2010/03/31 by Giorgi Japaridze · 2 citations
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #Logic, Reasoning, and Knowledge #Logic, programming, and type systems #acm:03D15 #acm:03D75 #acm:03F50 #acm:68Q10 #acm:68T27 #acm:68T30 #cs.CC #cs.LO #math.LO #math.NT #msc:03D15 #msc:03D75 #msc:03F50 #msc:68Q10 #msc:68T27 #msc:68T30
paper · pdf · doi:10.1016/j.ic.2011.07.002
published as Information and Computation 209 (2011), pp. 1312-1354
openalex publication_date 2011/08/03 · crossref created 2011/08/03 · arxiv created 2011/08/23 · arxiv updated 2011/08/24 · crossref issued 2011/10/01 · crossref published 2011/10/01 · crossref published-print 2011/10/01 · crossref deposited 2019/06/13 · crossref indexed 2022/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
"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.