A new face of the branching recurrence of computability logic

This letter introduces a new, substantially simplified version of the branching recurrence operation of computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html), and proves its equivalence to the old, “canonical” version.

💡 Research Summary

The paper addresses a long‑standing technical obstacle in Computability Logic (CoL), namely the complexity of the branching recurrence operator ◦| (also written as “circle‑bar”). In CoL, logical operators are interpreted as operations on games, and ◦| is the operation that allows a game A to be played in a potentially infinite family of parallel copies, with the player ⊤ (the machine) required to win in every copy. The traditional, “tight” definition of ◦| (denoted ◦| T in the paper) is rather involved: the environment (player ⊥) may at any moment perform a “replicative move” w: that splits the current thread w into two new threads w0 and w1. After such a split, ordinary moves of A can be made only in existing threads, and further splits can be performed recursively. Formally this is captured by a bit‑string tree (BT) structure that must be maintained throughout the play. While mathematically sound, this definition makes syntactic analysis, axiomatization, and proof‑theoretic work extremely cumbersome.

The author proposes a radically simpler counterpart, called the “loose” version (◦| L). In this version there are no replicative moves at all. Instead, from the very beginning the game is imagined to contain a continuum of threads, each identified by an infinite bit‑string. A legal move is any finite prefix w followed by a move α of A, written w.α. This move is understood to be simultaneously performed in every thread whose address begins with w. Consequently, the set of threads is static; the player can act in any thread at any time without first having to create it. The definition of winning is also straightforward: ⊤ wins ◦| L A iff for every infinite bit‑string v the projection of the run onto thread v (denoted Γ ↾ v) is a ⊤‑won run of A.

The paper first proves that both versions preserve the “static” property of games, i.e., the class of static games is closed under ◦| T and ◦| L. This is essential because many of CoL’s meta‑results (e.g., the soundness of various proof systems) rely on staticity. The proof proceeds by induction on the length of the shortest illegal prefix of a delayed run, using lemmas from the foundational CoL monograph (reference