## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 54, Number 1 (2013), 61-78.

### The Parallel versus Branching Recurrences in Computability Logic

Wenyan Xu and Sanyang Liu

#### Abstract

This paper shows that the basic logic induced by the parallel recurrence of computability logic (i.e., the one in the signature ) 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 logically implies , the reverse does not hold.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 54, Number 1 (2013), 61-78.

**Dates**

First available in Project Euclid: 14 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1355494523

**Digital Object Identifier**

doi:10.1215/00294527-1731389

**Mathematical Reviews number (MathSciNet)**

MR3007962

**Zentralblatt MATH identifier**

1291.03072

**Subjects**

Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}

Secondary: 03B70: Logic in computer science [See also 68-XX] 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) [See also 68Q85] 68T27: Logic in artificial intelligence 68T15: Theorem proving (deduction, resolution, etc.) [See also 03B35]

**Keywords**

computability logic cirquent calculus interactive computation game semantics resource semantics

#### Citation

Xu, Wenyan; Liu, Sanyang. The Parallel versus Branching Recurrences in Computability Logic. Notre Dame J. Formal Logic 54 (2013), no. 1, 61--78. doi:10.1215/00294527-1731389. https://projecteuclid.org/euclid.ndjfl/1355494523