A binary word is called $q$-decreasing, for $q>0$, if every of its length maximal factors of the form $0^a1^b$, $a>0$, satisfies $q \cdot a > b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx$. We show that the number of $q$-decreasing words of length $n$ grows as $\Phi(q)^{n} C_q $ for some constant $C_q$ which depends on $q$ but not on $n$. We demonstrate that $\Phi(1)$ is the golden ratio, $\Phi(2)$ is equal to the tribonacci constant, $\Phi(k)$ is $(k+1)$-bonacci constant. Furthermore, we prove that the function $\Phi(q)$ is strictly increasing, discontinuous at every positive rational point, exhibits a fractal structure related to the Stern--Brocot tree and Minkowski's question mark function.
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Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_\alpha(G):=\alpha A(G)+(1-\alpha)D(G),$ where $\alpha\in[0,1]$. In this paper, we present some bounds for the eigenvalues of $A_\alpha(G)$ and for the largest and smallest eigenvalues of $A_\alpha(l(G))$. Extremal graphs attaining some of these bounds are characterized.
In this paper I will develop a lambda-term calculus, lambda-2Int, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry-Howard correspondence, which has been well-established between the simply typed lambda-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing's bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed lambda-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.
Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal{S}$-adic representation where the morphisms in $\mathcal{S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper we give an $\mathcal{S}$-adic characterization of this family by means of two finite graphs. As an application, we are able to decide whether a shift space generated by a uniformly recurrent morphic word is (eventually) dendric.
We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of such frames are (discrete) probabilistic CI structures, semi-graphoids and structural semi-graphoids. We concentrate on those frames which are closed under the operation of set-theoretical intersection because, for these, the respective families of CI models are lattices. This allows one to apply the results from lattice theory and formal concept analysis to describe such families in terms of implications among CI statements. The central concept of this paper is that of self-adhesivity defined in algebraic terms, which is a combinatorial reflection of the self-adhesivity concept studied earlier in context of polymatroids and information theory. The generalization also leads to a self-adhesivity operator defined on the hyper-level of CI frames. We answer some of the questions related to this approach and raise other open questions. The core of the paper is in computations. The combinatorial approach to computation might overcome some memory and space limitation of software packages based on polyhedral geometry, in particular, if SAT solvers are utilized. We characterize some basic CI families over 4 variables in terms of canonical implications among CI statements. We apply our method in information-theoretical context to the task of entropic region demarcation over 5 variables.
Genome assembly is a prominent problem studied in bioinformatics, which computes the source string using a set of its overlapping substrings. Classically, genome assembly uses assembly graphs built using this set of substrings to compute the source string efficiently, having a tradeoff between scalability and avoiding information loss. The scalable de Bruijn graphs come at the price of losing crucial overlap information. The complete overlap information is stored in overlap graphs using quadratic space. Hierarchical overlap graphs [IPL20] (HOG) overcome these limitations, avoiding information loss despite using linear space. After a series of suboptimal improvements, Khan and Park et al. simultaneously presented two optimal algorithms [CPM2021], where only the former was seemingly practical. We empirically analyze all the practical algorithms for computing HOG, where the optimal algorithm [CPM2021] outperforms the previous algorithms as expected, though at the expense of extra memory. However, it uses non-intuitive approach and non-trivial data structures. We present arguably the most intuitive algorithm, using only elementary arrays, which is also optimal. Our algorithm empirically proves even better for both time and memory over all the algorithms, highlighting its significance in both theory and practice. We further explore the applications of hierarchical overlap graphs to solve various forms of suffix-prefix queries on a set of strings. Loukides et al. [CPM2023] recently presented state-of-the-art algorithms for these queries. However, these algorithms require complex black-box data structures and are seemingly impractical. Our algorithms, despite failing to match the state-of-the-art algorithms theoretically, answer different queries ranging from 0.01-100 milliseconds for a data set having around a billion characters.
The $k$-Maximum Inner Product Search ($k$MIPS) serves as a foundational component in recommender systems and various data mining tasks. However, while most existing $k$MIPS approaches prioritize the efficient retrieval of highly relevant items for users, they often neglect an equally pivotal facet of search results: \emph{diversity}. To bridge this gap, we revisit and refine the diversity-aware $k$MIPS (D$k$MIPS) problem by incorporating two well-known diversity objectives -- minimizing the average and maximum pairwise item similarities within the results -- into the original relevance objective. This enhancement, inspired by Maximal Marginal Relevance (MMR), offers users a controllable trade-off between relevance and diversity. We introduce \textsc{Greedy} and \textsc{DualGreedy}, two linear scan-based algorithms tailored for D$k$MIPS. They both achieve data-dependent approximations and, when aiming to minimize the average pairwise similarity, \textsc{DualGreedy} attains an approximation ratio of $1/4$ with an additive term for regularization. To further improve query efficiency, we integrate a lightweight Ball-Cone Tree (BC-Tree) index with the two algorithms. Finally, comprehensive experiments on ten real-world data sets demonstrate the efficacy of our proposed methods, showcasing their capability to efficiently deliver diverse and relevant search results to users.
Let $A$ be an $s$-sparse Hermitian matrix, $f(x)$ be a univariate function, and $i, j$ be two indices. In this work, we investigate the query complexity of approximating $\bra{i} f(A) \ket{j}$. We show that for any continuous function $f(x):[-1,1]\rightarrow [-1,1]$, the quantum query complexity of computing $\bra{i} f(A) \ket{j}\pm \varepsilon/4$ is lower bounded by $\Omega(\widetilde{\deg}_\varepsilon(f))$. The upper bound is at most quadratic in $\widetilde{\deg}_\varepsilon(f)$ and is linear in $\widetilde{\deg}_\varepsilon(f)$ under certain mild assumptions on $A$. Here the approximate degree $\widetilde{\deg}_\varepsilon(f)$ is the minimum degree such that there is a polynomial of that degree approximating $f$ up to additive error $\varepsilon$ in the interval $[-1,1]$. We also show that the classical query complexity is lower bounded by $\widetilde{\Omega}((s/2)^{(\widetilde{\deg}_{2\varepsilon}(f)-1)/6})$ for any $s\geq 4$. Our results show that the quantum and classical separation is exponential for any continuous function of sparse Hermitian matrices, and also imply the optimality of implementing smooth functions of sparse Hermitian matrices by quantum singular value transformation. The main techniques we used are the dual polynomial method for functions over the reals, linear semi-infinite programming, and tridiagonal matrices.
We present a KE-tableau-based procedure for the main TBox and ABox reasoning tasks for the description logic $\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle(\mathbf{D})$, in short $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$. The logic $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$, representable in the decidable multi-sorted quantified set-theoretic fragment $\mathsf{4LQS^R}$, combines the high scalability and efficiency of rule languages such as the Semantic Web Rule Language (SWRL) with the expressivity of description logics. Our algorithm is based on a variant of the KE-tableau system for sets of universally quantified clauses, where the KE-elimination rule is generalized in such a way as to incorporate the $\gamma$-rule. The novel system, called KE$^\gamma$-tableau, turns out to be an improvement of the system introduced in \cite{RR2017} and of standard first-order KE-tableau \cite{dagostino94}. Suitable benchmark test sets executed on C++ implementations of the three mentioned systems show that the performances of the KE$^\gamma$-tableau-based reasoner are often up to about 400% better than the ones of the other two systems. This a first step towards the construction of efficient reasoners for expressive OWL ontologies based on fragments of computable set-theory.
A C++ reasoner for the description logic $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ (Extended Version)
We present an ongoing implementation of a KE-tableau based reasoner for a decidable fragment of stratified elementary set theory expressing the description logic $\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle(\mathbf{D})$ (shortly $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$). The reasoner checks the consistency of $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$-knowledge bases (KBs) represented in set-theoretic terms. It is implemented in \textsf{C++} and supports $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$-KBs serialized in the OWL/XML format. To the best of our knowledge, this is the first attempt to implement a reasoner for the consistency checking of a description logic represented via a fragment of set theory that can also classify standard OWL ontologies.
We present a KE-tableau-based implementation of a reasoner for a decidable fragment of (stratified) set theory expressing the description logic $\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle(\mathbf{D})$ ($\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$, for short). Our application solves the main TBox and ABox reasoning problems for $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$. In particular, it solves the consistency problem for $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$-knowledge bases represented in set-theoretic terms, and a generalization of the \emph{Conjunctive Query Answering} problem in which conjunctive queries with variables of three sorts are admitted. The reasoner, which extends and optimizes a previous prototype for the consistency checking of $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$-knowledge bases (see \cite{cilc17}), is implemented in \textsf{C++}. It supports $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$-knowledge bases serialized in the OWL/XML format, and it admits also rules expressed in SWRL (Semantic Web Rule Language).
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