From formulas to cirquents in computability logic

TL;DR

This paper advances computability logic by introducing cirquents, circuit-style structures that enhance expressive power and enable sharing of subgames, thereby broadening the logic's applicability and connection to independence-friendly logic.

Contribution

It generalizes formulas to cirquents in computability logic, allowing subgame sharing and capturing independence-friendly logic as a conservative fragment.

Findings

Cirquents extend the expressive power of CoL.

Subgame sharing is enabled by cirquents.

Independence-friendly logic is a conservative fragment of the extended CoL.

Abstract

Computability logic (CoL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently introduced semantical platform and ambitious program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Its expressions represent interactive computational tasks seen as games played by a machine against the environment, and "truth" is understood as existence of an algorithmic winning strategy. With logical operators standing for operations on games, the formalism of CoL is open-ended, and has already undergone series of extensions. This article extends the expressive power of CoL in a qualitatively new way, generalizing formulas (to which the earlier languages of CoL were limited) to circuit-style structures termed cirquents. The latter, unlike formulas, are able to account for subgame/subtask sharing between different parts of the overall game/task. Among the many advantages offered by this ability is that it allows us to capture, refine and generalize the well known independence-friendly logic which, after the present leap forward, naturally becomes a conservative fragment of CoL, just as classical logic had been known to be a conservative fragment of the formula-based version of CoL. Technically, this paper is self-contained, and can be read without any prior familiarity with CoL.