Questions of definedness are ubiquitous in mathematics. Informally, these involve reasoning about expressions which may or may not have a value. This paper surveys work on logics in which such reasoning can be carried out directly, especially in computational contexts. It begins with a general logic of partial terms, continues with partial combinatory and lambda calculi, and concludes with an expressively rich theory of partial functions and polymorphic types, where termination of functional programs can be established in a natural way.

In this paper we introduce applicative theories which characterize the polynomial hierarchy of time and its levels. These theories are based on a characterization of the functions in the polynomial hierarchy using monotonicity constraints, introduced by Ben-Amram, Loff, and Oitavem.