Notre Dame Journal of Formal Logic

The Parallel versus Branching Recurrences in Computability Logic

Wenyan Xu and Sanyang Liu

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This paper shows that the basic logic induced by the parallel recurrence $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ of computability logic (i.e., the one in the signature $\{\neg,$\wedge$,\vee,\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}},\hspace {-2pt}\mbox {\raisebox {0.12cm}{\@setfontsize \small {7}{8}$\vee$}\hspace {-3.6pt}\raisebox {0.02cm}{\tiny $\mid$}\hspace {2pt}}\}$) is a proper superset of the basic logic induced by the branching recurrence $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ (i.e., the one in the signature $\{\neg,$\wedge$,\vee,\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}},\mbox {\raisebox {0.12cm}{$\circ$}\hspace {-0.115cm}\raisebox {0.02cm}{\tiny $\mid$}\hspace {2pt}}\}$). The latter is known to be precisely captured by the cirquent calculus system CL15, conjectured by Japaridze to remain sound—but not complete—with $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ instead of $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$. The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ is strictly weaker than $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ in the sense that, while $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}F$ logically implies $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}F$, the reverse does not hold.

Article information

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

First available in Project Euclid: 14 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

computability logic cirquent calculus interactive computation game semantics resource semantics


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.

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  • [1] Japaridze, G., “Introduction to computability logic,” Annals of Pure and Applied Logic, vol. 123 (2003), pp. 1–99.
  • [2] Japaridze, G., “Introduction to cirquent calculus and abstract resource semantics,” Journal of Logic and Computation, vol. 16 (2006), pp. 489–532.
  • [3] Japaridze, G., “The logic of interactive Turing reduction,” Journal of Symbolic Logic, vol. 72 (2007), pp. 243–76.
  • [4] Japaridze, G., “Cirquent calculus deepened,” Journal of Logic and Computation, vol. 18 (2008), pp. 983–1028.
  • [5] Japaridze, G., “In the beginning was game semantics,” pp. 249–350 in Games: Unifying Logic, Language, and Philosophy, edited by O. Majer, A.-V. Pietarinen, and T. Tulenheimo, vol. 15 of Logic, Epistemology, and the Unity of Science, Springer, New York, 2009.
  • [6] Japaridze, G., “Many concepts and two logics of algorithmic reduction,” Studia Logica 91 (2009), pp. 1–24.
  • [7] Japaridze, G., “From formulas to cirquents in computability logic,” Logical Methods in Computer Science, vol. 7 (2011), paper 2:1.
  • [8] Japaridze, G., “A new face of the branching recurrence of computability logic,” Applied Mathematics Letters, vol. 25 (2012), pp. 1585–9. DOI 10.1016/j.aml.2011.11.023
  • [9] Japaridze, G., “Separating the basic logics of the basic recurrences,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 377–89.
  • [10] Japaridze, G., “The taming of recurrences in computability logic through cirquent calculus, Part I,” preprint, arXiv:1105.3853v4 [cs.LO]
  • [11] Japaridze, G., “The taming of recurrences in computability logic through cirquent calculus, Part II,” preprint, arXiv:1106.3705v1 [cs.LO]
  • [12] Mezhirov, I., and N. Vereshchagin, “On abstract resource semantics and computability logic,” Journal of Computer and System Sciences, vol. 76 (2010), pp. 356–72.
  • [13] Xu, W., and S. Liu, “Deduction theorem for symmetric cirquent calculus,” pp. 121–26 in Quantitative Logic and Soft Computing 2010, vol. 82 of Advances in Intelligent and Soft Computing, Springer, Berlin, 2010.
  • [14] Xu, W., and S. Liu, “Soundness and completeness of the cirquent calculus system CL6 for computability logic,” Logic Journal of the IGPL, vol. 20 (2012), pp. 317–30.