# Lambda calculus

### 1930

It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing lambda terms and performing reduction operations on them.

Lambda calculus is also a current research topic in Category theory. == History == The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.

### 1935

The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J.

### 1936

Rosser developed the Kleene–Rosser paradox. Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus.

### 1940

In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus. Until the 1960s when its relation to programming languages was clarified, the lambda calculus was only a formalism.

### 1960

In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus. Until the 1960s when its relation to programming languages was clarified, the lambda calculus was only a formalism.

### 1964

According to Cardone and Hindley (2006): By the way, why did Church choose the notation “λ”? In [an unpublished 1964 letter to Harald Dickson] he stated clearly that it came from the notation “\hat{x}” used for class-abstraction by Whitehead and Russell, by first modifying “\hat{x}” to “∧x” to distinguish function-abstraction from class-abstraction, and then changing “∧” to “λ” for ease of printing. This origin was also reported in [Rosser, 1984, p.338].

### 1984

According to Cardone and Hindley (2006): By the way, why did Church choose the notation “λ”? In [an unpublished 1964 letter to Harald Dickson] he stated clearly that it came from the notation “\hat{x}” used for class-abstraction by Whitehead and Russell, by first modifying “\hat{x}” to “∧x” to distinguish function-abstraction from class-abstraction, and then changing “∧” to “λ” for ease of printing. This origin was also reported in [Rosser, 1984, p.338].

### 1988

Other process calculi have been developed for describing communication and concurrency. === Optimal reduction === In Lévy's 1988 paper "Sharing in the Evaluation of lambda Expressions", he defines a notion of optimal sharing, such that no work is duplicated.

### 1997

The Bulletin of Symbolic Logic, Volume 3, Number 2, June 1997. Barendregt, Hendrik Pieter, The Type Free Lambda Calculus pp1091–1132 of Handbook of Mathematical Logic, North-Holland (1977) Cardone and Hindley, 2006.

### 2006

The Bulletin of Symbolic Logic, Volume 3, Number 2, June 1997. Barendregt, Hendrik Pieter, The Type Free Lambda Calculus pp1091–1132 of Handbook of Mathematical Logic, North-Holland (1977) Cardone and Hindley, 2006.

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