scholarly journals QUASIPOLYNOMIAL SIZE FREGE PROOFS OF FRANKL’S THEOREM ON THE TRACE OF SETS

2016 ◽  
Vol 81 (2) ◽  
pp. 687-710 ◽  
Author(s):  
JAMES AISENBERG ◽  
MARIA LUISA BONET ◽  
SAM BUSS
Keyword(s):  
Pigeonhole Principle ◽  
Polynomial Size

AbstractWe extend results of Bonet, Buss and Pitassi on Bondy’s Theorem and of Nozaki, Arai and Arai on Bollobás’ Theorem by proving that Frankl’s Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parametert, we prove that Frankl’s Theorem has polynomial size AC0-Frege proofs from instances of the pigeonhole principle.

scholarly journals Polynomial size proofs of the propositional pigeonhole principle

1987 ◽  
Vol 52 (4) ◽  
pp. 916-927 ◽  
Author(s):  
Samuel R. Buss
Keyword(s):  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Polynomial Size ◽  
Frege Systems

AbstractCook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

scholarly journals Small Circuits and Dual Weak PHP in the Universal Theory of p-time Algorithms

2021 ◽  
Vol 22 (2) ◽  
pp. 1-4
Author(s):  
Jan Krajíček
Keyword(s):  
Computational Complexity ◽  
Time Function ◽  
Truth Table ◽  
Universal Theory ◽  
Pigeonhole Principle ◽  
Polynomial Size ◽  
Complexity Hypothesis

We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending bits to bits violates the dual weak pigeonhole principle: Every string equals the value of the function for some . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size .

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
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