Concepts inAlpha-structural recursion and induction
Recursion (computer science)
Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. The approach can be applied to many types of problems, and is one of the central ideas of computer science. "The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement.
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Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.
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Mathematical object
A mathematical object is an abstract object arising in philosophy of mathematics and mathematics. Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations. Geometry as a branch of mathematics has such objects as hexagons, points, lines, triangles, circles, spheres, polyhedra, topological spaces and manifolds. Algebra, another branch, has groups, rings, fields, group-theoretic lattices and order-theoretic lattices.
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Structural induction
Structural induction is a proof method that is used in mathematical logic (e.g. , in the proof of ¿o¿' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction. Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction.
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Abstract syntax
The abstract syntax of data is its structure described as a data type (possibly, but not necessarily, an abstract data type), independent of any particular representation or encoding. To be implemented either for computation or communications, a mapping from the abstract syntax to specific machine representations and encodings must be defined; these may be called the "concrete syntax" (in language implementation) or the "transfer syntax" (in communications).
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Abstract syntax tree
while b ¿ 0 if a > b a := a ¿ b else b := b ¿ a return a ]] In computer science, an abstract syntax tree (AST), or just syntax tree, is a tree representation of the abstract syntactic structure of source code written in a programming language. Each node of the tree denotes a construct occurring in the source code.
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Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). One might define an anagram of a word as a permutation of its letters.
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Equivalence class
This article is about equivalency in mathematics; for equivalency in music see equivalence class (music). In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element x in X is the subset of all elements in X which are equivalent to x.
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