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A Useful λNotation
, 1996
"... 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 firedexes become simpl ..."
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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 firedexes 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 ; fi which extends ! fi , 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 ø (\Gamma; A) of a term A in context \Gamma. This canonical type plays the role of a preference type and can be used to split \Gamma ` A : B in the two parts: \Gamma ` A and ø (\Gamma; A) = B. This means that the questio...
Redexes in Item Notation Classical Notation Item Notation
, 2009
"... • I(λx.B) = [x]I(B) and I(AB) = (I(B))I(A) • I((λx.(λy.xy))z) ≡ (z)[x][y](y)x. The items are (z), [x], [y] and (y). • applicator wagon (z) and abstractor wagon [x] occur NEXT to each other. • A term is a wagon followed by a term. • (β) ..."
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• I(λx.B) = [x]I(B) and I(AB) = (I(B))I(A) • I((λx.(λy.xy))z) ≡ (z)[x][y](y)x. The items are (z), [x], [y] and (y). • applicator wagon (z) and abstractor wagon [x] occur NEXT to each other. • A term is a wagon followed by a term. • (β)
Operated by Universities Space Research Association
"... CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak ..."
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CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
Parameters in Pure Type Systems
"... Abstract. In this paper we study the addition of parameters to typed�calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in pra ..."
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Abstract. In this paper we study the addition of parameters to typed�calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in practice in theorem provers and programming languages. 1 What are parameters? Parameters occur when functions are only allowed to occur when provided with arguments. As we will show below, both in mathematics and in programming languages the use of parameters is abundant and closely connected to the use of constants and definitions. If we want to be able to use type systems in accordance with practice and yet described in a precise manner, we therefore need parameters, constants, and definitions in type theory as well. Parameters, constants and and���� � ��������� definitions in theorem proving It is interesting to note that
: Notation classique ↦ → Notation de de Bruijn
, 2010
"... • Deduction modulo versus reduction modulo ..."
Linguas y Modeles por el Formalisatione y el Automation del Matemáticas y el Informatica Fairouz Kamareddine (Universidad de HeriotWatt, Edimbourgo, RU)
"... • In less than a half a century, computers have revolutionised the way we all live. • Google, Wikipedia, and other information and search engines have changed the way we store and exchange information. • Computerisation also enables excellent collaborations between different disciplines (think of Bi ..."
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• In less than a half a century, computers have revolutionised the way we all live. • Google, Wikipedia, and other information and search engines have changed the way we store and exchange information. • Computerisation also enables excellent collaborations between different disciplines (think of BioInformatics) and enables new discoveries in different disciplines. • This computerisation of information is only at its beginning. We need a lot of investments in research methods that enable faster, correct, and efficient information storage and retrieval. • Information here means every aspect of information (mathematical, medical, social, educational, law, etc). • Calculators process numbers, computers process information. Brasilià, Novembre 2009 1The languages of Mathematics Usually, mathematicians ignore formal logic and write mathematics using a certain language style which we call Cml. Advantages of Cml: • Expressivity: We can express all sorts of notions. • Acceptability: Cml is accepted by most mathematicians. • Traditionality: Cml exists since very long and has been refined with the time.
From the Foundation of Mathematics to the Birth of Computation
, 2011
"... deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back ..."
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deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back centuries? Algorithms existed in anciant Egypt at the time of Hypatia. The word is named after AlKhawarizmi. • But even more impressively, the following important 20th century (un)computability result was known to Aristotle. • Assume a problem Π, – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. HAPOC11: History and Philosophy of Computing 1 • But, this result was already known to Aristotle: • Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs: – program = algorithm = computable function = λterm. – By the PAT principle: Proofs are λterms.
(Typed) λCalculi à la de Bruijn
, 2012
"... Lambda2012 The two previous speakers discussing the origin of λ in Church’s writing Lambda2012 1De Bruijn’s typed λcalculi started with his Automath • In 1967, an internationally renowned mathematician called N.G. de Bruijn wanted to do something never done before: use the computer to formally chec ..."
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Lambda2012 The two previous speakers discussing the origin of λ in Church’s writing Lambda2012 1De Bruijn’s typed λcalculi started with his Automath • In 1967, an internationally renowned mathematician called N.G. de Bruijn wanted to do something never done before: use the computer to formally check the correctness of mathematical books. • Such a task needs a good formalisation of mathematics, a good competence in implementation, and extreme attention to all the details so that nothing is left informal. • Implementing extensive formal systems on the computer was never done before. • De Bruijn, an extremely original mathematician, did every step his own way. • He proudly announced at the ceremony of the publications of the collected Automath work: I did it my way. • Dirk van Dalen said at the ceremony: The Germans have their 3 B’s, but we Dutch too have our 3 B’s: Beth, Brouwer and de Bruijn.