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5646 results in Programming Languages and Applied Logic

Page 104 of 283

Nominal Sets
  • Names and Symmetry in Computer Science
  • Andrew M. Pitts
  • Published online:
    05 July 2013
    Print publication:
    30 May 2013
  • Nominal sets provide a promising new mathematical analysis of names in formal languages based upon symmetry, with many applications to the syntax and semantics of programming language constructs that involve binding, or localising names. Part I provides an introduction to the basic theory of nominal sets. In Part II, the author surveys some of the applications that have developed in programming language semantics (both operational and denotational), functional programming and logic programming. As the first book to give a detailed account of the theory of nominal sets, it will be welcomed by researchers and graduate students in theoretical computer science.

  • Introduction

  • Henk Barendregt, Radboud Universiteit Nijmegen, Wil Dekkers, Radboud Universiteit Nijmegen, Richard Statman
  • Book:
    Lambda Calculus with Types
    Published online:
    05 August 2013
    Print publication:
    20 June 2013, pp xv-xxii

  • Summary

    The rise of lambda calculus

    Lambda calculus is a formalism introduced by Church in 1932 that was intended to be used as a foundation for mathematics, including its computational aspects. Supported by his students Kleene and Rosser – who showed that the prototype system was inconsistent – Church distilled a consistent computational part and ventured in 1936 the Thesis that exactly the intuitively computable functions could be captured by it. He also presented a function that could not be captured by the λ-calculus. In that same year Turing introduced another formalism, describing what are now called Turing Machines, and formulated the related Thesis that exactly the mechanically computable functions are able to be captured by these machines. Turing also showed in the same paper that the question of whether a given statement could be proved(from a given setofaxioms) using the rules of any reasonable system of logic is not computable in this mechanical way. Finally Turing showed that the formalism of λ-calculus and Turing Machines define the same class of functions.

    Together Church's Thesis, concerning computability by homo sapiens, and Turing's Thesis, concerning computability by mechanical devices, using formalisms that are equally powerful and that have their computational limitations, made a deep impact on the 20th century philosophy of the power and limitations of the human mind. So far, cognitive neuropsychology has not been able to refute the combined Church-Turing Thesis. On the contrary, that discipline also shows the limitation of human capacities.

Page 104 of 283