27 research outputs found
Propositional computability logic I
In the same sense as classical logic is a formal theory of truth, the
recently initiated approach called computability logic is a formal theory of
computability. It understands (interactive) computational problems as games
played by a machine against the environment, their computability as existence
of a machine that always wins the game, logical operators as operations on
computational problems, and validity of a logical formula as being a scheme of
"always computable" problems. The present contribution gives a detailed
exposition of a soundness and completeness proof for an axiomatization of one
of the most basic fragments of computability logic. The logical vocabulary of
this fragment contains operators for the so called parallel and choice
operations, and its atoms represent elementary problems, i.e. predicates in the
standard sense. This article is self-contained as it explains all relevant
concepts. While not technically necessary, however, familiarity with the
foundational paper "Introduction to computability logic" [Annals of Pure and
Applied Logic 123 (2003), pp.1-99] would greatly help the reader in
understanding the philosophy, underlying motivations, potential and utility of
computability logic, -- the context that determines the value of the present
results. Online introduction to the subject is available at
http://www.cis.upenn.edu/~giorgi/cl.html and
http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi
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
A new face of the branching recurrence of computability logic
This letter introduces a new, substantially simplified version of the
branching recurrence operation of computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html), and proves its equivalence to the
old, "canonical" version
Intuitionistic computability logic
Computability logic (CL) is a systematic formal theory of computational tasks
and resources, which, in a sense, can be seen as a semantics-based alternative
to (the syntactically introduced) linear logic. With its expressive and
flexible language, where formulas represent computational problems and "truth"
is understood as algorithmic solvability, CL potentially offers a comprehensive
logical basis for constructive applied theories and computing systems
inherently requiring constructive and computationally meaningful underlying
logics.
Among the best known constructivistic logics is Heyting's intuitionistic
calculus INT, whose language can be seen as a special fragment of that of CL.
The constructivistic philosophy of INT, however, has never really found an
intuitively convincing and mathematically strict semantical justification. CL
has good claims to provide such a justification and hence a materialization of
Kolmogorov's known thesis "INT = logic of problems". The present paper contains
a soundness proof for INT with respect to the CL semantics. A comprehensive
online source on CL is available at http://www.cis.upenn.edu/~giorgi/cl.htm
The logic of interactive Turing reduction
The paper gives a soundness and completeness proof for the implicative
fragment of intuitionistic calculus with respect to the semantics of
computability logic, which understands intuitionistic implication as
interactive algorithmic reduction. This concept -- more precisely, the
associated concept of reducibility -- is a generalization of Turing
reducibility from the traditional, input/output sorts of problems to
computational tasks of arbitrary degrees of interactivity. See
http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on
computability logic