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Induction–recursion and Initial Algebras by Peter Dybjer; Anton Setzer is a Computer Science article available to read on EtoBox.

What is Induction–recursion and Initial Algebras about?

Induction-recursion is a powerful deÿnition method in intuitionistic type theory. It extends (generalized) inductive deÿnitions and allows us to deÿne all standard sets of Martin-L of type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals below the ÿrst Mahlo cardinal. In this article we give a new compact formalization of inductive-recursive deÿnitions by modeling them as initial algebras in slice categories. We give generic formation, introduction, elimination, and equality rules generalizing the usual rules of type theory. Moreover, we prove that the elimination and equality rules are equivalent to the principle of the existence of initial algebras for certain endofunctors. We also show the equivalence of the current formulation with the formulation of induction-recursion as a re ection principle given in Dybjer and Setzer (Lecture Notes in Comput. Sci. 2183Sci. (2001) 93)) 93). Finally, we discuss two type-theoretic analogues of Mahlo cardinals in set theory: an external Mahlo universe which is deÿned by induction-recursion and

Who reads Induction–recursion and Initial Algebras?

Author

Peter Dybjer; Anton Setzer

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0168-0072)

Published

2003

Language

EN

Field

Computer Science (Physical Sciences)