Computability logic

Introduced by Giorgi Japaridze in 2003, Computability logic is a research programme and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth. In this approach logical formulas represent computational problems (or, equivalently, computational resources), and their validity means being "always computable".

Computational problems and resources are understood in their most general - interactive sense. They are formalized as games played by a machine against its environment, and computability means existence of a machine that wins the game against any possible behavior by the environment. Defining what such game-playing machines mean, computability logic provides a generalization of the Church-Turing thesis to the interactive level.

The classical concept of truth turns out to be a special, zero-interactivity-degree case of computability. This makes classical logic a special fragment of computability logic. Being a conservative extension of the former, computability logic is, at the same time, by an order of magnitude more expressive, constructive and computationally meaningful. Providing a systematic answer to the fundamental question "what (and how) can be computed?", it has a wide range of potential application areas. Those include constructive applied theories, knowledge base systems, systems for planning and action.

Besides classical logic, linear logic (understood in a relaxed sense) and intuitionistic logic also turn out to be natural fragments of computability logic. Hence meaningful concepts of "intuitionistic truth" and "linear-logic truth" can be derived from the semantics of computability logic.

Being semantically constructed, as yet computability logic does not have a fully developed proof theory. Finding deductive systems for various fragments of it and exploring their syntactic properties is an area of ongoing research.