11 research outputs found
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
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
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
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 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
From formulas to cirquents in computability logic
Computability logic (CoL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a
recently introduced semantical platform and ambitious program for redeveloping
logic as a formal theory of computability, as opposed to the formal theory of
truth that logic has more traditionally been. Its expressions represent
interactive computational tasks seen as games played by a machine against the
environment, and "truth" is understood as existence of an algorithmic winning
strategy. With logical operators standing for operations on games, the
formalism of CoL is open-ended, and has already undergone series of extensions.
This article extends the expressive power of CoL in a qualitatively new way,
generalizing formulas (to which the earlier languages of CoL were limited) to
circuit-style structures termed cirquents. The latter, unlike formulas, are
able to account for subgame/subtask sharing between different parts of the
overall game/task. Among the many advantages offered by this ability is that it
allows us to capture, refine and generalize the well known
independence-friendly logic which, after the present leap forward, naturally
becomes a conservative fragment of CoL, just as classical logic had been known
to be a conservative fragment of the formula-based version of CoL. Technically,
this paper is self-contained, and can be read without any prior familiarity
with CoL.Comment: LMCS 7 (2:1) 201