An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
Logical operations and Kolmogorov complexity
Theoretical Computer Science
On Game Semantics of the Affine and Intuitionistic Logics
WoLLIC '08 Proceedings of the 15th international workshop on Logic, Language, Information and Computation
On abstract resource semantics and computability logic
Journal of Computer and System Sciences
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It is unknown, whether the logic of propositional formulas that are realizable in the sense of Kleene has a finite or recursive axiomatization. In this paper another approach to realizability of propositional formulas is studied. This approach is based on the following informal idea: a formula is realizable if it has a "simple" realization for each substitution. More precisely, logical connectives are interpreted as operations on sets of natural numbers and a formula is interpreted as a combined operation; if some sets are substituted for variables, then elements of the result are called realizations. A realization (a natural number) is simple if it has low Kolmogorov complexity, and a formula is called realizable if it has at least one simple realization whatever sets are substituted. Similar definitions may be formulated in arithmetical terms. A few "realizabilities" of this kind are considered and it is proved that all of them give the same finitely axiomatizable logic, namely, the logic of the weak law of excluded middle.