Related papers: Evaluating Matrix Circuits
This thesis investigates the extent to which the optimal value of a constraint satisfaction problem (CSP) can be approximated by some sentence of fixed point logic with counting (FPC). It is known that, assuming $\mathsf{P} \neq…
We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of…
It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate $n\in \mathbb{N}$; that is, each element in the group has a unique matrix…
We investigate the language classes recognized by group automata over matrix groups. We present a summary of the results obtained so far together with a number of new results. We look at the computational power of time-bounded group…
We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix. The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative…
Deep-learning-based NLP models are found to be vulnerable to word substitution perturbations. Before they are widely adopted, the fundamental issues of robustness need to be addressed. Along this line, we propose a formal framework to…
Anisimov and Seifert show that a group has a regular word problem ifand only if it is finite. Muller and Schupp (together with Dunwoody's accessibility result) show that a group has context free word problem if and only if it is virtually…
Large Language Models (LLMs) excel at various tasks, including problem-solving and question-answering. However, LLMs often find Math Word Problems (MWPs) challenging because solving them requires a range of reasoning and mathematical…
We consider the class of conditional graph patterns (\emph{CGPs}) that allow user to query data graphs with complex patterns that contain negation and predicates. To overcome the prohibitive cost of subgraph isomorphism, we consider…
We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The…
It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper we prove that the set of…
Certificates of polynomial nonnegativity can be used to obtain tight dual bounds for polynomial optimization problems. We consider Sums of Nonnegative Circuit (SONC) polynomials certificates, which are well suited for sparse problems since…
We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem…
The reflections in a Coxeter group are defined as conjugates of a single generator, and thus admit palindromic expressions as products of generators. Our main result gives closed formulas providing a palindromic reduced expression for each…
We show that the Identity Problem is decidable in polynomial time for finitely generated sub-semigroups of the group $\mathsf{UT}(4, \mathbb{Z})$ of $4 \times 4$ unitriangular integer matrices. As a byproduct of our proof, we also show the…
We study algorithmic complexity and expressive power of fusion grammars, a novel formalism introduced in [Kreowski, Kuske, and Lye 2017], which extends hyperedge replacement grammars. In the first part of the work, we prove that the…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup…
We present a definition of the class NP in combinatorial context as the set of languages of structures defined by finitely many forbidden lifted substructures. We apply this to special syntactically defined subclasses and show how they…
We show that derandomizing polynomial identity testing over an arbitrary finite field implies that NEXP does not have polynomial size boolean circuits. In other words, for any finite field F(q) of size q, $PIT_q\in NSUBEXP\Rightarrow…