Logics for computability are formulations of logic that
capture some aspect of computability as a basic notion. This usually involves a mix
of special logical connectives as well as a semantics that explains how the logic is to be interpreted in a computational way.
Probably the first formal treatment of logic for computability is the realizability interpretation by Stephen Kleene in 1945, who gave an interpretation of intuitionistic number theory in terms of Turing machine computations. His motivation was to make precise the Heyting–Brouwer–Kolmogorov (BHK) interpretation of intuitionism, according to which proofs of mathematical statements are to be viewed as constructive procedures.
With the rise of many other kinds of logic, such as modal logic and linear logic, and novel semantic models, such as game semantics, logics for computability have been formulated in several contexts. Here we mention two.
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