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The construction of robust computational models integrating reasoning and learning is a key research challenge for artificial intelligence. Recently, this challenge was also put forward as a fundamental problem in computer science [255]. Such a challenge intersects with another long-standing entry in the research agenda of artificial intelligence: the integration of its symbolic and connectionist paradigms. Such integration has long been a standing enterprise, with implications for and applications in cognitive science and neuroscience [51, 66, 130, 178, 179, 238, 240, 247, 248, 250]. Further, the importance of efforts to bridge the gap between the connectionist and symbolic paradigms of artificial intelligence has also been widely recognised (see e.g. [51, 66, 229, 242, 243]). [ABSTRACT FROM AUTHOR]

This chapter reviews the neural-symbolic approach presented in this book and provides a summary of the overall neural-symbolic cognitive model. The book deals with how to represent, learn, and compute expressive forms of symbolic knowledge using neural networks. We believe this is the way forward towards the provision of an integrated system of expressive reasoning and robust learning. The provision of such a system, integrating the two most fundamental phenomena of intelligent cognitive behaviour, has been identified as a key challenge for computer science [255]. Our goal is to produce computational models with integrated reasoning and learning capability, in which neural networks provide the machinery necessary for cognitive computation and learning, while logic provides practical reasoning and explanation capabilities to the neural models, facilitating the necessary interaction with the outside world. [ABSTRACT FROM AUTHOR]

Formal models of argumentation have been studied in several areas, notably in logic, philosophy, decision making, artificial intelligence, and law [25, 31, 39, 48, 83, 111, 153, 210, 212, 267]. In artificial intelligence, models of argumentation have been one of the approaches used in the representation of commonsense, nonmonotonic reasoning. They have been particularly successful in modelling chains of defeasible arguments so as to reach a conclusion [194, 209]. Although symbolic logic-based models have been the standard for the representation of argumentative reasoning [31, 108], such models are intrinsically related to artificial neural networks, as we shall show in this chapter. [ABSTRACT FROM AUTHOR]

Neural networks have been very successful as robust, massively parallel learning systems [125]. On the other hand, they have been severely criticised as being essentially propositional. In [176], John McCarthy argued that neural networks use unary predicates only, and that the concepts they compute are ground instances of these predicates. Thus, he claimed, neural networks could not produce concept descriptions, only discriminations. [ABSTRACT FROM AUTHOR]

This chapter presents some benchmark distributed-knowledge-representation applications of connectionist modal and intuitionistic reasoning. It shows how CML can be used for distributed knowledge representation and reasoning, illustrating the capabilities of the proposed connectionist model. It also compares the CML representation of a distributed knowledge representation problem with the representation of the same problem in connectionist intuitionistic logic (CIL), the type of reasoning presented in Chap. 7. We begin with a simple card game, as described in [87]. [ABSTRACT FROM AUTHOR]

In this chapter, we present a computational model combining intuitionistic reasoning and neural networks. We make use of ensembles of neural networks to represent intuitionistic theories, and show that for each intuitionistic theory and intuitionistic modal theory, there exists a corresponding neural-network ensemble that computes a fixed-point semantics of the theory. This provides a massively parallel model for intuitionistic reasoning. As usual, the neural networks can be trained from examples to adapt to new situations using standard neural learning algorithms, thus providing a unifying foundation for intuitionistic reasoning, knowledge representation, and learning. [ABSTRACT FROM AUTHOR]

In this chapter, we show that artificial neural networks can reason about probabilities, thus being able to integrate reasoning about uncertainty with modal, temporal, and epistemic logics, which have found a large number of applications, notably in game theory and in models of knowledge and interaction in multiagent systems [87,103,207]; artificial intelligence and computer science have made extensive use of decidable modal logics, including in the analysis and model checking of distributed and multiagent systems, in program verification and specification, and in hardware model checking. Finally, the combination of knowledge, time, and probability in a connectionist system provides support for integrated knowledge representation and learning in a distributed environment, dealing with the various dimensions of reasoning of an idealised agent [94, 202]. [ABSTRACT FROM AUTHOR]

As we have seen in Chaps. 4 to 7, neural networks can deal with a number of reasoning mechanisms. In many applications these need to be combined (fibred) into a system capable of dealing with the different dimensions of a reasoning agent. In this chapter, we introduce a methodology for combining neural-network architectures based on the idea of fibring logical systems [101]. Fibring allows one to combine different logical systems in a principled way. Fibred neural networks may be composed not only of interconnected neurons but also of other networks, forming a recursive architecture. A fibring function then defines how this recursive architecture behaves by defining how the networks in the ensemble relate to each other, typically by allowing the activation of neurons in one network (A) to influence the change of weights in another network (B). Intuitively, this can be seen as training network B at the same time as network A is running. Although both networks are simple, standard networks, we can show that, in addition to being universal approximators like standard feedforward networks, fibred neural networks can approximate any polynomial function to any desired degree of accuracy, thus being more expressive than standard feedforward networks. [ABSTRACT FROM AUTHOR]