8,343 research outputs found
Introduction to clarithmetic I
"Clarithmetic" is a generic name for formal number theories similar to Peano
arithmetic, but based on computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional
classical or intuitionistic logics. Formulas of clarithmetical theories
represent interactive computational problems, and their "truth" is understood
as existence of an algorithmic solution. Imposing various complexity
constraints on such solutions yields various versions of clarithmetic. The
present paper introduces a system of clarithmetic for polynomial time
computability, which is shown to be sound and complete. Sound in the sense that
every theorem T of the system represents an interactive number-theoretic
computational problem with a polynomial time solution and, furthermore, such a
solution can be efficiently extracted from a proof of T. And complete in the
sense that every interactive number-theoretic problem with a polynomial time
solution is represented by some theorem T of the system. The paper is written
in a semitutorial style and targets readers with no prior familiarity with
computability logic
A logical basis for constructive systems
The work is devoted to Computability Logic (CoL) -- the
philosophical/mathematical platform and long-term project for redeveloping
classical logic after replacing truth} by computability in its underlying
semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article
elaborates some basic complexity theory for the CoL framework. Then it proves
soundness and completeness for the deductive system CL12 with respect to the
semantics of CoL, including the version of the latter based on polynomial time
computability instead of computability-in-principle. CL12 is a sequent calculus
system, where the meaning of a sequent intuitively can be characterized as "the
succedent is algorithmically reducible to the antecedent", and where formulas
are built from predicate letters, function letters, variables, constants,
identity, negation, parallel and choice connectives, and blind and choice
quantifiers. A case is made that CL12 is an adequate logical basis for
constructive applied theories, including complexity-oriented ones
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
Tackling the infrared problem of thermal QCD
Perturbative calculations of corrections to the behavior of an ideal gas of
quarks and gluons, the limit that is formally realized at infinite temperature,
are obstructed by severe infrared divergences. The limits to computability that
the infrared problem poses can be overcome in the framework of dimensionally
reduced effective theories. Here, we give details on the evaluation of the
highest perturbative coefficient needed for this setup, in the continuum.Comment: 3 pages, Lattice2003(nonzero
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
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