The Pigeonhole Principle

2001 ◽  
pp. 49-54
Keyword(s):  
Pigeonhole Principle

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals FRAGMENTS OF APPROXIMATE COUNTING

2014 ◽  
Vol 79 (2) ◽  
pp. 496-525 ◽  
Author(s):  
SAMUEL R. BUSS ◽  
LESZEK ALEKSANDER KOŁODZIEJCZYK ◽  
NEIL THAPEN
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Approximate Counting ◽  
Image Position

AbstractWe study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the $\forall {\rm{\Sigma }}_1^b$ Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FPNP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of $T_2^1$ augmented with the surjective weak pigeonhole principle for polynomial time functions.

The Pigeonhole Principle

2015 ◽  
pp. 19-23
Author(s):  
Mark Joshi
Keyword(s):  
Pigeonhole Principle

The Pigeonhole Principle

Easy as π? ◽  
1999 ◽  
pp. 103-117
Author(s):  
O. A. Ivanov
Keyword(s):  
Pigeonhole Principle

The Pigeonhole Principle

2020 ◽  
pp. 151-162
Author(s):  
Peter Winkler
Keyword(s):  
Pigeonhole Principle

scholarly journals Proof Complexity of Modal Resolution

2021 ◽  
Author(s):  
Sarah Sigley ◽  
Olaf Beyersdorff
Keyword(s):  
Artificial Intelligence ◽  
Automated Reasoning ◽  
Proof Complexity ◽  
Pigeonhole Principle ◽  
Analytic Tableaux ◽  
Frege Systems ◽  
International Joint ◽  
Resolution Proofs ◽  
Exponential Size ◽  
Sixth International

AbstractWe investigate the proof complexity of modal resolution systems developed by Nalon and Dixon (J Algorithms 62(3–4):117–134, 2007) and Nalon et al. (in: Automated reasoning with analytic Tableaux and related methods—24th international conference, (TABLEAUX’15), pp 185–200, 2015), which form the basis of modal theorem proving (Nalon et al., in: Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI’17), pp 4919–4923, 2017). We complement these calculi by a new tighter variant and show that proofs can be efficiently translated between all these variants, meaning that the calculi are equivalent from a proof complexity perspective. We then develop the first lower bound technique for modal resolution using Prover–Delayer games, which can be used to establish “genuine” modal lower bounds for size of dag-like modal resolution proofs. We illustrate the technique by devising a new modal pigeonhole principle, which we demonstrate to require exponential-size proofs in modal resolution. Finally, we compare modal resolution to the modal Frege systems of Hrubeš (Ann Pure Appl Log 157(2–3):194–205, 2009) and obtain a “genuinely” modal separation.

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