Related papers: On Interpretations in B\"uchi Arithmetics

Let $\ord$ be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in $\ord$ is the sum of pairwise distinct units, if the unit equation $u+v=2$ has a non-trivial solution $u,v\in\ord^*$. We…

We propose $\omega$MSO$\Join$BAPA, an expressive logic for describing countable structures, which subsumes and transcends both Counting Monadic Second-Order Logic (CMSO) and Boolean Algebra with Presburger Arithmetic (BAPA). We show that…

Interpretation of a structure $\mathbb A$ in $\mathbb B$ allows to produce structures elementarily equivalent to $\mathbb A$ given those elementarily equivalent to $\mathbb B$. In particular, interpretation of the free group in $\mathbb N$…

Due to the widespread use of complex machine learning models in real-world applications, it is becoming critical to explain model predictions. However, these models are typically black-box deep neural networks, explained post-hoc via…

I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a…

We study a certain truncation of the ring of arithmetical functions with unitary convolution, consisting of functions vanishing on arguments >n. The truncations are artinian monomial quotients of a polynomial ring in finitely many…

Let $BV_p[0,1]$, $1\le p<\infty$, be the Banach algebra of functions of bounded $p$-variation in the sense of Wiener. Recently, Kowalczyk and Turowska \cite{KT19} proved that the multiplication in $BV_1[0,1]$ is an open bilinear mapping. We…

This short note exhibits an example of a Sigma^1_1-complete language that can be recognised by a one blind counter B\"uchi automaton (or equivalently a B\"uchi VASS with only one place).

Mechanistic Interpretability aims to understand neural networks through causal explanations. We argue for the Explanatory View Hypothesis: that Mechanistic Interpretability research is a principled approach to understanding models because…

We show that the classical interpretations of Tarski's inductive definitions actually allow us to define the satisfaction and truth of the quantified formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers…

While concept-based interpretability methods have traditionally focused on local explanations of neural network predictions, we propose a novel framework and interactive tool that extends these methods into the domain of mechanistic…

In this paper we consider the set ${\mathbb Z}^{\pm\omega}_{6}$ of two-way infinite words $\xi$ over the alphabet $\{0,1,2,3,4,5\}$ with the integer left part $\lfloor\xi\rfloor$ and the fractional right part $\{\xi\}$ separated by a radix…

A necessary and sufficient condition for an operator space to support a multiplication making it completely isometric and isomorphic to a unital operator algebra is proved. The condition involves only the holomorphic structure of the Banach…

High-dimensional representations for words, text, images, knowledge graphs and other structured data are commonly used in different paradigms of machine learning and data mining. These representations have different degrees of…

We consider the ring I_n of polynomial invariants over weighted graphs on n vertices. Our primary interest is the use of this ring to define and explore algebraic versions of isomorphism problems of graphs, such as Ulam's reconstruction…

We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an…

We examine the complexity of basic regular operations on languages represented by Boolean and alternating finite automata. We get tight upper bounds m+n and m+n+1 for union, intersection, and difference, 2^m+n and 2^m+n+1 for concatenation,…

Let $B^n$ be the $n$-dimensional unit complex ball and let $a$ and $b$ be two distinct points in its closure. Let $f$ be a real-analytic function on the complex unit sphere $\partial B^n.$ Suppose that for any complex line $L,$ meeting the…

The paper discusses the role of interpretations, understood as multifunctions that preserve and reflect logical consequence, as refinement witnesses in the general setting of pi-institutions. This leads to a smooth generalization of the…

Solutions of some partial differential equations are obtained as critical points of a real funtional. Then the Banach space where this functional is defined has to be real, otherwise, it is not differentiable. It follows that the equation…