Research into the processing of symbolic knowledge by means of connectionist networks aims at systems which combine the declarative nature of logic-based artificial intelligence with the robustness and trainability of artificial neural networks. This endeavour has been addressed quite successfully in the past for propositional knowledge representation and reasoning tasks. However, as soon as these tasks are extended beyond propositional logic, it is not obvious at all what 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 – which we present here – aims at such an integration. It is a method for connectionist model generation using recurrent networks with feedforward core. These networks can be trained by standard algorithms to learn symbolic knowledge, and they can be used for reasoning about this knowledge.

Artificial neural networks can be trained to perform excellently in many application areas. While they can learn from raw data to solve sophisticated recognition and analysis problems, the acquired knowledge remains hidden within the network architecture and is not readily accessible for analysis or further use: Trained networks are black boxes. Recent research efforts therefore investigate the possibility to extract symbolic knowledge from trained networks, in order to analyze, validate, and reuse the structural insights gained implicitly during the training process. In this paper, we will study how knowledge in form of propositional logic programs can be obtained in such a way that the programs are as simple as possible --- where simple is being understood in some clearly defined and meaningful way.

Abstract. Whereas symbol–based systems, like deductive reasoning devices, knowledge bases, planning systems, or tools for solving constraint satisfaction problems, presuppose (more or less) the consistency of data and the consistency of results of internal computations, this is far from being plausible in real–world applications,

Abstract. The paper formalises the famous algorithm of first-order unification by Robinson by means of the error-correction learning in neural networks. The significant achievement of this formalisation is that, for the first time, the first-order unification of two arbitrary first-order atoms is performed by finite (two-neuron) network. 1

Currently, neural-symbolic integration covers – at least in theory – a whole bunch of types of reasoning: neural representations (and partially also neural-inspired learning approaches) exist for modeling propositional logic (programs), whole classes of manyvalued logics, modal logic, temporal logic, and epistemic logic, just to mention some important examples [2,4]. Besides these propositional variants of logical theories, also first proposals exist for approximating “infinity” with neural means, in particular, theories of first-order logic. An example is the core method intended to learn the semantics of the single-step operator TP for first-order logic (programs) with a neural network [1]. Another example is the neural approximation of variable-free first-order logic by learning representations of arrow constructions (which represent logical expressions) in the R n using Topos constructions [3]. Although these examples show a certain success of neural-symbolic learning and reasoning research, there are several non-trivial challenges. First, there exist

Abstract. There is an obvious tension between symbolic and subsymbolic theories, because both show complementary strengths and weaknesses in corresponding applications and underlying methodologies. The resulting gap in the foundations and the applicability of these approaches is theoretically unsatisfactory and practically undesirable. We sketch a theory that bridges this gap between symbolic and subsymbolic approaches by the introduction of a Topos-based semi-symbolic level used for coding logical first-order expressions in a homogeneous framework. This semi-symbolic level can be used for neural learning of logical firstorder theories. Besides a presentation of the general idea of the framework, we sketch some challenges and important open problems for future research with respect to the presented approach and the field of neurosymbolic integration, in general. Keywords. Neuro–Symbolic Integration, Topos Theory, First–Order Logic 1

Abstract. The construction of computational cognitive models integrating the connectionist and symbolic paradigms of artificial intelligence is a standing research issue in the field. The combination of logic-based inference and connectionist learning systems may lead to the construction of semantically sound computational cognitive models in artificial intelligence, computer and cognitive sciences. Over the last decades, results regarding the computation and learning of classical reasoning within neural networks have been promising. Nonetheless, there still remains much do be done. Artificial intelligence, cognitive and computer science are strongly based on several non-classical reasoning formalisms, methodologies and logics. In knowledge representation, distributed systems, hardware design, theorem proving, systems specification and verification classical and non-classical logics have had a great impact on theory and real-world applications. Several challenges for neural-symbolic computation are pointed out, in particular for classical and non-classical computation in connectionist systems. We also analyse myths about neural-symbolic computation and shed new light on them considering recent research advances.