Type 2 recursion theory

Extracting Recursion Operators in Nuprl’s Type Theory

In this paper we describe the extraction of “efficient” recursion schemes from proofs of well-founded induction principles. This is part of a larger methodology; when these well-founded induction principles are used in proofs, the structure of the program extracted from the proof is determined by the recursion scheme inhabiting the induction principle. Our development is based on Paulson’s pape...

Modelling general recursion in type theory

Constructive type theory is a very expressive programming language. However, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that do not satisfy any syntactic condition that guarantees termination. We present a method to formalise general recursive algorithms in type theory that uses an inductive predicate to characterise termination a...

Constructing Recursion Operators in Intuitionistic Type Theory

Martin-Löf’s Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and rela...

Simple General Recursion in Type Theory

General recursive algorithms are such that the recursive calls are performed on arguments satisfying no condition that guarantees termination. Hence, there is no direct way of formalising them in type theory. The standard way of handling general recursion in type theory uses a well-founded recursion principle. Unfortunately, this way of formalising general recursive algorithms often produces un...

Fix-Point Equations for Well-Founded Recursion in Type Theory