Representing symbolic knowledge into a connectionist network is the key element for the integration of scalable learning and sound reasoning. Most of the previous studies focus on discriminative neural networks which unnecessarily require a separation of input output variables. Recent development of generative neural networks such as restricted Boltzmann machines (RBMs) has shown a capability of learning semantic abstractions directly from data, posing a promise for general symbolic learning and reasoning. Previous work on Penalty logic show a link between propositional logic and symmetric connectionist networks, however it is not applicable to RBMs. This paper proposes a novel method to represent propositional formulas into RBMs stack of RBMs where Gibbs sampling can be seen as maximising satisfiability. It also shows a promising use of RBMs to learn symbolic knowledge through maximum likelihood estimation.
In artificial neural networks, symbolic knowledge representation is based on the equivalence between feed-forward inference of the networks and modus ponens of logical rules. One of the earliest work is Knowledge-based artificial neural network @cite_20 which encodes if-then rules in a hierarchy of perceptrons. In another approach @cite_0 an 1-hidden layer neural networks with recurrent connections is proposed to support more complex rules. An extension of this system, called CILP++, uses bottom clause propositionalisation technique to work with first-order logic @cite_7 . Logic tensor network @cite_15 employs neural embedding to transform symbols to a vector space where logical inference is carried out through matrix tensor computation.
@cite_0 @cite_15 @cite_7 @cite_20
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We introduce neural networks for end-to-end differentiable proving of queries to knowledge bases by operating on dense vector representations of symbols. These neural networks are constructed recursively by taking inspiration from the backward chaining algorithm as used in Prolog. Specifically, we replace symbolic unification with a differentiable computation on vector representations of symbols using a radial basis function kernel, thereby combining symbolic reasoning with learning subsymbolic vector representations. By using gradient descent, the resulting neural network can be trained to infer facts from a given incomplete knowledge base. It learns to (i) place representations of similar symbols in close proximity in a vector space, (ii) make use of such similarities to prove queries, (iii) induce logical rules, and (iv) use provided and induced logical rules for multi-hop reasoning. We demonstrate that this architecture outperforms ComplEx, a state-of-the-art neural link prediction model, on three out of four benchmark knowledge bases while at the same time inducing interpretable function-free first-order logic rules. The effective integration of learning and reasoning is a well-known and challenging area of research within artificial intelligence. Neural-symbolic systems seek to integrate learning and reasoning by combining neural networks and symbolic knowledge representation. In this paper, a novel methodology is proposed for the extraction of relational knowledge from neural networks which are trainable by the efficient application of the backpropagation learning algorithm. First-order logic rules are extracted from the neural networks, offering interpretable symbolic relational models on which logical reasoning can be performed. The wellknown knowledge extraction algorithm TREPAN was adapted and incorporated into the first-order version of the neural-symbolic system CILP++. Empirical results obtained in comparison with a probabilistic model for relational learning, Markov Logic Networks, and a state-of-the-art Inductive Logic Programming system, Aleph, indicate that the proposed methodology achieves competitive accuracy results consistently in all datasets investigated, while either Markov Logic Networks or Aleph show considerably worse results in at least one dataset. It is expected that effective knowledge extraction from neural networks can contribute to the integration of heterogeneous knowledge representations. We marry two powerful ideas: deep representation learning for visual recognition and language understanding, and symbolic program execution for reasoning. Our neural-symbolic visual question answering (NS-VQA) system first recovers a structural scene representation from the image and a program trace from the question. It then executes the program on the scene representation to obtain an answer. Incorporating symbolic structure as prior knowledge offers three unique advantages. First, executing programs on a symbolic space is more robust to long program traces; our model can solve complex reasoning tasks better, achieving an accuracy of 99.8 on the CLEVR dataset. Second, the model is more data- and memory-efficient: it performs well after learning on a small number of training data; it can also encode an image into a compact representation, requiring less storage than existing methods for offline question answering. Third, symbolic program execution offers full transparency to the reasoning process; we are thus able to interpret and diagnose each execution step. Deep neural networks are known to be vulnerable to adversarial examples, i.e., images that are maliciously perturbed to fool the model. Generating adversarial examples has been mostly limited to finding small perturbations that maximize the model prediction error. Such images, however, contain artificial perturbations that make them somewhat distinguishable from natural images. This property is used by several defense methods to counter adversarial examples by applying denoising filters or training the model to be robust to small perturbations. In this paper, we introduce a new class of adversarial examples, namely "Semantic Adversarial Examples," as images that are arbitrarily perturbed to fool the model, but in such a way that the modified image semantically represents the same object as the original image. We formulate the problem of generating such images as a constrained optimization problem and develop an adversarial transformation based on the shape bias property of human cognitive system. In our method, we generate adversarial images by first converting the RGB image into the HSV (Hue, Saturation and Value) color space and then randomly shifting the Hue and Saturation components, while keeping the Value component the same. Our experimental results on CIFAR10 dataset show that the accuracy of VGG16 network on adversarial color-shifted images is 5.7 .