A coherence theorem for Martin-Löf's type theory

1998 ◽  
Vol 8 (4) ◽  
pp. 413-436 ◽  
Author(s):  
MICHAEL HEDBERG
Keyword(s):  
Large Class ◽  
Equivalence Relation ◽  
Type Theory ◽  
Category Structure ◽  
Arbitrary Type ◽  
Unique Identity ◽  
Equality Relation ◽  
Coherence Theorem ◽  
Initial Category

In type theory a proposition is represented by a type, the type of its proofs. As a consequence, the equality relation on a certain type is represented by a binary family of types. Equality on a type may be conventional or inductive. Conventional equality means that one particular equivalence relation is singled out as the equality, while inductive equality – which we also call identity – is inductively defined as the ‘smallest reflexive relation’. It is sometimes convenient to know that the type representing a proposition is collapsed, in the sense that all its inhabitants are identical. Although uniqueness of identity proofs for an arbitrary type is not derivable inside type theory, there is a large class of types for which it may be proved. Our main result is a proof that any type with decidable identity has unique identity proofs. This result is convenient for proving that the class of types with decidable identities is closed under indexed sum. Our proof of the main result is completely formalized within a kernel fragment of Martin-Löf's type theory and mechanized using ALF. Proofs of auxiliary lemmas are explained in terms of the category theoretical properties of identity. These suggest two coherence theorems as the result of rephrasing the main result in a context of conventional equality, where the inductive equality has been replaced by, in the former, an initial category structure and, in the latter, a smallest reflexive relation.

scholarly journals Quotienting the delay monad by weak bisimilarity

2017 ◽  
Vol 29 (1) ◽  
pp. 67-92 ◽  
Author(s):  
JAMES CHAPMAN ◽  
TARMO UUSTALU ◽  
NICCOLÒ VELTRI
Keyword(s):  
Computer Science ◽  
Equivalence Relation ◽  
Type Theory ◽  
Partial Functions ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Alternative Approach ◽  
The One ◽  
Axiom Of Countable Choice

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

General models and extensionality

1972 ◽  
Vol 37 (2) ◽  
pp. 395-397 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
General Model ◽  
Type Theory ◽  
Axiom Schema ◽  
Nonstandard Model ◽  
Equality Relation ◽  
Definition Of

It is well known that equality is definable in type theory. Thus, in the language of [2], the equality relation between elements of type α is definable as , i.e., xα = yα iff every set which contains xα also contains yα. However, in a nonstandard model of type theory, the sets may be so sparse that the wff above does not denote the true equality relation. We shall use this observation to construct a general model in the sense of [2] in which the Axiom of Extensionality is not valid. Thus Theorem 2 of [2] is technically incorrect. However, it is easy to remedy the situation by slightly modifying the definition of general model.Our construction will show that the Axiom Schema of Extensionality is independent even if one takes as an axiom schema.We shall assume familiarity with, and use the notation of, [2] and §§2–3 of [1].

A construction of non-well-founded sets within Martin-Löf's type theory

1989 ◽  
Vol 54 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ingrid Lindström
Keyword(s):  
Equivalence Relation ◽  
Type Theory ◽  
Additional Condition ◽  
Complete System ◽  
Definition Of ◽  
Induction Scheme

AbstractIn this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnäs' limit definition of these within Martin-Löfs theory of types. A system is a type W together with an assignment of and to each ∝ ∈ W. We show that for any system W we can define an equivalence relation =w such that ∝ =w ß ∈ U and =w is the maximal bisimulation. Aczel's proof that CZF can be interpreted in the type V of iterative sets shows that if the system W satisfies an additional condition (*), then we can interpret CZF minus the set induction scheme in W. W is then extended to a complete system W* by taking limits of approximation chains. We show that in W* the antifoundation axiom AFA holds as well as the axioms of CFZ−.

Equivalence of codes for countable sets of reals

2020 ◽  
pp. 1-11
Author(s):  
William Chan
Keyword(s):  
Equivalence Relation ◽  
Borel Set ◽  
Borel Sets ◽  
Generic Absoluteness ◽  
Equality Relation ◽  
Continuous Images

Abstract A set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

An application of PER models to program extraction

1993 ◽  
Vol 3 (3) ◽  
pp. 309-331 ◽  
Author(s):  
Stefano Berardi
Keyword(s):  
Equivalence Relation ◽  
Type Theory ◽  
Lambda Calculus ◽  
Higher Order ◽  
Constructive Proof ◽  
Program Extraction ◽  
Relation Model

Type theory allows us to extract from a constructive proof that a specification is satisfiable a program that satisfies the specification. Algorithms for optimization of such programs are currently the object of research.In this paper we consider one such algorithm, which was described in Beeson (1985) and which we will call ‘Harrop’. This algorithm greatly simplifies programs extracted from proofs in the Pure Construction Calculus. We use a Partial Equivalence Relation model for higher order lambda calculus, to check that t and Harrop(t) return the same outputs from the same inputs, i.e. that they are extensionally equal.As a corollary, we show that it is correct (and, of course, useful) to replace a program t with Harrop(t). Such a correctness result has already been proved by Möhring (Möhring 1989a, 1989b) using realizability semantics, but we obtain it as a corollary of a new result, the extensional equality between t and Harrop(t). Also the semantic method we use is interesting in its own right.

scholarly journals Multisets in type theory

2019 ◽  
Vol 169 (1) ◽  
pp. 1-18 ◽  
Author(s):  
HÅKON ROBBESTAD GYLTERUD
Keyword(s):  
Set Theory ◽  
Equivalence Relation ◽  
Type Theory ◽  
Constructive Set Theory

AbstractA multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative multisets, where multisets are iteratively built up from other multisets, in the context Martin–Löf Type Theory, in the presence of Voevodsky’s Univalence Axiom.In his 1978 paper, “the type theoretic interpretation of constructive set theory” Aczel introduced a model of constructive set theory in type theory, using a W-type quantifying over a universe, and an inductively defined equivalence relation on it. Our investigation takes this W-type and instead considers the identity type on it, which can be computed from the univalence axiom. Our thesis is that this gives a model of multisets. In order to demonstrate this, we adapt axioms of constructive set theory to multisets, and show that they hold for our model.

scholarly journals Constructing a small category of setoids

2011 ◽  
Vol 22 (1) ◽  
pp. 103-121 ◽  
Author(s):  
OLOV WILANDER
Keyword(s):  
Type Theory ◽  
Order Theory ◽  
Small Category ◽  
First Order ◽  
First Order Theory ◽  
Unique Identity

Consider the first-order theory of a category.d It has a sort of objects, and a sort of arrows (so we may think of it as a small category). We show that, assuming the principle of unique substitutions, the setoids inside a type theoretic universe provide a model for this first-order theory. We also show that the principle of unique substitutions is not derivable in type theory, but that it is strictly weaker than the principle of unique identity proofs.

scholarly journals An Ulm-type classification theorem for equivalence relations in Solovay model

1997 ◽  
Vol 62 (4) ◽  
pp. 1333-1351 ◽  
Author(s):  
Vladimir Kanovei
Keyword(s):  
Equivalence Relation ◽  
Binary Sequences ◽  
Classification Theorem ◽  
Equivalence Relations ◽  
Equality Relation ◽  
Solovay Model ◽  
Type Classification

AbstractWe prove that in the Solovay model, every OD equivalence relation, Ε, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length < ω1) binary sequences, or continuously embeds Ε0, the Vitali equivalence.If Ε is a (resp. ) relation then the reduction above can be chosen in the class of all Δ1 (resp. Δ2) functions.The proofs are based on a topology generated by OD sets.

Setoids in type theory

2003 ◽  
Vol 13 (2) ◽  
pp. 261-293 ◽  
Author(s):  
GILLES BARTHE ◽  
VENANZIO CAPRETTA ◽  
OLIVIER PONS
Keyword(s):  
Type Theory ◽  
Type Systems ◽  
The Other ◽  
Dependent Type Theory ◽  
Common Definition ◽  
Equality Relation ◽  
The Common ◽  
Definition Of ◽  
Dependent Type

Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach.

scholarly journals Neural network-based models of binomial time series in data analysis problems

2021 ◽  
Vol 65 (6) ◽  
pp. 654-660
Author(s):  
Yu. S. Kharin
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Time Series ◽  
Pattern Recognition ◽  
Data Analysis ◽  
Equivalence Relation ◽  
Model Parameters ◽  
Computer Data ◽  
Arbitrary Type ◽  
New Family

This article is devoted to constructing neural network-based models for discrete-valued time series and their use in computer data analysis. A new family of binomial time series based on neural networks is presented, which makes it possible to approximate the arbitrary-type stochastic dependence in time series. Ergodicity conditions and an equivalence relation for these models are determined. Consistent statistical estimators for model parameters and algorithms for computer data analysis (including forecasting and pattern recognition) are developed.

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