We study the public-coin randomized communication complexity of the equality function. The communication complexity of this function is known to be low when the error probability is constant and the players have access to many random bits. The complexity grows, however, if the allowed error probability and the amount of randomness are restricted. We show that public-coin randomized protocols for equality and error-correcting codes are essentially the same object. That is, given a protocol for equality, we can construct a code, and vice versa. We substantially extend the protocol-implies-code direction: any protocol computing a function with a large fooling set can be converted into an error-correcting code. As a corollary, we show that among functions with a fooling set of size s, equality on log s bits has the least randomized communication complexity, regardless of the restrictions on the error probability and the amount of randomness. Finally, we use the connection to error-correcting codes to analyze the randomized communication complexity of equality for varying restrictions on the error probability and the amount of randomness. In most cases, we provide tight bounds. We pinpoint the setting in which tight bounds are still unknown.

A nearest neighbor representation of a Boolean function f is a set of vectors (anchors) labeled by 0 or 1 such that f(x) = 1 if and only if the closest anchor to x is labeled by 1. This model was introduced by Hajnal, Liu and Turan [2022], who studied bounds on the minimum number of anchors required to represent Boolean functions under different choices of anchors (real vs. Boolean vectors) as well as the analogous model of kappa-nearest neighbors representations. We initiate a systematic study of the representational power of nearest and kappa-nearest neighbors through Boolean circuit complexity. To this end, we establish a close connection between Boolean functions with polynomial nearest neighbor complexity and those that can be efficiently represented by classes based on linear inequalities - min-plus polynomial threshold functions - previously studied in relation to threshold circuits. This extends an observation of Hajnal et al. [2022]. Next, we further extend the connection between nearest neighbor representations and circuits to the k-nearest neighbors case. As an outcome of these connections we obtain exponential lower bounds on the kappa-nearest neighbors complexity of explicit n-variate functions, assuming k <= n(1-epsilon). Previously, no superlinear lower bound was known for any kappa > 1. At the same time, we show that proving superpolynomial lower bounds for the kappa-nearest neighbors complexity of an explicit function for arbitrary kappa would require a breakthrough in circuit complexity. In addition, we prove an exponential separation between the nearest neighbor and k-nearest neighbors complexity (for unrestricted kappa) of an explicit function. These results address questions raised by [17] of proving strong lower bounds for kappa-nearest neighbors and understanding the role of the parameter k. Finally, we devise new bounds on the nearest neighbor complexity for several families of Boolean functions.

Let $A \in \{0,1\}^{n \times n}$ be a matrix with $z$ zeroes and $u$ ones and $x$ be an $n$-dimensional vector of formal variables over a semigroup $(S, \circ)$. How many semigroup operations are required to compute the linear operator $Ax$? As we observe in this paper, this problem contains as a special case the well-known range queries problem and has a rich variety of applications in such areas as graph algorithms, functional programming, circuit complexity, and others. It is easy to compute $Ax$ using $O(u)$ semigroup operations. The main question studied in this paper is: can $Ax$ be computed using $O(z)$ semigroup operations? We prove that in general this is not possible: there exists a matrix $A \in \{0,1\}^{n \times n}$ with exactly two zeroes in every row (hence $z=2n$) whose complexity is $\Theta(n\alpha(n))$ where $\alpha(n)$ is the inverse Ackermann function. However, for the case when the semigroup is commutative, we give a constructive proof of an $O(z)$ upper bound. This implies that in commutative settings, complements of sparse matrices can be processed as efficiently as sparse matrices (though the corresponding algorithms are more involved). Note that this covers the cases of Boolean and tropical semirings that have numerous applications, e.g., in graph theory. As a simple application of the presented linear-size construction, we show how to multiply two $n\times n$ matrices over an arbitrary semiring in $O(n^2)$ time if one of these matrices is a 0/1-matrix with $O(n)$ zeroes (i.e., a complement of a sparse matrix).

It is known that the Wadge reducibility of regular co-languages is efficiently decidable (Krishnan et al., 1995), (Wilke, Yoo, 1995). In this paper we study analogous problem for regular k-partitions of co-languages. In the series of previous papers (Selivanov, 2011), (Alaev, Selivanov, 2021), (Selivanov, 2012) there was a partial progress towards obtaining an efficient algorithm for deciding the Wadge reducibility in this setting as well. In this paper we finalize this line of research providing a quadratic algorithm (in RAM model). For this we construct a quadratic algorithm to decide a preorder relation on iterated posets. Additionally, we discuss the size of the representation of regular co-languages and suggest a more compact way to represent them. The algorithm we provide is efficient for the more compact representation as well.

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of $F$ is closely related to parity decision tree complexity of $f$. Montanaro and Osbourne (2009) observed that one-sided communication complexity $D_{cc}^{\rightarrow}(F)$ of $F$ is exactly equal to nonadaptive parity decision tree complexity $NADT^{\oplus}(f)$ of $f$. Hatami et al. (2018) showed that unrestricted communication complexity of $F$ is polynomially related to parity decision tree complexity of $f$. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of $f$. On the one hand, if $D_{cc}^{\rightarrow}(F)=t$ and $f$ is undefined on at most $O(\frac{2^{n-t}}{\sqrt{n-t}})$, then $NADT^{\oplus}(f)=t$. On the other hand, for a wide range of values of $D_{cc}^{\rightarrow}(F)$ and $NADT^{\oplus}(f)$ (from constant to $n-2$) we provide partial functions for which $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$. In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

In this paper, we study the problem of computing the majority function by low-depth monotone circuits and a related problem of constructing low-depth sorting networks. We consider both the classical setting with elementary operations of arity $2$ and the generalized setting with operations of arity $k$, where $k$ is a parameter. For both problems and both settings, there are various constructions known, the minimal known depth being logarithmic. However, there is currently no known construction that simultaneously achieves sub-log-squared depth, effective constructability, simplicity, and has a potential to be used in practice. In this paper we make progress towards resolution of this problem. For computing majority by standard monotone circuits (gates of arity 2) we provide an explicit monotone circuit of depth $O(\log_2^{5/3} n)$. The construction is a combination of several known and not too complicated ideas. For arbitrary arity of gates $k$ we provide a new sorting network architecture inspired by representation of inputs as a high-dimensional cube. As a result we provide a simple construction that improves previous upper bound of $4 \log_k^2 n$ to $2 \log_k^2 n$. We prove the similar bound for the depth of the circuit computing majority of $n$ bits consisting of gates computing majority of $k$ bits. Note, that for both problems there is an explicit construction of depth $O(\log_k n)$ known, but the construction is complicated and the constant hidden in $O$-notation is huge.

We contribute to the study of formal languages that can be recognized by transformer encoders. We focus on two self-attention mechanisms: (1) UHAT (Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention Transformers). UHAT encoders are known to recognize only languages inside the circuit complexity class ${\sf AC}^0$, i.e., accepted by a family of poly-sized and depth-bounded boolean circuits with unbounded fan-ins. On the other hand, AHAT encoders can recognize languages outside ${\sf AC}^0$), but their expressive power still lies within the bigger circuit complexity class ${\sf TC}^0$, i.e., ${\sf AC}^0$-circuits extended by majority gates. We first show a negative result that there is an ${\sf AC}^0$-language that cannot be recognized by an UHAT encoder. On the positive side, we show that UHAT encoders can recognize a rich fragment of ${\sf AC}^0$-languages, namely, all languages definable in first-order logic with arbitrary unary numerical predicates. This logic, includes, for example, all regular languages from ${\sf AC}^0$. We then show that AHAT encoders can recognize all languages of our logic even when we enrich it with counting terms. We apply these results to derive new results on the expressive power of UHAT and AHAT up to permutation of letters (a.k.a. Parikh images).

In this paper, we address sorting networks that are constructed from comparators of arity $k > 2$. That is, in our setting the arity of the comparators -- or, in other words, the number of inputs that can be sorted at the unit cost -- is a parameter. We study its relationship with two other parameters -- $n$, the number of inputs, and $d$, the depth. This model received considerable attention. Partly, its motivation is to better understand the structure of sorting networks. In particular, sorting networks with large arity are related to recursive constructions of ordinary sorting networks. Additionally, studies of this model have natural correspondence with a recent line of work on constructing circuits for majority functions from majority gates of lower fan-in. Motivated by these questions, we obtain the first lower bounds on the arity of constant-depth sorting networks. More precisely, we consider sorting networks of depth $d$ up to 4, and determine the minimal $k$ for which there is such a network with comparators of arity $k$. For depths $d=1,2$ we observe that $k=n$. For $d=3$ we show that $k = \lceil \frac n2 \rceil$. For $d=4$ the minimal arity becomes sublinear: $k = \Theta(n^{2/3})$. This contrasts with the case of majority circuits, in which $k = O(n^{2/3})$ is achievable already for depth $d=3$.

Our concern is the problem of efficiently determining the data complexity of answering queries mediated by description logic ontologies and constructing their optimal rewritings to standard database queries. Originated in ontology-based data access and datalog optimisation, this problem is known to be computationally very complex in general, with no explicit syntactic characterisations available. In this article, aiming to understand the fundamental roots of this difficulty, we strip the problem to the bare bones and focus on Boolean conjunctive queries mediated by a simple covering axiom stating that one class is covered by the union of two other classes. We show that, on the one hand, these rudimentary ontology-mediated queries, called disjunctive sirups (or d-sirups), capture many features and difficulties of the general case. For example, answering d-sirups is Pi^p_2-complete for combined complexity and can be in AC0 or LogSpace-, NL-, P-, or coNP-complete for data complexity (with the problem of recognising FO-rewritability of d-sirups being 2ExpTime-hard); some d-sirups only have exponential-size resolution proofs, some only double-exponential-size positive existential FO-rewritings and single-exponential-size nonrecursive datalog rewritings. On the other hand, we prove a few partial sufficient and necessary conditions of FO- and (symmetric/linear-) datalog rewritability of d-sirups. Our main technical result is a complete and transparent syntactic AC0/NL/P/coNP tetrachotomy of d-sirups with disjoint covering classes and a path-shaped Boolean conjunctive query. To obtain this tetrachotomy, we develop new techniques for establishing P- and coNP-hardness of answering non-Horn ontology-mediated queries as well as showing that they can be answered in NL.

We suggest a generalization of Karchmer - Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables. This resolves a conjecture of Cohen et al. (CRYPTO 2013).

For $S \subseteq \{0,1\}^n$ a Boolean function $f \colon S \to \{-1,1\}$ is a polynomial threshold function (PTF) of degree $d$ and weight $W$ if there is a polynomial $p$ with integer coefficients of degree $d$ and with sum of absolute coefficients $W$ such that $f(x) = \text{sign}(p(x))$ for all $x \in S$. We study a representation of decision lists as PTFs over Boolean cubes $\{0,1\}^n$ and over Hamming balls $\{0,1\}^{n}_{\leq k}$. As our first result, we show that for all $d = O\left( \left( \frac{n}{\log n}\right)^{1/3}\right)$ any decision list over $\{0,1\}^n$ can be represented by a PTF of degree $d$ and weight $2^{O(n/d^2)}$. This improves the result by Klivans and Servedio [Klivans, Servedio, 2006] by a $\log^2 d$ factor in the exponent of the weight. Our bound is tight for all $d = O\left( \left( \frac{n}{\log n}\right)^{1/3}\right)$ due to the matching lower bound by Beigel [Beigel, 1994]. For decision lists over a Hamming ball $\{0,1\}^n_{\leq k}$ we show that the upper bound on weight above can be drastically improved to $n^{O(\sqrt{k})}$ for $d = \Theta(\sqrt{k})$. We also show that similar improvement is not possible for smaller degrees by proving the lower bound $W = 2^{\Omega(n/d^2)}$ for all $d = O(\sqrt{k})$. \end{abstract}

Его отец Адиян Иван Аракелович 1908 г. рождения был сыном пастуха. Не имея возможности закончить среднюю школу, он научился ремеслу плотника и работал на местных строительных площадках. В 1930 г. он женился на семнадцатилетней Лусик, дочери местного крестьянина Константина Трузьяна. Через два года родители Сережи переехали в г. Кировабад, где отец устроился на работу плотником. Сначала они снимали комнату, и только к концу 30-х годов отец купил участок в центре города и построил там небольшой домик с одной жилой комнатой, балконом и подвальчиком. Будучи сам строителем, отец планировал достроить в доме второй этаж, но помешала война. К этому времени в семье уже было четверо детей. Мама не работала, но родители завершили свое среднее образование в вечерней рабочей армянской школе. Хотя Сережа, как и его родители, не умел говорить по-русски, в 1938 г. его отдали

We show that deciding boundedness (aka FO-rewritability) of monadic single rule datalog programs (sirups) is 2\Exp-hard, which matches the upper bound known since 1988 and finally settles a long-standing open problem. We obtain this result as a byproduct of an attempt to classify monadic 'disjunctive sirups'---Boolean conjunctive queries $\q$ with unary and binary predicates mediated by a disjunctive rule $T(x) łor F(x) łeftarrow A(x)$---according to the data complexity of their evaluation. Apart from establishing that deciding FO-rewritability of disjunctive sirups with a dag-shaped $\q$ is also 2\Exp-hard, we make substantial progress towards obtaining a complete FO/Ł-hardness dichotomy of disjunctive sirups with ditree-shaped $\q$.

In this paper we study decision tree models with various types of queries. For a given function it is usually not hard to determine the complexity in the standard decision tree model (each query evaluates a variable). However in more general settings showing tight lower bounds is substantially harder. Threshold functions often have non-trivial complexity in such models and can be used to provide interesting examples. Standard decision trees can be viewed as a computational model in which each query depends on only one input bit. In the first part of the paper we consider natural generalization of standard decision tree model: we address decision trees that are allowed to query any function depending on two input bits. We show the first lower bound of the form n − o(n) for an explicit function (namely, the majority function) in this model. We also show that in the decision tree model with AND and OR queries of arbitrary fan-in the complexity of the majority function is n − 1. In the second part of the paper we address parity decision trees that are allowed to query arbitrary parities of input bits. There are various lower bound techniques for parity decision trees complexity including analytical techniques (degree over 𝔽_2 , Fourier sparsity, granularity) and combinatorial techniques (generalizations of block sensitivity and certificate complexity). These techniques give tight lower bounds for many natural functions. We give a new inductive argument tailored specifically for threshold functions. A combination of this argument with granularity lower bound allows us to provide a simple example of a function for which all previously known lower bounds are not tight.

A major goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems f :({0, 1} n ) k → {0, 1} with k > log n parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every k ≥ log n , showing in both cases that Θ(1 + ⌈log n ⌉/ log ⌈1 + k / log n ⌉) bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is k ≥ n ϵ for some constant ϵ > 0.

Aiming to understand the data complexity of answering conjunctive queries mediated by an axiom stating that a class is covered by the union of two other classes, we show that deciding their first-order rewritability is PSPACE-hard and obtain a number of sufficient conditions for membership in AC0, L, NL, and P. Our main result is a complete syntactic AC0/NL/P/CONP tetrachotomy of path queries under the assumption that the covering classes are disjoint.

We study the following computational problem: for which values of k, the majority of n bits MAJn can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJk ∘ MAJk. We observe that the minimum value of k for which there exists a MAJk ∘ MAJk circuit that has high correlation with the majority of n bits is equal to Θ(n1/2). We then show that for a randomized MAJk ∘ MAJk circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n2/3 + o(1). We show a worst case lower bound: if a MAJk ∘ MAJk circuit computes the majority of n bits correctly on all inputs, then k ≥ n13/19 + o(1).

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz and moreover we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems.