Introduction to Cirquent Calculus and Abstract Resource Semantics

Abstract

This article introduces a refinement of the sequent calculus approach called cirquent calculus. Roughly speaking, the difference between the two is that, while in Gentzen-style proof trees sibling (or cousin, etc.) sequents are disjoint and independent sequences of formulas, in cirquent calculus they are permitted to share elements. Explicitly allowing or disallowing shared resources and thus taking to a more subtle level the resource-awareness intuitions underlying substructural logics, cirquent calculus offers much greater flexibility and power than sequent calculus does. A need for substantially new deductive tools came with the birth of computability logic — the semantically constructed formal theory of computational resources, which has stubbornly resisted all axiomatization attempts within the framework of traditional syntactic approaches. Cirquent calculus breaks the ice. Removing contraction from the full (‘classical’) collection of its rules yields a sound and complete system for the basic fragment CL5 of computability logic, previously thought to be ‘most unaxiomatizable’. Deleting the offending rule of contraction in ordinary sequent calculus, on the other hand, throws out the baby with the bath water, resulting in the strictly weaker affine logic. An implied claim of computability logic is that it is CL5 rather than affine logic that adequately materializes the resource philosophy traditionally associated with the latter. To strengthen this claim, the article further introduces an abstract resource semantics and shows the soundness and completeness of CL5 with respect to it. Unlike the semantics of computability logic, which understands resources in a special — computational — sense, abstract resource semantics can be seen as a direct formalization of the more general yet naive intuitions in the ‘can you get both a candy and an apple for one dollar?’ style. The inherent incompleteness of affine or linear logics, resulting from the fundamental limitations of the underlying sequent calculus approach, is apparently the reason why such intuitions and examples, while so heavily relied on in the popular linear-logic literature, have never really found a good explication in the form of a mathematically strict and intuitively convincing semantics. The article is written in a style accessible to a wide range of readers. Some basic familiarity with computability logic, sequent calculus or linear logic is desirable only as much as to be able to duly appreciate the import of the present contribution. * This material is based upon work supported by the National Science Foundation under Grant No. 0208816, and 2005 Summer Research Grant from Villanova University.