Combinatory logic - Combinatory logic in mathematics

Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. Combinatory logic - Combinatory logic in mathematics. Combinatory logic was intended as a simple 'pre-logic' which would clarify the meaning of variables in logical notation, and indeed eliminate the need for ...

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• Combinatory logic - Combinatory logic in mathematics
• Combinatory logic - Combinatory logic in computing
• Combinatory logic - Summary of the lambda calculus
• Combinatory logic - Combinatory calculi
• Combinatory logic - Combinatory terms
• Combinatory logic - Examples of combinators
• Combinatory logic - Completeness of the S-K basis
• Combinatory logic - Simplifications of the transformation
• Combinatory logic - Reverse conversion
• Combinatory logic - Undecidability of combinatorial calculus
• Combinatory logic - Combinatory terms as graphs
• Combinatory logic - Applications
• Combinatory logic - Compilation of functional languages
• Combinatory logic - Logic

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Combinatory logic was intended as a simple 'pre-logic' which would clarify the meaning of variables in logical notation, and indeed eliminate the need for them. See Curry, 1958-72. ...

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Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Combinatory logic in mathematics

Since abstraction is the only way to manufacture functions in the lambda calculus, something must replace it in the combinatory calculus. Instead of abstraction, combinatory calculus provides a limited set of primitive functions out of which other functions may be built. Combinatory logic - Combinatory terms. A combinatory term has one of the following forms: vSee also:

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Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Combinatory calculi

It is undecidable whether a general combinatory term has a normal form; whether two combinatory terms are equivalent, etc. This is equivalent to the undecidability of the corresponding problems for lambda terms. However, a direct proof is as follows: First, observe that the term Ω = (S I I (S I I)) has no normal form, because it reduces to itself after three steps, as follows: (S I I (S I I)) ...

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Combinatory logic, Combinatory logic - Combinatory logic in mathematics, Combinatory logic - Combinatory logic in computing, Combinatory logic - Summary of the lambda calculus, Combinatory logic - Combinatory calculi, Combinatory logic - Combinatory terms, Combinatory logic - Examples of combinators, Combinatory logic - Completeness of the S-K basis, Combinatory logic - Simplifications of the transformation, Combinatory logic - Reverse conversion, Combinatory logic - Undecidability of combinatorial calculus, Combinatory logic - Combinatory terms as graphs, Combinatory logic - Applications, Combinatory logic - Compilation of functional languages, Combinatory logic - Logic

Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Undecidability of combinatorial calculus

For complete details about the lambda calculus, see the article under that head. We will summarize here. The lambda calculus is concerned with objects called lambda-terms, which are strings of symbols of one of the following forms: v λv.E1 (E1 E2) where v is a variable name drawn from a predefined infinite set of variable names, and E1 and E2 are lambda-terms. Terms of the form λv.E1 are called abstractions. The variable v is called the formal parameter of t ...

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Combinatory logic, Combinatory logic - Combinatory logic in mathematics, Combinatory logic - Combinatory logic in computing, Combinatory logic - Summary of the lambda calculus, Combinatory logic - Combinatory calculi, Combinatory logic - Combinatory terms, Combinatory logic - Examples of combinators, Combinatory logic - Completeness of the S-K basis, Combinatory logic - Simplifications of the transformation, Combinatory logic - Reverse conversion, Combinatory logic - Undecidability of combinatorial calculus, Combinatory logic - Combinatory terms as graphs, Combinatory logic - Applications, Combinatory logic - Compilation of functional languages, Combinatory logic - Logic

Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Summary of the lambda calculus

In computer science, combinatory logic is used as a simplified model of computation, used in computability theory (the study of what can be computed) and proof theory (the study of what can be mathematically proven.) The theory, despite its simplicity, captures many essential features of the nature of computation. Combinatory logic can be looked at as a variation of the lambda calculus, in which lambda expressions (used to allow for functional abstraction) are replaced by a limited set of combinators, primitive functions which ...

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Combinatory logic, Combinatory logic - Combinatory logic in mathematics, Combinatory logic - Combinatory logic in computing, Combinatory logic - Summary of the lambda calculus, Combinatory logic - Combinatory calculi, Combinatory logic - Combinatory terms, Combinatory logic - Examples of combinators, Combinatory logic - Completeness of the S-K basis, Combinatory logic - Simplifications of the transformation, Combinatory logic - Reverse conversion, Combinatory logic - Undecidability of combinatorial calculus, Combinatory logic - Combinatory terms as graphs, Combinatory logic - Applications, Combinatory logic - Compilation of functional languages, Combinatory logic - Logic

Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Combinatory logic in computing