Pigeonhole Principle Research Articles

Maximal increasing sequences in fillings of almost-moon polyominoes

It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino that do not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey's proof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces the result for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijective proof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removing one of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chain of the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sum of the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, the row sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijective proof of Rubey's result but also two refinements of it.

Goldbach's Pigeonhole

SummaryGoldbach's conjecture states that every even integer greater than two can be written as the sum of two primes. For some even integers, we can prove the existence of two such primes nonconstructively via the pigeonhole principle. Using a computer search and asymptotic bounds on classic number-theoretic functions, we determine the greatest value for which this approach works.

Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Jeřábek introduced fragments of bounded arithmetic which are axiomatized with weak surjective pigeonhole principles and support a robust notion of approximate counting. We extend these fragments to accommodate modular counting quantifiers. These theories can formalize and prove the relativized versions of Toda’s theorem on the collapse of the polynomial hierarchy with modular counting. We introduce a version of the Paris-Wilkie translation for converting formulas and proofs of bounded arithmetic with modular counting quantifiers into constant depth propositional logic with modular counting gates. We also define Paris-Wilkie translations to Nullstellensatz and polynomial calculus refutations. As an application, we prove that constant depth propositional proofs that use connectives AND, OR, and mod p gates, for p p a prime, can be translated, with quasipolynomial increase in size, into propositional proofs containing only propositional formulas of depth three in which the top level is Boolean, the middle level consists of mod p gates, and the bottom level subformulas are small conjunctions. These results are improved to depth two by using refutations from the weak surjective pigeonhole principles.

Quasipolynomial size proofs of the propositional pigeonhole principle

Cook and Reckhow proved in 1979 that the propositional pigeonhole principle has polynomial size extended Frege proofs. Buss proved in 1987 that it also has polynomial size Frege proofs; these Frege proofs used a completely different proof method based on counting. This paper shows that the original Cook and Reckhow extended Frege proofs can be formulated as quasipolynomial size Frege proofs. The key point is that st-connectivity can be used to define the Cook–Reckhow construction.

Space Complexity in Polynomial Calculus

During the last 10 to 15 years, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important concern in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably well-understood from this point of view. For other proof systems of interest, however, such as polynomial calculus or cutting planes, progress has been more limited. Essentially nothing has been known about space complexity in cutting planes, and for polynomial calculus the only lower bound has been for conjunctive normal form (CNF) formulas of unbounded width in [Alekhnovich et al., SIAM J. Comput., 31 (2002), pp. 1184--1211], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any $k$-CNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied by Alekhnovich et al.: (1) We prove an $\omega(n)$ space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas $PHP_{m}^{n}$ with $m$ pigeons and $n$ holes, and show that this is tight. (2) For PCR, we prove an $\omega(n)$ space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over Alekhnovich et al. measured in the width of the formulas. (3) We then present another encoding of the pigeonhole principle that has constant width, and prove an $\omega(n)$ space lower bound in PCR for these formulas as well. (4) Finally, we prove that any $k$-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity.

Maximal increasing sequences in fillings of almost-moon polyominoes

It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino thatdo not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey’sproof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces theresult for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijectiveproof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removingone of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chainof the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sumof the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, therow sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijectiveproof of Rubey’s result but also two refinements of it. Rubey a montré que le nombre de remplissages d’un polyomino lunaire donné par des zéros et des uns quine contiennent pas de chaîne nord-est d’une taille fixée ne dépend que de l’ensemble des longueurs des colonnesdu polyomino. La preuve de Rubey utilise une adaptation du jeu de taquin et de la promotion sur des remplissagesarbitraires de polyominos lunaires et déduit le résultat pour les remplissages 0/1 par inclusion-exclusion. Dans cetarticle, nous présentons la première preuve bijective de ce résultat en considérant des remplissages de polyominospresque lunaires, qui sont des polyominos lunaires dont on a supprimé une ligne. Plus précisément, nous construisonsune bijection simple qui préserve la taille de la plus longue chaîne nord-est des remplissages lorsque deux lignesadjacentes du polyomino sont échangées. Cette bijection préserve aussi la somme des colonnes des remplissages. Enoutre, nous présentons aussi une bijection simple qui préserve la taille de la plus longue chaîne nord-est, la sommedes lignes et la somme des colonnes si chaque ligne du remplissage contient au plus un 1. Nous fournissons donc nonseulement une preuve bijective du résultat de Rubey, mais aussi deux raffinements de celui-ci.

Application of the Aviation Black Box Principle in Pediatric Cardiac Surgery: Tracking All Failures in the Pediatric Cardiac Operating Room

Cardiac surgical procedures are complex and require the coordinated action of many. This creates the potential for small failures that could be the substrate for subsequent morbidity or mortality. High-reliability science suggests that preoccupation with small failures can lead to improved outcomes. Failures of all magnitudes (ie, events) were captured within the pediatric cardiac operating room starting with a single surgeon in April 2008. As the surgical team became more familiar with the process, failure recording was extended to all surgeons and all surgical procedures performed until the conclusion of the study in December 2010. New recording processes were developed and used on a rolling basis during this study. With systematic capture, event rates increased (from occurring within 20% to 50% of operative procedures). Although we identified 9 recurrent patterns, 2 categories (ie, Equipment and Patient Instability) accounted for almost half of the events (45%). The greatest number of events occurred during the prebypass period (40.2%), compared with bypass (20.1%) and postbypass (32.3%) periods. These events were mainly difficulties in access (31.8%), equipment (42.4%), and patient instability (33.3%) in each of the epochs, respectively. Of all events, 7.3% occurred during nonbypass cases, 30.6% of these were communication events. Implementation of this initiative led to recognition of major system-wide issues (eg, need for change in the blood-product acquisition process). Preoccupation with all failures in the operating room can reveal important information about the operating room and perioperative microenvironment that can prompt substantive process changes both locally and within the larger health system.

A counterexample to a geometric Hales-Jewett type conjecture

Por and Wood conjectured that for all \(k,l \ge 2\) there exists \(n \ge 2\) with the following property: whenever \(n\) points, no \(l + 1\) of which are collinear, are chosen in the plane and each of them is assigned one of \(k\) colours, then there must be a line (that is, a maximal set of collinear points) all of whose points have the same colour. The conjecture is easily seen to be true for \(l = 2\) (by the pigeonhole principle) and in the case \(k = 2\) it is an immediate corollary of the Motzkin-Rabin theorem. In this note we show that the conjecture is false for \(k, l \ge 3\).

The Ordering Principle in a Fragment of Approximate Counting

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 T 1 2 . 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.

O-MINIMALISM

AbstractThis paper is devoted to o-minimalism, the study of the first-order properties of o-minimal structures. The main protagonists are the pseudo-o-minimal structures, that is to say, the models of the theory of all o-minimal L-structures, but we start with a more in-depth analysis of the well-known fragment DCTC (Definable Completeness/Type Completeness), and show how it already admits many of the properties of o-minimal structures: dimension theory, monotonicity, Hardy structures, and quasi-cell decomposition, provided one replaces finiteness by discreteness in all of these. Failure of cell decomposition leads to the related notion of a eukaryote structure, and we give a criterium for a pseudo-o-minimal structure to be eukaryote.To any pseudo-o-minimal structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a nontrivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics. As an application, we study certain analytic subsets, called Taylor sets.

FRAGMENTS OF APPROXIMATE COUNTING

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 FPNPfunctions. 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.

An Energy-Efficient Outlier Detection Based on Data Clustering in WSNs

Sensor nodes in wireless sensor networks are prone to malfunction because they are exposed to the nearby environment directly. Consequently, wrong sensor readings occurred from sensor nodes and these readings are called an outlier. Commonly, since an outlier deviates from normal sensor readings and it can bring about some problems, various techniques to detect the outliers have been proposed. In this paper, we propose an efficient outlier detection technique based on data clustering. In order to decide the width of the cluster that consists of the sensor readings, we applied the Pigeonhole Principle and then detected the outliers based on clusters. In experiments, we demonstrate the efficiency of our proposed technique compared to other outlier detection techniques.

Twins in graphs

A basic pigeonhole principle insures an existence of two objects of the same type if the number of objects is larger than the number of types. Can such a principle be extended to a more complex combinatorial structure? Here, we address such a question for graphs. We call two disjoint subsets A,B of vertices twins if they have the same cardinality and induce subgraphs of the same size. Let t(G) be the largest k such that G has twins on k vertices each. We provide the bounds on t(G) in terms of the number of edges and vertices using discrepancy results for induced subgraphs. In addition, we give conditions under which t(G)=|V(G)|/2 and show that if G is a forest then t(G)≥|V(G)|/2−1.

Approximate Search in Very Large Files Using the Pigeonhole Principle

This paper presents a new technique for efficient searching with fuzzy criteria in very large information systems. The suggested technique uses the Pigeonhole Principle approach. This approach can be utilized with different embodiments, but the most effective realization would be to amplify some already given intrinsic approximate matching capabilities, like those in the FuzzyFind method [1][2]. Considering the following problem, a data to be searched is presented as a bit-attribute vector. The searching operation consists of finding a subset of this bit-attribute vector that is within particular Hamming distance. Normally, this search with approximate matching criteria requires sequential lookup for the whole collection of the attribute vector. This process can be easily parallelized, but in very large information systems this still would be slow and energy consuming. The suggested method, in this paper, of approximate search in very large files using the Pigeonhole Principle, circumvents the sequential search operations and reduces the calculations tremendously

The Pigeonhole Principle, Two Centuries Before Dirichlet

The name of Dirichlet is commonly associated with the pigeonhole principle, because it is widely believed that he was the first to state it. In the writings of Dirichlet, the application of the principle is to be found in 1842 in [7] and [5] (both reproduced in [8], see pp. 579–580 and 633–638) and, later, in [6]. It seems that no one knows any precise previous reference, even if 1834 is frequently mentioned as the year of the discovery (for some details on later references, see [15]). In [7] and [5], the pigeonhole principle is used to prove complex and multidimensional versions of Equation 1. In the second reference, of 1863 (hence published 4 years after Dirichlet’s death), it is used to provide a proof of the existence of infinitely many integers x and y such that x yD\1 þ 2 ffiffiffiffi D p (for D integer and not a perfect square) which does not rely on continued fractions. In these publications Dirichlet does not assign any name to the principle, nor does he pretend that this principle is new. In a later work he calls it the ‘‘Schubfachprinzip.’’ Anyway, it appears that Dirichlet was not the first to make use of the principle that bears his name. The pigeonhole principle appeared no less than two centuries before him.

Parameterized Complexity of DPLL Search Procedures

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game that models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k -clique requires n Ω(k) steps for a nontrivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked by Beyersdorff et al. [2012] of understanding the Resolution complexity of this family of formulas.

Large restricted sumsets in general Abelian groups

Let A, B and S be subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A∧SB={a+b:a∈A,b∈Banda−b∉S}. Let LS=maxz∈G|{(x,y):x,y∈G,x+y=zandx−y∈S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+LS implies A∧SB=G. We then prove that if |A|+|B|=|G|+LS then |A∧SB|≥|G|−2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+LS and |A∧SB|=|G|−2|S|. Moreover, in this case, we also provide the structure of the set G∖(A∧SB).

A characterization of tree-like Resolution size

We explain an asymmetric Prover–Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas (Beyersdorff et al. (2013) [2]) and for the classical pigeonhole principle (Beyersdorff et al. (2010) [1]). The main point of this note is to show that the asymmetric game in fact characterizes tree-like Resolution proof size, i.e. in principle our proof method allows to always achieve the optimal lower bounds. This is in contrast with previous techniques described in the literature. We also provide a very intuitive information-theoretic interpretation of the game.

To Lie or to Comply: Defending against Flood Attacks in Disruption Tolerant Networks

Disruption Tolerant Networks (DTNs) utilize the mobility of nodes and the opportunistic contacts among nodes for data communications. Due to the limitation in network resources such as contact opportunity and buffer space, DTNs are vulnerable to flood attacks in which attackers send as many packets or packet replicas as possible to the network, in order to deplete or overuse the limited network resources. In this paper, we employ rate limiting to defend against flood attacks in DTNs, such that each node has a limit over the number of packets that it can generate in each time interval and a limit over the number of replicas that it can generate for each packet. We propose a distributed scheme to detect if a node has violated its rate limits. To address the challenge that it is difficult to count all the packets or replicas sent by a node due to lack of communication infrastructure, our detection adopts claim-carry-and-check: each node itself counts the number of packets or replicas that it has sent and claims the count to other nodes; the receiving nodes carry the claims when they move, and cross-check if their carried claims are inconsistent when they contact. The claim structure uses the pigeonhole principle to guarantee that an attacker will make inconsistent claims which may lead to detection. We provide rigorous analysis on the probability of detection, and evaluate the effectiveness and efficiency of our scheme with extensive trace-driven simulations.

A SMALL VALUE ESTIMATE FOR

A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product Ga x Gm whose underlying group of complex points is C x C*. It shows that if a certain sequence of non-zero polynomials in Z[X,Y] take small values at a point (xi,eta) together with their first derivatives with respect to the invariant derivation of the group, then both xi and eta are algebraic over Q. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon's criterion for algebraic independence and compares favorably with constructions coming from Dirichlet's box principle.