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In the distributional Twenty Questions game, Bob chooses a number $x$ from $1$ to $n$ according to a distribution $\mu$, and Alice (who knows $\mu$) attempts to identify $x$ using Yes/No questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for $\mu$, yet this strategy could potentially uses all $2^n$ possible questions. Dagan et al. constructed a set of $1.25^{n+o(n)}$ questions which suffice to construct an optimal strategy for all $\mu$, and showed that this number is optimal (up to sub-exponential factors) for infinitely many $n$. We determine the optimal size of such a set of questions for all $n$ (up to sub-exponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the $d$-ary setting, obtaining similar results with $1.25$ replaced by $1 + (d-1)/d^{d/(d-1)}$.

We define various monoid versions of the R. Thompson group $V$, and prove connections with monoids of acyclic digital circuits. We show that the monoid $M_{2,1}$ (based on partial functions) is not embeddable into Thompson's monoid ${\sf tot}M_{2,1}$, but that ${\sf tot}M_{2,1}$ has a submonoid that maps homomorphically onto $M_{2,1}$. This leads to an efficient completion algorithm for partial functions and partial circuits. We show that the union of partial circuits with disjoint domains is an element of $M_{2,1}$, and conversely, every element of $M_{2,1}$ can be decomposed efficiently into a union of partial circuits with disjoint domains.

We investigate completions of partial combinatory algebras (pcas), in particular of Kleene's second model $\mathcal{K}_2$ and generalizations thereof. We consider weak and strong notions of embeddability and completion that have been studied before. By a result of Klop it is known that not every pca has a strong completion. The study of completions of $\mathcal{K}_2$ has as corollaries that weak and strong embeddings are different, and that every countable pca has a weak completion. We then consider generalizations of $\mathcal{K}_2$ for larger cardinals, and use these to show that it is consistent that every pca has a weak completion.

We introduce efficient algorithms for approximate sampling from symmetric Gibbs distributions on the sparse random (hyper)graph. The examples we consider include (but are not restricted to) important distributions on spin systems and spin-glasses such as the q state antiferromagnetic Potts model for $q\geq 2$, including the colourings, the uniform distributions over the Not-All-Equal solutions of random k-CNF formulas. Finally, we present an algorithm for sampling from the spin-glass distribution called the k-spin model. To our knowledge this is the first, rigorously analysed, efficient algorithm for spin-glasses which operates in a non trivial range of the parameters. Our approach builds on the one that was introduced in [Efthymiou: SODA 2012]. For a symmetric Gibbs distribution $\mu$ on a random (hyper)graph whose parameters are within an certain range, our algorithm has the following properties: with probability $1-o(1)$ over the input instances, it generates a configuration which is distributed within total variation distance $n^{-\Omega(1)}$ from $\mu$. The time complexity is $O((n\log n)^2)$. The algorithm requires a range of the parameters which, for the graph case, coincide with the tree-uniqueness region, parametrised w.r.t. the expected degree d. For the hypergraph case, where uniqueness is less restrictive, we go beyond uniqueness. Our approach utilises in a novel way the notion of contiguity between Gibbs distributions and the so-called teacher-student model.

The merit factor of a $\{-1, 1\}$ binary sequence measures the collective smallness of its non-trivial aperiodic autocorrelations. Binary sequences with large merit factor are important in digital communications because they allow the efficient separation of signals from noise. It is a longstanding open question whether the maximum merit factor is asymptotically unbounded and, if so, what is its limiting value. Attempts to answer this question over almost sixty years have identified certain classes of binary sequences as particularly important: skew-symmetric sequences, symmetric sequences, and anti-symmetric sequences. Using only elementary methods, we find an exact formula for the mean and variance of the reciprocal merit factor of sequences in each of these classes, and in the class of all binary sequences. This provides a much deeper understanding of the distribution of the merit factor in these four classes than was previously available. A consequence is that, for each of the four classes, the merit factor of a sequence drawn uniformly at random from the class converges in probability to a constant as the sequence length increases.

We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\geq 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\geq 1$. Recently, Bespalov proved nonexistence of extended $1$-perfect codes for $q=3$, $4$, $n>q+2$. In this work, we characterize all positive integers $n$, $r$ and prime $p$, for which there exist such a code in $H(n,p^r)$.

We investigate the R\'enyi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the R\'enyi entropy, for Poisson-Bernoulli variables. As applications we prove that a discrete ``min-entropy power'' is super additive on independent variables up to a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the ``Poisson regime''.

In this paper, we propose two algorithms for a hybrid construction of all $n\times n$ MDS and involutory MDS matrices over a finite field $\mathbb{F}_{p^m}$, respectively. The proposed algorithms effectively narrow down the search space to identify $(n-1) \times (n-1)$ MDS matrices, facilitating the generation of all $n \times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. To the best of our knowledge, existing literature lacks methods for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. In our approach, we introduce a representative matrix form for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. The determination of these representative MDS matrices involves searching through all $(n-1)\times (n-1)$ MDS matrices over $\mathbb{F}_{p^m}$. Our contributions extend to proving that the count of all $3\times 3$ MDS matrices over $\mathbb{F}_{2^m}$ is precisely $(2^m-1)^5(2^m-2)(2^m-3)(2^{2m}-9\cdot 2^m+21)$. Furthermore, we explicitly provide the count of all $4\times 4$ MDS and involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=2, 3, 4$.

A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.

The logics $\mathsf{CS4}$ and $\mathsf{IS4}$ are the two leading intuitionistic variants of the modal logic $\mathsf{S4}$. Whether the finite model property holds for each of these logics have been long-standing open problems. It was recently shown that $\mathsf{IS4}$ has the finite frame property and thus the finite model property. In this paper, we prove that $\mathsf{CS4}$ also enjoys the finite frame property. Additionally, we investigate the following three logics closely related to $\mathsf{IS4}$. The logic $\mathsf{GS4}$ is obtained by adding the G\"odel--Dummett axiom to $\mathsf{IS4}$; it is both a superintuitionistic and a fuzzy logic and has previously been given a real-valued semantics. We provide an alternative birelational semantics and prove strong completeness with respect to this semantics. The extension $\mathsf{GS4^c}$ of $\mathsf{GS4}$ corresponds to requiring a crisp accessibility relation on the real-valued semantics. We give a birelational semantics corresponding to an extra confluence condition on the $\mathsf{GS4}$ birelational semantics and prove strong completeness. Neither of these two logics have the finite model property with respect to their real-valued semantics, but we prove that they have the finite frame property for their birelational semantics. Establishing the finite birelational frame property immediately establishes decidability, which was previously open for these two logics. Our proofs yield NEXPTIME upper bounds. The logic $\mathsf{S4I}$ is obtained from $\mathsf{IS4}$ by reversing the roles of the modal and intuitionistic relations in the birelational semantics. We also prove the finite frame property, and thereby decidability, for $\mathsf{S4I}$