Experience using type theory as a foundation for computer science

This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences with the notion of set as codified in the classical theory Zermelo–Fraenkel. In the current article we illustrate in some detail the notion of set as expressed in Martin–L¨of type theory and present the essential characters of this theory. In a second article we shall explore a distinct notion of set, as arising in the context of intuitionistic versions of Zermelo–Fraenkel set theory. The theory we shall analyse there is Aczel’s CZF (Constructive Zermelo– Fraenkel) and we shall supplement its exposition by a succinct account of Aczel’s interpretation of CZF in type theory. This will enable us to compare the two notions in a more precise sense.

Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory, that offers an alternative to the foundations provided by ZFC set theory and category theory. This paper explains and motivates an account of how to define, justify and think about HoTT in a way that is self-contained, and argues that it can serve as an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be expected to answer, and find that the standard formulation of HoTT as presented in the ``HoTT Book'' does not answer many of them. More importantly, the way HoTT is developed in the HoTT Book suggests that it is not a candidate \emph{autonomous} foundation since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon sophisticated ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation). Our elaboration of HoTT is based on a new interpretation of types as \emph{mathematical concepts}, which accords with the intensional nature of the type theory.

In this paper, we show that Markov's principle is not derivable in dependent type theory with natural numbers and one universe. One tentative way to prove this would be to remark that Markov's principle does not hold in a sheaf model of type theory over Cantor space, since Markov's principle does not hold for the generic point of this model. It is however not clear how to interpret the universe in a sheaf model. Instead we design an extension of type theory, which intuitively extends type theory by the addition of a generic point of Cantor space. We then show the consistency of this extension by a normalization argument. Markov's principle does not hold in this extension, and it follows that it cannot be proved in type theory.

Constructive type theory was first presented in 1970, by the Swedish logician Per Martin-Lof. It has become one of the main approaches in the foundations of mathematics and computer science. But it has remained relatively unknown among linguists and philosophers, although it provides a comsiderable extension of the concepts and techniques of logic. The book contains an introduction to type theory from the point of view of linguistics and the philosophy of language. Type theory is then applied in the areas of quantification, anaphora, temporal reference, and the structure of text and discourse. In virtue of the type theoretical concepts of proof object and context, various phenomena of dependence and progression in language can be discussed in precise terms, and several well-known problems can be solved.

A unified approach to type theory through a refined λ-calculus

In this paper, I am concerned with registerial variation in selections from the experiential system of the grammar of the clause - i.e. the system of transitivity, focussing one of its main component systems, the system of process type: drawing on extensive manual analysis of English texts instantiating different registers, I will present some quantitative findings regarding the relative frequencies of the terms of the system - of ‘material’, ‘behavioural’, ‘mental’, ‘verbal’, ‘relational’ and ‘existential’ process types - and also qualitative findings regarding characteristic discourse uses of these process types in different registers. This second aspect of the study constitutes the bulk of the paper. I will suggest that characteristic discourse uses of different process types correlate with the field of activity within context, i.e. with “what’s going on” in a given context. Here I will draw on our work on a context-based typology of register, particularly the differentiation of different fields of activity (e.g.Matthiessen, 2014a,2015a,b,2019;Matthiessen & Teruya, 2015;Steiner, 2024). The quantitative and qualitative findings are, of course, part of the same picture of the system of process type - complementary aspects of its profile of use. By bringing the two together, I am continuing the work within the SAL (Systemics Across Languages) research programme, initiated by Leila Barbara, and developed by Natalia Ignatieva, other Systemic Functional Linguists in Latin America, and a few of us in other parts of the world.

The Internet of Things designates a vision of an informationalized world, in which everyday objects are modified through computer technology such as RFID, sensors and sensor systems, tracking systems, or small computer units, giving them additional information or functions and integrating them with their environment. This paper explores the phenomenon of Internet of Things by applying some of the key concepts found in the philosophical works of Martin Heidegger and conducts a critical analysis using Martin Heidegger’s reflections on “world” and “technology” from Sein und Zeit (Being and Time) of 1957, and from the paper “Die Frage nach der Technik” (“The Question Concerning Technology”). The concept of “Ge-stell” introduced by Martin Heidegger to the philosophy of technology describes the situation of mankind and technology in a world shaped by modern technology. Both, mankind and technology, are tied to one dimension and share the same world. The world has been shaped and redesigned by technology, and the individual is completely tied into the technologized world. The basis for all Internet of Things concepts and installations is the expression of real-world things and processes as mathematical quantities and concepts, and the connecting of individual values and concepts with each other. The basis for all information technology is the reduction of reality to figures and calculable structures. Mathematics is the foundation of computer science, and thus the foundation of ubiquitous computing.

We show that Nechiporuk's method [I. Wegener, The Complexity of Boolean Functions, Teubner-Wiley, New York, 1987] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an $\Omega(n^2/\log^2 n)$ lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [A. Yao, Proceedings of 34th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 352--361] states that the majority function does not have a linear-size quantum formula. We also show that quantum formulas can be simulated by Boolean circuits of almost the same size.

Third Advanced Course on Foundations of Computer Science

45th Annual IEEE Symposium on Foundations of Computer Science

Special FOCS issue—twenty-second annual IEEE symposium on foundations of computer science

24th Annual Symposium on Foundations of Computer Science (sfcs 1983)

16th Annual Symposium on Foundations of Computer Science (sfcs 1975)

§1. Introduction.Russell's type-theory can be seen as a theory of properties, relations, and propositions (PRPs) (in short, a property theory). It relies on rigid type distinctions at the grammatical level to circumvent the property theorist's major problem, namely Russell's paradox, or, more generally, the paradoxes of predication. Type theory has arguably been the standard property theory for years, often taken for granted, and used in many applications. In particular, Montague [27] has shown how to use a type-theoretical property-theory as a foundation for natural language semantics.In recent years, it has been persuasively argued that many linguistic and ontological data are best accounted for by using a type-free property theory. Several type-free property theories, typically with fine-grained identity conditions for PRPs, have therefore been proposed as potential candidates to play a foundational role in natural language semantics, or for related applications in formal ontology and the foundations of mathematics (Bealer [6], Cocchiarella [18], Turner [35], etc.).Attempts have then been made to combine some such property theory with a Montague-style approach in natural language semantics. Most notably, Chierchia and Turner [15] propose a Montague-style semantic analysis of a fragment of English, by basically relying on the type-free system of Turner [35]. For a similar purpose Chierchia [14] relies on one of the systems based on homogeneous stratification due to Cocchiarella. Cocchiarella's systems have also been used for applications in formal ontology, inspired by Montague's account of quantifier phrases as, roughly, properties of properties (see, e.g., Cocchiarella [17], [19], Landini [25], Orilia [29]).

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Godel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.