Fibrational induction rules for initial algebras

Neil Ghani, Patricia Johann, Clement Fumex

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

14 Citations (Scopus)
58 Downloads (Pure)

Abstract

This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs’ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, an induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobs’ work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set.
Original languageEnglish
Title of host publicationComputer Science Logic
Subtitle of host publication24th International Workshop, CSL 2010, 19th Annual Conference of the EACSL, Brno, Czech Republic, August 23-27, 2010. Proceedings
EditorsAnuj Dawar , Helmut Veith
PublisherSpringer
Pages336-350
Number of pages15
Volume6247
ISBN (Print)978-3-642-15204-7
DOIs
Publication statusPublished - 2010

Publication series

NameLecture Notes In Computer Science
PublisherSpringer
Volume6247
ISSN (Print)0302-9743

Keywords

  • fibrational Induction Rules
  • initial algebras
  • computer science

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