Abstract
The ordering principle states that every finite linear order has a least element. We show that, in the relativized setting, the surjective weak pigeonhole principle for polynomial time functions does not prove a Herbrandized version of the ordering principle over T12. This answers an open question raised in Buss et al. [2012] and completes their program to compare the strength of Jeřábek's bounded arithmetic theory for approximate counting with weakened versions of it.
- S. Buss. 1986. Bounded Arithmetic. Bibliopolis, Naples.Google Scholar
- S. Buss, L. Kołodziejczyk, and N. Thapen. 2012. Fragments of approximate counting. Retrieved July 5, 2014, from www.math.cas.cz/~thapen/.Google Scholar
- S. Buss and J. Krajíček. 1994. An application of Boolean complexity to separation problems in bounded arithmetic. Proceedings of the London Mathematical Society 69, 1--21.Google ScholarCross Ref
- M. Chiari and J. Krajíček. 1998. Witnessing functions in bounded arithmetic and search problems. Journal of Symbolic Logic 63, 3, 1095--1115.Google ScholarCross Ref
- J. Hanika. 2004. Search Problems and Bounded Arithmetic. Ph.D. Dissertation. Charles University, Prague. Available at eccc.hpi-web.de/static/books/theses/.Google Scholar
- R. Impagliazzo and J. Krajíček. 2002. A note on conservativity relations among bounded arithmetic theories. Mathematical Logic Quarterly 48, 3, 375--377.Google ScholarCross Ref
- E. Jeřábek. 2006. The strength of sharply bounded induction. Mathematical Logic Quarterly 52, 6, 613--624.Google ScholarCross Ref
- E. Jeřábek. 2007. On independence of variants of the weak pigeonhole principle. Journal of Logic and Computation 17, 3, 587--604.Google ScholarCross Ref
- E. Jeřábek. 2009. Approximate counting by hashing in bounded arithmetic. Journal of Symbolic Logic 74, 3, 829--860.Google ScholarCross Ref
- J. Krajíček. 1995. Bounded Arithmetic, Propositional Logic and Computational Complexity. Cambridge University Press, Cambridge. Google ScholarDigital Library
- J. Krajíček. 2001. On the weak pigeonhole principle. Fundamenta Mathematicae, 123--140.Google Scholar
- M. Lauria. 2012. Short Res*(Polylog) Refutations If and Only If Narrow Res Refutations. Retrieved July 5, 2014, from arXiv:1310.5714Google Scholar
- N. Thapen. 2002. A model-theoretic characterization of the weak pigeonhole principle. Annals of Pure and Applied Logic, 175--195.Google ScholarCross Ref
Index Terms
- The Ordering Principle in a Fragment of Approximate Counting
Recommendations
Nisan-Wigderson generators in proof complexity: new lower bounds
CCC '22: Proceedings of the 37th Computational Complexity ConferenceA map g : {0, 1}n → {0, 1}m (m > n) is a hard proof complexity generator for a proof system P iff for every string b ∈ {0, 1}m \ Rng(g), formula τb(g) naturally expressing b ∉ Rng(g) requires superpolynomial size P-proofs. One of the well-studied maps ...
Some Subsystems of Constant-Depth Frege with Parity
We consider three relatively strong families of subsystems of AC0[2]-Frege proof systems, i.e., propositional proof systems using constant-depth formulas with an additional parity connective, for which exponential lower bounds on proof size are known. ...
On the complexity of parity games
VoCS'08: Proceedings of the 2008 international conference on Visions of Computer Science: BCS International Academic ConferenceParity games underlie the model checking problem for the modal µ-calculus, the complexity of which remains unresolved after more than two decades of intensive research. The community is split into those who believe this problem - which is known to be ...
Comments