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A block in a linear order is an equivalence class when factored by the block relation B(x,y), satisfied by elements that are finitely far apart. We show that every computable linear order with dense condensation-type (i.e. a dense…

We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two…

Recursive definitions of predicates are usually interpreted either inductively or coinductively. Recently, a more powerful approach has been proposed, called flexible coinduction, to express a variety of intermediate interpretations,…

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence…

This paper examines the characterization and learning of grammars defined with enriched representational models. Model-theoretic approaches to formal language theory traditionally assume that each position in a string belongs to exactly one…

These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following…

Continuous word representations learned separately on distinct languages can be aligned so that their words become comparable in a common space. Existing works typically solve a least-square regression problem to learn a rotation aligning a…

Motivated by an application where we try to make proofs for Description Logic inferences smaller by rewriting, we consider the following decision problem, which we call the small term reachability problem: given a term rewriting system $R$,…

This paper is motivated by the theory of sequential dynamical systems, developed as a basis for a mathematical theory of computer simulation. It contains a classification of finite dynamical systems on binary strings, which are obtained by…

We consider existential rules (aka Datalog+) as a formalism for specifying ontologies. In recent years, many classes of existential rules have been exhibited for which conjunctive query (CQ) entailment is decidable. However, most of these…

We study an alternative model of infinitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions. We consider both a weak and a strong notion of convergence and…

A theory of recursive and corecursive definitions has been developed in higher-order logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive…

Automata-logic connections are pillars of the theory of regular languages. Such connections are harder to obtain for transducers, but important results have been obtained recently for word-to-word transformations, showing that the three…

Given an infinite word ${\bf w}$ on a finite alphabet, an immediate question arises:~can we understand the frequency of letters in ${\bf w}$\,? For words that are the fixed points of substitutions, the answer to this question is often `yes'…

It seems that the index theory for non-compact spaces has found its ultimate formulation in realm of coarse spaces and $K$-theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and…

We introduce a coinductive definition of infinitary term rewriting. The setup is surprisingly simple, and has in contrast to the usual definitions of infinitary rewriting, neither need for ordinals nor for metric convergence. While the idea…

In this paper, we investigate the structure of the most general kind of substitution shifts, including non-minimal ones, and allowing erasing morphisms. We prove the decidability of many properties of these morphisms with respect to the…

Conventional word embeddings represent words with fixed vectors, which are usually trained based on co-occurrence patterns among words. In doing so, however, the power of such representations is limited, where the same word might be…

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions…

We establish various complexity results for the entailment problem between formulas in Separation Logic with user-defined predicates denoting recursive data structures. The considered fragments are characterized by syntactic conditions on…