Notion Of Monad Research Articles

Monadic Expressions and Their Derivatives

There are several well-known ways to compute derivatives of regular expressions due to Brzozowski, Antimirov or Lombardy and Sakarovitch. We propose another one which abstracts the underlying data structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization advantage, we first introduce a new derivation technique based on the graded module monad and then show an application of this technique to generalize the parsing of expressions with capture groups and back references. We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad. Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.

Monadic Expressions and their Derivatives

We propose another interpretation of well-known derivatives computations from regular expressions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization advantage, we introduce a new derivation technique based on the graded module monad. We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad. Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.

The Sound Monad: A Philosophical Perspective on Sound Design

Abstract This article aims at sketching a philosophical theory of sound based on the perspective of sound designers: unique agents blurring the boundaries between engineering, music, acoustics and sound-based art. After having introduced the general framing in Section 1, focusing on a short history of the theory and practice of sound design, in Section 2 we propose a reading of sound as monad. We derive such intuition from the technology of digital sampling of audio signals, based on the decomposition of complex sound waves in a number of elementary sinusoidal waves. Thus, in Section 3, we attempt at grounding the resulting “sound-atom” on Leibniz’s notion of monad, intended both as a “simple substance without parts” and as a “nucleus of forces in statu possibilitatis.” The insight is resumed and further discussed in Section 4, where we draw our conclusions by demonstrating the fitness of such framing with regards to the standpoint of sound design, while accounting for the work of sound artists Carsten Nicolai and Ryoji Ikeda.

Leibniz e Hegel: em torno da questão dos princípios de Razão Suficiente e Identidade

This article attempts to bring some questions proposed by Hegel on the philosophy of Leibniz, in view of the considerations of this with respect to Principles of Philosophy or Monadology. Specifically, the question to be addressed is about reading proposed by Hegel around the whole notion of Monad in order to clarify some thoughts on this concept and how this comes within the second part of the Science of Logic, namely, the Doctrine of the Concept. The question posed is affiliated with two separate issues: the principle of sufficient reason and principle of identity. We should consider ultimately the thought of Leibniz proposes a philosophy for the reasoning of the individual, and the experience of identity and difference are still, albeit insufficient author Monadology.

From theFlos Substantiaeto the Vortex: A Shift in Leibniz's Early Philosophy

In February 1676, one of Leibniz's main concerns is with the problem of the seat of the soul and its relationship with the body, to which, in two very short papers, he provides two different solutions: the doctrine of the flos substantiae and the vortex theory. By analyzing the former, I suggest that, despite what other scholars claim, it is far from being an earlier exposition of the notion of monad. I argue that this doctrine is entertained by Leibniz only for a period, but is rejected later on and excluded from the final monadic system. This hypothesis seems to be supported by the shift to the notion of a vortex, which – despite having some evident pantheistic and monistic implications – offers a different solution to the problem of mind-body union, by identifying the soul as the only cement of matter. In this article, by following the progress of such a shift, we discover some fascinating nuances in the young Leibniz's development.

Notions of Monad Strength

Over the past two decades the notion of a strong monad has found wide applicability in computing. Arising out of a need to interpret products in computational and semantic settings, different approaches to this concept have arisen. In this paper we introduce and investigate the connections between these approaches and also relate the results to monad composition. We also introduce new methods for checking and using the required laws associated with such compositions, as well as provide examples illustrating problems and issues that arise.

Category Theoretic Understandings of Universal Algebra and its Dual: Monads and Lawvere Theories, Comonads and What?

Universal algebra is often known within computer science in the guise of algebraic specification or equational logic. In 1963, it was given a category theoretic characterisation in terms of what are now called Lawvere theories. Unlike operations and equations, a Lawvere theory is uniquely determined by its category of models. Except for a caveat about nullary operations, the notion of Lawvere theory is equivalent to the universal algebraistʼs notion of an abstract clone. Lawvere theories were soon followed by a further characterisation of universal algebra in terms of monads, the latter quickly becoming preferred by category theorists but not by universal algebraists. In the 1990ʼs began a systematic attempt to dualise the situation. The notion of monad dualises to that of comonad, providing a framework for studying transition systems in particular. Constructs in universal algebra have begun to be dualised too, with different leading examples. But there is not yet a definitive dual of the concept of Lawvere theory, or that of abstract clone, or even a definitive dual of operations and equations. We explore the situation here.

Countable Lawvere Theories and Computational Effects

Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.

Variations on Algebra: Monadicity and Generalisations of Equational Therories

Abstract. This is a largely tutorial paper about the categorical notion of monad and the ways in which monads on different categories correspond to variations on the standard notion of algebraic theory.

A new notation for arrows

The categorical notion of monad, used by Moggi to structure denotational descriptions, has proved to be a powerful tool for structuring combinator libraries. Moreover, the monadic programming style provides a convenient syntax for many kinds of computation, so that each library defines a new sublanguage. Recently, several workers have proposed a generalization of monads, called variously "arrows" or Freyd-categories. The extra generality promises to increase the power, expressiveness and efficiency of the embedded approach, but does not mesh as well with the native abstraction and application. Definitions are typically given in a point-free style, which is useful for proving general properties, but can be awkward for programming specific instances. In this paper we define a simple extension to the functional language Haskell that makes these new notions of computation more convenient to use. Our language is similar to the monadic style, and has similar reasoning properties. Moreover, it is extensible, in the sense that new combining forms can be defined as expressions in the host language.

Modularity and Dyads

We study modularity in denotational semantics. We define the notion of a dyad, generalising the notion of monad, and we define the notion of symmetric premonoidal dyad, generalising the notion of strong monad. We show how dyads give rise to semantic categories for modelling computational effects, generalising the Kleisli construction for a monad, and we show that symmetric premonoidal dyads give a symmetric premonoidal structure on the semantic category, allowing us to model contexts. We give an outline, supported by technical results, how this may be used to give a modular account of adding computational effects to the semantics of an idealised programming language.