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Domain specific languages have successfully been used in a variety of fields to cleanly express scientific problems as well as to simplify implementation and performance opti- mization on different computer architectures. Although a large…
Existentially closed groups are, informally, groups that contain solutions to every consistent finite system of equations and inequations. They were introduced in 1951 in an algebraic context and subsequent research elucidated deep…
We consider the class of groups whose word problem is poly-context-free; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free…
Let $A^*$ denote the free monoid generated by a finite nonempty set $A.$ In this paper we introduce a new measure of complexity of languages $L\subseteq A^*$ defined in terms of the semigroup structure on $A^*.$ For each $L\subseteq A^*,$…
Full set of autonomous completely solvable differential systems of equations in total differentials is built by basis of infinitesimal operators, universal invariant, and structure constants of admited multiparametric Lie group (abelian and…
We construct a finitely presented group with undecidable word problem and with Dehn function bounded by a quadratic function on an infinite set of positive integers.
This article is an introduction to formal languages from the point of view of combinatorial group theory. Group theoretic applications are included and language classes are defined algebraically.
A special inverse monoid is one defined by a presentation where all the defining relations have the form $r = 1$. By a result of Ivanov Margolis and Meakin the word problem for such an inverse monoid can often be reduced to the word problem…
We consider grammar-restricted exact learning of formulas and terms in finite variable logics. We propose a novel and versatile automata-theoretic technique for solving such problems. We first show results for learning formulas that…
A class of algebras is constructed using free fermions and the invariant antisymmetric tensors associated with irreducible holonomy groups. (This version contains minor typographical corrections and some additional references. )
A wide range of constraints can be compactly specified using automata or formal languages. In a sequence of recent papers, we have shown that an effective means to reason with such specifications is to decompose them into primitive…
It is shown that there is $N$ such that there is no algorithm to decide for identities in at most $N$ variables validity in the class of finite modular lattices. This is based on Slobodskoi's result that the Restricted Word Problem is…
Let $FP(X)$ be the free paratopological group over a topological space $X$. For each non-negative integer $n\in\mathbb{N}$, denote by $FP_{n}(X)$ the subset of $FP(X)$ consisting of all words of reduced length at most $n$, and $i_{n}$ by…
The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on categories with finite limits and colimits. As an…
The inexpressive Description Logic (DL) $\mathcal{FL}_0$, which has conjunction and value restriction as its only concept constructors, had fallen into disrepute when it turned out that reasoning in $\mathcal{FL}_0$ w.r.t. general TBoxes is…
We introduce Metatheory.jl: a lightweight and performant general purpose symbolics and metaprogramming framework meant to simplify the act of writing complex Julia metaprograms and to significantly enhance Julia with a native term rewriting…
We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…
We construct words with small image in a given finite alternating or unimodular group. This shows that word width in these groups is unbounded in general.
This work addresses the challenge of democratizing advanced Large Language Models (LLMs) by compressing their mathematical reasoning capabilities into sub-billion parameter Small Language Models (SLMs) without compromising performance. We…
A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements require the use of projectors,…