1,721,078 research outputs found
The Algebra of Partial Equivalence Relations
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1611339.pdf (Publisher’s version ) (Open Access
A Predicate/State Transformer Semantics for Bayesian Learning
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161290.pdf (Publisher’s version ) (Open Access
Rewriting in Free Hypegraph Categories
Full text linkWe study rewriting for equational theories in the context of symmetric monoidal categories where there is a separable Frobenius monoid on each object. These categories, also called hypergraph categories, are increasingly relevant: Frobenius structures recently appeared in cross-disciplinary applications, including the study of quantum processes, dynamical systems and natural language processing. In this work we give a combinatorial characterisation of arrows of a free hypergraph category as cospans of labelled hypergraphs and establish a precise correspondence between rewriting modulo Frobenius structure on the one hand and double-pushout rewriting of hypergraphs on the other. This interpretation allows to use results on hypergraphs to ensure decidability of confluence for rewriting in a free hypergraph category. Our results generalise previous approaches where only categories generated by a single object (props) were considered
Functorial semantics as a unifying perspective on logic programming
Full text linkLogic programming and its variations are widely used for formal reasoning in various areas of Computer Science, most notably Artificial Intelligence. In this paper we develop a systematic and unifying perspective for (ground) classical, probabilistic, weighted logic programs, based on categorical algebra. Our departure point is a formal distinction between the syntax and the semantics of programs, now regarded as separate categories. Then, we are able to characterise the various variants of logic program as different models for the same syntax category, i.e. structure-preserving functors in the spirit of Lawvere’s functorial semantics. As a first consequence of our approach, we showcase a series of semantic constructs for logic programming pictorially as certain string diagrams in the syntax category. Secondly, we describe the correspondence between probabilistic logic programs and Bayesian networks in terms of the associated models. Our analysis reveals that the correspondence can be phrased in purely syntactical terms, without resorting to the probabilistic domain of interpretation
Income and educational differences in grandparental childcare: evidence from English grandmothers and grandfathers
Full text linkGrandparents are actively involved in grandchildren's lives, but there is little research concerning socio-economic differences in the content of the relationship. This study explores the socio-economic gradient in childcare provided by grandparents, touching on the intensity of care, the activities performed with grandchildren and the motives driving this involvement, by grandparents’ gender. We explore two dimensions of socio-economic status, education and family income, pertaining to different dimensions of grandparents’ and grandchildren's relationship: child development versus parental childcare needs. Using the English Longitudinal Study of Ageing (ELSA 2016–2017, 2018–2019), logistic regression models show that intensive care is more common for grandfathers in the lowest income tercile. A high income decreases involvement in physical care activities (i.e. preparing meals); instead, the involvement is driven by motives to help children financially. Higher education is a good predictor of support with homework, driven by motives to ‘help grandchildren develop as people’. Even though grandfathers show an involvement in grandchildren's upbringing, highly-educated grandmothers remain the most inclined to offer support. Overall, the study suggests that grandparents’ involvement in grandchildren's lives could be among the mechanisms structuring the intergenerational transmission of inequality
Categories of Differentiable Polynomial Circuits for Machine Learning
Full text linkReverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular model classes: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose polynomial circuits as a suitable machine learning model. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values
String Diagrams for Layered Explanations
Full text linkWe propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems biology, electrical circuit theory, and concurrency. We demonstrate how three explanatory models in these seemingly diverse areas can be all understood uniformly via a graphical calculus of layered props. Layered props allow for a compact visual presentation of the same phenomenon at different levels of precision, as well as the translation between these levels. Notably, our approach allows for partial explanations, that is, for translating just one part of a diagram while keeping the rest of the diagram untouched. Furthermore, our approach paves the way for formal reasoning about counterfactual models in systems biology
A string diagrammatic axiomatisation of finite-state automata
Full text linkWe develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite— a result which is provably impossible for the one-dimensional syntax of regular expressions. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks
Coalgebraic semantics for probabilistic logic programming
Full text linkProbabilistic logic programming is increasingly important in artificial intelligence and related fields as a formalism to reason about uncertainty. It generalises logic programming with the possibility of annotating clauses with probabilities. This paper proposes a coalgebraic semantics on probabilistic logic programming. Programs are modelled as coalgebras for a certain functor F, and two semantics are given in terms of cofree coalgebras. First, the cofree F-coalgebra yields a semantics in terms of derivation trees. Second, by embedding F into another type G, as cofree G-coalgebra we obtain a 'possible worlds' interpretation of programs, from which one may recover the usual distribution semantics of probabilistic logic programming. Furthermore, we show that a similar approach can be used to provide a coalgebraic semantics to weighted logic programming
Reverse derivative ascent: A categorical approach to learning boolean circuits
Full text linkWe introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets
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