Induction–recursion and initial algebras

Dybjer, Peter; Setzer, Anton

doi:10.1016/s0168-0072(02)00096-9

Cited by 53 publications

(56 citation statements)

References 16 publications

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“…In this paper we have introduced the theory IR + of positive inductive-recursive definitions as a generalization of Dybjer and Setzer's theory IR of inductive-recursive definitions [DS99,DS03,DS06], different from the fibrational generalization explored in Ghani et al [GMNFS13]: by modifying both syntax and semantics of IR we have been able to broaden the semantics to all of Fam(C) and not just Fam |C|. The theory of IR + , with IR as a subtheory, paves the way to the analysis of more sophisticated data types which allow not only for the simultaneous definition of an inductive type X and of a recursive function f : X → D, but also takes the intrinsic structure between objects in the target type D into account.…”

Section: Discussionmentioning

confidence: 99%

Positive Inductive-Recursive Definitions

Ghani

Malatesta

Forsberg

2013

Algebra and Coalgebra in Computer Science

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Abstract.A new theory of data types which allows for the definition of types as initial algebras of certain functors Fam(C) → Fam(C) is presented. This theory, which we call positive inductive-recursive definitions, is a generalisation of Dybjer and Setzer's theory of inductive-recursive definitions within which C had to be discrete -our work can therefore be seen as lifting this restriction. This is a substantial endeavour as we need to not only introduce a type of codes for such data types (as in Dybjer and Setzer's work), but also a type of morphisms between such codes (which was not needed in Dybjer and Setzer's development). We show how these codes are interpreted as functors on Fam(C) and how these morphisms of codes are interpreted as natural transformations between such functors. We then give an application of positive inductive-recursive definitions to the theory of nested data types and we give concrete examples of recursive functions defined on universes by using their elimination principle. Finally we justify the existence of positive inductiverecursive definitions by adapting Dybjer and Setzer's set-theoretic model to our setting.

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Section: Discussionmentioning

confidence: 99%

Positive Inductive-Recursive Definitions

Ghani

Malatesta

Forsberg

2013

Algebra and Coalgebra in Computer Science

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“…P pnq so that the eliminator for pN, r0, sucsq becomes P : N Ñ Set step 0 : 1 Ñ P p0q step suc : pn : Nq Ñ P pnq Ñ P psucpnqq elim 1 X pP, step 0 , step suc q : px : Nq Ñ P pxq For polynomial functors F , l F can be defined inductively over the structure of F as is given in e.g. Dybjer and Setzer [8]. However, l F and dmap F can be defined for any functor F : Set Ñ Set by defining l F pP, xq : ty : F pΣ z : A. P pzqq|Fpπ 0 qpyq xu dmap F pP, step c , xq : F pλy.xy, step c pyqyqpxq .…”

Section: Warm-up: a Generic Eliminator For An Inductive Definitionmentioning

confidence: 99%

“…One could imagine that that inductive-inductive definitions could be described by functors mapping families of sets to families of sets (similar to the situation for induction-recursion [8]), but this fails to take into account that the constructors for B should be able to refer to the constructors for A. Thus, we will see that the constructor for B can be described by an operation Arg B : pA : SetqpB : A Ñ Setqpc : Arg A pA, Bq Ñ Aq Ñ Arg A pA, Bq Ñ Set where c : Arg A pA, Bq Ñ A refers to the already defined constructor for A.…”

Section: Introductionmentioning

confidence: 99%

A Categorical Semantics for Inductive-Inductive Definitions

Altenkirch

Morris

Forsberg

et al. 2011

Algebra and Coalgebra in Computer Science

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Abstract. Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A Ñ 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 data types 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 rules.

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“…Induction-recursion (IR), developed in the seminal works of Peter Dybjer and Anton Setzer [9,10,11], remedies this deficiency. The key feature of an inductive-recursive definition is the simultaneous inductive definition of a small type X of indices together with the recursive definition of a function T : X !…”

Section: Introductionmentioning

confidence: 99%

Small Induction Recursion

Hancock

McBride

Ghani

et al. 2013

Lecture Notes in Computer Science

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Abstract. There are several di↵erent approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in which the data have an associated indices which can be used to store important computational information. The crucial and salient feature of dependent polynomials and indexed containers is that the index types are defined in advance of the data. At the most sophisticated level, induction-recursion allows us to define data and indices simultaneously.This work investigates the relationship between the theory of small inductive recursive definitions and the theory of dependent polynomials and indexed containers. Our central result is that the expressiveness of small inductive recursive definitions is exactly the same as that of dependent polynomials and indexed containers. A second contribution of this paper is the definition of morphisms of small inductive recursive definitions. This allows us to extend our main result to an equivalence between the category of small inductive recursive definitions and the category of dependent polynomials/indexed containers. We comment on both the theoretical and practical ramifications of this result.

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Induction–recursion and initial algebras

Cited by 53 publications

References 16 publications

Positive Inductive-Recursive Definitions

Positive Inductive-Recursive Definitions

A Categorical Semantics for Inductive-Inductive Definitions

Small Induction Recursion

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