An algebraic view of structural induction

  • Claudio Hermida
  • Bart Jacobs
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 933)

Abstract

We propose a uniform, category-theoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial algebras for certain kind of endofunctors T: \(\mathbb{B} \to \mathbb{B}\)on a bicartesian/distributive category \(\mathbb{B}\). Regarding a predicate logic as a fibration p: \(\mathbb{P} \to \mathbb{B}\)over \(\mathbb{B}\), we consider a logical predicate lifting of T to the total category \(\mathbb{P}\). Then, a predicate is inductive precisely when it carries an algebra structure for such lifted endofunctor. The validity of the induction principle is formulated by requiring that the ‘truth’ predicate functor ⊤: \(\mathbb{B} \to \mathbb{P}\)preserve initial algebras. We then show that when the fibration admits a comprehension principle, analogous to the one in set theory, it satisfies the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. the ‘stability’ property of the induction principle.

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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Claudio Hermida
    • 1
  • Bart Jacobs
    • 2
  1. 1.Computer Science DepartmentAarhus UniversityDenmark
  2. 2.CWISJ AmsterdamThe Netherlands

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