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Completeness and Herbrand Theorems for Nominal Logic
 Journal of Symbolic Logic
, 2006
"... Nominal logic is a variant of firstorder logic in which abstract syntax with names and binding is formalized in terms of two basic operations: nameswapping and freshness. It relies on two important principles: equivariance (validity is preserved by nameswapping), and fresh name generation ("ne ..."
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Nominal logic is a variant of firstorder logic in which abstract syntax with names and binding is formalized in terms of two basic operations: nameswapping and freshness. It relies on two important principles: equivariance (validity is preserved by nameswapping), and fresh name generation ("new" or fresh names can always be chosen).
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
"... ..."
century: A journey through Frege, Russell and
, 2004
"... The evolution of types and logic in the 20th ..."
Parts of this talk are based on Kamareddine [2001]; Kamareddine et al. [2002]; Kamaredine and Nederpelt [2004], and on joint work with Maarek and Wells in Kamaredine et al. [2004b,a] University of LeipzigA Century of Complexity
, 1900
"... The formalisation and computerization of ..."
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.
The Triumph of Types: Principia Mathematica’s Impact on Computer Science
"... Types now play an essential role in computer science; their ascent originates from Principia Mathematica. Type checking and type inference algorithms are used to prevent semantic errors in programs, and type theories are the native language of several major interactive theorem provers. Some of these ..."
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Types now play an essential role in computer science; their ascent originates from Principia Mathematica. Type checking and type inference algorithms are used to prevent semantic errors in programs, and type theories are the native language of several major interactive theorem provers. Some of these trace key features back to Principia. This lecture examines the influence of Principia Mathematica on modern type theories implemented in software systems known as interactive proof assistants. These proof assistants advance daily the goal for which Principia was designed: to provide a comprehensive formalization of mathematics. For instance, the definitive formal proof of the Four Color Theorem was done in type theory. Type theory is considered seriously now more than ever as an adequate foundation for both classical and constructive mathematics as well as for computer science. Moreover, the seminal work in the history of formalized mathematics is the Automath project of N.G. de Bruijn whose formalism is type theory. In addition we explain how type theories have enabled the use of formalized mathematics as a practical programming language, a connection entirely unanticipated at the time of Principia Mathematica’s creation.