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A categorical semantics for inductive-inductive definitions
 

Summary: A categorical semantics for inductive-inductive
definitions
Thorsten Altenkirch1,, Fredrik Nordvall Forsberg2, Peter Morris1, and
Anton Setzer2
1
School of Computer Science, University of Nottingham
2
Department of Computer Science, Swansea University
Abstract. Induction-induction is a principle for defining datatypes in
Martin-Lof Type Theory. An inductive-inductive definition consists of a
set A, together with an A-indexed family B : A Set, where both A and
B are inductively defined in such a way that the constructors for A can
refer to B and vice versa. In addition, the constructors for B can refer
to the constructors for A. We extend the usual initial algebra semantics
for ordinary inductive datatypes to the inductive-inductive setting by
considering dialgebras instead of ordinary algebras. This gives a new and
compact formalisation of inductive-inductive definitions, which we prove
is equivalent to the usual formulation with elimination and computation
rules.
1 Introduction

  

Source: Altenkirch, Thorsten - School of Computer Science, University of Nottingham

 

Collections: Computer Technologies and Information Sciences