduction using the cyclic Y rule; the correctness of normal reduction is an immediate consequence. Our proof avoids the use of infinite trees to explain cyclic graphs. Instead, we show how to consider reduction with cycles as an optimization of reduction without cycles.