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Logic Programs and Connectionist Networks
 Journal of Applied Logic
, 2004
"... One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic p ..."
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Cited by 62 (22 self)
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One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic programs can be computed by feedforward connectionist networks, and that the same semantic operators for firstorder normal logic programs can be approximated by feedforward connectionist networks. Turning the networks into recurrent ones allows one also to approximate the models associated with the semantic operators. Our methods depend on a wellknown theorem of Funahashi, and necessitate the study of when Funahasi's theorem can be applied, and also the study of what means of approximation are appropriate and significant.
Computational Logic and Human Thinking: How to be Artificially Intelligent
, 2011
"... The mere possibility of Artificial Intelligence (AI) – of machines that can think and act as intelligently as humans – can generate strong emotions. While some enthusiasts are excited by the thought that one day machines may become more intelligent than people, many of its critics view such a prosp ..."
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Cited by 39 (10 self)
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The mere possibility of Artificial Intelligence (AI) – of machines that can think and act as intelligently as humans – can generate strong emotions. While some enthusiasts are excited by the thought that one day machines may become more intelligent than people, many of its critics view such a prospect with horror. Partly because these controversies attract so much attention, one of the most important accomplishments of AI has gone largely unnoticed: the fact that many of its advances can also be used directly by people, to improve their own human intelligence. Chief among these advances is Computational Logic. Computational Logic builds upon traditional logic, which was originally developed to help people think more effectively. It employs the techniques of symbolic logic, which has been used to build the foundations of mathematics and computing. However, compared with traditional logic, Computational Logic is much more powerful; and compared with symbolic logic, it is much simpler and more practical. Although the applications of Computational Logic in AI require the use of mathematical notation, its human applications do not. As a consequence, I have written the main part of this book informally, to reach as wide an audience as possible. Because human thinking is also the subject of study in many other fields, I have drawn upon related studies in Cognitive Psychology, Linguistics, Philosophy, Law, Management Science and English
The Connectionist Inductive Learning and Logic Programming System
, 1999
"... This paper presents the Connectionist Inductive Learning and Logic Programming System (CIL²P). CIL²P is a new massively parallel computational model based on a feedforward Artificial Neural Network that integrates inductive learning from examples and background knowledge, with deductive learning ..."
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Cited by 28 (8 self)
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This paper presents the Connectionist Inductive Learning and Logic Programming System (CIL²P). CIL²P is a new massively parallel computational model based on a feedforward Artificial Neural Network that integrates inductive learning from examples and background knowledge, with deductive learning from Logic Programming. Starting with the background knowledge represented by a propositional logic program, a translation algorithm is applied generating a neural network that can be trained with examples. The results obtained with this refined network can be explained by extracting a revised logic program from it. Moreover, the neural network computes the stable model of the logic program inserted in it as background knowledge, or learned with the examples, thus functioning as a parallel system for Logic Programming. We have successfully applied CIL2Ptotwo realworld problems of computational biology, specifically DNA sequence analyses. Comparisons with the results obtained by some of the main neural, symbolic, and hybrid inductive learning systems, using the same domain knowledge, show the effectiveness of CIL²P.
Dimensions of neuralsymbolic integration – a structural survey
 We Will Show Them: Essays in Honour of Dov Gabbay
"... Research on integrated neuralsymbolic systems has made significant progress in the recent past. In particular the understanding of ways to deal with symbolic knowledge within connectionist systems (also called artificial neural networks) has reached a critical mass which enables the community to ..."
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Cited by 25 (8 self)
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Research on integrated neuralsymbolic systems has made significant progress in the recent past. In particular the understanding of ways to deal with symbolic knowledge within connectionist systems (also called artificial neural networks) has reached a critical mass which enables the community to
Fibring Neural Networks
, 2004
"... Neuralsymbolic systems are hybrid systems that integrate symbolic logic and neural networks. The goal of neuralsymbolic integration is to benefit from the combination of features of the symbolic and connectionist paradigms of artificial intelligence. This paper introduces a new neural network ..."
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Cited by 20 (7 self)
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Neuralsymbolic systems are hybrid systems that integrate symbolic logic and neural networks. The goal of neuralsymbolic integration is to benefit from the combination of features of the symbolic and connectionist paradigms of artificial intelligence. This paper introduces a new neural network architecture based on the idea of fibring logical systems. Fibring allows one to combine di#erent 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 must behave 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 that one runs network A. We 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.
Logic Programs, Iterated Function Systems, and Recurrent Radial Basis Function Networks
 Journal of Applied Logic
, 2004
"... Graphs of the singlestep operator for firstorder logic programs  displayed in the real plane  exhibit selfsimilar structures known from topological dynamics, i.e. they appear to be fractals, or more precisely, attractors of iterated function systems. We show that this observation can be ..."
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Cited by 19 (14 self)
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Graphs of the singlestep operator for firstorder logic programs  displayed in the real plane  exhibit selfsimilar structures known from topological dynamics, i.e. they appear to be fractals, or more precisely, attractors of iterated function systems. We show that this observation can be made mathematically precise. In particular, we give conditions which ensure that those graphs coincide with attractors of suitably chosen iterated function systems, and conditions which allow the approximation of such graphs by iterated function systems or by fractal interpolation. Since iterated function systems can easily be encoded using recurrent radial basis function networks, we eventually obtain connectionist systems which approximate logic programs in the presence of function symbols.
Extracting Symbolic Rules from Trained Neural Network Ensembles
 AI COMMUNICATIONS
, 2003
"... Neural network ensemble can significantly improve the generalization ability of neural network based systems. However, its comprehensibility is even worse than that of a single neural network because it comprises a collection of individual neural networks. In this paper, an approach named REFNE is p ..."
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Cited by 17 (5 self)
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Neural network ensemble can significantly improve the generalization ability of neural network based systems. However, its comprehensibility is even worse than that of a single neural network because it comprises a collection of individual neural networks. In this paper, an approach named REFNE is proposed to improve the comprehensibility of trained neural network ensembles that perform classification tasks. REFNE utilizes the trained ensembles to generate instances and then extracts symbolic rules from those instances. It gracefully breaks the ties made by individual neural networks in prediction. It also employs specific discretization scheme, rule form, and fidelity evaluation mechanism. Experiments show that with different configurations, REFNE can extract rules with good fidelity that well explain the function of trained neural network ensembles, or rules with strong generalization ability that are even better than the trained neural network ensembles in prediction.
Combining Abductive Reasoning and Inductive Learning to Evolve Requirements Specifications
 IEE Proceedings  Software
, 2003
"... The development of requirements specifications inevitably involves modification and evolution. To support modification while preserving particular requirements goals and properties, we propose the use of a cycle composed of two phases: analysis and revision. In the analysis phase, a desirable prop ..."
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Cited by 15 (6 self)
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The development of requirements specifications inevitably involves modification and evolution. To support modification while preserving particular requirements goals and properties, we propose the use of a cycle composed of two phases: analysis and revision. In the analysis phase, a desirable property of the system is checked against a partial specification. Should the property be violated, diagnostic information is provided. In the revision phase, the diagnostic information is used to help modify the specification in such a way that the new specification no longer violates the original property.
The logical way to be artificially intelligent
 Computational Logic in MultiAgent Systems (CLIMA VI PostProceedings), LNAI 3900
, 2006
"... Abstract. Abductive logic programming (ALP) can be used to model reactive, proactive and preactive thinking in intelligent agents. Reactive thinking assimilates observations of changes in the environment, whereas proactive thinking reduces goals to subgoals and ultimately to candidate actions. Pre ..."
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Cited by 13 (3 self)
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Abstract. Abductive logic programming (ALP) can be used to model reactive, proactive and preactive thinking in intelligent agents. Reactive thinking assimilates observations of changes in the environment, whereas proactive thinking reduces goals to subgoals and ultimately to candidate actions. Preactive thinking generates logical consequences of candidate actions, to help in deciding between the alternatives. These different ways of thinking are compatible with any way of deciding between alternatives, including the use of both decision theory and heuristics. The different forms of thinking can be performed as they are needed, or they can be performed in advance, transforming highlevel goals and beliefs into lowerlevel conditionaction rule form, which can be implemented in neural networks. Moreover, the higherlevel and lowerlevel representations can operate in tandem, as they do in dualprocess models of thinking. In dual process models, intuitive processes form judgements rapidly, subconsciously and in parallel, while deliberative processes form and monitor judgements slowly, consciously and serially. ALP used in this way can not only provide a framework for constructing artificial agents, but can also be used as a cognitive model of human agents. As a cognitive model, it combines both a descriptive model of how humans actually think with a normative model of humans can think more effectively. 1
The integration of connectionism and firstorder knowledge representation and reasoning as a challenge for artificial intelligence
 In Proceedings of the Third International Conference on Information
, 2006
"... Intelligent systems based on firstorder logic on the one hand, and on artificial neural networks (also called connectionist systems) on the other, differ substantially. It would be very desirable to combine the robust neural networking machinery with symbolic knowledge representation and reasoning ..."
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Cited by 12 (8 self)
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Intelligent systems based on firstorder logic on the one hand, and on artificial neural networks (also called connectionist systems) on the other, differ substantially. It would be very desirable to combine the robust neural networking machinery with symbolic knowledge representation and reasoning paradigms like logic programming in such a way that the strengths of either paradigm will be retained. Current stateoftheart research, however, fails by far to achieve this ultimate goal. As one of the main obstacles to be overcome we perceive the question how symbolic knowledge can be encoded by means of connectionist systems: Satisfactory answers to this will naturally lead the way to knowledge extraction algorithms and to integrated neuralsymbolic systems. 1