A Robust Class of Linear Recurrence Sequences

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers

Ambiguity, Weakness, and Regularity in Probabilistic B\"uchi Automata

Probabilistic B\"uchi automata are a natural generalization of PFA to infinite words, but have been studied in-depth only rather recently and many interesting questions are still open. PBA are known to accept, in general, a class of languages that goes beyond the regular languages. In this work we extend the known classes of restricted PBA which are still regular, strongly relying on notions concerning ambiguity in classical omega-automata. Furthermore, we investigate the expressivity of the not yet considered but natural class of weak PBA, and we also show that the regularity problem for weak PBA is undecidable

Automata with One-way Jumping Mode (Algebraic system, Logic, Language and Related Areas in Computer Sciences II)

Recently, new types of non-sequential machine models have been introduced and studied, such as jumping automata and one-way jumping automata. We study the abilities and limitations of automata with these two jumping modes of tape heads with respect to how they affect the class of accepted languages. We give several methods to determine whether a language is accepted by a machine with jumping mode. We also consider relationships among the classes of languages defined by the new machines and their classical counterparts

Security Engineering of Patient-Centered Health Care Information Systems in Peer-to-Peer Environments: Systematic Review

Background: Patient-centered health care information systems (PHSs) enable patients to take control and become knowledgeable about their own health, preferably in a secure environment. Current and emerging PHSs use either a centralized database, peer-to-peer (P2P) technology, or distributed ledger technology for PHS deployment. The evolving COVID-19 decentralized Bluetooth-based tracing systems are examples of disease-centric P2P PHSs. Although using P2P technology for the provision of PHSs can be flexible, scalable, resilient to a single point of failure, and inexpensive for patients, the use of health information on P2P networks poses major security issues as users must manage information security largely by themselves. Objective: This study aims to identify the inherent security issues for PHS deployment in P2P networks and how they can be overcome. In addition, this study reviews different P2P architectures and proposes a suitable architecture for P2P PHS deployment. Methods: A systematic literature review was conducted following PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) reporting guidelines. Thematic analysis was used for data analysis. We searched the following databases: IEEE Digital Library, PubMed, Science Direct, ACM Digital Library, Scopus, and Semantic Scholar. The search was conducted on articles published between 2008 and 2020. The Common Vulnerability Scoring System was used as a guide for rating security issues. Results: Our findings are consolidated into 8 key security issues associated with PHS implementation and deployment on P2P networks and 7 factors promoting them. Moreover, we propose a suitable architecture for P2P PHSs and guidelines for the provision of PHSs while maintaining information security. Conclusions: Despite the clear advantages of P2P PHSs, the absence of centralized controls and inconsistent views of the network on some P2P systems have profound adverse impacts in terms of security. The security issues identified in this study need to be addressed to increase patients\u27 intention to use PHSs on P2P networks by making them safe to use

Boolean Algebras from Trace Automata

We consider trace automata. Their vertices are Mazurkiewicz traces and they accept finite words. Considering the length of a trace as the length of its Foata normal form, we define the operations of level-length synchronization and of superposition of trace automata. We show that if a family F of trace automata is closed under these operations, then for any deterministic automaton H in F, the word languages accepted by the deterministic automata of F that are length-reducible to H form a Boolean algebra. We show that the family of trace suffix automata with level-regular contexts and the subfamily of vector addition systems satisfy these closure properties. In particular, this yields various Boolean algebras of word languages accepted by deterministic vector addition systems

On the Complexity of BWT-Runs Minimization via Alphabet Reordering

The Burrows-Wheeler Transform (BWT) has been an essential tool in text compression and indexing. First introduced in 1994, it went on to provide the backbone for the first encoding of the classic suffix tree data structure in space close to the entropy-based lower bound. Recently, there has been the development of compact suffix trees in space proportional to "$r$", the number of runs in the BWT, as well as the appearance of $r$ in the time complexity of new algorithms. Unlike other popular measures of compression, the parameter $r$ is sensitive to the lexicographic ordering given to the text's alphabet. Despite several past attempts to exploit this, a provably efficient algorithm for finding, or approximating, an alphabet ordering which minimizes $r$ has been open for years. We present the first set of results on the computational complexity of minimizing BWT-runs via alphabet reordering. We prove that the decision version of this problem is NP-complete and cannot be solved in time $2^{o(\sigma + \sqrt{n})}$ unless the Exponential Time Hypothesis fails, where $\sigma$ is the size of the alphabet and $n$ is the length of the text. We also show that the optimization problem is APX-hard. In doing so, we relate two previously disparate topics: the optimal traveling salesperson path and the number of runs in the BWT of a text, providing a surprising connection between problems on graphs and text compression. Also, by relating recent results in the field of dictionary compression, we illustrate that an arbitrary alphabet ordering provides a $O(\log^2 n)$-approximation. We provide an optimal linear-time algorithm for the problem of finding a run minimizing ordering on a subset of symbols (occurring only once) under ordering constraints, and prove a generalization of this problem to a class of graphs with BWT like properties called Wheeler graphs is NP-complete