Structural induction is a proof method that is used in mathematical logic (e.g. , in the proof of Łoś' 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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Recursive definition
In mathematical logic and computer science, a recursive definition (or inductive definition) is used to define an object in terms of itself . A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs. For example, the factorial function n! is defined by the rules 0! = 1. (n+1)! = (n+1)·n!. This definition is valid because, for all n, the recursion eventually reaches the base case of 0.
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Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from linguistics to logic.
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First-order logic
First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic (a less precise term). First-order logic is distinguished from propositional logic by its use of quantified variables.
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Dialectica interpretation
In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic arithmetic into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic. The name of the interpretation comes from the journal Dialectica, where Gödel's paper was published in a special issue dedicated to Paul Bernays on his 70th birthday.
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Partial function
In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y (only some subset X' of X). If X' = X, then ƒ is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, X', is not known (e.g. many functions in computability theory).
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Scope (computer science)
In computer programming, a scope is the context within a computer program in which a variable name or other identifier is valid and can be used, or within which a declaration has effect. Outside of the scope of a variable name, the variable's value may still be stored, and may even be accessible in some way, but the name does not refer to it; that is, the name is not bound to the variable's storage.
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Nominal techniques
Nominal techniques are a range of techniques, based on nominal sets, for handling names and binding, e.g. in abstract syntax. Research into nominal sets gave rise to nominal terms, a metalanguage for embedding object languages with name binding constructs into.
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