Conference Paper

A Unified Sequent Calculus for Focused Proofs.

DOI: 10.1109/LICS.2009.47 Conference: Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, LICS 2009, 11-14 August 2009, Los Angeles, CA, USA
Source: DBLP

ABSTRACT We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicative-additive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classical-linear hybrid logics.

0 Bookmarks
 · 
92 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicative–additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut-elimination holds in such fragments. From cut-elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classical–linear hybrid logics.
    Annals of Pure and Applied Logic 01/2011; · 0.50 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Existing focused proof systems for classical and intuitionistic logic allow contraction for exactly those formulas chosen for focus. For proof-search applications, contraction is undesirable, as we risk traversing the same path multiple times. We present here a contraction-free focused sequent calculus for classical propositional logic, called LKF CF , which is a modification of the recently developed proof system LKF. We prove that our system is sound and complete with respect to LKF, and therefore it is also sound and complete with respect to propositional classical logic. LKF can be justified with a compilation into focused proofs for linear logic; in this work we show how to do a similar compilation for LKF CF , but into focused proofs for linear logic with subexponentials instead. We use two subexponentials, neither allowing contraction but one allowing weakening. We show how the focused proofs for linear logic can then simulate proofs in LKF CF . Returning to proof-search, we end this work with a small experimental study showing that a proof-search implementation based on LKF CF performs well compared to implementations based on leanT A P and several variants and optimizations on LK and LKF.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: While logic was once developed to serve philosophers and mathematicians, it is increasingly serving the varied needs of computer scientists. In fact, recent decades have witnessed the creation of the new discipline of Computational Logic. While Computation Logic can claim involvement in diverse areas of computing, little has been done to systematize the foundations of this new discipline. Here, we envision a unity for Computational Logic organized around the proof theory of the sequent calculus: recent results in the area of focused proof systems will play a central role in developing this unity.

Full-text (2 Sources)

Download
72 Downloads
Available from
Jun 2, 2014