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Combining a standard proof search method, such as resolution or tableaux, and rewriting is a powerful way to cut off search space in automated theorem proving, but proving the completeness of such combined methods may be challenging. It may…
We study induction on the program structure as a proof method for bisimulation-based compiler correctness. We consider a first-order language with mutually recursive function definitions, system calls, and an environment semantics. The…
We propose a new definition of effective formulas for problems in enumerative combinatorics. We outline the proof of the fact that every linear recurrence sequence of integers has such a formula. It follows from a lower bound that can be…
Applying machine learning to mathematical terms and formulas requires a suitable representation of formulas that is adequate for AI methods. In this paper, we develop an encoding that allows for logical properties to be preserved and is…
Large language models (LLMs) have achieved state-of-the-art performance in natural language processing; however, their high computational cost remains a major bottleneck. In this study, we target computational efficiency by focusing on a…
Let $X$ be a fs logarithmic scheme that is generically logarithmically smooth, and that admits a strict closed embedding into a logarithmically smooth scheme $Y$ over a field $\kk$ of characteristic zero. We construct a simple and fast…
Automated theorem proving in first-order logic is an active research area which is successfully supported by machine learning. While there have been various proposals for encoding logical formulas into numerical vectors -- from simple…
This paper builds and extends on the authors' previous work related to the algorithmic tool, Cylindrical Algebraic Decomposition (CAD), and one of its core applications, Real Quantifier Elimination (QE). These topics are at the heart of…
In this paper, a practicable simulation-free model order reduction method by nonlinear moment matching is developed. Based on the steady-state interpretation of linear moment matching, we comprehensively explain the extension of this…
The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
The Szemer\'edi Regularity Lemma, in combination with the Blow-up Lemma, form the Regularity Method, a fundamental tool in graph embeddings, albeit restricted to very large and dense graphs. We propose an alternative vertex-partitioning…
An algorithm for a family of self-starting high-order implicit time integration schemes with controllable numerical dissipation is proposed for both linear and nonlinear transient problems. This work builds on the previous works of the…
Many a concrete theorem of abstract algebra admits a short and elegant proof by contradiction but with Zorn's Lemma (ZL). A few of these theorems have recently turned out to follow in a direct and elementary way from the Principle of Open…
A new algorithm is presented for computing a direct solution to a system of consistent linear equations. It produces a minimum norm particular solution, a generalized inverse (of type {124}), and a null space projection operator. In…
Many iterative algorithms in optimization, computational geometry, computer algebra, and other areas of computer science require repeated computation of some algebraic expression whose input changes slightly from one iteration to the next.…
Transformer-based Large Language Models (LLMs) have become a fixture in modern machine learning. Correspondingly, significant resources are allocated towards research that aims to further advance this technology, typically resulting in…
The Nemhauser-Trotter local optimization theorem applies to the NP-hard Vertex Cover problem and has applications in approximation as well as parameterized algorithmics. We present a framework that generalizes Nemhauser and Trotter's result…
This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and…
We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending ${\sf RCA}_0$ and axiomatizable by a…