Computability logic¶

Computability logic (CoL) is a research program 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.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8616
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Mathematical Logic, Computability Theory → Mathematics

Core Idea¶

Computability logic is treated here as the recurring computer science and information systems identity summarized by this source-grounded definition: Computability logic (CoL) is a research program 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. Computability logic (CoL) is a research program 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. It was introduced and so named by Giorgi Japaridze in 2003.

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Can-a-Computer-Do-It Logic

Most logic asks one question: is this true or false? Computability logic asks a different question: can a computer always actually get the job done? It is a way of doing logic about solving problems instead of about being true.

Logic About Solving, Not Truth

Logic is the study of good reasoning. Ordinary (classical) logic is about truth: its sentences are true or false, and an argument is valid when true starting sentences always lead to a true conclusion. Computability logic, started by Giorgi Japaridze in 2003, rebuilds logic around a different question — what can be computed. In computability logic, something counts as valid when it is always computable, meaning a machine can always successfully handle it.

Logic of Computability

Computability logic (CoL) is a research program and mathematical framework, introduced and named by Giorgi Japaridze in 2003, that rebuilds logic as a formal theory of computability rather than of truth. In classical logic, formulas stand for true-or-false statements, and validity depends only on an argument's form: it tells you when the truth of some statements always follows from the truth of others. CoL keeps the idea of a systematic logic but changes what validity means: a formula is valid when it is always computable. So instead of asking whether truth is preserved, CoL asks whether computability is guaranteed. This makes it a logic about what machines can reliably do, not just about what is so.

Computability logic (CoL) is a research program and mathematical framework, introduced and so named by Giorgi Japaridze in 2003, that redevelops logic as a systematic formal theory of computability, in contrast with classical logic as a formal theory of truth. In classical logic, formulas represent true/false statements, validity is a matter of form alone, and the logic characterizes when the truth of a statement always follows from the truth of a set of premises. CoL reinterprets the central notion of the discipline: validity means being always computable. The program is thus a wholesale reconstruction of logic around computational rather than truth-theoretic semantics, not merely an application of logic to computing. What makes something an instance of CoL is this foundational replacement of truth by computability as the organizing concept; sharing vocabulary with computation or simply studying algorithms is not enough. It sits at the boundary of mathematics and computer science, and its methods and tests are anchored there.

Scope of Application¶

As a problem solving tool. It typically includes all Peano axioms, and adds to them one or two extra-Peano axioms such as ⊓ x ⊔ y(y=x) expressing the computability of the successor function.
As a problem solving tool. So, such a theory can be used for finding not merely algorithmic solutions, but also efficient ones on demand, such as solutions that run in polynomial time or logarithmic space.
Documented setting. CoL systematically answers the fundamental question of what can be computed and how; thus CoL has many applications, such as constructive applied theories, knowledge base systems, systems for planning and action.
Documented setting. This has necessitated developing alternative, more general and flexible methods of proof, such as cirquent calculus.
Documented setting. Out of these, only applications in constructive applied theories have been extensively explored so far: a series of CoL-based number theories, termed "clarithmetics", have been constructed as computationally and complexity-theoretically meaningful.

Clarity¶

A clear use of Computability logic names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Computability logic (CoL) is a research program 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.

Manages Complexity¶

Computability logic compresses multiple computer science and information systems details into a stable diagnostic relation. The source shows both the central mechanism—both semantically and syntactically, classical logic is nothing but the fragment of CoL obtained by forbidding general atoms in its language, and forbidding all operators other than ¬, ∧, ∨, →, ∀, ∃.—and the practical consequence—the parallel implication operator → ("pimplication") is defined by A→B = ¬A∨B.

Abstract Reasoning¶

Type the carrier. Identify the computer science and information systems entities to which the claim applies.
State the relation. Use the source-grounded identity: Computability logic (CoL) is a research program 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.
Check operation and conditions. Each run (play) is won by one of these players and lost by the other.
Demand recognition evidence.

Knowledge Transfer¶

Within the home domain. Knowledge about Computability logic transfers literally when a new case preserves the same carrier type, relation, and recognition test. It typically includes all Peano axioms, and adds to them one or two extra-Peano axioms such as ⊓ x ⊔ y(y=x) expressing the computability of the successor function. So, such a theory can be used for finding not merely algorithmic solutions, but also efficient ones on demand, such.

Neighborhood in Abstraction Space¶

Computability logic sits in a moderately populated region (40^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Formal Logic & Semantic Systems (18 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08