The Computational Complexity of Propositional Cirquent Calculus

Introduced in 2006 by Japaridze, cirquent calculus is a refinement of sequent calculus. The advent of cirquent calculus arose from the need for a deductive system with a more explicit ability to reason about resources. Unlike the more traditional proof-theoretic approaches that manipulate tree-like objects (formulas, sequents, etc.), cirquent calculus is based on circuit-style structures called cirquents, in which different "peer" (sibling, cousin, etc.) substructures may share components. It is this resource sharing mechanism to which cirquent calculus owes its novelty (and its virtues). From its inception, cirquent calculus has been paired with an abstract resource semantics. This semantics allows for reasoning about the interaction between a resource provider and a resource user, where resources are understood in the their most general and intuitive sense. Interpreting resources in a more restricted computational sense has made cirquent calculus instrumental in axiomatizing various fundamental fragments of Computability Logic, a formal theory of (interactive) computability. The so-called "classical" rules of cirquent calculus, in the absence of the particularly troublesome contraction rule, produce a sound and complete system CL5 for Computability Logic. In this paper, we investigate the computational complexity of CL5, showing it is $\Sigma_2^p$-complete. We also show that CL5 without the duplication rule has polynomial size proofs and is NP-complete

Introduction to Cirquent Calculus and Abstract Resource Semantics

This paper introduces a refinement of the sequent calculus approach called cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.) sequents are disjoint 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 (see http://www.cis.upenn.edu/~giorgi/cl.html) - the semantically constructed formal theory of computational resources, which has stubbornly resisted any axiomatization attempts within the framework of traditional syntactic approaches. Cirquent calculus breaks the ice. Removing contraction from the full collection of its rules yields a sound and complete system for the basic fragment CL5 of computability logic. Doing the same in 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 paper further introduces an abstract resource semantics and shows the soundness and completeness of CL5 with respect to it.Comment: To appear in Journal of Logic and Computatio

Cirquent calculus deepened

Cirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuits, 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, is 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, 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.Comment: Significant improvements over the previous version

From truth to computability I

The recently initiated approach called computability logic is a formal theory of interactive computation. See a comprehensive online source on the subject at http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a soundness and completeness proof for the deductive system CL3 which axiomatizes the most basic first-order fragment of computability logic called the finite-depth, elementary-base fragment. Among the potential application areas for this result are the theory of interactive computation, constructive applied theories, knowledgebase systems, systems for resource-bound planning and action. This paper is self-contained as it reintroduces all relevant definitions as well as main motivations.Comment: To appear in Theoretical Computer Scienc

The intuitionistic fragment of computability logic at the propositional level

This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible -- and hence most natural -- sense, disjunction and conjunction as deterministic-choice combinations of problems (disjunction = machine's choice, conjunction = environment's choice), and "absurd" as a computational problem of universal strength. See http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on computability logic