Exp-Function Method for Finding Exact Solutions of Nonlinear Evolution Equations

We applied Exp-function method to some nonlinear evolution equations to obtain its exact solution. The solution procedure of this method, by the help of symbolic computation of mathematical software, is of utter simplicity. The prominent merit of this method is to facilitate the process of solving systems of partial differential equations. These methods are straightforward and concise by themselves; moreover their applications are promising to obtain exact solutions of various partial differential equations. The obtained results show that Exp-function method is very powerful and convenient mathematical tool for nonlinear evolution equations in science and engineering

Symbolic software for symmetry reduction and computation of invariant solutions of differential equations

Problems involving partial or ordinary differential equations arise in various fields of science. Therefore, the task of obtaining exact solutions of differential equations is of primary importance, and attracts high attention. The main purpose of the current thesis is the development of a Maple-based, symbolic software package for symmetry reduction of differential equations and computation of symmetry-invariant solutions. The package developed in the current thesis is compatible with and can be viewed as an extension of the package GeM for symbolic symmetry analysis, developed by Prof. Alexei Cheviakov. The reduction procedure is based on the Lie's classical symmetry reduction method involving canonical coordinates. The developed package is applicable for obtaining solutions arising from extension of Lie's method, in particular, nonlocal and approximate symmetries. The developed software is applied to a number of PDE problems to obtain exact invariant solutions. The considered equations include the one-dimensional nonlinear heat equation, the potential Burgers' equation, as well as equations arising in nonlinear elastostatics and elastodynamics

A Computer Verified Theory of Compact Sets

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and displaying images with a computer. In this paper, I build upon existing work about complete metric spaces to define compact sets as the completion of the space of finite sets under the Hausdorff metric. This definition allowed me to quickly develop a computer verified theory of compact sets. I applied this theory to compute provably correct plots of uniformly continuous functions.Comment: This paper is to be part of the proceedings of the Symbolic Computation in Software Science Austrian-Japanese Workshop (SCSS 2008

A neural-symbolic system for temporal reasoning with application to model verification and learning

The effective integration of knowledge representation, reasoning and learning into a robust computational model is one of the key challenges in Computer Science and Artificial Intelligence. In particular, temporal models have been fundamental in describing the behaviour of Computational and Neural-Symbolic Systems. Furthermore, knowledge acquisition of correct descriptions of the desired system’s behaviour is a complex task in several domains. Several efforts have been directed towards the development of tools that are capable of learning, describing and evolving software models. This thesis contributes to two major areas of Computer Science, namely Artificial Intelligence (AI) and Software Engineering. Under an AI perspective, we present a novel neural-symbolic computational model capable of representing and learning temporal knowledge in recurrent networks. The model works in integrated fashion. It enables the effective representation of temporal knowledge, the adaptation of temporal models to a set of desirable system properties and effective learning from examples, which in turn can lead to symbolic temporal knowledge extraction from the corresponding trained neural networks. The model is sound, from a theoretical standpoint, but is also tested in a number of case studies. An extension to the framework is shown to tackle aspects of verification and adaptation under the SE perspective. As regards verification, we make use of established techniques for model checking, which allow the verification of properties described as temporal models and return counter-examples whenever the properties are not satisfied. Our neural-symbolic framework is then extended to deal with different sources of information. This includes the translation of model descriptions into the neural structure, the evolution of such descriptions by the application of learning of counter examples, and also the learning of new models from simple observation of their behaviour. In summary, we believe the thesis describes a principled methodology for temporal knowledge representation, learning and extraction, shedding new light on predictive temporal models, not only from a theoretical standpoint, but also with respect to a potentially large number of applications in AI, Neural Computation and Software Engineering, where temporal knowledge plays a fundamental role

Neurons and symbols: a manifesto

We discuss the purpose of neural-symbolic integration including its principles, mechanisms and applications. We outline a cognitive computational model for neural-symbolic integration, position the model in the broader context of multi-agent systems, machine learning and automated reasoning, and list some of the challenges for the area of neural-symbolic computation to achieve the promise of effective integration of robust learning and expressive reasoning under uncertainty