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Proof terms in term rewriting are a representation means for reduction sequences, and more in general for contraction activity, allowing to distinguish e.g simultaneous from sequential reduction. Proof terms for finitary, first-order,…
While the reasoning capabilities of Large Language Models (LLMs) excel in analytical tasks such as mathematics and code generation, their utility for abstractive summarization remains widely assumed but largely unverified. To bridge this…
All current investigations to analyze the derivational complexity of term rewrite systems are based on a single termination method, possibly preceded by transformations. However, the exclusive use of direct criteria is problematic due to…
Kan extensions provide a natural general framework for a variety of combinatorial problems. We have developed rewriting procedures for Kan extensions (over the category of sets) and this enables one program to address a wide range of…
In the trace reconstruction problem, the goal is to reconstruct an unknown string $x$ of length $n$ from multiple traces obtained by passing $x$ through the deletion channel. In the relaxed problem of $approximate$ trace reconstruction, the…
In recent months, substantial progress has been made in complex reasoning of Large Language Models, particularly through the application of test-time scaling. Notable examples include o1/o3/o4 series and DeepSeek-R1. When responding to a…
Solving an integer least squares (ILS) problem usually consists of two stages: reduction and search. This thesis is concerned with the reduction process for the ordinary ILS problem and the ellipsoid-constrained ILS problem. For the…
Covering and elimination inequalities are central to combinatorial optimization, yet their role has largely been studied in problem-specific settings or via no-good cuts. This paper introduces a unified perspective that treats these…
Listwise reranking utilizing Large Language Models (LLMs) has achieved state-of-the-art retrieval effectiveness. Recently, reasoning-enhanced models have further pushed these boundaries by employing Chain-of-Thought (CoT) to perform deep…
It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not.…
We give three different computations of the total number of runs of length $i$ in binary $n$-strings, and we discuss the connection of this problem with the compositions of $n$.
We describe arithmetic computations in terms of operations on some well known free algebras (S1S, S2S and ordered rooted binary trees) while emphasizing the common structure present in all them when seen as isomorphic with the set of…
In this paper, we study renormalization, that is, the procedure for eliminating singularities, for a special model using both combinatorial techniques in the framework of working with formal series, and using a limit transition in a…
This paper examines restart strategies for algorithms whose successful termination depends on an unknown parameter $\lambda$. After each restart, $\lambda$ is increased, until the algorithm terminates successfully. It is assumed that there…
The well-known trace reconstruction problem is the problem of inferring an unknown source string $x \in \{0,1\}^n$ from independent "traces", i.e. copies of $x$ that have been corrupted by a $\delta$-deletion channel which independently…
We define the regular Euclidean algorithm and the general form which leads to the method of least absolute remainders and also the method of negative remainders. We are going to show that if looked from the perspective of subtraction, the…
In this text I present a couple of new principles and thereon based iterative methods for numerical solution of sequences of systems of linear equations with fixed system matrix and changing right-hand-sides. The use of the new methods is…
A class of random recursive sequences (Y_n) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X\stackrel{L}{=}X. For…
The theory of finite and infinitary term rewriting is extensively developed for orthogonal rewrite systems, but to a lesser degree for weakly orthogonal rewrite systems. In this note we present some contributions to the latter case of weak…
General prescriptions of differential renormalization are presented. It is shown that renormalization group functions are straightforwardly expressed through some constants that naturally arise within this approach. The status of the action…