## Recursive Models of General Inductive Types (1993)

Venue: | Fundam. Inf |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Fu93recursivemodels,

author = {Yuxi Fu and Yuxi Fu},

title = {Recursive Models of General Inductive Types},

journal = {Fundam. Inf},

year = {1993},

volume = {26}

}

### OpenURL

### Abstract

We give an interpretation of Martin-Lof's type theory (with universes) extended with generalized inductive types. The model is an extension of the recursive model given by Beeson. By restricting our attention to PER model, we show that the strictness of positivity condition in the definition of generalized inductive types can be dropped. It therefore gives an interpretation of general inductive types in Martin-Lof's type theory. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Science, Manchester University, are available by anonymous ftp from m1.cs.man.ac.uk (130.88.13.4) in the directory /pub/TR. The files are stored as PostScript, in compressed form, with the report number as filename. Alternative...

### Citations

342 | Intuitionistic type theory - Martin-Löf - 1984 |

264 | Foundations of Constructive Mathematics - Beeson - 1985 |

261 | Programming in Martin-Löf’s Type Theory, An introduction - Nordström, Peterson, et al. - 1990 |

177 |
An introduction to inductive definition
- Aczel
- 1977
(Show Context)
Citation Context ...is the total relation. In fact, the category Per is a locally cartesian closed category with finite colimits. In the sequel we need an `effective version' of the following definition. Definition 2.2 (=-=[Acz77]-=-) A rule on the set U is a pair u v such that u ae U and v 2 U . A rule set on U is a set of rules on U . Given a rule set R on U , a set A is R-closed if for any rule u v 2 R, u ae A implies v 2 A. T... |

170 |
N.: Elementary induction on abstract structures. North
- Moschovakis
- 1974
(Show Context)
Citation Context ...son is that the -continuous functors are no longer available. The interpretation using rule sets ([Dyb91]) also breaks down because the negativity makes it impossible to give an inductive definition (=-=[Mos74]-=-). As a consequence, the !-Set model of the generalized inductive types given in [Fu92b] can not be modified to interpret general inductive types in the Calculus of Constructions. 6 Acknowledgement Th... |

169 |
The category-theoretic solution of recursive domain equations
- Smyth, Plotkin
- 1982
(Show Context)
Citation Context ..., The University, Oxford Road, Manchester M13 9PL, U.K. Interest in inductively defined types has been around for some time. An early work on a categorical approach to recursively defined types is in =-=[SP82]-=-, where the authors demonstrate that the initial T -algebra approach is a good alternative to what was proposed by ADJ-group. Later on, several researchers have investigated the idea of extending the ... |

119 | The type theoretic interpretation of constructive set theory - Aczel - 2001 |

109 |
Inductively defined types
- Coquand, Paulin-Mohring
- 1990
(Show Context)
Citation Context ...ime. An old approach is to interpret an inductive type by the initial T -algebra of a -continuous functor on Set, the category of sets. This method works also for the generalized inductive types, see =-=[CPM90]-=-. Unfortunately, a monotonic and -bounded functor on the category !-Set of !-sets ([Luo90, Ore92]) in general does not possess an initial T -algebra, the reason being that !-Set does not have all (fil... |

95 | Inductive types and type constraints in the second-order lambda calculus - Mendler |

92 |
Algebraically complete categories
- Freyd
- 1990
(Show Context)
Citation Context ....cit., there are interesting examples of how to use this kind of inductive types. Categorical formulation of inductive (coinductive) types can be found in [Men90, Men91b]. Related to this is the work =-=[Fre91] where som-=-e properties of "algebraic complete" categories are obtained. The model theory of the inductive types has been a research topic for some time. An old approach is to interpret an inductive ty... |

84 | An Extended Calculus of Constructions - Luo - 1990 |

76 | Inductive sets and families in Martin-Löfs Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löfs type theory
- Dybjer
- 1991
(Show Context)
Citation Context ...he classical set theoretical model can not be extended to that of the general inductive types. The reason is that the -continuous functors are no longer available. The interpretation using rule sets (=-=[Dyb91]-=-) also breaks down because the negativity makes it impossible to give an inductive definition ([Mos74]). As a consequence, the !-Set model of the generalized inductive types given in [Fu92b] can not b... |

65 |
Recursive Definition in Type Theory
- Mendler
- 1987
(Show Context)
Citation Context ...ed by ADJ-group. Later on, several researchers have investigated the idea of extending the existing typed calculi with inductively (coinductively) defined types. In [Men87a, Men87b, Men91a], see also =-=[CM85]-=-, the author considers a particular version of inductive and coinductive types. Two combinators and two rules are added to the second order -calculus to capture recursion and computational rules. In c... |

43 | Inductively Defined Types in the Calculus of Constructions - Pfenning, Paulin-Mohring - 1990 |

24 | A higher-order calculus and theory abstraction
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- 1991
(Show Context)
Citation Context ...dom([Γ])−→Per such that m[Γ]n implies [A](m) = [A](n). The context Γ, x : A is interpreted by σ([Γ], [A]). [Γ ⊢ Πx:A.B ] is the map π [Γ ]([A], [B ]) that sends α ∈ [Γ] to π([A](α, ), [B ](α, )). See =-=[17, 19]-=- for more on Per models. In Per the rule set that corresponds to ˙ RΘj can be defined as RΘj , except that Vκ is replaced by the ‘largest’ per ω×ω: ˙RΘ j def = ⎧ ⎨ ⎩ ∪k=1···mrange(fk) 〈nΘ j , m, f1, .... |

23 |
Constructive natural deduction and its modest interpretation
- Longo, Moggi
- 1988
(Show Context)
Citation Context ...dom([Γ])−→Per such that m[Γ]n implies [A](m) = [A](n). The context Γ, x : A is interpreted by σ([Γ], [A]). [Γ ⊢ Πx:A.B ] is the map π [Γ ]([A], [B ]) that sends α ∈ [Γ] to π([A](α, ), [B ](α, )). See =-=[17, 19]-=- for more on Per models. In Per the rule set that corresponds to ˙ RΘj can be defined as RΘj , except that Vκ is replaced by the ‘largest’ per ω×ω: ˙RΘ j def = ⎧ ⎨ ⎩ ∪k=1···mrange(fk) 〈nΘ j , m, f1, .... |

19 | A note on categorical datatypes - Wraith - 1989 |

16 | Predicative type universes and primitive recursion - Mendler - 1991 |

15 |
On the meaning and construction of the rules in Martin-Löf’s theory of types
- Backhouse
- 1987
(Show Context)
Citation Context ...egree of conceptual clarity if one has in mind a constructive set-theoretical semantics. A general scheme for adding inductively defined types in Martin-Löf’s type theory and in ECC are considered in =-=[3, 4, 10, 11]-=- and in [8, 20]. In both cases, the motivating example is the well-known W -types. The basic idea is to formalize the introductions of, and inductions on, data types. This method is more intuitive, an... |

14 | Recursive models for constructive set theories - Beeson - 1982 |

14 |
A unifying theory of dependent types: the schematic approach
- Luo
- 1992
(Show Context)
Citation Context ...bjects are defined and manipulated, then the correct notion of equality seems to be the intensional one. When formulated appropriately, such intensional equality is internally extensional as shown in =-=[Luo92]-=-. Second, the meta-theory of the extensional theory is hard. As a matter of fact, useful results in this area are almost non-existent. There are many conjectures though! Third, the encodings of induct... |

8 | Inductive data types: Well-ordering types revisited - Goguen, Luo |

8 |
The extended calculus of constructions (ECC) with inductive types
- Ore
- 1992
(Show Context)
Citation Context ...that the encodings are impossible in an intensional type theory, are convincing enough for us to prefer the traditional method. The formulation of inductive types as initial T -algebras is studied in =-=[Ore92]-=-. The construction of functors from type constructors is used to code up some T -algebras. In loc.cit., there are interesting examples of how to use this kind of inductive types. Categorical formulati... |

7 | An inversion principle for Martin-Lof's type theory. Talk presented at the Workshop on Programming Logic - Dybjer - 1989 |

6 |
Inductively defined sets in Martin-Lof's set theory
- Dybjer
- 1987
(Show Context)
Citation Context ...ct, useful results in this area are almost non-existent. There are many conjectures though! Third, the encodings of inductive types via W -types look messy. Those reasons, together with the result in =-=[Dyb88]-=- which shows that the encodings are impossible in an intensional type theory, are convincing enough for us to prefer the traditional method. The formulation of inductive types as initial T -algebras i... |

6 |
Inductively defined sets in Martin-Lof's type theory
- Dybjer
- 1987
(Show Context)
Citation Context ...ct, useful results in this area are almost non-existent. There are many conjectures though! Third, the encodings of inductive types via W -types look messy. Those reasons, together with the result in =-=[9]-=- which shows that the encodings are impossible in an intensional type theory, are convincing enough for us to prefer the traditional method. The formulation of inductive types as initial T -algebras i... |

6 |
Programming in Martin-Lof's Type Theory.An Introduction
- Nordstrom, Petersson, et al.
- 1989
(Show Context)
Citation Context ...s, say, λx 1 kk1 :K1 kk1 [tk1, . . . , tkk1−1/xk1, . . . , xkk1−1]. · · · . λx m kk1 :Km kk1 [tk1, . . . , tkk1−1/xk1, . . . , xkk1−1].recµ( � f)(tkk1 x1 kk1 · · · xm kk1 ). Example 2.3 The W -types (=-=[26, 28]-=-). This example is meant to bring out the intuition behind the above rules. There is only one constructor type : Θ = Πx:A.Πy:B(x) → X.X. Here φ1(X) = A and φ2(X) = B(x)→X. The introduction rule is the... |

5 | Do-it-yourself type theory (part 1 - Backhouse, Chisholm, et al. - 1989 |

4 |
Some Semantic Issues In Type Theory
- Fu
- 1992
(Show Context)
Citation Context ... types given in [Fu92b] can not be modified to interpret general inductive types in the Calculus of Constructions. 6 Acknowledgement The material in this paper is taken from the author's Ph.D thesis (=-=[Fu92a]-=-). My supervisor David Rydeheard has been very helpful all the way over the last two years. The examiners Peter Aczel and Martin Hyland have raised some interesting points and made quite a few comment... |

2 | Understanding Inductive Types in Constructions
- Fu
- 1993
(Show Context)
Citation Context ...ule sets ([Dyb91]) also breaks down because the negativity makes it impossible to give an inductive definition ([Mos74]). As a consequence, the !-Set model of the generalized inductive types given in =-=[Fu92b]-=- can not be modified to interpret general inductive types in the Calculus of Constructions. 6 Acknowledgement The material in this paper is taken from the author's Ph.D thesis ([Fu92a]). My supervisor... |

1 | Inductive Types via Initial Algebras - Mendler - 1990 |

1 | Encodings in Polymorphism, Revisited
- Fu
- 1993
(Show Context)
Citation Context ...nts in addition the disjoint sums and the unit type at least. A different methodology is to code up all inductive types in an extensional type theory with enough type constructors. This is adopted in =-=[32, 30, 16, 14]-=-. The argument against this approach is threefold. First, encoded inductive types are sometimes not good enough. For instance, the polymorphic encoding of the natural numbers type verifies only a weak... |