Related papers: A Note on Ordinal DFAs

Ramsey's theorem states that for all finite colorings of an infinite set, there exists an infinite homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let $S$ be a linearly ordered set…

A generate and test algorithm is described which parses a surface form into one or more lexical entries using linearly ordered phonological rules. This algorithm avoids the exponential expansion of search space which a naive parsing…

In the lecture notes it is shown that an ordinal $\psi_{\Omega}(\varepsilon_{\mathbb{S}^{+}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory ${\sf KP}\omega+(M\prec_{\Sigma_{1}}V)$. In this note we show that ${\sf…

We show a theorem on monadic second-order k-ary queries on finite words. It may be illustrated by the following example: if the number of results of a query on binary strings is O(number of 0s $\times$ number of 1s), then each result can be…

(1) Let 1\leq k\leq \omega. Call an atom structure \alpha weakly k neat representable, the term algebra is in \RCA_n\cap \Nr_n\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra…

We introduce deterministic suffix-reading automata (DSA), a new automaton model over finite words. Transitions in a DSA are labeled with words. From a state, a DSA triggers an outgoing transition on seeing a word ending with the…

Most language modeling methods rely on large-scale data to statistically learn the sequential patterns of words. In this paper, we argue that words are atomic language units but not necessarily atomic semantic units. Inspired by HowNet, we…

We consider first-order logic over the subword ordering on finite words, where each word is available as a constant. Our first result is that the $\Sigma_1$ theory is undecidable (already over two letters). We investigate the decidability…

We consider the minimization of submodular functions subject to ordering constraints. We show that this optimization problem can be cast as a convex optimization problem on a space of uni-dimensional measures, with ordering constraints…

Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I${\it-reducible} if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An {\it obstruction} in a…

We study the expressive power and succinctness of order-invariant sentences of first-order (FO) and monadic second-order (MSO) logic on structures of bounded tree-depth. Order- invariance is undecidable in general and, thus, one strives for…

We explore the Weihrauch degree of the problems ``find a bad sequence in a non-well quasi order'' ($\mathsf{BS}$) and ``find a descending sequence in an ill-founded linear order'' ($\mathsf{DS}$). We prove that $\mathsf{DS}$ is strictly…

Dependency distance minimization (DDm) is a well-established principle of word order. It has been predicted theoretically that DDm implies compression, namely the minimization of word lengths. This is a second order prediction because it…

Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[<,MOD] with order comparison $x<y$ and predicates for $x \equiv i \mod n$ has been investigated by Barrington, Compton,…

Reasoning is a fundamentally algorithmic task. Yet current work on LLM-based reasoning relies on free-form generation whose theoretical guarantees (soundness, completeness, complexity, optimality) remain poorly understood. We argue that we…

Reordering is a challenge to machine translation (MT) systems. In MT, the widely used approach is to apply word based language model (LM) which considers the constituent units of a sentence as words. In speech recognition (SR), some phrase…

A zero-one language L is a regular language whose asymptotic probability converges to either zero or one. In this case, we say that L obeys the zero-one law. We prove that a regular language obeys the zero-one law if and only if its…

Despite large language models' (LLMs) recent advancements, their bias and hallucination issues persist, and their ability to offer consistent preferential rankings remains underexplored. This study investigates the capacity of LLMs to…

Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$…

Let $b$ be an integer strictly greater than $1$. Each set of nonnegative integers is represented in base $b$ by a language over $\{0, 1, \dots, b - 1\}$. The set is said to be $b$-recognisable if it is represented by a regular language. It…