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

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened homotopy type theory . In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, called the Univalence Axiom , which has a number of striking consequences. He has subsequently advocated a program, which he calls univalent foundations , of developing mathematics in the setting of type theory with the Univalence Axiom and possibly other additional axioms motivated by the simplicial set model. Because type theory possesses good computational properties, this program can be carried out in a computer proof assistant. In this paper we give an introduction to homotopy type theory in Voevodsky’s setting, paying attention to both theoretical and practical issues. In particular, the paper serves as an introduction to both the general ideas of homotopy type theory as well as to some of the concrete details of Voevodsky’s work using the well-known proof assistant Coq. The paper is written for a general audience of mathematicians with basic knowledge of algebraic topology; the paper does not assume any preliminary knowledge of type theory, logic, or computer science. Because a defining characteristic of Voevodsky’s program is that the Coq code has fundamental mathematical content, and many of the mathematical concepts which are efficiently captured in the code cannot be explained in standard mathematical English without a lengthy detour through type theory, the later sections of this paper (beginning with Section \ref{sec2}) make use of code; however, all notions are introduced from the beginning and in a self-contained fashion.

Three versions of the lambda calculus are discussed: typed lambda calculus, untyped lambda calculus, and polymorphically typed lambda calculus. It is shown that the syntax for these calculi can be described as initial algebras for suitable endofunctors. Models of the first two calculi are described as in [Lambek and Scott 1985] and the Per model of the third is recalled. Algebraic types are described in terms of sketches. A main contribution of the paper is the internalization of the theory of sketches in the category Per. Sketches themselves, models of sketches, categories of models of sketches and the category of all models of all sketches can all be realized in this setting. Next, dinatural transformations are reviewed in the context of the category Per and the relation between certain end formulas and initial algebras for finite signatures is discussed. The point of the theory of algebraic semantics in Per is to provide a setting for thinking about algebraic types and algebraic operations in conjunction with logical types and polymorphic operations. Two specific examples are discussed and a general principle for combining algebraic and polymorphic operations is proposed.KeywordsNatural TransformationCategory TheoryLogical TypeDenotational SemanticGraph ObjectThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recently a certain q-Painlevé type system has been obtained from a reduction of the q-Garnier system. In this paper it is shown that the q-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type E₇⁽¹⁾ and is different from the well-known q-Painlevé system of type E₇⁽¹⁾ from the point of view of evolution directions. We also study a connection between the q-Painlevé type system and the q-Painlevé system of type E₇⁽¹⁾. Furthermore determinant formulas of particular solutions for the q-Painlevé type system are constructed in terms of the terminating q-hypergeometric function.

Revisiting the notion of function

data type systems

This paper describes a fibrational categorical semantics for the modal necessity-only fragment of constructive modal type theory, both with and without dependent types. Constructive type theory does not usually discuss logical modalities, and modalities tend to be mostly studied within classical logic, not within type theory. But modalities should be very useful in type theory, as they are very useful in modelling theoretical computing systems. Providing constructive versions of modal logics and their associated Curry-Howard modal type theories is also a very productive program, e.g. helpful when dealing with computational effects, staged computation, and functional reactive types, for example. There seems to be renewed interest in the notion of constructive modal type theory (and in notions of linear type theory), in part because of the interest in homotopy type theory. The modal type theory presented here uses dependent types, in the style of Ritter's categorical models of the Calculus of Constructions. To build up to these, we first discuss the kinds of constructive modal type theory in the literature. Then we provide a non-dependent modal type theory, introduced in previous work, that we generalize to dependent types in the following section. Dependent type theories are usually but not always given categorical semantics in terms of fibrations. We provide semantics in terms of fibrations for both the non-dependent and the dependent type systems discussed and prove them sound and complete, thereby providing evidence that the type theory is meaningful. These fibrational models should be also applicable to the homotopy type theory setting.

We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level n can be constructed in HoTT for any externally fixed natural number n. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where n will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky’s Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be assumed by postulating that the inner and outer natural numbers types are isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of $(\infty,1)$ -category and give some examples.

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.

Type theory is an elegant organisation of the fundamental principles of a foundational theory of computing, with theory taken in the sense of a scientific theory as well as a deductive theory. This theory generates a research programme. I examine the elements of this programme and assess progress. A large number of people world wide have been pursuing the type theory aspects of this research programme, so we can survey a large body of work created over a 20 year period for hints of success and failure and challenge. I first look at a few successes. Some of the applications we have attempted have not worked out as expected, and we don't know whether the fault lies with the type theory or elsewhere. I first describe a failure that is clearly not the type theory, but the state of the foundations of computational mathematics. Then we look at problems closer to the structure of modern type theories-problems suggested by the success of classical set theory.

We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified proof of standardization for beta reduction that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.

A unifier of two terms s and t is a substitution sigma such that ssigma=tsigma and for first-order terms there exists a most general unifier sigma in the sense that any other unifier delta can be composed from sigma with some substitution lambda, i.e. delta=sigmacirclambda. This notion can be generalised to E-unification , where E is an equational theory, =_{E} is equality under E andsigmaa is an E-unifier if ssigma =_{E}tsigma. Depending on the equational theory E, the set of most general unifiers is always a singleton (as above), or it may have more than one, either finitely or infinitely many unifiers and for some theories it may not even exist, in which case we call the theory of type nullary. String unification (or Lob's problem, Markov's problem, unification of word equations or Makanin's problem as it is often called in the literature) is the E-unification problem, where E = {f(x,f(y,z))=f(f(x,y),z)}, i.e. unification under associativity or string unification once we drop the fs and the brackets. It is well known that this problem is infinitary and decidable. Essential unifiers, as introduced by Hoche and Szabo, generalise the notion of a most general unifier and have a dramatically pleasant effect on the set of most general unifiers: the set of essential unifiers is often much smaller than the set of most general unifiers. Essential unification may even reduce an infinitary theory to an essentially finitary theory. The most dramatic reduction known so far is obtained for idempotent semigroups or bands as they are called in computer science: bands are of type nullary, i.e. there exist two unifiable terms s and t, but the set of most general unifiers is not enumerable. This is in stark contrast to essential unification: the set of essential unifiers for bands always exists and is finite. We show in this paper that the early hope for a similar reduction of unification under associativity is not justified: string unification is essentially infinitary. But we give an enumeration algorithm for essential unifiers. And beyond, this algorithm terminates when the considered problem is finitary.

We present the type theory LTT, intended to form a basis for typed target languages, providing an internal notion of logical proposition and proof. The inclusion of explicit proofs allows the type system to guarantee properties that would otherwise be incompatible with decidable type checking. LTT also provides linear facilities for tracking ephemeral properties that hold only for certain program states.Our type theory allows for re-use of typechecking software by casting a variety of type systems within a single language. We illustrate our methodology of representation by means of two examples, one functional and one stateful, and describe the associated operational semantics and proofs of type safety.

In the paper the evolutionary strategy (μ, λ) is applied for learning flexible neuro-fuzzy systems [13]–[15]. In the process of evolution we determine: (i) fuzzy inference (Mamdani type or logical type - described by an S-implication), (ii) concrete fuzzy implication, if the logical type system is found in the process of evolution or concrete t-norm connecting antecedents and consequences, if the Mamdani type system is found in the process of evolution, (iii) concrete t-norm for aggregation of antecedents in each rule, (iv) concrete triangular norm describing aggregation operator, (v) shapes and parameters of fuzzy membership functions, (vi) weights describing importance of antecedents of rules and weights describing importance of rules, (vii) parameters of adjustable triangular norms. It should be noted that the crossover and mutation operators are chosen in a self-adaptive way. The method is tested using well known benchmarks.

"This paper is about our hobby." That is the first sentence of [MP93], the first report on our formal development of lambda calculus and type theory, written in autumn 1992. We have continued to pursue this hobby on and off ever since, and have developed a substantial body of formal knowledge, including Church-Rosser and standardization<br />theorems for beta reduction, and the basic theory of<br />Pure Type Systems (PTS) leading to the strengthening theorem and type checking algorithms for PTS. Some of this work is reported in [MP93, vBJMP94, Pol94b, Pol95]. In the present paper we survey this work, including some new proofs, and point out what we feel has been learned about the general issues of formalizing mathematics. On the technical side, we describe an abstract, and simplified, proof of standardization for beta reduction, not previously published, that does<br />not mention redex positions or residuals. On the general issues, we emphasize the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts. The LEGO Proof Development System [LP92] was used to check the work in an implementation of the Extended Calculus of Constructions<br />(ECC) with inductive types [Luo94]. LEGO is a refinement style<br />proof checker, publicly available by ftp and WWW, with a User's Manual [LP92] and a large collection of examples. Section 1.3 contains information on accessing the formal development described in this paper. Other interesting examples formalized in LEGO include program specification and data refinement [Luo91], strong normalization of System F [Alt93], synthetic domain theory [Reu95, Reu96], and operational semantics for imperative programs [Sch97].

The main idea of the theory of discrete-time single-input single-output linear time-invariant systems (LTI-system) is that every such system has an input-output map that can be represented by the familiar convolution. Such a system is called system of convolution type. It was recently discovered by Boyd and Sandberg, that not all LTI-systems are systems of convolution type. Since not all LTI-systems are of convolution type it would be interesting to characterize the systems of convolution type. In this paper such a characterization is given with the help of the approximation behaviour of the system with respect to FIR-systems. A necessary and sufficient condition is given, such that the system is a system of convolution type. Some applications for the filter design are also described.