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Dive into the research topics where Rp Rob Nederpelt is active.

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Featured researches published by Rp Rob Nederpelt.


Studies in logic and the foundations of mathematics | 1994

Strong normalization in a typed lambda calculus with lambda structured types

Rp Rob Nederpelt

Publisher Summary This chapter discusses the strong normalization in a typed lambda calculus with lambda structured types. The importance of lambda calculus for the development of recursive functions is also discussed. The calculus has also been brought in relation with the theory of ordinal numbers, predicate calculus, and other theories. The important issue in lambda calculus is the question of the normalization of terms. In lambda calculus, which allows all functions as arguments of functions, such a termination of the reduction is not guaranteed. If there is some reduction sequence which terminates, a term in lambda calculus is called normalizable. Normalization problems also arise in systems of typed lambda calculus investigated a lambda calculus with types and found all terms in this calculus to be strongly normalizable. A typed lambda calculus, in which the types themselves have lambda structure, is described. The typed lambda calculus Λ, has a large overlap with the mathematical language Automath. Normalization and strong normalization for the system is also discussed. An introduction of a method for deriving strong normalization from normalization together with the uniqueness of normal forms is also provided.


International Journal of Foundations of Computer Science | 1993

On stepwise explicit substitution

Fairouz Kamareddine; Rp Rob Nederpelt

This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and application. This item notation proves to be a powerful device for the representation of basic substitution steps, giving rise to different versions of β-reduction including local and global β-reduction. In other words substitution, thanks to the new notation, can be easily formalised as an object language notion rather than remaining a meta language one. Such formalisation will have advantages with respect to various areas including functional application and the partial unfolding of definitions. Moreover our substitution is, we believe, the most general to date. This is shown by the fact that our framework can accommodate most of the known reduction strategies, which range from local to global. Finally, we show how the calculus of substitution of Abadi et al., can be embedded into our calculus. We show moreover that many of the rules of Abadi et al. are easily derivable in our calculus.


Journal of Logic, Language and Information | 2004

A Refinement of de Bruijn's Formal Language of Mathematics

Fairouz Kamareddine; Rp Rob Nederpelt

We provide a syntax and a derivation system fora formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples.WTT is a refinement of de Bruijns Mathematical Vernacular (MV) and hence:– WTT is faithful to the mathematicians language yet isformal and avoids ambiguities.– WTT is close to the usualway in which mathematicians express themselves in writing.– WTT has a syntaxbased on linguistic categories instead of set/type theoretic constructs.More so than MV however, WTT has a precise abstractsyntax whose derivation rules resemble those of modern typetheory enabling us to establish important desirable properties of WTT such as strong normalisation, decidability of type checking andsubject reduction. The derivation system allows one to establish thata book written in WTT is well-formed following the syntax ofWTT, and has great resemblance with ordinary mathematics books.WTT (like MV) is weak as regardscorrectness: the rules of WTT only concern linguisticcorrectness, its types are purely linguistic sothat the formal translation into WTT is satisfactory as areadable, well-organized text. In WTT, logico-mathematical aspects of truth are disregarded. This separates concerns and means that WTT– can be easily understood by either a mathematician, a logician or a computerscientist, and– acts as an intermediary between thelanguage of mathematicians and that of logicians.


Theoretical Computer Science | 1996

A useful l-notation

Fairouz Kamareddine; Rp Rob Nederpelt

In this article, we introduce a ?-notation that is useful for many concepts of the ?-calculus. The new notation is a simple translation of the classical one. Yet, it provides many nice advantages. First, we show that definitions such as compatibility, the heart of a term and s-redexes become simpler in item notation. Second, we show that with this item notation, reduction can be generalised in a nice way. We find a relation s which extends ?s, which is Church-Rosser and strongly normalising. This reduction relation may be the way to new reduction strategies. In classical notation, it is much harder to present this generalised reduction in a convincing manner. Third, we show that the item notation enables one to represent in a very simple way the canonical type t(G,A) of a term A in context G. This canonical type plays the role of a preference type and can be used to split G A : B into the two parts G A and t(G,A) = B. This means that the question is A typable with a type B is divided into two questions: is A typable and is B in the class of types of A. It turns out that calculating this preference type of A in item notation is a straightforward operation. One just goes through A from left to right performing very trivial steps on the items till the end variable (or heart) of A is reached. Fourth, we can with this item notation, find the parts of a term t relevant for a variable occurrence x° in terms of binding, typing and substitution. Again, this part of t, t | x°, is very easy to find in item notation. Just take the part of t to the left of x° and remove all unmatched parentheses. Fifth, we reflect on the status of variables and show that indeed it is easy to study this status in item notation. Finally, we show that for a substitution calculus a la de Bruijn with open terms, it is simpler to describe normal forms using item notation. There are further advantages of item notation that are studied elsewhere. For example, in [9], we show that explicit substitution is easily built in item notation and that global and local strategies of substitution can be accommodated. In [10], we show that with item notation, one can give a unified approach to type theory. An implementation of this item notation with most of the concepts discussed in this paper can be found in [15].


Information & Computation | 1996

The Barendregt Cube with Definitions and Generalised Reduction

Cj Roel Bloo; Fairouz Kamareddine; Rp Rob Nederpelt

In this paper, we propose to extend the Barendregt Cube by generalisings-reduction and by adding definition mechanisms. Generalised reduction allows contracting more visible redexes than usual, and definitions are an important tool to allow for a more flexible typing system. We show that this extension satisfies most of the original properties of the Cube including Church-Rosser, Subject Reduction and Strong Normalisation.


The Bulletin of Symbolic Logic | 2002

Types in Logic and Mathematics before 1940

Fairouz Kamareddine; Tdl Laan; Rp Rob Nederpelt

In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whiteheads Principia Mathematica ([71], 1910-1912) and Churchs simply typed ?-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Freges Grundgesetze der Arithmetik for which Russell applied his famous paradox and this led him to introduce the first theory of types, the Ramified Type Theory (RTT). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Churchs own simply typed ?-calculus (? ? C) and we finish by comparing RTT, STT and ? ?C.


Annals of Pure and Applied Logic | 1999

On Π-conversion in the λ-cube and the combination with abbreviations

Fairouz Kamareddine; Cj Roel Bloo; Rp Rob Nederpelt

Abstract Typed λ-calculus uses two abstraction symbols (λ and Π) which are usually treated in different ways: λ x:∗ .x has as type the abstraction Π x:∗. ∗ , yet Π x:∗ .∗ has type □ rather than an abstraction; moreover, (λx:A.B)C is allowed and β-reduction evaluates it, but (Πx:A.B)C is rarely allowed. Furthermore, there is a general consensus that λ and Π are different abstraction operators. While we agree with this general consensus, we find it nonetheless important to allow Π to act as an abstraction operator. Moreover, experience with AUTOMATH and the recent revivals of Π-reduction as in [11, 14], illustrate the elegance of giving Π-redexes a status similar to λ-redexes. However, Π-reduction in the λ-cube faces serious problems as shown in [11, 14]: it is not safe as regards subject reduction, it does not satisfy type correctness, it loses the property that the type of an expression is well-formed and it fails to make any expression that contains a Π-redex well-formed. In this paper, we propose a solution to all those problems. The solution is to use a concept that is heavily present in most implementations of programming languages and theorem provers: abbreviations (viz. by means of a definition) or let-expressions. We will show that the λ-cube extended with Π-conversion and abbreviations satisfies all the desirable properties of the cube and does not face any of the serious problems of Π-reduction. We believe that this extension of the λ-cube is very useful: it gives a full formal study of two concepts (Π-reduction and abbreviations) that are useful for theorem proving and programming languages.


Journal of Functional Programming | 1996

Canonical typing and

Fairouz Kamareddine; Rp Rob Nederpelt

In this article, we extend the Barendregt Cube with ?-conversion (which is the analogue of s-conversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term.


Theoretical Computer Science | 1994

\Pi

Fairouz Kamareddine; Rp Rob Nederpelt

Abstract In the area of foundations of mathematics and computer science, three related topics dominate. These are λ-calculus, type theory and logic. There are moreover, many versions of λ-calculi and type theories. In these versions, the presence of logic ranges from the non-existent to the dominant. In fact, the three subjects of λ-calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregts cube, for example, is an attempt to unify various type systems and his associated logic cube is an attempt to find connections between type theories and logic. We devise a new λ-notation which enables categorising most of the known systems in a unified way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automath-like systems to singly typed pure type systems. The system and the notation that we propose have far reaching advantages than just being used as a generalisation formalism. These advantages range from generalising reduction and substitution to representing Mathematics and are investigated in detail in various articles cited in the bibliography.


international symposium on functional and logic programming | 2001

-conversion in the Barendregt cube

Fairouz Kamareddine; Tdl Laan; Rp Rob Nederpelt

The Barendregt Cube (introduced in [3]) is a framework in which eight important typed λ-calculi are described in a uniform way. Moreover, many type systems (like Automath [18], LF [11], ML [17], and system F [10]) can be related to one of these eight systems. Furthermore, via the propositions-as-types principle, many logical systems can be described in the Barendregt Cube as well (see for instance [9]). However, there are important systems (including AUTOMATH, LF and ML) that cannot be adequately placed in the Barendregt Cube or in the larger framework of Pure Type Systems. In this paper we add a parameter mechanism to the systems of the Barendregt Cube. In doing so, we obtain a refinement of the Cube. In this refined Barendregt Cube, systems like AUTOMATH, LF, and ML can be described more naturally and accurately than in the original Cube.

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Dive into the Rp Rob Nederpelt's collaboration.

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Herman Geuvers

Radboud University Nijmegen

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Tdl Laan

Eindhoven University of Technology

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Cj Roel Bloo

Eindhoven University of Technology

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Jh Herman Geuvers

Radboud University Nijmegen

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Francien Dechesne

Delft University of Technology

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Gueorgui I. Jojgov

Eindhoven University of Technology

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Lex Bijlsma

Eindhoven University of Technology

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M. Scheffer

Eindhoven University of Technology

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