11 research outputs found
The taming of recurrences in computability logic through cirquent calculus, Part I
This paper constructs a cirquent calculus system and proves its soundness and
completeness with respect to the semantics of computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html). The logical vocabulary of the system
consists of negation, parallel conjunction, parallel disjunction, branching
recurrence, and branching corecurrence. The article is published in two parts,
with (the present) Part I containing preliminaries and a soundness proof, and
(the forthcoming) Part II containing a completeness proof
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
-complete. We also show that CL5 without the duplication rule has
polynomial size proofs and is NP-complete
Build your own clarithmetic I: Setup and completeness
Clarithmetics are number theories based on computability logic (see
http://www.csc.villanova.edu/~japaridz/CL/ ). Formulas of these theories
represent interactive computational problems, and their "truth" is understood
as existence of an algorithmic solution. Various complexity constraints on such
solutions induce various versions of clarithmetic. The present paper introduces
a parameterized/schematic version CLA11(P1,P2,P3,P4). By tuning the three
parameters P1,P2,P3 in an essentially mechanical manner, one automatically
obtains sound and complete theories with respect to a wide range of target
tricomplexity classes, i.e. combinations of time (set by P3), space (set by P2)
and so called amplitude (set by P1) complexities. Sound in the sense that every
theorem T of the system represents an interactive number-theoretic
computational problem with a solution from the given tricomplexity class and,
furthermore, such a solution can be automatically extracted from a proof of T.
And complete in the sense that every interactive number-theoretic problem with
a solution from the given tricomplexity class is represented by some theorem of
the system. Furthermore, through tuning the 4th parameter P4, at the cost of
sacrificing recursive axiomatizability but not simplicity or elegance, the
above extensional completeness can be strengthened to intensional completeness,
according to which every formula representing a problem with a solution from
the given tricomplexity class is a theorem of the system. This article is
published in two parts. The present Part I introduces the system and proves its
completeness, while Part II is devoted to proving soundness