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A Refinement of de Bruijn's Formal Language of Mathematics

Fairouz Kamareddine, Rob Nederpelt
2004 Journal of Logic, Language and Information  
WTT is a refinement of de Bruijn's Mathematical Vernacular (MV) and hence: WTT is faithful to the mathematician's language yet is formal and avoids ambiguities.  ...  We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples.  ...  We also take ideas from de Bruijn's languages SEMIPAL and PAL [7] used to represent the administrative structure of mathematical texts.  ... 
doi:10.1023/b:jlli.0000028393.47593.b8 fatcat:4fvkefihbngpddqzcjc5qb7q7a

Flexible Encoding of Mathematics on the Computer [chapter]

Fairouz Kamareddine, Manuel Maarek, J. B. Wells
2004 Lecture Notes in Computer Science  
Thus, MathLang now supports the ability to capture the essential mathematical structure of mathematics written using natural language text.  ...  This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text.  ...  Like N.G. de Bruijn's MV and Z. Luo and P. Callaghan's Mathematical Vernacular  ... 
doi:10.1007/978-3-540-27818-4_12 fatcat:inor6ggk4jh2npx4k6rzk3ybye

CINNI - A Generic Calculus of Explicit Substitutions and its Application to λ- ς- and π-Calculi

Mark- Oliver Stehr
2000 Electronical Notes in Theoretical Computer Science  
The solution we propose is based on CINNI, a new calculus of explicit substitutions that makes use of a term representation that contains both the standard named notation and de Bruijn's indexed notation  ...  As a practical result we obtain executable formal representations of these object languages in Maude with a representational distance close to zero.  ...  We also appreciate the comments of several referees which helped to improve the paper.  ... 
doi:10.1016/s1571-0661(05)80125-2 fatcat:ggrxnrgmpnhb5ie6rxdwfw2w5q

Page 2383 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews  
Levy [in Conference record of the Seventeenth Annual ACM Symposium on Principles of Programming Languages (San Fran- cisco, CA, 1990), 16-30, ACM, New York, 1990; CCA 1991:2819], is a refinement of A-calculus  ...  (in de Bruijn’s notation), extended with an explicit operator applying substitutions to Ao-terms.  ... 

Toward an Object-Oriented Structure for Mathematical Text [chapter]

Fairouz Kamareddine, Manuel Maarek, J. B. Wells
2006 Lecture Notes in Computer Science  
For this reason, we have been developing MathLang, a language for representing mathematical texts that has weak type checking and support for the special mathematical use of natural language.  ...  ., L A T E X) to full formalization (e.g., Mizar, Coq, etc.).  ...  Generally, one does not expect formal checking of mathematics encoded in OMDoc. Full formalization.  ... 
doi:10.1007/11618027_15 fatcat:zkeydsxphrgt5npmc3gmkrrhwe

Pure Type Systems in Rewriting Logic: Specifying Typed Higher-Order Languages in a First-Order Logical Framework [chapter]

Mark-Oliver Stehr, José Meseguer
2004 Lecture Notes in Computer Science  
Furthermore, we use rewriting logic itself to give a formal operational description of type checking, that directly serves as an efficient type checking algorithm.  ...  This paper is a detailed study on the ease and naturalness with which a family of higher-order formal systems, namely pure type systems (PTSs) [6, 50] , can be represented in the first-order logical framework  ...  Acknowledgements Support for this work by DARPA and NASA (Contract NAS2-98073), by Office of Naval Research (Contract N00014-96-C-0114), by National Science Foundation Grant (CCR-9633363), and by a DAAD  ... 
doi:10.1007/978-3-540-39993-3_16 fatcat:67pi5kbczbae3iqakokze2j4t4

Faithfully Reflecting the Structure of Informal Mathematical Proofs into Formal Type Theories

G.I. Jojgov, R.P. Nederpelt, M. Scheffer
2004 Electronical Notes in Theoretical Computer Science  
In order to reduce the possibility of formalization errors and therefore increase the reliability of the translation of CML texts into type theories, we use a version of Nederpelt's formal language WTT  ...  Mathematical proofs, as written in every-day Common Mathematical Language (CML), are informal and many details are left implicit.  ...  WTT was first introduced in [5] and is a refinement of de Bruijn's Mathematical Vernacular [1] (MV). • WTT is faithful to the mathematician's language yet is formal and avoids ambiguities. • WTT is  ... 
doi:10.1016/j.entcs.2003.12.030 fatcat:xzn2ocxvvbdp5ia76pwhdzuela

The Gradual Computerisation of Mathematics in MathLang

Fairouz Kamareddine
2007 Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2007)  
We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text.  ...  In the MathLang-Mizar project, we create from the dependency graph, the roles of the nodes of the graph, and the (preamble) of the CGa encoding, a Mizar Formal Proof Sketch (FPS) skeleton.  ...  We call this principle de Bruijn's path. The work may be subdivided.  ... 
doi:10.1109/synasc.2007.85 dblp:conf/synasc/Kamareddine07 fatcat:qb35zdkwhvaixe2mvt5fvisyn4

A Formal System for the Universal Quantification of Schematic Variables [article]

Ferruccio Guidi
2021 arXiv   pre-print
We advocate the use of de Bruijn's universal abstraction λ^∞ for the quantification of schematic variables in the predicative setting and we present a typed λ-calculus featuring the quantifier λ^∞ accompanied  ...  This calculus belongs to the λδ family of formal systems, which borrow some features from the pure type systems and some from the languages of the Automath tradition, but stand outside both families.  ...  languages) because of their undeniable importance in the practice of mathematical development.  ... 
arXiv:1911.12749v5 fatcat:nrkuh36h7ngfldoqnf2gqdbf4u

The SAD System: Deductive Assistance in an Intelligent Linguistic Environment

Alexander Lyaletski, Andrei Paskevich, Konstantin Verchinine
2006 2006 3rd International IEEE Conference Intelligent Systems  
We outline specific features of SAD -a handy input language, powerful reasoning strategy, opportunity to use various low level inference engines.  ...  Formal methods are widely used in the computer science community. Formal verification and certification is an important component of any formal approach.  ...  G. de Bruijn's Automath Project [2] . Actually, one can clearly observe three "pure" branches in the domain of proof assistance.  ... 
doi:10.1109/is.2006.348446 fatcat:ypbwfvryt5d7fdgjnj25rtct5u

Mechanized Metatheory for the Masses: The PoplMark Challenge [chapter]

Brian E. Aydemir, Aaron Bohannon, Matthew Fairbairn, J. Nathan Foster, Benjamin C. Pierce, Peter Sewell, Dimitrios Vytiniotis, Geoffrey Washburn, Stephanie Weirich, Steve Zdancewic
2005 Lecture Notes in Computer Science  
We have also experimented with lightweight tools designed to make it easier to define and typeset both formal and informal mathematics.  ...  of Featherweight Java, proving type soundness of variants of the simply typed λ-calculus and F <: , and a substantial formalization of the behavior of TCP, UDP, and the Sockets API.  ...  Another popular concrete representation is de Bruijn's nameless representation.  ... 
doi:10.1007/11541868_4 fatcat:6eprs3kxvjg67ak2zhuipl5qyu

Residual theory in λ-calculus: a formal development

Gérard Huet
1994 Journal of functional programming  
Gallina is the specification language of the Coq Proof Assistant (Doweket al., 1991; Huet 1992b).  ...  The full transcript is available as a standard library in the distribution of Coq.  ...  of coordination of a community of competent users contributing to add to a general library of Formalized Mathematics.  ... 
doi:10.1017/s0956796800001106 fatcat:rvlumdmsmvcgha663kezkoiwtm

MathLang: Experience-driven Development of a New Mathematical Language

Fairouz Kamareddine, Manuel Maarek, J.B. Wells
2004 Electronical Notes in Theoretical Computer Science  
In this paper we report on the design of a new mathematical language and our method of designing it, driven by the encoding of mathematical texts.  ...  At the same time, it is intended to allow the addition of fuller formalization to a document as time and effort permits.  ...  From MV to WTT to MathLang In 1979, de Bruijn developed a course on the Mathematical Vernacular (MV) intended to be a language to write mathematics.  ... 
doi:10.1016/j.entcs.2003.12.032 fatcat:fxn7zciw5rfshh3r7qxfnoknfu

Formalising Mathematics In Simple Type Theory [article]

Lawrence C. Paulson
2018 arXiv   pre-print
However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism.  ...  This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL.  ...  The development of HOL and Isabelle has been supported by numerous EPSRC grants. The ERC project ALEXAN-DRIA supports continued work on the topic of this paper.  ... 
arXiv:1804.07860v1 fatcat:zkrhn2d5nzawrfi2rh62fxghqy

LFTOP: An LF-Based Approach to Domain-Specific Reasoning

Jian-Min Pang, Paul Callaghan, Zhao-Hui Luo
2005 Journal of Computer Science and Technology  
or medium without the formal permission of the copyright holders.  ...  For domain-specific issues researchers focus mainly on the design and implementation of domain-specific languages (DSL) and pay little attention to the reasoning aspects.  ...  Just as implied in its name, PAL+ can be seen as a successor of de Bruijn's PAL for Automath [de Bruijn, 1980] .  ... 
doi:10.1007/s11390-005-0526-y fatcat:u67u3mr4jvgjfomcepekpffl5a
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