Abstract
This paper shows that the basic logic induced by the parallel recurrence of computability logic (i.e., the one in the signature $\{ ¬, ∧,∨,⅄,\small{𝖸} \}$) is a proper superset of the basic logic induced by the branching recurrence $⫰$ (i.e., the one in the signature $\{ ¬,∧,∨, ⫰, ⫯ \}$). The latter is known to be precisely captured by the cirquent calculus system CL15, conjectured by Japaridze to remain sound—but not complete—with $⅄$ instead of $⫰$. The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that $⅄$ is strictly weaker than $⫰$ in the sense that, while $⫰F$ logically implies $⅄F$, the reverse does not hold.
Citation
Wenyan Xu. Sanyang Liu. "The Parallel versus Branching Recurrences in Computability Logic." Notre Dame J. Formal Logic 54 (1) 61 - 78, 2013. https://doi.org/10.1215/00294527-1731389
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