Combinatory logic

This article deals with combinatory logic, with the direction which has this word in Logique mathematics and in theoretical Informatique. It should not be confused with what one calls combinatory logic in electronic.

The combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to remove the need for variables in mathematics and to minimize the number of operators necessary to define the calul predicates following Henry Mr. Sheffer. More recently it was used in data processing as ideal model of calculation and as bases for the design of functional computer programming languages.

It is based on the controllers, which are of the functions of a higher nature which use only the application of functions and possibly of other controllers to define a result starting from their arguments. It has very strong bonds with the Lambda calculation with the Logique intuitionalist thanks to the Correspondance of Curry-Howard.

Whole of controllers

The simplest example is the controller I (identity) defined by

( I X ) = X

for any term X .

Another simple controller is K , which manufactures constant functions: ( K X ) is the function which, for any parameter, turns over X , in other words

(( K X ) there ) = X

for all terms X and there . Or while following the same convention of notation as in the Lambda-calculation:

( K X there ) = X

The third controller S is a generalized version of the application:

( S X there Z ) = ( X Z ( there Z ))

S applies X to there after having applied them to Z

Notice that once one has S and K , I becomes useless since it can be rebuilt:

(( S K K ) X )

( S K K X )

( K X ( K X ))

X

for any term X .

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