Related papers: Self-Similar Algebras with connections to Run-leng…
Analogical reasoning is a fundamental capacity of human cognition that allows us to reason abstractly about novel situations by relating them to past experiences. While it is thought to be essential for robust reasoning in AI systems,…
Autoformalization, the task of automatically translating natural language descriptions into a formal language, poses a significant challenge across various domains, especially in mathematics. Recent advancements in large language models…
We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix…
The emergence of large-language models (LLMs) has enabled a new class of semantic data processing systems (SDPSs) to support declarative queries against unstructured documents. Existing SDPSs are, however, lacking a unified algebraic…
These are lecture notes on the algebraic approach to regular languages. The classical algebraic approach is for finite words; it uses semigroups instead of automata. However, the algebraic approach can be extended to structures beyond…
In the computational-mechanics structural analysis of one-dimensional cellular automata the following automata-theoretic analogue of the \emph{change-point problem} from time series analysis arises: \emph{Given a string $\sigma$ and a…
We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra…
We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in…
Recent large language models (LLMs) have achieved impressive reasoning milestones but continue to struggle with high computational costs, logical inconsistencies, and sharp performance degradation on high-complexity problems. While…
We consider an algebra with non-standard operations on the class of row monomial matrices (having one unit and rest of zeros in every row). The class of row monomial matrices is closed under multiplication, but not closed under ordinary…
Language models (LMs) have exhibited exceptional versatility in reasoning and in-depth financial analysis through their proprietary information processing capabilities. Previous research focused on evaluating classification performance…
Causal reasoning is a core component of intelligence. Large language models (LLMs) have shown impressive capabilities in generating human-like text, raising questions about whether their responses reflect true understanding or statistical…
Automatic assessment needs short answer based evaluation and automated assessment. Various techniques used are Ontology, Semantic similarity matching and Statistical methods. An automatic short answer assessment system is attempted in this…
We introduce a term algebra as a new formal specification language for the coordinating architectures of distributed systems consisting of a finite yet unbounded number of components. The language allows to describe infinite sets of systems…
The "similarity" degree of a unital operator algebra $A$ was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements:…
We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed $m$-letter alphabet) that form identities in the semigroup $\ut{n}$ of $n\times n$ upper triangular tropical…
Worked examples are step-by-step solutions to problems in a specific domain, offered to students to acquire domain-specific problem-solving skills. The effectiveness of worked examples could be enhanced by combining them with…
In this paper, we study a series of algorithmic problems related to the subsequences occurring in the strings of a given language, under the assumption that this language is succinctly represented by a grammar generating it, or an automaton…
We study polynomial identities of nonassociative algebras constructed by using infinite binary words and their combinatorial properties. Infinite periodic and Sturmian words were first applied for constructing examples of algebras with…
Analogical reasoning is the ability to detect parallels between two seemingly distant objects or situations, a fundamental human capacity used for example in commonsense reasoning, learning, and creativity which is believed by many…