Fairouz Kamareddine

A Lambda-Calculus `a la de Bruijn with Explicit Substitutions

The aim of this paper is to present the λs-calculus which is a very simple λ-calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of λ-terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the λσ-calculus [ACCL91], and therefore the relation between both calculi will be made explicit. The confluence of the λs-calculus is obtained by the “interpretation method” ([Har89], [CHL92]). The proof of the preservation of normalisation follows the lines of an analogous result for the λv-calculus (cf. [BBLRD95]). The relation between λs and λv is also studied.

Extending a λ-calculus with explicit substitution which preserves strong normalisation into a confluent calculus on open terms

The last 15 years have seen an explosion in work on explicit substitution, most of which is done in the style of the λσ-calculus. In Kamareddine and Rios (1995a), we extended the λ-calculus with explicit substitutions by turning de Bruijns meta-operators into object-operators offering a style of explicit substitution that differs from that of λσ. The resulting calculus, λs, remains as close as possible to the λ-calculus from an intuitive point of view and, while preserving strong normalisation (Kamareddine and Rios, 1995a), is extended in this paper to a confluent calculus on open terms: the λse-caculus. Since the establishment of these results, another calculus, λζ, came into being in Munoz Hurtado (1996) which preserves strong normalisation and is itself confluent on open terms. However, we believe that λse still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical β-reduction, whereas λζ is not. To prove confluence we introduce a generalisation of the interpretation method (cf. Hardin, 1989; Curien et al., 1992) to a technique which uses weak normal forms (instead of strong ones). We consider that this extended method is a useful tool to obtain confluence when strong normalisation of the subcalculus of substitutions is not available. In our case, strong normalisation of the corresponding subcalculus of substitutions se, is still a challenging open problem to the rewrite community, but its weak normalisation is established here via an effective strategy.

Computerizing Mathematical Text with MathLang

Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerizing mathematical texts and knowledge which is flexible enough to connect the different approaches to computerization, which allows various degrees of formalization, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Three Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), and Robert Lamar (since 2006)) and over a dozen masters degree and undergraduate students have worked on MathLang. The projects progress and design choices are driven by the needs for computerizing real representative mathematical texts chosen from various branches of mathematics. Currently, MathLang supports entry of mathematical text either in an XML format or using the editor. Methods are provided for adding, checking, and displaying various information aspects. One aspect is a kind of weak type system that assigns categories (term, statement, noun (class), adjective (class modifier), etc.) to parts of the text, deals with binding names to meanings, and checks that a kind of grammatical sense is maintained. Another aspect allows weaving together mathematical meaning and visual presentation and can associate natural language text with its mathematical meaning. Another aspect allows identifying chunks of text, marking their roles (theorem, definition, explanation, example, section, etc.), and indicating relationships between the chunks (A uses B, A contradicts B, A follows from B, etc.). Software tool support can use this aspect to check and explain the overall logical structure of a text. Further aspects are being designed to allow adding additional formality to a text such as proof structure and details of how a human-readable proof is encoded into a fully formalized version (so far this has only been done for Mizar and started for Isabelle). A number of mathematical texts have been computerized, helping with the development of these aspects, and indicating what additional work is needed for the future. This paper surveys the past and future work of the MathLang project.

On stepwise explicit substitution

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.

A Refinement of de Bruijn's Formal Language of Mathematics

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.

Restoring Natural Language as a Computerised Mathematics Input Method

Methods for computerised mathematics have found little appeal among mathematicians because they call for additional skills which are not available to the typical mathematician. We herein propose to reconcile computerised mathematics to mathematicians by restoring natural language as the primary medium for mathematical authoring. Our method associates portions of text with grammatical argumentation roles and computerises the informal mathematical style of the mathematician. Typical abbreviations like the aggregation of equations a = b > c, are not usually accepted as input to computerised languages. We propose specific annotations to explicate the morphology of such natural language style, to accept input in this style, and to expand this input in the computer to obtain the intended representation (i.e., a = b and b > c). We have named this method syntax souringin contrast to the usual syntax sugaring. All results have been implemented in a prototype editor developed on top of

Thirty Five Years of Automating Mathematics

{\rm\kern-.15em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}

The Barendregt Cube with Definitions and Generalised Reduction

_{{\rm {\sc MACS}}}

Types in Logic and Mathematics before 1940

as a GUI for the core grammatical aspect of MathLang, a framework developed by the ULTRA group to computerise and formalise mathematics.