Martin-Lof Type Research Articles

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.

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.

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.

Studies of a Theory of Specifications with Built-in Program Extraction

We present a Theory of Specifications based on Martin-Lof's type theory, with rules for simultaneously constructing programs and their correctness proofs. The theory contains types for representing specifications whose corresponding notion of implementation is that of a pair formed by a program and a correctness proof. The rules of the theory are such that in implementations the program parts appear mixed together with the proof parts. A confluent and normalizing computational relation performs the task of separating programs from proofs. As a consequence, every implementation computes to a pair composed of a program and a proof of its correctness, and so the program extraction procedure is immediate.

Extending Martin-Löf Type Theory by one Mahlo-universe

We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjen's theory KPM. This is achieved by replacing the universe in Martin-Lof's Type Theory by a new universe V having the property that for every function f, mapping families of sets in V to families of sets in V, there exists a universe inside V closed under f. We show that the proof theoretical strength of MLM is \(\geq \psi_{\Omega_1}\Omega_{{\rm M}+\omega}\). This is slightly greater than \(|{\rm KPM}|\), and shows that V can be considered to be a Mahlo-universe. Together with [Se96a] it follows \(|{\rm MLM}|=\psi_{\Omega_1}(\Omega_{{\rm M}+\omega})\).

Anti-Realist Semantics

I argue that the implementation of theDummettian program of an ``anti-realist'' semanticsrequires quite different conceptions of the technicalmeaning-theoretic terms used than those presupposed byDummett. Starting from obvious incoherences in anattempt to conceive truth conditions as assertibilityconditions, I argue that for anti-realist purposesnon-epistemic semantic notions are more usefully kept apart from epistemic ones rather than beingreduced to them. Embedding an anti-realist theory ofmeaning in Martin-Lof's Intuitionistic Type Theory(ITT) takes care, however, of many notorious problemsthat have arisen in trying to specify suitableintuitionistic notions of semantic value,truth-conditions, and validity, taking into accountthe so-called ``defeasibility of evidence'' forassertions in empirical discourses.

Formalizing Context in Intuitionistic Type Theory

The article discusses formal aspects of the notion of context as needed in AI applications. We advocate the use of Martin-Lof's intuitionistic type theory to formalize and implement contexts. Through many examples belonging to the domains of computational semantics and knowledge based systems, we show that the built-in notion of context in intuitionistic type theory is a structure rich enough for representing most of the features that characterize contexts in an AI perspective. The fact that many recent theorem provers are built on the different theories of types suggests new perspectives in the development of AI applications.

Intuitionistic choice and classical logic

Abstract. The effort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [4]. In this paper we show how to combine unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in constructively given models. These models are sheaf models over a $\sigma$ -complete Boolean algebra, whose topologies are generated by finite or countable covering relations. By a judicious choice of the Boolean algebra we can directly extract effective content from $\Pi_2^0$ -statements true in the model. All the arguments of the present paper can be formalised in Martin-Lof's constructive type theory with generalised inductive definitions.

Denotational semantics for intuitionistic type theory using a hierarchy of domains with totality

A modified version of Normann's hierarchy of domains with totality [9] is presented and is shown to be suitable for interpretation of Martin-Lof's intuitionistic type theory. This gives an interpretation within classical set theory, which is natural in the sense that \(\Sigma\)-types are interpreted as sets of pairs and \(\Pi\)-types as sets of choice functions. The hierarchy admits a natural definition of the total objects in the domains, and following an idea of Berger [3] this makes possible an interpretation where a type is defined to be true if its interpretation contains a total object. In particular, the empty type contains no total objects and will therefore be false (in any non-empty context). In addition, there is a natural equivalence relation on the total objects, so we derive a hierarchy of topological spaces (quotient spaces wrt. the Scott topology), and give a second interpretation using this hierarchy.

Optimized encodings of fragments of type theory in first-order logic

The paper presents sound and complete translations of several fragments of Martin-Lof's monomorphic type theory to first order predicate calculus. The translations are optimised for the purpose of automated theorem proving in the mentioned fragments. The implementation of the theorem prover Gandalf and several experimental results are described.

A type-theoretic interpretation of constructive domain theory

We present an interpreation of a constructive domain theory in Martin-Lof's type theory. More specifically, we construct a well-pointed Cartesian closed category of semilattices and approximable mappings. This construction is completely formalized and checked using the interactive proof assistant ALF.

RECURSIVE MODELS OF GENERAL INDUCTIVE TYPES

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.

The strength of some Martin-L�f type theories

One objective of this paper is the determination of the proof-theoretic strength of Martin-Lof's type theory with a universe and the type of well-founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with Δ 2 1 comprehension and bar induction. As Martin-Lof intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the proof-theoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman.

Type-theoretic interpretation of iterated, strictly positive inductive definitions

We interpret intuitionistic theories of (iterated) strictly positive inductive definitions (s.p.-ID i′ s) into Martin-Lof's type theory. The main purpose being to obtain lower bounds of the proof-theoretic strength of type theories furnished with means for transfinite induction (W-type, Aczel's set of iterative sets or recursion on (type) universes). Thes.p.-ID i′ s are essentially the wellknownID i -theories, studied in ordinal analysis of fragments of second order arithmetic, but the set variable in the operator form is restricted to occur only strictly positively. The modelling is done by constructivizing continuity notions for set operators at higher number classes and proving that strictly positive set operators are continuous in this sense. The existence of least fixed points, or more accurately, least sets closed under the operator, then easily follows.

Remarks on Martin-Löf's partial type theory

In this paper we discuss Martin-Lof's partial type theory, that is type theory with general recursion, and in particular the consequences of the presence of a fixed point operator. We model Martin-Lof's logical framework domain-theoretically in the category of conditional upper semi lattices and parametrizations thereof. An interpretation of a type of sets in the logical framework, which defines partial type theory with one universe, is finally described.

A Construction of Type: Type in Martin-Lof's Partial Type Theory with One Universe

In this note we construct Martin-Lof's inconsistent type theory, Type: Type (Martin-Lof [1971]), inside partial type theory with one universe. Thus adding a fixed point operator to type theory with one predicative universe gives impredicativity. We may describe the theory Type:Type as follows. It contains the rules for the product construction (II) of Martin-Lof [1984] except the η-rule and it contains the usual rules for definitional equality (=). Moreover it contains the following strongly impredicative universe This theory is inconsistent (i.e. every set is inhabited), and this is seen by proving a variant of the Burali—Forti paradox—Girard's paradox—cf. Troelstra and van Dalen [1988]. Coquand [199?] has shown that by adding the well-order type and the strong dependent sum to the universe, the fixed point operator becomes definable. It is an open problem whether it is definable without the well-order type. The present result could be seen as a converse, namely by adding the fixed point operator to type theory with one universe, Type:Type becomes definable and, as is already known, so does the well-order type.

Introducing data types in intuitionistic type theory

Martin-Lof's intuitionistic type theory (ITT) can be regarded as a programming logic. Yet its data type aspect has not been much addressed. In the present paper we show how to introduce the data types in ITT in a systematic way. The main tool is data logic. Data logics are introduced in [12], which are many-sorted second order theories. For a given data type we can establish its data logic, i.e. a set of non-logical axioms, then from the axioms its rules in ITT can be obtained.

The inconsistency of higher order extensions of Martin-L�f's type theory

Martin-Lof's constructive type theory forms the basis of this paper. His central notions of category and set, and their relations with Russell's type theories, are discussed. It is shown that addition of an axiom — treating the category of propositions as a set and thereby enabling higher order quantification — leads to inconsistency. This theorem is a variant of Girard's paradox, which is a translation into type theory of Mirimanoff's paradox (concerning the set of all well-founded sets). The occurrence of the contradiction is explained in set theoretical terms. Crucial here is the way a proof-object of an existential proposition is understood. It is shown that also Russell's paradox can be translated into type theory. The type theory extended with the axiom mentioned above contains constructive higher order logic, but even if one only adds constructive second order logic to type theory the contradictions arise.

Propositional functions and families of types.

When specifying the task of a computer program, it is often natural to use recursion on a data type. In Martin-Lof's type theory, a universe must be used when defining a propositional function by recursion. Using a logical framework for type theory, formulated by Martin-Lof, an extension of type theory is proposed by which propositional functions can be directly defined without using a universe

Terminating general recursion

In Martin-Lof's type theory, general recursion is not available. The only iterating constructs are primitive recursion over natural numbers and other inductive sets. The paper describes a way to allow a general recursion operator in type theory (extended with propositions). A proof rule for the new operator is presented. The addition of the new operator will not destroy the property that all well-typed programs terminate. An advantage of the new program construct is that it is possible to separate the termination proof of the program from the proof of other properties.