Constructive Type Theory Research Articles

Fins et moyens. La théorie constructive des types considérée comme guide pour former des modèles en théorie de l'esprit et de l'action

The following discussion aims at clarifying the formal relation between end and means in terms of constructive type theory. Philosophical tradition offers two opposed conceptions: a Machiavellian, on the one hand, dealing with end and means as to some extent independent items, and a Kantian, on the other, which regards the relation as analytical. The first seems to be both morally and logically unacceptable, but also the second one faces difficulties. I propose to resolve these difficulties along the lines of constructive type theory by considering the end of an action as the type of means which are apt to realise this end.

Modelling general recursion in type theory

Constructive type theory is an expressive programming language in which both algorithms and proofs can be represented. A limitation of constructive type theory as a programming language is that only terminating programs can be defined in it. Hence, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that satisfy no syntactic condition guaranteeing termination. In this work, we present a method to formalise general recursive algorithms in type theory. Given a general recursive algorithm, our method is to define an inductive special-purpose accessibility predicate that characterises the inputs on which the algorithm terminates. The type-theoretic version of the algorithm is then defined by structural recursion on the proof that the input values satisfy this predicate. The method separates the computational and logical parts of the definitions and thus the resulting type-theoretic algorithms are clear, compact and easy to understand. They are as simple as their equivalents in a functional programming language, where there is no restriction on recursive calls. Here, we give a formal definition of the method and discuss its power and its limitations.

Pretopologies and a uniform presentation of sup-lattices, quantales and frames

We introduce the notion of infinitary preorder and use it to obtain a predicative presentation of sup-lattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as “sup-lattices on monoids”, by using the notion of pretopology. Our presentation is then applied to frames, the link with Johnstone’s presentation of frames is spelled out, and his theorem on freely generated frames becomes a special case of our results on quantales. The main motivation of this paper is to contribute to the development of formal topology. That is why all our definitions and proofs can be expressed within an intuitionistic and predicative foundation, like constructive type theory.

Aspects of general topology in constructive set theory

Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures need to be allowed for in constructive set theory because the powerset axiom and the full separation scheme are necessarily missing from constructive set theory. We also consider the notion of a formal topology, usually treated in Intuitionistic type theory, and show that the category of set-generated locales is equivalent to the category of formal topologies. We exploit ideas of Palmgren and Curi to obtain versions of their results about when the class of formal points of a set-presentable formal topology form a set.

Normalisation and Language-Games1

The question of finding a suitable formal account of meaning for the (primitive) logical signs has troubled many philosophers and logicians since the early days of formal logic, whenever it is even recognised as a problem. Here I attempt to show how two operational (as opposed to denotational, or truth-based) approaches to the problem can still be shown to be ‘technically’ equivalent, despite having emerged from two different readings of a single philosophical account, and being essentially distinct with respect to the role of ‘will’ in the mathematical activity: on the one hand, the ‘semantics of use’, my own reformulation of P. Martin-Lof's Intuitionistic Type Theory canonical-values based semantics by taking the normalisation rules as the key semantical device; and, on the other hand, J. Hintikka's Game-Theoretical Semantics, where the meaning of logical signs is given via semantical games. The philosophical account from which both emerge is precisely Wittgenstein's later account of propositions, where the notion of ‘language-games’ is introduced as a key semantical device. Observing that the normalisation rules seem to be able to formalise the explanation of the (immediate) consequences one can draw from a proposition,3 thus showing the function/purpose/usefulness of its main connective in the calculus of language, it seems reasonable to advocate that such a meta-mathematical device can be a semantically useful notion which would lead to a more reasonable account of the problem of formulating the meaning of logical constants.

Records and Record Types in Semantic Theory

This paper explores possibilities for formulating linguistic semantics in terms of records and record types of the kind used in recent developments of Martin-Lof type theory. We will suggest that this gives us tools to develop a single theory which includes aspects of Montague semantics, Discourse Representation Theory, situation semantics and Head-Driven Phrase Structure Grammar. We will also argue that formulating these theories in terms of record types may provide us not only with a unified approach but also with certain improvements over the individual theories.

Computational semantics in type theory

This paper aims to show how Montague-style grammars can be completely formalized and thereby declaratively implemented by using the Grammatical Framework GF. The implementation covers the fundamental operations of Montague’s PTQ model: the construction of analysis trees, the linearization of trees into strings, and the interpretation of trees as logical formulas. Moreover, a parsing algorithm is derived from the grammar. Given that GF is a constructive type theory with dependent types, the technique extends from explain anaphoric reference. On the other hand, GF has a built-in compositionality requirement that is stronger than in PTQ and prevents us from formulating quantifying-in rules of Montague style. This leads us to alternative formulations of such rules in terms of combinators and discontinuous constituents. The PTQ fragment will moreover be presented as an example of how a GF grammar is modified by replacing English with another target language, French. The paper concludes by a discussion of the complementary rôles of logically and linguistically oriented syntax.

What is the world of mathematics?

It may be argued that the language of mathematics is about the category of sets, although the definite article requires some justification. As possible worlds of mathematics we may admit all models of type theory, by which we mean all local toposes. For an intuitionist, there is a distinguished local topos, namely the so-called free topos, which may be constructed as the Tarski–Lindenbaum category of intuitionistic type theory. However, for a classical mathematician, to pick a distinguished model may be as difficult as to define the notion of truth in classical type theory, which Tarski has shown to be impossible.

Induction–recursion and initial algebras

Induction–recursion is a powerful definition method in intuitionistic type theory. It extends (generalized) inductive definitions and allows us to define all standard sets of Martin-Löf type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals below the first Mahlo cardinal. In this article we give a new compact formalization of inductive–recursive definitions by modeling them as initial algebras in slice categories. We give generic formation, introduction, elimination, and equality rules generalizing the usual rules of type theory. Moreover, we prove that the elimination and equality rules are equivalent to the principle of the existence of initial algebras for certain endofunctors. We also show the equivalence of the current formulation with the formulation of induction–recursion as a reflection principle given in Dybjer and Setzer (Lecture Notes in Comput. Sci. 2183 (2001) 93). Finally, we discuss two type-theoretic analogues of Mahlo cardinals in set theory: an external Mahlo universe which is defined by induction–recursion and captured by our formalization, and an internal Mahlo universe, which goes beyond induction–recursion. We show that the external Mahlo universe, and therefore also the theory of inductive–recursive definitions, have proof-theoretical strength of at least Rathjen's theory KPM.

Realizing Mahlo set theory in type theory

After introducing the large set notion of Mahloness, this paper shows that constructive set theory with an axiom asserting the existence of a Mahlo set has a realizability interpretation in an extension of Martin-Lof type theory developed by A. Setzer.

A cartesian closed category in Martin-Löf's intuitionistic type theory

First, we briefly recall the main definitions of the theory of Information Bases and Translations. These mathematical structures are the basis to construct the cartesian closed category InfBas, which is equivalent to the category ScDom of Scott domains. Then, we will show that all the definitions and the proof of all the properties that one needs in order to show that InfBas is indeed a cartesian closed category can be formalized within Martin-Löf's intuitionistic type theory.

Erratic Fudgets: a semantic theory for an embedded coordination language

The powerful abstraction mechanisms of functional programming languages provide the means to develop domain-specific programming languages within the language itself. Typically, this is realised by designing a set of combinators (higher-order reusable programs) for an application area, and by constructing individual applications by combining and coordinating individual combinators. This paper is concerned with a successful example of such an embedded programming language, namely Fudgets, a library of combinators for building graphical user interfaces in the lazy functional language Haskell. The Fudget library has been used to build a number of substantial applications, including a web browser and a proof editor interface to a proof checker for constructive type theory. This paper develops a semantic theory for the non-deterministic stream processors that are at the heart of the Fudget concept. The interaction of two features of stream processors makes the development of such a semantic theory problematic: (i) the sharing of computation provided by the lazy evaluation mechanism of the underlying host language, and (ii) the addition of non-deterministic choice needed to handle the natural concurrency that reactive applications entail. We demonstrate that this combination of features in a higher-order functional language can be tamed to provide a tractable semantic theory and induction principles suitable for reasoning about contextual equivalence of Fudgets.

Extensionality Versus Constructivity

We analyze some extensions of Martin-Löf 's constructive type theory by means of extensional set constructors and we show that often the most natural requirements over them lead to classical logic or even to inconsistency.

External Rewriting for Skeptical Proof Assistants

This paper presents the design, the implementation, and experiments of the integration of syntactic, conditional possibly associative-commutative term rewriting into proof assistants based on constructive type theory. Our approach is called external because it consists in performing term rewriting in a specific and efficient environment and checking the computations later in a proof assistant. Two typical systems are considered in this work: ELAN, based on the rewriting calculus, as the term rewriting-based environment, and Coq, based on the calculus of inductive constructions as the proof assistant. We first formalize the proof terms for deduction by rewriting and strategies in ELAN using the rewriting calculus with explicit substitutions. We then show how these proof terms can soundly be translated into Coq syntax where they can be directly type checked. For the method to be applicable for rewriting modulo associativity and commutativity, we provide an effective method to prove equalities modulo these axioms in Coq using ELAN. These results have been integrated into an ELAN-based rewriting tactic in Coq.

Coercion completion and conservativity in coercive subtyping

Coercive subtyping offers a general approach to subtyping and inheritance by introducing a simple abbreviational mechanism to constructive type theories. In this paper, we study coercion completion in coercive subtyping and prove that the formal extension with coercive subtyping of a type theory such as Martin–Löf's type theory and UTT is a conservative extension. The importance of coherence conditions for the conservativity result is also discussed.

Formal specifications of debuggers

Programming in constructive type theory corresponds to theorem proving in mathematics: the specification plays the role of the proposition to be proved and the program is obtained from the proof. In this paper, we present an approach of using constructive type theory to derive a debugger of a given programming language from its denotational semantic definition. The development is supported by a proof development system called PowerEpsilon .

Denotational semantics of programming languages and compiler generation in PowerEpsilon

Programming in constructive type theory corresponds to theorem proving in mathematics: the specification plays the role of the proposition to be proved and the program is obtained from the proof. In this paper, we present an approach of using constructive type theory to derive a compiler of a given programming language from its denotational semantic definition. The development is supported by a proof development system called PowerEpsilon .

Special issue: Modalities in type theory

This special issue reports on some recent advances in the area of intuitionistic modal type theories and their application to Computer Science. It collects a selection of papers presented at the Logic in Computer Science (LICS'99) satellite workshop on Intuitionistic Modal Logics and Applications (IMLA'99) held at Trento, Italy in July 1999. All of the contributors to this one day workshop, which was widely attended, were invited to submit a full and revised version of their papers to this special issue of MSCS. The selection was based on a second round of peer reviewing.

The strength of Martin-Löf type theory with a superuniverse. Part II

Universes of types were introduced into constructive type theory by Martin-Lof [12]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say $\mathcal{C}$ . The universe then “reflects” $\mathcal{C}$ . This is the first part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf. [16, 18, 19]). It is proved that Martin-Lof type theory with a superuniverse, termed MLS, is a system whose proof-theoretic ordinal resides strictly above the Feferman-Schutte ordinal $\Gamma_0$ but well below the Bachmann-Howard ordinal. Not many theories of strength between $\Gamma_0$ and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory.

A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems

In Russell's Ramified Theory of Types RTT, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of RTT is predicative. The concept of order however, is almost dead since Ramsey eliminated it from RTT. This is why we find Church's simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science. An important example is the proof checker Nuprl, which is based on Martin-Lof's Type Theory which uses type universes. Those type universes, and also degrees of expressions in AUTOMATH, are closely related to orders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl. The novelty of our paper lies in: (1) a modest revival of Russell's orders, (2) the placing of the historical system RTT underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Lof's type theory and PTSs), and (3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.