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

Logics for complexity classes

A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered structures, and another form to define them on unordered non-Aristotelian structures. Using the canonical forms, logics are developed for complete problems in various complexity classes. Evidence is shown that there cannot be any complete problem on Aristotelian structures for several complexity classes. Our approach is extended beyond complete problems. Using a similar form, a logic is developed to capture the complexity class $NP\cap coNP$ which very likely contains no complete problem.Comment: This article has been accepted for publication in Logic Journal of IGPL Published by Oxford University Press; 23 pages, 2 figure

The Complexity of Prenex Separation Logic with One Selector

We first show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with $k\geq1$ selector fields (\seplogk{k}). Second, we show that this entails the decidability of the finite and infinite satisfiability problem for the class of prenex formulas of \seplogk{1}, by reduction to the first-order theory of one unary function symbol and unary predicate symbols. We also prove that the complexity is not elementary, by reduction from the first-order theory of one unary function symbol. Finally, we prove that the Bernays-Sch\"onfinkel-Ramsey fragment of prenex \seplogk{1} formulae with quantifier prefix in the language $\exists^*\forall^*$ is \pspace-complete. The definition of a complete (hierarchical) classification of the complexity of prenex \seplogk{1}, according to the quantifier alternation depth is left as an open problem

Extended RDF: Computability and Complexity Issues

ERDF stable model semantics is a recently proposed semantics for ERDF ontologies and a faithful extension of RDFS semantics on RDF graphs. In this paper, we elaborate on the computability and complexity issues of the ERDF stable model semantics. Based on the undecidability result of ERDF stable model semantics, decidability under this semantics cannot be achieved, unless ERDF ontologies of restricted syntax are considered. Therefore, we propose a slightly modified semantics for ERDF ontologies, called ERDF #n- stable model semantics. We show that entailment under this semantics is, in general, decidable and also extends RDFS entailment. Equivalence statements between the two semantics are provided. Additionally, we provide algorithms that compute the ERDF #n-stable models of syntax-restricted and general ERDF ontologies. Further, we provide complexity results for the ERDF #nstable model semantics on syntax-restricted and general ERDF ontologies. Finally, we provide complexity results for the ERDF stable model semantics on syntax-restricted ERDF ontologies

Tractability and the computational mind

We overview logical and computational explanations of the notion of tractability as applied in cognitive science. We start by introducing the basics of mathematical theories of complexity: computability theory, computational complexity theory, and descriptive complexity theory. Computational philosophy of mind often identifies mental algorithms with computable functions. However, with the development of programming practice it has become apparent that for some computable problems finding effective algorithms is hardly possible. Some problems need too much computational resource, e.g., time or memory, to be practically computable. Computational complexity theory is concerned with the amount of resources required for the execution of algorithms and, hence, the inherent difficulty of computational problems. An important goal of computational complexity theory is to categorize computational problems via complexity classes, and in particular, to identify efficiently solvable problems and draw a line between tractability and intractability. We survey how complexity can be used to study computational plausibility of cognitive theories. We especially emphasize methodological and mathematical assumptions behind applying complexity theory in cognitive science. We pay special attention to the examples of applying logical and computational complexity toolbox in different domains of cognitive science. We focus mostly on theoretical and experimental research in psycholinguistics and social cognition