We present NeurASP, a simple extension of answer set programs by embracing neural networks. By treating the neural network output as the probability distribution over atomic facts in answer set programs, NeurASP provides a simple and effective way to integrate sub-symbolic and symbolic computation. We demonstrate how NeurASP can make use of a pre-trained neural network in symbolic computation and how it can improve the neural network's perception result by applying symbolic reasoning in answer set programming. Also, NeurASP can make use of ASP rules to train a neural network better so that a neural network not only learns from implicit correlations from the data but also from the explicit complex semantic constraints expressed by the rules.

LPMLN is a recently introduced formalism that extends answer set programs by adopting the log-linear weight scheme of Markov Logic. This paper investigates the relationships between LPMLN and two other extensions of answer set programs: weak constraints to express a quantitative preference among answer sets, and P-log to incorporate probabilistic uncertainty. We present a translation of LPMLN into programs with weak constraints and a translation of P-log into LPMLN, which complement the existing translations in the opposite directions. The first translation allows us to compute the most probable stable models (i.e., MAP estimates) of LPMLN programs using standard ASP solvers. This result can be extended to other formalisms, such as Markov Logic, ProbLog, and Pearl's Causal Models, that are shown to be translatable into LPMLN. The second translation tells us how probabilistic nonmonotonicity (the ability of the reasoner to change his probabilistic model as a result of new information) of P-log can be represented in LPMLN, which yields a way to compute P-log using standard ASP solvers and MLN solvers.

Logic Programs with Ordered Disjunction (LPOD) is an extension of standard answer set programs to handle preference using the construct of ordered disjunction, and CR-Prolog 2 is an extension of standard answer set programs with consistency restoring rules and LPOD-like ordered disjunction. We present reductions of each of these languages into the standard ASP language, which gives us an alternative way to understand the extensions in terms of the standard ASP language. (The paper is under consideration for acceptance in TPLP.) ⊥ ← hotel (2), not med ⊥ ← hotel (2), not star3 ⊥ ← hotel(3), not tooF ar ⊥ ← hotel(3), not star4 Π 2 has 4 × 3 split programs but only the following three programs are consistent (The regular part of Π 2 is not listed).close med ← not close star2 ← not star4, not star3 star3 ← not star4 tooF ar ← not close, not med, not f ar star4

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The integration of low-level perception with high-level reasoning is one of the oldest problems in Artificial Intelligence. Recently, several proposals were made to implement the reasoning process in complex neural network architectures. While these works aim at extending neural networks with the capability of reasoning, a natural question that we consider is: can we extend answer set programs with neural networks to allow complex and high-level reasoning on neural network outputs? As a preliminary result, we propose NeurASP -a simple extension of answer set programs by embracing neural networks where neural network outputs are treated as probability distributions over atomic facts in answer set programs. We show that NeurASP can not only improve the perception accuracy of a pre-trained neural network, but also help to train a neural network better by giving regularization through logic rules. However, training with NeurASP implementation takes much more time than pure neural network training due to the internal use of a symbolic reasoning engine. For future work, we plan to investigate the potential ways to solve the scalability issue of NeurASP implementation. One way is to embed logic programs directly in neural networks. On this route, we plan to design a SAT solver using neural networks and extend such a solver to allow logic programs.