The past 25 years have witnessed a tremendous progress in the nature and scope of Logic and its application in Computing Science. At the same time, distributed learning systems and, in particular, Neural Networks, have been playing a central role in the development of Artificial Intelligence (AI). Notwithstanding, human beings do not use these techniques in isolation, but in an integrated way. The study of the integration of Logics and Neural Networks has now become crucial to the development of more effective AI and learning systems in computing.
In this special issue of the Journal of Applied Logic, we investigate how the symbolic and connectionist approaches to AI can be combined and integrated into Neural-Symbolic Systems. Our aim is to benefit from the advantages presented by each paradigm. By integrating Logic and Neural Networks, Neural-Symbolic Systems may provide (i) a logical characterisation of a connectionist system, (ii) a connectionist implementation of a logic, or (iii) a hybrid system bringing together advantages from connectionist systems and symbolic AI.
Firstly, a connectionist implementation of a logic can be produced with the use of a translation algorithm responsible for representing a logical theory in a neural network architecture. The translation must be based in a theorem showing the soundness of the algorithm, which allows the reasoning about the theory to be carried out in parallel in the network. If the network is a simple, standard connectionist model, knowledge acquisition and theory revision may take place as a result of learning from examples in the network, which can be performed with the help of a number of neural learning algorithms such as Backpropagation.
Secondly, a logical characterisation of a connectionist system may be provided by algorithms for rule extraction from neural networks. Rule extraction provides neural networks with explanation capability, the lack of which being frequently cited as neural nets main drawback. Generally speaking, rule extraction algorithms perform the inverse relation of that performed by the translation algorithms, once a trained neural network is given. As before, a proof of soundness of the rule extraction algorithm should be provided, whereas a proof of completeness would render the rules produced by the extraction algorithm equivalent to the original network. Equally important to these proofs, there are more practical issues of rule accuracy and comprehensibility, and issues of algorithmic complexity to be