Neuro-Symbolic Integration of Hopfield Neural Network for Optimal Maximum Random kSatisfiability (Maxrksat) Representation

  • Hamza Abubakar School of Mathematical Sciences, Universiti Sains Malaysia (USM), Pulau Pinang, Malaysia
  • Sagir Abdu Masanawa Department of Mathematics, Federal University Dutsin-Ma (FUD), Katsina State, Nigeria
  • Surajo Yusuf  Department of Mathematics, Isa Kaita College of Education, Dutsin-Ma, Katsina, Nigeria
Keywords: Artificial neural networks, hopfield neural networks, wan abdullahi method, boolean Satisfiability, random maximum ksatisfiabilit

Abstract

Boolean satisfiability logical representation is a programming paradigm that has its foundations in mathematical logic. It has been classified as an NP-complete problem that difficult practical combinatorial optimization and search problems can be easily converted into it. Random Maximum kSatisfiability (MAX-RkSAT) composed of the most consistent mapping in a Boolean formula that generates a maximum number of random satisfied clauses. Many optimization and search problems can be easily expressed by mapping the problem into a Hopfield neural network (HNN) to minimize the optimal configuration of the corresponding Lyapunov energy function. In this paper, a hybrid computational model hs been proposed that incorporates the Random Maximum kSatisfiability (MAX-RkSAT) into the Hopfield neural network (HNN) for optimal Random Maximum kSatisfiability representation (HNN-MAX-RkSAT). Hopfield neural network learning will be integrated with the random maximum satisfiability to enhance the correct neural state of the network model representation. The computer simulation using C+++⁣+ has been used to demonstrate the ability of MAX-RkSAT to be embedded optimally in Hopfield neural network to serve as Neuro-symbolic integration. The performance of the proposed hybrid HNN-MAXRkSAT model has been explored and compared with the existing model. The proposed HNN-MAXRkSAT demonstrates good agreement with the existing models measured in terms of Global minimum Ratio (Gm), Hamming Distance (HD), Mean Absolute Error (MAE) and network computation Time CPU time). The proposed framework explored that MAX-RkSAT can be optimally represented in HNN and subsequently provides an additional platform for neural-symbolic integration, representing the various types of satisfiability logic.

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Author Biographies

Hamza Abubakar, School of Mathematical Sciences, Universiti Sains Malaysia (USM), Pulau Pinang, Malaysia

Hamza Abubakar received both his B.Sc (Mathematics) and MSc (Financial Mathematics) from the University of Abuja, Nigeria in (2006) and (2014) respectively. He is currently pursuing a PhD degree in Mathematical Science, Universiti Sains Malaysia. Hamza joint the service of Isa Kaita College of Education, Dutsin-ma, Katsina, Nigeria in 2008 and raised from an assistant lecturer to a Senior lecturer in Mathematics and computers. He is an active member of the Nigerian Mathematical Society, Mathematical Association of Nigeria, Science Teachers Association of Nigeria and International Association of Engineers (OR and AI). His research interest include, Financial Mathematics, neural network modelling and metaheuristics optimization.

Sagir Abdu Masanawa, Department of Mathematics, Federal University Dutsin-Ma (FUD), Katsina State, Nigeria

Sagir Abdu Masanawa is a Senior Lecturer in the Department of Mathematical Sciences, Federal University Dutsin-ma, Nigeria. He received his BSc Mathematics, Executive PGD in Computer Studies, MSc Information Technology and MTech Mathematics from Bayero University Kano, A.T.B.U. Bauchi, National Open University of Nigeria and Federal University of Technology Minna respectively. He became a Doctor of Philosophy in Mathematics (Neural Network) from Universiti Sains Malaysia. His current research interest includes neural networks, Data mining, Metaheuristic algorithms, Information Technology and Numerical methods.

Surajo Yusuf , Department of Mathematics, Isa Kaita College of Education, Dutsin-Ma, Katsina, Nigeria

Surajo Yusuf obtained his BSc. (Mathematics) and PGD (Computer Science) both at Bayero University Kano, Nigeria. He also obtained PGD in Education at Federal College of Education Kano, Nigeria. Surajo received his MSc. In Mathematical Sciences from the Universiti Teknologi Malaysia. He is currently a Senior lecturer with the Department of Mathematics, Isa Kaita College of Education Dutsin-ma, Katsina state Nigeria. His research interest include, Topology and modelling of networks.

Published
2020-11-02
Section
Articles