Related papers: The Compositional Integral: A Brief Introduction
In this article, we explore a series of elementary yet insightful results involving integrals related to Gaussian sums. Using techniques rooted in classical calculus, we derive several identities and evaluate nontrivial definite integrals…
We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics.
Compositionality is believed to be fundamental to intelligence. In humans, it underlies the structure of thought, language, and higher-level reasoning. In AI, compositional representations can enable a powerful form of out-of-distribution…
Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the…
In this paper we introduce a framework for option model composition. Option models are temporal abstractions that, like macro-operators in classical planning, jump directly from a start state to an end state. Prior work has focused on…
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…
The contribution of this paper is threefold: first, it defines a framework for modelling component-based systems, as well as a formalization of integration rules to combine their behavior. This is based on finite state machines (FSM).…
Graph compositions generalize both integer compositions and partitions of a finite set. We develop formulas, generating functions and recurrence relations for composition counting functions for several families of graphs.
There are different meanings of the term "compositionality" within science: what one researcher would call compositional, is not at all compositional for another researcher. The most established conception is usually attributed to Frege,…
It is of course well known that the usual definitions of Riemann integration and Riemann integrals are equivalent to simpler definitions which can be expressed in terms of just one sequence of partitions, using dyadic intervals or dyadic…
Compositional Explanations is a method for identifying logical formulas of concepts that approximate the neurons' behavior. However, these explanations are linked to the small spectrum of neuron activations (i.e., the highest ones) used to…
A generally intelligent learner should generalize to more complex tasks than it has previously encountered, but the two common paradigms in machine learning -- either training a separate learner per task or training a single learner for all…
Compositional data analysis is concerned with multivariate data that have a constant sum, usually 1 or 100\%. These are data often found in biochemistry and geochemistry, but also in the social sciences, when relative values are of interest…
This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems…
Compositional generalization is a crucial step towards developing data-efficient intelligent machines that generalize in human-like ways. In this work, we tackle a challenging form of distribution shift, termed compositional shift, where…
Compositional data represent a specific family of multivariate data, where the information of interest is contained in the ratios between parts rather than in absolute values of single parts. The analysis of such specific data is…
An integral formula is developed which applies to an essentially arbitrary function. An application is made to the Riemann zeta function.
This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some…
This essay recounts my personal journey towards a deeper understanding of the mathematical foundations of algorithmic music composition. I do not spend much time on specific mathematical algorithms used by composers; rather, I focus on…
Compositionality is a cognitive mechanism that allows humans to systematically combine known concepts in novel ways. This study demonstrates how artificial neural agents acquire and utilize compositional generalization to describe…