Modular Counting Research Articles

On some $$\Sigma ^{B}_{0}$$-formulae generalizing counting principles over $$V^{0}$$

AbstractWe formalize various counting principles and compare their strengths over $$V^{0}$$ V 0 . In particular, we conjecture the following mutual independence between: a uniform version of modular counting principles and the pigeonhole principle for injections, a version of the oddtown theorem and modular counting principles of modulus p, where p is any natural number which is not a power of 2, and a version of Fisher’s inequality and modular counting principles. Then, we give sufficient conditions to prove them. We give a variation of the notion of PHP-tree and k-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have o(n)-evaluations. As for the remaining two, we utilize well-known notions of p-tree and k-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.

Modular counting management system for mall parking services

Motorcycle customers are a dominant consumer group in developing countries or regions around the world. Although motorcyclist parking is very important to the management and operation of the mall, the issue is rarely concerned. The existing parking management systems are mostly used for vehicle delivery management. At the same time, if the traditional parking management system needs to be updated, the large-scale construction of the entrance and exit is usually required, which will increase the construction cost and affect customer satisfaction, leading to the overall profit loss of the mall. Therefore, this paper develops a new system, named Modular Counting Management System (MCMS), which utilizes CCD video and digital image processing (DIP) and counting algorithms for real-time monitoring without resetting the entrance device. The system can be implemented easily, provide sufficient monitoring management information to effectively improve the utilization rate of motorcycle parking areas and combine with other identification systems for the mall. This study has three main contributions: (1) A MCMS has been developed and deployed for a large shopping mall in Taiwan to improve the satisfaction of motorcyclists and increase profits; (2) The proposed MCMS provides adequate monitoring management information to effectively improve the usage rate of motorcycle parking areas; and (3) A survey of motorcyclist opinion showed an 82% satisfaction rate from a total of 479 effective samples and revenues increased 13.8% in the year following the implementation of this system. These have proven the system's contribution to practice and theory.

Counting Homomorphisms to Trees Modulo a Prime

Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of # p H OMS T O H , the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p . Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree H and every prime p the problem # p H OMS T O H is either polynomial time computable or # p P-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of # p H OMS T O H are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p . These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p .

On the triple correlations of fractional parts of

AbstractFor fixed$\alpha \in [0,1]$, consider the set$S_{\alpha ,N}$of dilated squares$\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $modulo$1$. Rudnick and Sarnak conjectured that, for Lebesgue, almost all such$\alpha $the gap-distribution of$S_{\alpha ,N}$is consistent with the Poisson model (in the limit asNtends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of$S_{\alpha ,N}$in a random interval of length$L/N$, provided that$L> N^{1/4+\varepsilon }$. The threshold of$\tfrac {1}{4}$is substantially smaller than the threshold of$\tfrac {1}{2}$(which is the threshold that would be given by a naïve discrepancy estimate).Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations$(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $for a nonlacunary sequence$(a_n)_{n=1}^{\infty } $of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

Uniform proofs of ACC representations

We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with $$\hbox {MOD}_{m}$$ gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular counting quantifiers. This allows the main proofs to be expressed in terms of formula classes instead of Boolean circuits. The uniform version of the Beigel–Tarui theorem is then obtained automatically via the Furst–Saxe–Sipser and Paris–Wilkie translations. As a special case, we obtain a uniform version of Razborov and Smolensky’s representation of $$\hbox {AC}^{0}[p]$$ circuits. The paper is partly expository, but is also motivated by the desire to recast Toda’s theorem, the Beigel–Tarui theorem, and their proofs into the language of bounded arithmetic. However, no knowledge of bounded arithmetic is needed.

Counting Homomorphisms to Square-Free Graphs, Modulo 2

We study the problem ⊕H oms T o H of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H . A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, ⊕H oms T o H is either in polynomial time or is ⊕P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

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.

The complexity of counting homomorphisms to cactus graphs modulo 2

A homomorphism from a graph G to a graph H is a function from V ( G ) to V ( H ) that preserves edges. Many combinatorial structures that arise in mathematics and in computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this article, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H , counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H , the problem is complete in the complexity class ⊕ P , which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.

On the complexity of solving linear congruences and computing nullspaces modulo a constant

We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k>1. These problems are known to be complete for the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case that k is prime (Buntrock et al, 1992). By considering variants of standard logspace function classes --- related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k --- we show that these problems of linear algebra are also complete for {coMod_k L} for any constant k>1. Our results are obtained by defining a class of functions FUL_k which are low for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1 reductions (including {Mod_k L} oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL_k for arbitrary moduli k to the class {F.coMod_k L} of functions whose output symbols are verifiable by {coMod_k L} algorithms; and consider what consequences such a comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L} = {Mod_k L} for composite k.

THE THOMPSON–HIGMAN MONOIDS Mk,i: THE ${\mathcal J}$-ORDER, THE ${\mathcal D}$-RELATION, AND THEIR COMPLEXITY

The Thompson–Higman groups Gk,ihave a natural generalization to monoids, called Mk,i, and inverse monoids, called Invk,i. We study some structural features of Mk,iand Invk,iand investigate the computational complexity of related decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity.The maximal subgroups of Mk,1and Invk,1are isomorphic to the groups Gk,j(1 ≤ j ≤ k - 1); so we rediscover all the Thompson–Higman groups within Mk,1.Deciding the Green relations [Formula: see text] and [Formula: see text] of Mk,1, when the inputs are words over a finite generating set of Mk,1, is in P.When a circuit-like generating set is used for Mk,1then deciding [Formula: see text] is coDP-complete (where DP is the complexity class consisting of differences of sets in NP). The multiplier search problem for [Formula: see text] is xNPsearch-complete, whereas the multiplier search problems of [Formula: see text] and [Formula: see text] are not in xNPsearch unless NP = coNP. The class of search problems xNPsearch is introduced as a slight generalization of NPsearch.Deciding [Formula: see text] for Mk,1when the inputs are words over a circuit-like generating set, is ⊕k-1• NP -complete; for any h ≥ 2, ⊕h•NP is a modular counting complexity class, whose verification problems are in NP. Related problems for partial circuits are the image size problem (which is # • NP-complete), and the image size modulo h problem (which is ⊕h•NP-complete). For Invk,1over a circuit-like generating set, deciding [Formula: see text] is ⊕k-1P-complete. It is interesting that the little known complexity classes coDP and ⊕k-1•NP play a central role in Mk,1.

A New Characterization of ACC0 and Probabilistic CC0

Barrington, Straubing and Therien (1990) conjectured that the Boolean AND function can not be computed by polynomial size constant depth circuits built from modular counting gates, i.e., by CC^0 circuits. In this work we show that the AND function can be computed by uniform probabilistic CC^0 circuits that use only O(log n) random bits. This may be viewed as evidence contrary to the conjecture. As a consequence of our construction we get that all of ACC^0 can be computed by probabilistic CC^0 circuits that use only O(log n) random bits. Thus, if one were able to derandomize such circuits, we would obtain a collapse of circuit classes giving ACC^0=CC^0. We present a derandomization of probabilistic CC^0 circuits using AND and OR gates to obtain ACC^0 = AND o OR o CC^0 = OR o AND o CC^0. AND and OR gates of sublinear fan-in suffice. Both these results hold for uniform as well as non-uniform circuit classes. For non-uniform circuits we obtain the stronger conclusion that ACC^0 = rand-ACC^0 = rand-CC^0 = rand(log n)-CC^0, i.e., probabilistic ACC^0 circuits can be simulated by probabilistic CC^0 circuits using only O(log n) random bits. As an application of our results we obtain a characterization of ACC^0 by constant width planar nondeterministic branching programs, improving a previous characterization for the quasipolynomial size setting.

Modular Counting and Substitution of Structures

The aim of this paper is to point to a difference between binary and hyperary structures. The modular counting functions of a class of structures defined by a sentence of second-order monadic logic with equality, based on binary relations, are ultimately periodic. However, this is not the case for sentences based on quaternary relations.

Algebraic proof systems over formulas

We introduce two algebraic propositional proof systems F - NS and F - PC . The main difference of our systems from (customary) Nullstellensatz and polynomial calculus is that the polynomials are represented as arbitrary formulas (rather than sums of monomials). Short proofs of Tseitin's tautologies in the constant-depth version of F - NS provide an exponential separation between this system and Polynomial Calculus. We prove that F - NS (and hence F - PC ) polynomially simulates Frege systems, and that the constant-depth version of F - PC over finite field polynomially simulates constant-depth Frege systems with modular counting. We also present a short constant-depth F - PC (in fact, F - NS ) proof of the propositional pigeon-hole principle. Finally, we introduce several extensions of our systems and pose numerous open questions.

Languages Defined with Modular Counting Quantifiers

We prove that a regular language defined by a boolean combination of generalized Σ 1 -sentences built using modular counting quantifiers can be defined by a boolean combination of Σ 1 -sentences in which only regular numerical predicates appear. The same statement, with “Σ 1 ” replaced by “first-order,” is equivalent to the conjecture that the nonuniform circuit complexity class ACC is strictly contained in NC 1 . The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case.

Counting modulo finite semigroups

Rudimentary relations are those relations over natural integers that are defined by a first-order arithmetical formula, in which all quantifications are bounded by some variable. Paris and Wilkie conjectured that the class R of rudimentary relations is not closed under modular counting or constant-bounded recursion. A consequence would be the proper containment of R in LINSPACE. It is now known that the closure of R under counting modulo an unsolvable non-abelian finite group, or modulo the semigroup of all binary relations on a finite set, contains ALINTIME. We characterize here the semigroups for which the counting is reducible to modular counting, or contains counting modulo an unsolvable non-abelian finite group.

Lower bounds for modular counting by circuits with modular gates

We prove that constant depth circuits, with one layer of MOD m gates at the inputs, followed by a fixed number of layers of MOD p gates, where p is prime, require exponential size to compute the MOD q function, if q is a prime that divides neither p nor q.

A lower bound on the mod 6 degree of the OR function

We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in Boolean variables over Z m . In particular, we say that a polynomial P weakly represents a Boolean function f (both have n variables) if for any inputs x and y in {0,1}n, we have $P(x)\neq P(y)$ whenever $f(x)\neq f(y)$ . Barrington et al. (1994) investigated the minimal degree of a polynomial representing the OR function in this way, proving an upper bound of O(n 1/ r ) (where r is the number of distinct primes dividing m) and a lower bound of $\omega (1)$ . Here, we show a lower bound of $\Omega (\log n)$ when m is a product of two primes and $\Omega ((\log n)^{1/(r-1)})$ in general. While many lower bounds are known for a much stronger form of representation of a function by a polynomial (Barrington et al. 1994, Tsai 1996), very little is known using this liberal (and, we argue, more natural) definition. While the degree is known to be $\Omega (\log n)$ for the generalized inner product because of its high communication complexity (Grolmusz 1995), our bound is the best known for any function of low communication complexity and any modulus not a prime power.

Lower bounds for the majority communication complexity of various graph accessibility problems

We investigate the modular communication complexity of the graph accessibility problem GAP and its modular counting versions MODk-GAP, k≥2. Due to arguments concerning variation ranks and certain projection reductions, we prove that, for any partition of the input variables and for any moduls k and m, GAP and MODk-GAP have MOD m -communication complexity Ω(n), where n denotes the number of nodes of the graphs under consideration.

Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

We prove a lower bound of the formNΩ(1) on the degree of polynomials in a Nullstellensatz refutation of theCount q polynomials over ℤ m , whereq is a prime not dividingm. In addition, we give an explicit construction of a degreeN Ω(1) design for theCount q principle over ℤ m . As a corollary, using Beameet al. (1994) we obtain a lower bound of the form 2NΩ(1) for the number of formulas in a constant-depth Frege proof of the modular counting principleCount q N from instances of the counting principleCount m M .

Lower Bounds on Hilbert's Nullstellensatz and Propositional Proofs

The so-called weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Q i ( x ¯ ) = 0 , does not have a solution in the algebraic closure if and only if 1 is in theideal generated by the polynomials Q i ( x ¯ ) . We shall prove a lower bound on the degrees of polynomials P i ( x ¯ ) such that ∑ i P i ( x ¯ ) Q i ( x ¯ ) = 1 . This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into q-element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count Count q N ( x e , … ) with underlying variables xe, where e ranges over q-element subsets of N. Ajtai [4] proved recently that, whenever p,q are two different primes, the propositional formulas Count Count q q n + 1 do not have polynomial size, constant-depth Frege proofs from substitution instances of Count Count p m , where m ≢ 0 (mod p). We give a new proof of this theorem based on the lower bound for Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This improved lower bound together with new upper bounds yield an exact characterization of when Countq can be proved efficiently from Countp, for all values of p and q