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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 22 (6 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
On the Integration of Connectionist and LogicBased Systems
, 2004
"... We discuss the computation by neural networks of semantic operators TP determined by propositional logic programs P. We revisit and clarify the foundations of the relevant notions employed in approximating both TP and its fixed points when P is a firstorder program. ..."
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Cited by 17 (8 self)
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We discuss the computation by neural networks of semantic operators TP determined by propositional logic programs P. We revisit and clarify the foundations of the relevant notions employed in approximating both TP and its fixed points when P is a firstorder program.
Connectionist Model generation: A FirstOrder Approach
, 2007
"... Knowledge based artificial neural networks have been applied quite successfully to propositional knowledge representation and reasoning tasks. However, as soon as these tasks are extended to structured objects and structuresensitive processes as expressed e.g., by means of firstorder predicate log ..."
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Cited by 17 (4 self)
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Knowledge based artificial neural networks have been applied quite successfully to propositional knowledge representation and reasoning tasks. However, as soon as these tasks are extended to structured objects and structuresensitive processes as expressed e.g., by means of firstorder predicate logic, it is not obvious at all what neural symbolic systems would look like such that they are truly connectionist, are able to learn, and allow for a declarative reading and logical reasoning at the same time. The core method aims at such an integration. It is a method for connectionist model generation using recurrent networks with feedforward core. We show in this paper how the core method can be used to learn firstorder logic programs in a connectionist fashion, such that the trained network is able to do reasoning over the acquired knowledge. We also report on experimental evaluations which show the feasibility of our approach.
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 16 (11 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.
Integrating FirstOrder Logic Programs and Connectionist Systems  A Constructive Approach
 Proceedings of the IJCAI05 Workshop on NeuralSymbolic Learning and Reasoning, NeSy’05
, 2005
"... Significant advances have recently been made concerning the integration of symbolic knowledge representation with artificial neural networks (also called connectionist systems). However, while the integration with propositional paradigms has resulted in applicable systems, the case of firstord ..."
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Cited by 11 (6 self)
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Significant advances have recently been made concerning the integration of symbolic knowledge representation with artificial neural networks (also called connectionist systems). However, while the integration with propositional paradigms has resulted in applicable systems, the case of firstorder knowledge representation has so far hardly proceeded beyond theoretical studies which prove the existence of connectionist systems for approximating firstorder logic programs up to any chosen precision.
A fully connectionist model generator for covered firstorder logic programs
 Proceedings of the Twentieth International Joint Conference on Artificial Intelligence (IJCAI07), Hyderabad, India, Menlo Park CA, AAAI Press (2007) 666–671
, 2007
"... We present a fully connectionist system for the learning of firstorder logic programs and the generation of corresponding models: Given a program and a set of training examples, we embed the associated semantic operator into a feedforward network and train the network using the examples. This resu ..."
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Cited by 11 (4 self)
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We present a fully connectionist system for the learning of firstorder logic programs and the generation of corresponding models: Given a program and a set of training examples, we embed the associated semantic operator into a feedforward network and train the network using the examples. This results in the learning of firstorder knowledge while damaged or noisy data is handled gracefully. 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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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
Some aspects of the integration of connectionist and logicbased systems
 In Proceedings of the Third International Conference on Information, pages 297–300, International Information Institute
, 2004
"... We discuss the computation by neural networks of semantic operators TP determined by propositional logic programs P over quite general manyvalued logics T. We revisit and clarify the foundations of the relevant notions employed in approximating both TP and ..."
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Cited by 8 (3 self)
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We discuss the computation by neural networks of semantic operators TP determined by propositional logic programs P over quite general manyvalued logics T. We revisit and clarify the foundations of the relevant notions employed in approximating both TP and
The core method: Connectionist model generation
 In Proceedings of the 16th International Conference on Artificial Neural Networks (ICANN
, 2006
"... Abstract. Knowledge based artificial networks networks have been applied quite successfully to propositional knowledge representation and reasoning tasks. However, as soon as these tasks are extended to structured objects and structuresensitive processes it is not obvious at all how neural symbol ..."
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Cited by 6 (3 self)
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Abstract. Knowledge based artificial networks networks have been applied quite successfully to propositional knowledge representation and reasoning tasks. However, as soon as these tasks are extended to structured objects and structuresensitive processes it is not obvious at all how neural symbolic systems should look like such that they are truly connectionist and allow for a declarative reading at the same time. The core method aims at such an integration. It is a method for connectionist model generation using recurrent networks with feedforward core. After an introduction to the core method, this paper will focus on possible connectionist representations of structured objects and their use in structuresensitive reasoning tasks. 1
Computing firstorder logic programs by fibring artificial neural networks
 Proceedings of the Eighteenth International Florida Artificial Intelligence Research Symposium Conference
, 2005
"... The integration of symbolic and neuralnetworkbased artificial intelligence paradigms constitutes a very challenging area of research. The overall aim is to merge these two very different major approaches to intelligent systems engineering while retaining their respective strengths. For symbolic ..."
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Cited by 6 (3 self)
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The integration of symbolic and neuralnetworkbased artificial intelligence paradigms constitutes a very challenging area of research. The overall aim is to merge these two very different major approaches to intelligent systems engineering while retaining their respective strengths. For symbolic paradigms that use the syntax of some firstorder language this appears to be particularly difficult. In this paper, we will extend on an idea proposed by Garcez and Gabbay (2004) and show how firstorder logic programs can be represented by fibred neural networks. The idea is to use a neural network to iterate a global counter n. For each clause Ci in the logic program, this counter is combined (fibred) with another neural network, which determines whether Ci outputs an atom of level n for a given interpretation I. As a result, the fibred network approximates the singlestep operator TP of the logic program, thus capturing the semantics of the program.