Constructive Type Theory Research Articles

Viewing [formula omitted]-terms through maps

In this paper we introduce the notion of map, which is a notation for the set of occurrences of a symbol in a syntactic expression such as a formula or a λ term. We use binary trees over 0 and 1 as maps, but some well-formedness conditions are required. We develop a representation of lambda terms using maps. The representation is concrete (inductively definable in HOL or Constructive Type Theory) and canonical (one representative per λ term). We define substitution for our map representation, and prove the representation is in substitution preserving isomorphism with both nominal logic λ terms and de Bruijn nameless terms. These proofs are mechanically checked in Isabelle/HOL and Minlog respectively.The map representation has good properties. Substitution does not require adjustment of binding information: neither α conversion of the body being substituted into, nor de Bruijn lifting of the term being implanted. We have a natural proof of the substitution lemma of λ calculus that requires no fresh names, or index manipulation.Using the notion of map we study conventional raw λ syntax. E.g. we give, and prove correct, a decision procedure for α equivalence of raw λ terms that does not require fresh names.We conclude with a definition of β reduction for map terms, some discussion on the limitations of our current work, and suggestions for future work.

Intuitionistic completeness of first-order logic

We constructively prove completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable in iFOL if and only if it is uniformly valid in intuitionistic evidence semantics as defined in intuitionistic type theory extended with an intersection operator.Our completeness proof provides an effective procedure that converts any uniform evidence into a formal iFOL proof. Uniform evidence can involve arbitrary concepts from type theory such as ordinals, topological structures, algebras and so forth. We have implemented that procedure in the Nuprl proof assistant.Our result demonstrates the value of uniform validity as a semantic notion for studying logical theories, and it provides new techniques for showing that formulas are not intuitionistically provable. Here we demonstrate its value for minimal and intuitionistic first-order logic.

An open logical framework

The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of Star-modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and sub-structural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincare Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP. We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP : strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments. Moreover, we provide a canonical presentation of LFP, based on a suitable extension of the notion of βη-long normal form, allowing for smooth formulations of adequacy statements.

A model of type theory in simplicial sets: A brief introduction to Voevodsky's homotopy type theory

We describe how to interpret constructive type theory in the topos of simplicial sets where types appear as Kan complexes and families of types as Kan fibrations. Since Kan complexes may be understood as weak higher-dimensional groupoids this model generalizes and extends the (ordinary) groupoid model which was introduced by M. Hofmann and the author about 20 years ago. Finally, we discuss Voevodsky's Univalence Axiom which has been shown to hold in this model. This axiom roughly states that isomorphic types are equal. The type theoretic notion of isomorphism provided by this model coincides with homotopy equivalence of Kan complexes. For this reason it has become common to refer to it as Homotopy Type Theory.

Assembling the Proofs of Ordered Model Transformations

In model-driven development, an ordered model transformation is a nested set of transformations between source and target classes, in which each transformation is governed by its own pre and post- conditions, but structurally dependent on its parent. Following the proofs-as-model-transformations approach, in this paper we consider a formalisation in Constructive Type Theory of the concepts of model and model transformation, and show how the correctness proofs of potentially large ordered model transformations can be systematically assembled from the proofs of the specifications of their parts, making them easier to derive.

A Constructive Type-Theoretical Formalism for the Interpretation of Subatomically Sensitive Natural Language Constructions

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-Lof) type theory [8, 9] by stating rules for the formation, introduction, elimination and equality of atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs.

Are There Enough Injective Sets?

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Lof’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory.

Constructivist and structuralist foundations: Bishop’s and Lawvere’s theories of sets

Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of Martin-Löf. The theory, CETCS, provides a structuralist foundation for constructive mathematics in the style of Bishop.

Topological inductive definitions

In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that of finite products, while FSpi is complete (in particular has all products), but does not contain some naturally occurring classes of formal spaces. Moreover, completeness of FSpi only holds under the assumption of strong principles for the existence of inductively defined sets.In this paper, working in the context of constructive set theory, the category ImLoc of imaginary locales is introduced. ImLoc is complete already over weak formulations of constructive set theory, contains FSp and FSpi as full subcategories, and, as FSp and FSpi, is equivalent to the category of locales in IZF.Applications of the concept of imaginary locale to a proof of a point-free version of Tychonoff embedding theorem, and to set-representation theorems, are then presented.

Proof-relevance of families of setoids and identity in type theory

Families of types are fundamental objects in Martin-Lof type theory. When extending the notion of setoid (type with an equivalence relation) to families of setoids, a choice between proof-relevant or proof-irrelevant indexing appears. It is shown that a family of types may be canonically extended to a proof-relevant family of setoids via the identity types, but that such a family is in general proof-irrelevant if, and only if, the proof-objects of identity types are unique. A similar result is shown for fibre representations of families. The ubiquitous role of proof-irrelevant families is discussed.

Resumption-based big-step and small-step interpreters for While with interactive I/O

In this tutorial, we program big-step and small-step total interpreters for the While language extended with input and output primitives. While is a simple imperative language consisting of skip, assignment, sequence, conditional and loop. We first develop trace-based interpreters for While. Traces are potentially infinite nonempty sequences of states. The interpreters assign traces to While programs: for us, traces are denotations of While programs. The trace is finite if the program is terminating and infinite if the program is non-terminating. However, we cannot decide (i.e., write a program to determine), for any given program, whether its trace is finite or infinite, which amounts to deciding the halting problem. We then extend While with interactive input/output primitives. Accordingly, we extend the interpreters by generalizing traces to resumptions. The tutorial is based on our previous work with T. Uustalu on reasoning about interactive programs in the setting of constructive type theory.

An introduction to small scale reflection in Coq

This tutorial presents the Ssreflect extension to the Coq system. This extension consists of an extension to the Coq language of script, and of a set of libraries, originating from the formal proof of the Four Color theorem. This tutorial proposes a guided tour in some of the basic libraries distributed in the Ssreflect package. It focuses on the application of the small scale reflection methodology to the formalization of finite objects in intuitionistic type theory.

On the existence of Stone-Čech compactification

Introduction. In 1937 E. Čech and M.H. Stone, independently, introduced the maximal compactification of a completely regular topological space, thereafter called Stone-Čech compactification [8, 23]. In the introduction of [8] the non-constructive character of this result is so described: “It must be emphasized that β(S) [the Stone-Čech compactification of S] may be defined only formally (not constructively) since it exists only in virtue of Zermelo's theorem”.By replacing topological spaces with locales, Banaschewski and Mulvey [4, 5, 6], and Johnstone [14] obtained choice-free intuitionistic proofs of Stone-Čech compactification. Although valid in any topos, these localic constructions rely—essentially, as is to be demonstrated—on highly impredicative principles, and thus cannot be considered as constructive in the sense of the main systems for constructive mathematics, such as Martin-Löf's constructive type theory and Aczel's constructive set theory.In [10] I characterized the locales of which the Stone-Čech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of Dependent Choice.

Effective homology of bicomplexes, formalized in Coq

In this paper, we present a complete formalization in the Coq theorem prover of an important algorithm in computational algebra, namely the calculation of the effective homology of a bicomplex. As a necessary tool, we encode a hierarchy of algebraic structures in constructive type theory, including graded and infinite data structures. The experience shows how some limitations of the Coq proof assistant to deal with this kind of algebraic data can be overcome by applying a separation of concerns principle; more concretely, we propose to distinguish in the representation of an algebraic structure (such as a group or a module) a behavioural part, containing operation signatures and axioms, and a structural part determining if the algebraic data is free, of finite type and so on.

Typing logic contents using Lingua Cosmica

This paper informs how elements of constructive type theory can be used effectively for clarifying textual messages meant for communication with ETI. Within the setting of a suitable environment consisting of declared terms, it is shown how logical contents of texts can be modelled and codified.

Resumptions, Weak Bisimilarity and Big-Step Semantics for While with Interactive I/O: An Exercise in Mixed Induction-Coinduction

We look at the operational semantics of languages with interactive I/O through the glasses of constructive type theory. Following on from our earlier work on coinductive trace-based semantics for While, we define several big-step semantics for While with interactive I/O, based on resumptions and termination-sensitive weak bisimilarity. These require nesting inductive definitions in coinductive definitions, which is interesting both mathematically and from the point-of-view of implementation in a proof assistant. After first defining a basic semantics of statements in terms of resumptions with explicit internal actions (delays), we introduce a semantics in terms of delay-free resumptions that essentially removes finite sequences of delays on the fly from those resumptions that are responsive. Finally, we also look at a semantics in terms of delay-free resumptions supplemented with a silent divergence option. This semantics hinges on decisions between convergence and divergence and is only equivalent to the basic one classically. We have fully formalized our development in Coq.

Assertion and grounding: a theory of assertion for constructive type theory

Taking Per Martin-Löf’s constructive type theory as a starting-point a theory of assertion is developed, which is able to account for the epistemic aspects of the speech act of assertion, and in which it is shown that assertion is not a wide genus. From a constructivist point of view, one is entitled to assert, for example, that a proposition A is true, only if one has constructed a proof object a for A in an act of demonstration. One thereby has grounded the assertion by an act of demonstration, and a grounding account of assertion therefore suits constructive type theory. Because the act of demonstration in which such a proof object is constructed results in knowledge that A is true, the constructivist account of assertion has to ward off some of the criticism directed against knowledge accounts of assertion. It is especially the internal relation between a judgement being grounded and its being known that makes it possible to do so. The grounding account of assertion can be considered as a justification account of assertion, but it also differs from justification accounts recently proposed, namely in the treatment of selfless assertions, that is, assertions which are grounded, but are not accompanied by belief.

On some peculiar aspects of the constructive theory of point-free spaces

This paper presents several independence results concerning the topos-valid and the intuitionistic (generalised) predicative theory of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Completeness and Cut-elimination in the Intuitionistic Theory of Types--Part 2

Completeness and Cut-elimination in the Intuitionistic Theory of Types--Part 2

A Note on Forcing and Type Theory

The goal of this note is to show the uniform continuity of definable functional in intuitionistic type theory as an application of forcing with dependent type theory.