Concepts inFixed point semantics and partial recursion in Coq
Coq
In computer science, Coq is an interactive theorem prover. It allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions.
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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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Fixed point (mathematics)
Not to be confused with a stationary point where f'(x) = 0. In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. For example, if f is defined on the real numbers by then 2 is a fixed point of f, because f(2) = 2.
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Semantics
Semantics (from Greek: s¿mantiká, neuter plural of s¿mantikós) is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata. Linguistic semantics is the study of meaning that is used to understand human expression through language. Other forms of semantics include the semantics of programming languages, formal logics, and semiotics.
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Divergence (computer science)
In computer science, a computation is said to diverge if it does not terminate or terminates in an (unobservable) exceptional state. Otherwise it is said to converge. In domains where computations are expected to be infinite, such as process calculi, a computation is said to diverge if it fails to be productive (always produces an action within a finite amount of time.)
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Least fixed point
In order theory, a branch of mathematics, the least fixed point (lfp or LFP) of a function is the fixed point which is less than or equal to all other fixed points, according to some partial order. For example, the least fixed point of the real function f(x) = x is x = 0 with the usual order on the real numbers (since the only other fixpoint is 1 and 0 < 1). Many fixed-point theorems yield algorithms for locating the least fixed point.
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Alfred Tarski
Alfred Tarski (January 14, 1901 ¿ October 26, 1983) was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death.
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Calculus of constructions
The calculus of constructions (CoC) is a formal language in which both computer programs and mathematical proofs can be expressed. This language forms the basis of theory behind the Coq proof assistant, which implements the derivative calculus of inductive constructions.
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