Related papers: Fundamentals of Compositional Rewriting Theory
In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to…
In traditional rewriting theory, one studies a set of terms up to a set of rewriting relations. In algebraic rewriting, one instead studies a vector space of terms, up to a vector space of relations. Strikingly, although both theories are…
Additive composition (Foltz et al, 1998; Landauer and Dumais, 1997; Mitchell and Lapata, 2010) is a widely used method for computing meanings of phrases, which takes the average of vector representations of the constituent words. In this…
We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying…
Classically, any structure for a signature $\Sigma$ may be completed to a model of a desired regular theory $T$ by means of the chase construction or small object argument. Moreover, this exhibits $\mathrm{Mod}(T)$ as weakly reflective in…
So far, a very large amount of work in Natural Language Processing (NLP) rely on trees as the core mathematical structure to represent linguistic informations (e.g. in Chomsky's work). However, some linguistic phenomena do not cope properly…
Compositional generalization refers to the ability to generalize to novel combinations of previously observed words and syntactic structures. Since it is regarded as a desired property of neural models, recent work has assessed…
We develop a general model theoretic semantics to rewriting beyond the usual confluence and termination assumptions. This is based on preordered algebra which is a model theory that extends many sorted algebra. In this framework we…
We discuss an extension of the standard logical rules (functional application and abstraction) in Categorial Grammar (CG), in order to deal with some specific cases of polysemy. We borrow from Generative Lexicon theory which proposes the…
We provide an overview of the hybrid compositional distributional model of meaning, developed in Coecke et al. (arXiv:1003.4394v1 [cs.CL]), which is based on the categorical methods also applied to the analysis of information flow in…
String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and…
We describe a mathematical framework for equational reasoning about infinite families of string diagrams which is amenable to computer automation. The framework is based on context-free families of string diagrams which we represent using…
Systematic generalization refers to the capacity to understand and generate novel combinations from known components. Despite recent progress by large language models (LLMs) across various domains, these models often fail to extend their…
We introduce a notion of compatibility between constraint encoding and compositional structure. Phrased in the language of category theory, it is given by a "composable constraint encoding". We show that every composable constraint encoding…
Generalization of models to out-of-distribution (OOD) data has captured tremendous attention recently. Specifically, compositional generalization, i.e., whether a model generalizes to new structures built of components observed during…
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of unicity of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence…
Eklund et al. (2002) present a graphical technique aimed at simplifying the verification of various category-theoretic constructions, notably the composition of monads. In this note we take a different approach involving string rewriting.…
Compositional generalization is a key ability of humans that enables us to learn new concepts from only a handful examples. Neural machine learning models, including the now ubiquitous Transformers, struggle to generalize in this way, and…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are…