From formulas to cirquents in computability logic

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.

💡 Research Summary

The paper “From formulas to cirquents in computability logic” presents a major extension of the formal language of Computability Logic (CoL) by replacing the traditional formula‑based syntax with a circuit‑style structure called a cirquent. CoL, originally introduced as a game‑theoretic semantics for logic, interprets logical expressions as interactive computational tasks (games) between a machine (the prover) and its environment, and declares a statement true precisely when the machine possesses an algorithmic winning strategy. In its earlier incarnations CoL’s language was limited to formulas, which are essentially tree‑shaped expressions. While adequate for many purposes, this restriction forces the same sub‑game to be duplicated wherever it appears, preventing the language from expressing resource sharing or the reuse of strategies across different parts of a larger task.

A cirquent generalizes a formula by allowing its components to be arranged as a directed acyclic graph rather than a tree. Nodes (called “ports”) represent atomic games, and gates correspond to the logical operators of CoL (∧, ∨, →, ¬, etc.). The crucial innovation is the distinction between shared gates and duplicate gates. A shared gate connects several higher‑level components to a single sub‑game, thereby modelling the situation where the same sub‑task is simultaneously available to different parts of the overall computation. Duplicate gates behave like the usual formula nodes, creating independent copies of a sub‑game. This structural flexibility directly captures the notion of sub‑game sharing, which is essential for modelling parallelism, concurrency, and resource‑sensitive reasoning.

The authors develop a rigorous semantics for cirquents. Each gate inherits the game‑combination rule of its corresponding logical operator, and the semantics of a shared gate ensures that a single strategy can be used simultaneously in all its incident edges. Consequently, the truth condition for a cirquent is defined as the existence of a uniform winning strategy that respects the sharing constraints. The paper shows that many phenomena previously treated only informally in CoL become transparent in the cirquent framework.

One of the most striking applications is to Independence‑Friendly (IF) logic. In IF logic, the slash notation (∃/∀) expresses quantifier independence, i.e., a quantifier’s choice must be made without knowledge of certain other choices. The authors demonstrate that any IF‑formula can be translated into a cirquent where the independence is realized by appropriate sharing of sub‑games. In this translation, the “independent” quantifier becomes a duplicate gate, while the “dependent” context is represented by a shared gate, thereby providing a clean, game‑theoretic account of IF semantics within CoL. As a result, IF logic emerges as a conservative fragment of the cirquent‑based CoL, just as classical logic is a conservative fragment of the original formula‑based CoL.

The paper also establishes two conservativity theorems. First, every classical propositional formula can be represented as a cirquent that uses only duplicate gates, showing that classical logic embeds faithfully. Second, every formula of the earlier CoL language can be transformed into an equivalent cirquent without loss of expressive power. These results guarantee that the new framework does not overturn existing results but rather extends them.

Beyond theoretical elegance, the authors argue that cirquents bring concrete advantages. By allowing strategy reuse, proof sizes can be dramatically reduced, and algorithms derived from cirquents can exploit shared sub‑computations, leading to more efficient implementations. The paper sketches how cirquents could be incorporated into automated theorem provers and interactive proof assistants, where the graph structure can be stored and manipulated directly, avoiding the combinatorial blow‑up caused by naive duplication.

Finally, the paper outlines future research directions: exploring richer forms of sharing (e.g., cyclic sharing for fixed‑point constructions), integrating cirquents with complexity‑aware fragments of CoL, and applying the framework to real‑world concurrent systems such as distributed protocols and multi‑agent planning. In sum, the work provides a qualitatively new expressive layer for Computability Logic, unifies several previously disparate logical systems under a common game‑theoretic umbrella, and opens a path toward more resource‑sensitive and modular reasoning about computation.