10.2168/LMCS-12(3:8)2016
Japaridze, Giorgi
Giorgi
Japaridze
Build your own clarithmetic I: Setup and completeness
episciences.org
2017
Computer Science - Logic in Computer Science
03F50, 03D75, 03D15, 68Q10, 68T27, 68T30
F.1.1
F.1.2
F.1.3
contact@episciences.org
episciences.org
2015-10-30T00:00:00+01:00
2016-09-27T12:30:35+02:00
2017-04-27
eng
Journal article
https://lmcs.episciences.org/2020
arXiv:1510.08564
1860-5974
PDF
1
Logical Methods in Computer Science ; Volume 12, Issue 3 ; 1860-5974
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.