In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another.
Realizability can be seen as a formalization of the BHK interpretation of intuitionistic logic; in realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly provide.
Beyond giving insight into intuitionistic provability, realizability can be applied to prove the disjunction and existence properties for intuitionistic theories and to extract programs from proofs, as in proof mining. It is also related to topos theory via realizability topoi.
Kleene's original version of realizability uses natural numbers as realizers for formulas in Heyting arithmetic. A few pieces of notation are required: first, an ordered pair (n,m) is treated as a single number using a fixed primitive recursive pairing function; second, for each natural number n, φn is the computable function with index n. The following clauses are used to define a relation "n realizes A" between natural numbers n and formulas A in the language of Heyting arithmetic, known as Kleene's 1945-realizability relation:
Any number n realizes an atomic formula s=t if and only if s=t is true. Thus every number realizes a true equation, and no number realizes a false equation.
A pair (n,m) realizes a formula A∧B if and only if n realizes A and m realizes B.
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In constructive mathematics, Church's thesis is an axiom stating that all total functions are computable functions. The similarly named Church–Turing thesis states that every effectively calculable function is a computable function, thus collapsing the former notion into the latter. is stronger in the sense that with it every function is computable. The constructivist principle is fully formalizable, using formalizations of "function" and "computable" that depend on the theory considered.
vignette|250x250px|Une représentation artistique d'une machine de Turing. Le principe de Markov dit que s'il est impossible qu'une machine de Turing ne s'arrête pas, alors elle doit s'arrêter. Le principe de Markov, nommé d'après Andreï Markov Jr, est une déclaration d'existence conditionnelle pour laquelle il existe de nombreuses formulations, ainsi qu'il est discuté ci-dessous. Ce principe est utilisé dans la validité logique classique, mais pas dans les mathématiques intuitionniste constructives.
In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic logic (Heyting 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 1958 special issue dedicated to Paul Bernays on his 70th birthday.