相关论文: Self-Similar Algebras with connections to Run-leng…
We introduce "synchronous algebras", an algebraic structure tailored to recognize automatic relations (aka. synchronous relations, or regular relations). They are the equivalent of monoids for regular languages, however they conceptually…
The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical…
Let $\mathcal{P}(\Sigma^*)$ be the semiring of languages, and consider its subset $\mathcal{P}(\Sigma)$. In this paper we define the language recognized by a weighted automaton over $\mathcal{P}(\Sigma)$ and a one-letter alphabet.…
Rational word languages can be defined by several equivalent means: finite state automata, rational expressions, finite congruences, or monadic second-order (MSO) logic. The robust subclass of aperiodic languages is defined by: counter-free…
We define the arithmetic self-similarity (AS) of a one-sided infinite sequence sigma to be the set of arithmetic progressions through sigma which are a vertical shift of sigma. We study the AS of several famlies of sequences, viz.…
On a (pseudo-)Riemannian manifold (MM,g), some fields of endomorphisms i.e. sections of End(TMM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra,…
We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced.…
We give an algebraic characterisation of first-order logic with the neighbour relation, on finite words. For this, we consider languages of finite words over alphabets with an involution on them. The natural algebras for such languages are…
The ability of Large Language Models (LLMs) to encode syntactic and semantic structures of language is well examined in NLP. Additionally, analogy identification, in the form of word analogies are extensively studied in the last decade of…
A bialgebra is a structure which is simultaneously an algebra and a coalgebra, such that the algebraic and coalgebraic parts are "compatible". Bialgebras are normally studied over a field or commutative ring. In this paper, we show how to…
Statistical properties of the taxonomic classification of human languages are studied. It is shown that, at the highest levels of the taxonomic hierarchy, the frequency of taxon members as a function of the number of languages belonging to…
Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all…
Analogical reasoning -- the capacity to identify and map structural relationships between different domains -- is fundamental to human cognition and learning. Recent studies have shown that large language models (LLMs) can sometimes match…
Rule-based reasoning is an essential part of human intelligence prominently formalized in artificial intelligence research via logic programs. Describing complex objects as the composition of elementary ones is a common strategy in computer…
We show that three fixed point structures equipped with (sequential) composition, a sum operation, and a fixed point operation share the same valid equations. These are the theories of (context-free) languages, (regular) tree languages, and…
This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the…
Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon's congruence $\sim_k$ is one of the most classical approaches. Two words are $\sim_k$-equivalent if they have the…
The association between language and (non-linguistic) thinking ability in humans has long been debated, and recently, neuroscientific evidence of brain activity patterns has been considered. Such a scientific context naturally raises an…
Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of…
Autoformalization is the task of automatically translating mathematical content written in natural language to a formal language expression. The growing language interpretation capabilities of Large Language Models (LLMs), including in…