Lambda Calculus

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Lambda Calculus

• Introduction

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History
Gottfried Wilhelm Leibniz (1646 - 1716)
"It is unworthy of excellent men to lose hours
like slaves in the labour of calculation which
could safely be regulated to anyone else if
machines were used."
Stepped Reckoner
Sought a formal language for a machine to
determine the truth of mathematical statements.
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Entscheidungsproblem
The decision problem (1928) Find an algorithm
which takes as input a description of a formal
language and a mathematical statement expressed
in that language and outputs true or false
depending on the mathematical validity of the
statement.
David Hilbert
William Ackermann
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Alonzo Church

• Churchs Theorem (1936)
• Answered the decision problem in the negative

Alonzo Church 1903-1995
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Alonzo Church

• Defined the Lambda (l) Calculus - a language
  foundation for computing
• Led to family of functional programming languages

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Alan Turing

• Independently answered the decision problem in
  the negative
• Defined the Turing Machine a machine foundation
  for computing
• Led to Von Neumann computers and family of
  imperative programming languages

Alan Turing 1912-1954

• Work at Bletchley Park in WW2
• Died by suicide (apple laced with cyanide)

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Bletchley Park
Turing designed the Bombe machine to decrypt
Enigma messages.
The Enigma Cryptographic Device
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The l Calculus

• ltexpressiongt ltnamegt
• ltfunctiongt
• ltapplicationgt
• ltfunctiongt l ltnamegt . ltexpressiongt
• ltapplicationgt ltexpressiongt ltexpressiongt