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Encyclopedia > 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". Giorgi Japaridze is a logician, at Villanova University in Villanova, Pennsylvania. ... Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...

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. In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ...

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. // Definition A logical theory T2 is a conservative extension of theory T1 if any consequence of T2 involving symbols of T1 only is already a consequence of T1. ...

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. In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...

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. Results from FactBites:

 Computability logic (334 words) Computability logic is a formal theory of computability, introduced by Giorgi Japaridze in 2003. Computability logic is a conservative extension of classical logic and, being more expressive, constructive and computationally meaningful, it has a wide range of application areas. Computability logic is an attempt to lay foundations for a comprehensive and systematic theory of interactive computation.
 Computability logic - Wikipedia, the free encyclopedia (315 words) 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. 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. Besides classical logic, linear logic (understood in a relaxed sense) and intuitionistic logic also turn out to be natural fragments of computability logic.
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