1,436 research outputs found
Introduction to clarithmetic II
The earlier paper "Introduction to clarithmetic I" constructed an axiomatic
system of arithmetic based on computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html), and proved its soundness and
extensional completeness with respect to polynomial time computability. The
present paper elaborates three additional sound and complete systems in the
same style and sense: one for polynomial space computability, one for
elementary recursive time (and/or space) computability, and one for primitive
recursive time (and/or space) computability
From truth to computability II
Computability logic is a formal theory of computational tasks and resources.
Formulas in it represent interactive computational problems, and "truth" is
understood as algorithmic solvability. Interactive computational problems, in
turn, are defined as a certain sort games between a machine and its
environment, with logical operators standing for operations on such games.
Within the ambitious program of finding axiomatizations for incrementally rich
fragments of this semantically introduced logic, the earlier article "From
truth to computability I" proved soundness and completeness for system CL3,
whose language has the so called parallel connectives (including negation),
choice connectives, choice quantifiers, and blind quantifiers. The present
paper extends that result to the significantly more expressive system CL4 with
the same collection of logical operators. What makes CL4 expressive is the
presence of two sorts of atoms in its language: elementary atoms, representing
elementary computational problems (i.e. predicates, i.e. problems of zero
degree of interactivity), and general atoms, representing arbitrary
computational problems. CL4 conservatively extends CL3, with the latter being
nothing but the general-atom-free fragment of the former. Removing the blind
(classical) group of quantifiers from the language of CL4 is shown to yield a
decidable logic despite the fact that the latter is still first-order. A
comprehensive online source on computability logic can be found at
http://www.cis.upenn.edu/~giorgi/cl.htm
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
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
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
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
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
Ptarithmetic
The present article introduces ptarithmetic (short for "polynomial time
arithmetic") -- a formal number theory similar to the well known Peano
arithmetic, but based on the recently born computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html) instead of classical logic. The
formulas of ptarithmetic represent interactive computational problems rather
than just true/false statements, and their "truth" is understood as existence
of a polynomial time solution. The system of ptarithmetic elaborated in this
article 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 effectively 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 self-contained, and can be read without any previous familiarity
with computability logic.Comment: Substantially better versions are on their way. Hence the present
article probably will not be publishe
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
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