episciences.org_2028_1628056773
1628056773
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
Logical Methods in Computer Science
1860-5974
04
27
2017
Volume 12, Issue 3
Build your own clarithmetic II: Soundness
Giorgi
Japaridze
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 previous Part I has introduced the system and
proved its completeness, while the present Part II is devoted to proving
soundness.
04
27
2017
2028
arXiv:1510.08566
10.2168/LMCS-12(3:12)2016
https://lmcs.episciences.org/2028