We show that a deep neural network can learn the semantics of linear-time temporal logic (LTL). As a challenging task that requires deep understanding of the LTL semantics, we show that our network can solve the trace generation problem for LTL: given a satisfiable LTL formula, find a trace that satisfies the formula. We frame the trace generation problem for LTL as a translation task, i.e., to translate from formulas to satisfying traces, and train an off-the-shelf implementation of the Transformer, a recently introduced deep learning architecture proposed for solving natural language processing tasks. We provide a detailed analysis of our experimental results, comparing multiple hyperparameter settings and formula representations. After training for several hours on a single GPU the results were surprising: the Transformer returns the syntactically equivalent trace in 89% of the cases on a held-out test set. Most of the "mispredictions", however, (and overall more than 99% of the predicted traces) still satisfy the given LTL formula.
In the last few years the systematic adoption of deep learning to visual generation has produced impressive results that, amongst others, definitely benefit from the massive exploration of convolutional architectures. In this paper, we propose a general approach to visual generation that combines learning capabilities with logic descriptions of the target to be generated. The process of generation is regarded as a constrained satisfaction problem, where the constraints describe a set of properties that characterize the target. Interestingly, the constraints can also involve logic variables, while all of them are converted into real-valued functions by means of the t-norm theory. We use deep architectures to model the involved variables, and propose a computational scheme where the learning process carries out a satisfaction of the constraints. We propose some examples in which the theory can naturally be used, including the modeling of GAN and auto-encoders, and report promising results in problems with the generation of handwritten characters and face transformations.
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A fundamental challenge in developing high-impact machine learning technologies is balancing the need to model rich, structured domains with the ability to scale to big data. Many important problem areas are both richly structured and large scale, from social and biological networks, to knowledge graphs and the Web, to images, video, and natural language. In this paper, we introduce two new formalisms for modeling structured data, and show that they can both capture rich structure and scale to big data. The first, hinge-loss Markov random fields (HL-MRFs), is a new kind of probabilistic graphical model that generalizes different approaches to convex inference. We unite three approaches from the randomized algorithms, probabilistic graphical models, and fuzzy logic communities, showing that all three lead to the same inference objective. We then define HL-MRFs by generalizing this unified objective. The second new formalism, probabilistic soft logic (PSL), is a probabilistic programming language that makes HL-MRFs easy to define using a syntax based on first-order logic. We introduce an algorithm for inferring most-probable variable assignments (MAP inference) that is much more scalable than general-purpose convex optimization methods, because it uses message passing to take advantage of sparse dependency structures. We then show how to learn the parameters of HL-MRFs. The learned HL-MRFs are as accurate as analogous discrete models, but much more scalable. Together, these algorithms enable HL-MRFs and PSL to model rich, structured data at scales not previously possible.