Cirquent calculus deepened

A new version of the paper is now available at
                                      http://arxiv.org/abs/0709.1308
A substantial change of perspective has taken place: classical logic can be seen as a natural conservative fragment of resource logics in the form of deep cirquent calculus. It is an extreme special case of cirquent-based resource logics where "everything is shared". And the approach of linear logic is another imperfect extreme where "nothing is shared". Cirquent calculus is thus a general unifying framework.

Those who earlier took interest in this paper, might want to at least read the (totally new) introductory section of the new version.

                                                      G.Japaridze
                                         Cirquent calculus deepened
                                                       Abstract

Cirquent calculus is a new proof-theoretic and  semantic framework, whose main distinguishing feature is being based on circuit-style structures (called cirquents), as opposed   to the more traditional approaches that deal with tree-like objects such as formulas or sequents. Among its advantages are greater efficiency, flexibility and expressiveness. This paper presents a detailed elaboration of a deep-inference cirquent logic, which is naturally and inherently resource conscious. It shows that classical logic, both syntactically and semantically, can be seen to be just a special, conservative fragment of this more general and, in a sense, more basic logic --- the logic of resources in the form of cirquent calculus. The reader will find various arguments in favor of switching to the new framework, such as arguments showing the insufficiency of the expressive power of linear logic or other formula-based approaches to developing resource logics, exponential improvements over the traditional approaches in both representational and proof complexities offered by cirquent calculus (including the existence of polynomial size cut-, substitution- and extension-free cirquent calculus proofs for the notoriously hard pigeonhole principle), and more. Among the main purposes of this paper is to provide an introductory-style  starting point for what, as the author wishes to hope, might have a chance to become a new line of research in proof theory --- a proof theory based on circuits instead of formulas.

Giorgi Japridze
http://www.csc.villanova.edu/~japaridz/