realizable

Realizable is an adjective meaning capable of being realized or achieved. In everyday usage it denotes feasibility under given conditions or resources. In mathematical logic and theoretical computer science, realizability has a more precise technical sense: a statement is realizable if there exists a constructive witness—often an algorithm or concrete object—that demonstrates its truth.

The central idea of realizability is to interpret logical formulas as statements about computational witnesses. Originating with Stephen Kleene in the 1950s, Kleene’s realizability assigns to each formula a condition or code (such as a natural number coding a program) that "realizes" the formula. This provides a bridge from proofs to computation, enabling the extraction of executable content from constructive proofs. Realizability is closely tied to intuitionistic logic and the Curry–Howard correspondence, which connects proofs with programs.

Several variants and extensions exist. Modified realizability, developed by Kreisel and others, refines the interpretation to handle a broader class of propositions. Relative realizability and other categorical forms, such as realizability toposes, place realizability within a topos-theoretic framework. These approaches often use structures like partial combinatory algebras to model computation inside a rich mathematical setting, providing semantic tools for constructive mathematics and type theory.

Applications of realizability include proving the constructive validity of mathematical principles, producing programs from proofs, and giving semantic models for intuitionistic and constructive logics. In short, realizable concepts emphasize that mathematical truth can be witnessed by concrete computational or constructive data.