Keywords: Calculus of Constructions, Higher Order Logic, Typed Lambda Calculus, Equational Logic, Rewriting Logic Rewriting logic together with it's membership equational sublogic favors the use of abstract specifications. It has a flexible computation system based on conditional rewriting modulo equations, and via its initial semsntics it supports a very liberal notion of inductive definitions. Pure type systems, on the other hand, in particular the calculus of constructions, provide higher-order (dependent) types, but they are based on a fixed notion of computation, namely beta-reduction. To close this gap we investigate (OCC) an equational variant of the calculus of constructions. The formal executable specification of OCC exploits the reflective capabilities of Maude, yielding an computational efficiency that brings us closer the the goal of a unified tool for programming, specification and verification integrating equational logic, higher-order logic and dependent types.
To Dissect a Mockingbird:A Graphical Notation for the Lambda Calculus with Animated Reduction David C Keenan, 27-Aug-1996 last updated 10-May-200 The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation. Introduction In the 1930s and 40s, around the birth of the "automatic computer", mathematicians wanted to formalise what we mean when we say some result or some function is "effectively computable", whether by machine or human. A "computer", originally, was a person who performed arithmetic calculations. The "effectively" part is included to indicate that we are not concerned with the time any particular computer might take to produce the result, so long as it would get there eventually. They wanted to find the simplest possible system that could be said to compute.
Notes on the Combinator Birds: 1. The combinatory birds were borrowed from To Mock A MockingBird, by Raymond Smullyan. 2. Some additional information about combinator birds can be found in To Dissect a Mockingbird by David C Keenan. 3. Some of the SK Combinatory terms were first reduced using the Combinatory Logic Tutorial by Chris Barker.