Related papers: Representing permutations without permutations, or…
Sequential Constraint Grammar (SCG) (Karlsson, 1990) and its extensions have lacked clear connections to formal language theory. The purpose of this article is to lay a foundation for these connections by simplifying the definition of…
Diffusion language models, especially masked discrete diffusion models, have achieved great success recently. While there are some theoretical and primary empirical results showing the advantages of latent reasoning with looped transformers…
Coinductive reasoning about infinitary structures such as streams is widely applicable. However, practical frameworks for developing coinductive proofs and finding reasoning principles that help structure such proofs remain a challenge,…
We study two notions of expressiveness, which have appeared in abstraction theory for model checking, and find them incomparable in general. In particular, we show that according to the most widely used notion, the class of Kripke Modal…
We revisit the problem of computing (robust) controlled invariant sets for discrete-time linear systems. Departing from previous approaches, we consider implicit, rather than explicit, representations for controlled invariant sets.…
Transformer components such as non-linear activations and normalization are inherently non-injective, suggesting that different inputs could map to the same output and prevent exact recovery of the input from a model's representations. In…
A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations,…
Non-deductive reasoning systems are often {\em representation dependent}: representing the same situation in two different ways may cause such a system to return two different answers. Some have viewed this as a significant problem. For…
The expression problem describes a fundamental tradeoff between two types of extensibility: extending a type with new operations, such as by pattern matching on an algebraic data type in functional programming, and extending a type with new…
In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in $\mathbb{Z}^d$. The minimal shifts are those shifts in which all configurations contain exactly the…
Linear extended top-down tree transducers (or synchronous tree-substitution grammars) are popular formal models of tree transformations. The expressive power of compositions of such transducers with and without regular look-ahead is…
In the impredicative type theory of System F ({\lambda}2), it is possible to create inductive data types, such as natural numbers and lists. It is also possible to create coinductive data types such as streams. They work well in the sense…
This paper defines a linear representation for nonlinear maps $F:\mathbb{F}^n\rightarrow\mathbb{F}^n$ where $\mathbb{F}$ is a finite field, in terms of matrices over $\mathbb{F}$. This linear representation of the map $F$ associates a…
Designing and implementing typed programming languages is hard. Every new type system feature requires extending the metatheory and implementation, which are often complicated and fragile. To ease this process, we would like to provide…
There are two well-known formulations of recursive types: iso-recursive and equi-recursive types. Abadi and Fiore [1996] have shown that iso- and equi-recursive types have the same expressive power. However, their encoding of equi-recursive…
In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are…
Abductive explanations (AXp's) are widely used for understanding decisions of classifiers. Existing definitions are suitable when features are independent. However, we show that ignoring constraints when they exist between features may lead…
In this work, we study the concept of self-overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a…
This paper explores the characterization of equivariant linear layers for representations of permutations and related groups. Unlike traditional approaches, which address these problems using parameter-sharing, we consider an alternative…
We show that for any permutation $\pi$ there exists an integer $k_{\pi}$ such that every permutation avoiding $\pi$ as a pattern is a product of at most $k_{\pi}$ separable permutations. In other words, every strict class $\mathcal C$ of…