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We describe an equivalent formulation of algebraic weak factorisation systems, not involving monads and comonads, but involving double categories of morphisms equipped with a lifting operation satisfying lifting and factorisation axioms.
Morphisms, structure preserving maps, are everywhere in Mathematics as useful tools for thinking and problem solving, or as objects to study. Here, we argue that the idea of operations being compatible across two domains goes beyond its…
Many tools used to process programs, like compilers, analyzers, or verifiers, perform transformations on their intermediate program representation, like abstract syntax trees. Implementing such program transformations is a non-trivial task,…
Tractability results for the model checking problem of logics yield powerful algorithmic meta theorems of the form: Every computational problem expressible in a logic $L$ can be solved efficiently on every class $\mathscr{C}$ of structures…
Inspired by distributed algorithms, we introduce a new class of finite graph automata that recognize precisely the graph languages definable in monadic second-order logic. For the cases of words and trees, it has been long known that the…
Much theoretical work has described the ability of transformers to represent formal languages. However, linking theoretical results to empirical performance is not straightforward due to the complex interplay between the architecture, the…
We introduce layered automata, a subclass of alternating parity automata that generalises deterministic automata. Assuming a consistency property, these automata are history deterministic and 0-1 probabilistic. We show that every…
Large language models (LLMs) are increasingly evaluated on reasoning tasks, yet their logical abilities remain contested. To address this, we study LLMs' reasoning in a well-defined fragment of logic: syllogistic reasoning. We cast the…
Regular languages -- the languages accepted by deterministic finite automata -- are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
In this paper, we propose several dictionary learning algorithms for sparse representations that also impose specific structures on the learned dictionaries such that they are numerically efficient to use: reduced number of…
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…
Weighted monadic second-order logic is a weighted extension of monadic second-order logic that captures exactly the behaviour of weighted automata. Its semantics is parameterized with respect to a semiring on which the values that weighted…
We define a notion of morphism between combinatorial codes, making the class of all combinatorial codes into a category $\mathbf{Code}$. We show that morphisms can be used to remove redundant information from a code, and that morphisms…
A regular language is almost fully characterized by its right congruence relation. Indeed, a regular language can always be recognized by a DFA isomorphic to the automaton corresponding to its right congruence, henceforth the Rightcon…
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…
Recently, language models (LMs) have shown impressive proficiency in code generation tasks, especially when fine-tuned on code-specific datasets, commonly known as Code LMs. However, our understanding of the internal decision-making…
Computational models of human language often involve combinatorial problems. For instance, a probabilistic parser may marginalize over exponentially many trees to make predictions. Algorithms for such problems often employ dynamic…
Research on information extraction from Web pages (wrapping) has seen much activity recently (particularly systems implementations), but little work has been done on formally studying the expressiveness of the formalisms proposed or on the…
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…