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Computational Complexity and Induction for Partial Computable Functions in Type Theory

by Robert L. Constable, Karl Crary

In: Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman.

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An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in type theory, and consequently such concepts are not available in applications of type theory where they are greatly needed. It is also not clear how to provide a practical and adequate account in programming logics based on set theory.

This paper provides a practical theory supporting all these concepts in the setting of constructive type theories. We first introduce an extensional theory of partial computable functions in type theory. We then add support for intensional reasoning about programs by explicitly reflecting the essential properties of the underlying computation system. We use the resulting intensional reasoning tools to justify computational induction and to define computational complexity classes. Complexity classes take the form of complexity-constrained function types. These function types are also used in conjunction with the propositions-as-types principle to define a resource-bounded logic in which proofs of existence can guarantee feasibility of construction.

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