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In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in the study of the denotational semantics of computer programs, because it is a continuous function on complete partial orders. Apply is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can be seen as those that are "continuous" in the Scott topology.
In computer science, lambda calculi are said to have explicit substitutions if they pay special attention to the formalization of the process of substitution. This is in contrast to the standard lambda calculus where substitutions are performed by beta reductions in an implicit manner which is not expressed within the calculus; the "freshness" conditions in such implicit calculi are a notorious source of errors. The concept has appeared in a large number of published papers in quite different fields, such as in abstract machines, predicate logic, and symbolic computation.
The categorical abstract machine (CAM) is a model of computation for programs that preserves the abilities of applicative, functional, or compositional style. It is based on the techniques of applicative computing.
References
- ↑ Wolfengagen V.E. Methods and means for computations with objects. Applicative Computational Systems. — M.: JurInfoR Ltd., «Center JurInfoR», 2004. — xvi+789 pp. ISBN 5-89158-100-0.
- ↑ Backus, J. (1978). "1977 Turing Award Lecture: Can Programming Be Liberated from the von Neumann Style? A Functional Style and Its Algebra of Programs". Comm. Of the ACM. 2 (8): 613–641. doi: 10.1145/359576.359579 . S2CID 16367522.