Combinator
In December 1920, M. Schönfinkel presented in a report to the Mathematical Society in Göttingen a new type of formal logic based on the concept of a generalized
function whose argument is also a function (Schönfinkel 1924). This mathematical
discipline was subsequently termed combinatory logic
by Curry and "
-conversion" or "
-calculus"
by Church. Combinators can be used in the study of algebra, topology, and category
theory, and have found application in the study of programs in algorithmic languages.
SEE ALSO: Church Numeral,
Combinatory
Logic,
Lambda Calculus
REFERENCES:
Barendregt, H. P. The
Lambda Calculus. Amsterdam, Netherlands: North-Holland, 1981.
Curry, H. B. Foundations
of Mathematical Logic. New York: Dover, pp. 118-119, 1977.
Curry, H. and Feys, R. Combinatory
Logic, Vol. 1. Amsterdam, Netherlands: North-Holland, 1958.
Hindley, J. R.; Lercher, B.; Seldin, J. P. Introduction
to Combinatory Logic. London: Cambridge University Press, 1972.
Hindley, J. R. and Seldin, J. P. Introduction to Combinators and lambda-Calculus. Cambridge, England: Cambridge University
Press, 1986.
Holmes, M. R. "Systems of Combinatory Logic Related to Quine's 'New Foundations.'
" Annals Pure Appl. Logic 53, 103-133, 1991.
Quine, W. V. "New Foundations for Mathematical Logic." Amer. Math.
Monthly 44, 70-80, 1937.
Révész, G. E. Lambda-Calculus, Combinators, and Functional Programming. Cambridge, England: Cambridge University
Press, 1988.
Seldin, J. P. and Hindley, J. R. (Eds.). To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism.
New York: Academic Press, 1980.
Smullyan, R. To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory
Logic. New York: Knopf, 1985.
Statman, R. "The Word Problem for Smullyan's Lark Combinator Is Decidable."
J. Symb. Comput. 7, 103-112, 1989.
Schönfinkel, M. "Über die Bausteine der mathematischen Logik."
Math. Ann. 92, 305-316, 1924.
Schönfinkel, M. "Sur les éléments de construction de la logique mathématique." Math. Inform. Sci. Humaines, No. 112, 5-26
and 59, 1990. [French translation with commentary.]
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 711-714
and 1121-1123,
2002.
Referenced on Wolfram|Alpha:
Combinator
CITE THIS AS:
Weisstein, Eric W. "Combinator." From
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