A Note on Efficient Computation of All Abelian Periods in a String

We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all `long' Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems

Asymptotic approximation for the quotient complexities of atoms

In a series of papers, Brzozowski together with Tamm, Davies, and Szykuła studied the quotient complexities of atoms of regular languages [6, 7, 3, 4]. The authors obtained precise bounds in terms of binomial sums for the most complex situations in the following five cases: (G): general, (R): right ideals, (L): left ideals, (T): two-sided ideals and (S): suffix-free languages. In each case let κc(n) be the maximal complexity of an atom of a regular language L, where L has complexity n ≥ 2 and belongs to the class C ϵ {G, R, L, T , S}. It is known that κT(n) ≤ κL(n) = κR(n) ≤ κG(n) 3 if and only if κC(n+1)/κC(n) < 3

Learnability and Overgeneration in Computational Syntax

This paper addresses the hypothesis that unnatural patterns generated by grammar formalisms can be eliminated on the grounds that they are unlearnable. I consider three examples of formal languages thought to represent dependencies unattested in natural language syntax, and show that all three can be learned by grammar induction algorithms following the Distributional Learning paradigm of Clark and Eyraud (2007). While learnable language classes are restrictive by necessity (Gold, 1967), these facts suggest that learnability alone may be insufficient for addressing concerns of overgeneration in syntax

Place-Labeled Petri Net Controlled Grammars

A place-labeled Petri net (pPN) controlled grammar is a context-free grammar equipped with a Petri net and a function which maps places of the net to the productions of the grammar. The language consists of all terminal strings that can be obtained by simultaneously applying of the rules of multisets which are the images of the sets of the input places of transitions in a successful occurrence sequence of the Petri net. In this paper, we study the generative power and structural properties of pPN controlled grammars. We show that pPN controlled grammars have the same generative power as matrix grammars. Moreover, we prove that for each pPN controlled grammar, we can construct an equivalent place-labeled ordinary net controlled grammar

Trust, Accountability, and Autonomy in Knowledge Graph-based AI for Self-determination

Knowledge Graphs (KGs) have emerged as fundamental platforms for powering intelligent decision-making and a wide range of Artificial Intelligence (AI) services across major corporations such as Google, Walmart, and AirBnb. KGs complement Machine Learning (ML) algorithms by providing data context and semantics, thereby enabling further inference and question-answering capabilities. The integration of KGs with neuronal learning (e.g., Large Language Models (LLMs)) is currently a topic of active research, commonly named neuro-symbolic AI. Despite the numerous benefits that can be accomplished with KG-based AI, its growing ubiquity within online services may result in the loss of self-determination for citizens as a fundamental societal issue. The more we rely on these technologies, which are often centralised, the less citizens will be able to determine their own destinies. To counter this threat, AI regulation, such as the European Union (EU) AI Act, is being proposed in certain regions. The regulation sets what technologists need to do, leading to questions concerning: How can the output of AI systems be trusted? What is needed to ensure that the data fuelling and the inner workings of these artefacts are transparent? How can AI be made accountable for its decision-making? This paper conceptualises the foundational topics and research pillars to support KG-based AI for self-determination. Drawing upon this conceptual framework, challenges and opportunities for citizen self-determination are illustrated and analysed in a real-world scenario. As a result, we propose a research agenda aimed at accomplishing the recommended objectives

Word-representability of triangulations of grid-covered cylinder graphs

A graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y, x ≠ y, alternate in w if and only if (x,y) ∈ E. Halldórsson, Kitaev and Pyatkin have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary of this result is that any 3-colorable graph is word-representable. Akrobotu, Kitaev and Masàrovà have shown that a triangulation of a grid graph is word-representable if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder graphs; indeed, there are such word-representable graphs with chromatic number 4. In this paper we show that word-representability of triangulations of grid-covered cylinder graphs with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel graphs W5 and W7)

Word-representability of face subdivisions of triangular grid graphs

A graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if (x, y) ∈ E. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph K4. Inspired by a recent elegant result of Akrobotu et al., who classified wordrepresentable triangulations of grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpi´nski gasket graph is wordrepresentable

Trust, Accountability, and Autonomy in Knowledge Graph-Based AI for Self-Determination

Knowledge Graphs (KGs) have emerged as fundamental platforms for powering intelligent decision-making and a wide range of Artificial Intelligence (AI) services across major corporations such as Google, Walmart, and AirBnb. KGs complement Machine Learning (ML) algorithms by providing data context and semantics, thereby enabling further inference and question-answering capabilities. The integration of KGs with neuronal learning (e.g., Large Language Models (LLMs)) is currently a topic of active research, commonly named neuro-symbolic AI. Despite the numerous benefits that can be accomplished with KG-based AI, its growing ubiquity within online services may result in the loss of self-determination for citizens as a fundamental societal issue. The more we rely on these technologies, which are often centralised, the less citizens will be able to determine their own destinies. To counter this threat, AI regulation, such as the European Union (EU) AI Act, is being proposed in certain regions. The regulation sets what technologists need to do, leading to questions concerning How the output of AI systems can be trusted? What is needed to ensure that the data fuelling and the inner workings of these artefacts are transparent? How can AI be made accountable for its decision-making? This paper conceptualises the foundational topics and research pillars to support KG-based AI for self-determination. Drawing upon this conceptual framework, challenges and opportunities for citizen self-determination are illustrated and analysed in a real-world scenario. As a result, we propose a research agenda aimed at accomplishing the recommended objectives