82 research outputs found
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
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
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
Separating the basic logics of the basic recurrences
This paper shows that, even at the most basic level, the parallel, countable
branching and uncountable branching recurrences of Computability Logic (see
http://www.cis.upenn.edu/~giorgi/cl.html) validate different principles
Introduction to Cirquent Calculus and Abstract Resource Semantics
This paper introduces a refinement of the sequent calculus approach called
cirquent calculus. While in Gentzen-style proof trees sibling (or cousin, etc.)
sequents are disjoint sequences of formulas, in cirquent calculus they are
permitted to share elements. Explicitly allowing or disallowing shared
resources and thus taking to a more subtle level the resource-awareness
intuitions underlying substructural logics, cirquent calculus offers much
greater flexibility and power than sequent calculus does. A need for
substantially new deductive tools came with the birth of computability logic
(see http://www.cis.upenn.edu/~giorgi/cl.html) - the semantically constructed
formal theory of computational resources, which has stubbornly resisted any
axiomatization attempts within the framework of traditional syntactic
approaches. Cirquent calculus breaks the ice. Removing contraction from the
full collection of its rules yields a sound and complete system for the basic
fragment CL5 of computability logic. Doing the same in sequent calculus, on the
other hand, throws out the baby with the bath water, resulting in the strictly
weaker affine logic. An implied claim of computability logic is that it is CL5
rather than affine logic that adequately materializes the resource philosophy
traditionally associated with the latter. To strengthen this claim, the paper
further introduces an abstract resource semantics and shows the soundness and
completeness of CL5 with respect to it.Comment: To appear in Journal of Logic and Computatio
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