Knowledge representation and reasoning in neural networks have been a long-standing endeavor which has attracted much attention recently. The principled integration of reasoning and learning in neural networks is a main objective of the area of neurosymbolic Artificial Intelligence (AI). In this chapter, a simple energy-based neurosymbolic AI system is described that can represent and reason formally about any propositional logic formula. This creates a powerful combination of learning from data and knowledge and logical reasoning. We start by positioning neurosymbolic AI in the context of the current AI landscape that is unsurprisingly dominated by Large Language Models (LLMs). We identify important challenges of data efficiency, fairness and safety of LLMs that might be addressed by neurosymbolic reasoning systems with formal reasoning capabilities. We then discuss the representation of logic by the specific energy-based system, including illustrative examples and empirical evaluation of the correspondence between logical reasoning and energy minimization using Restricted Boltzmann Machines (RBM). Learning from data and knowledge is also evaluated empirically and compared with a symbolic, neural and a neurosymbolic system. Results reported in this chapter in an accessible way are expected to reignite the research on the use of neural networks as massively-parallel models for logical reasoning and promote the principled integration of reasoning and learning in deep networks. We conclude the chapter with a discussion of the importance of positioning neurosymbolic AI within a broader framework of formal reasoning and accountability in AI, discussing the challenges for neurosynbolic AI to tackle the various known problems of reliability of deep learning.

The idea of representing symbolic knowledge in connectionist systems has been a long-standing endeavour which has attracted much attention recently with the objective of combining machine learning and scalable sound reasoning. Early work has shown a correspondence between propositional logic and symmetrical neural networks which nevertheless did not scale well with the number of variables and whose training regime was inefficient. In this paper, we introduce Logical Boltzmann Machines (LBM), a neurosymbolic system that can represent any propositional logic formula in strict disjunctive normal form. We prove equivalence between energy minimization in LBM and logical satisfiability thus showing that LBM is capable of sound reasoning. We evaluate reasoning empirically to show that LBM is capable of finding all satisfying assignments of a class of logical formulae by searching fewer than 0.75% of the possible (approximately 1 billion) assignments. We compare learning in LBM with a symbolic inductive logic programming system, a state-of-the-art neurosymbolic system and a purely neural network-based system, achieving better learning performance in five out of seven data sets.

Most supervised text classification approaches assume a closed world, counting on all classes being present in the data at training time. This assumption can lead to unpredictable behaviour during operation, whenever novel, previously unseen, classes appear. Although deep learning-based methods have recently been used for novelty detection, they are challenging to interpret due to their black-box nature. This paper addresses \emph{interpretable} open-world text classification, where the trained classifier must deal with novel classes during operation. To this end, we extend the recently introduced Tsetlin machine (TM) with a novelty scoring mechanism. The mechanism uses the conjunctive clauses of the TM to measure to what degree a text matches the classes covered by the training data. We demonstrate that the clauses provide a succinct interpretable description of known topics, and that our scoring mechanism makes it possible to discern novel topics from the known ones. Empirically, our TM-based approach outperforms seven other novelty detection schemes on three out of five datasets, and performs second and third best on the remaining, with the added benefit of an interpretable propositional logic-based representation.

The Regression Tsetlin Machine (RTM) addresses the lack of interpretability impeding state-of-the-art nonlinear regression models. It does this by using conjunctive clauses in propositional logic to capture the underlying non-linear frequent patterns in the data. These, in turn, are combined into a continuous output through summation, akin to a linear regression function, however, with non-linear components and unity weights. Although the RTM has solved non-linear regression problems with competitive accuracy, the resolution of the output is proportional to the number of clauses employed. This means that computation cost increases with resolution. To reduce this problem, we here introduce integer weighted RTM clauses. Our integer weighted clause is a compact representation of multiple clauses that capture the same sub-pattern-N repeating clauses are turned into one, with an integer weight N. This reduces computation cost N times, and increases interpretability through a sparser representation. We further introduce a novel learning scheme that allows us to simultaneously learn both the clauses and their weights, taking advantage of so-called stochastic searching on the line. We evaluate the potential of the integer weighted RTM empirically using six artificial datasets. The results show that the integer weighted RTM is able to acquire on par or better accuracy using significantly less computational resources compared to regular RTMs. We further show that integer weights yield improved accuracy over real-valued ones.

The Tsetlin Machine (TM) is an interpretable mechanism for pattern recognition that constructs conjunctive clauses from data. The clauses capture frequent patterns with high discriminating power, providing increasing expression power with each additional clause. However, the resulting accuracy gain comes at the cost of linear growth in computation time and memory usage. In this paper, we present the Weighted Tsetlin Machine (WTM), which reduces computation time and memory usage by \emph{weighting} the clauses. Real-valued weighting allows one clause to replace multiple and supports fine-tuning the impact of each clause. Our novel scheme simultaneously learns both the composition of the clauses and their weights. Furthermore, we increase training efficiency by replacing $k$ Bernoulli trials of success probability $p$ with a uniform sample of average size $p k$, the size drawn from a binomial distribution. In our empirical evaluation, the WTM achieved the same accuracy as the TM on MNIST, IMDb, and Connect-4, requiring only $1/4$, $1/3$, and $1/50$ of the clauses, respectively. With the same number of clauses, the WTM outperformed the TM, obtaining peak test accuracies of respectively $98.58\%$, $90.15\%$, and $87.49\%$. Finally, our novel sampling scheme reduced sample generation time by a factor of $7$.

The recently introduced Tsetlin Machine (TM) has provided competitive pattern classification accuracy in several benchmarks, composing patterns with easy-to-interpret conjunctive clauses in propositional logic. In this paper, we go beyond pattern classification by introducing a new type of TMs, namely, the Regression Tsetlin Machine (RTM). In all brevity, we modify the inner inference mechanism of the TM so that input patterns are transformed into a single continuous output, rather than to distinct categories. We achieve this by: (1) using the conjunctive clauses of the TM to capture arbitrarily complex patterns; (2) mapping these patterns to a continuous output through a novel voting and normalization mechanism; and (3) employing a feedback scheme that updates the TM clauses to minimize the regression error. The feedback scheme uses a new activation probability function that stabilizes the updating of clauses, while the overall system converges towards an accurate input-output mapping. The performance of the proposed approach is evaluated using six different artificial datasets with and without noise. The performance of the RTM is compared with the Classical Tsetlin Machine (CTM) and the Multiclass Tsetlin Machine (MTM). Our empirical results indicate that the RTM obtains the best training and testing results for both noisy and noise-free datasets, with a smaller number of clauses.

While knowledge representation and reasoning are considered the keys for human-level artificial intelligence, connectionist networks have been shown successful in a broad range of applications due to their capacity for robust learning and flexible inference under uncertainty. The idea of representing symbolic knowledge in connectionist networks has been well-received and attracted much attention from research community as this can establish a foundation for integration of scalable learning and sound reasoning. In previous work, there exist a number of approaches that map logical inference rules with feed-forward propagation of artificial neural networks (ANN). However, the discriminative structure of an ANN requires the separation of input/output variables which makes it difficult for general reasoning where any variables should be inferable. Other approaches address this issue by employing generative models such as symmetric connectionist networks, however, they are difficult and convoluted. In this paper we propose a novel method to represent propositional formulas in restricted Boltzmann machines which is less complex, especially in the cases of logical implications and Horn clauses. An integration system is then developed and evaluated in real datasets which shows promising results.

Although simple individually, artificial neurons provide state-of-the-art performance when interconnected in deep networks. Unknown to many, there exists an arguably even simpler and more versatile learning mechanism, namely, the Tsetlin Automaton. Merely by means of a single integer as memory, it learns the optimal action in stochastic environments. In this paper, we introduce the Tsetlin Machine, which solves complex pattern recognition problems with easy-to-interpret propositional formulas, composed by a collective of Tsetlin Automata. To eliminate the longstanding problem of vanishing signal-to-noise ratio, the Tsetlin Machine orchestrates the automata using a novel game. Our theoretical analysis establishes that the Nash equilibria of the game are aligned with the propositional formulas that provide optimal pattern recognition accuracy. This translates to learning without local optima, only global ones. We argue that the Tsetlin Machine finds the propositional formula that provides optimal accuracy, with probability arbitrarily close to unity. In four distinct benchmarks, the Tsetlin Machine outperforms both Neural Networks, SVMs, Random Forests, the Naive Bayes Classifier and Logistic Regression. It further turns out that the accuracy advantage of the Tsetlin Machine increases with lack of data. The Tsetlin Machine has a significant computational performance advantage since both inputs, patterns, and outputs are expressed as bits, while recognition of patterns relies on bit manipulation. The combination of accuracy, interpretability, and computational simplicity makes the Tsetlin Machine a promising tool for a wide range of domains, including safety-critical medicine. Being the first of its kind, we believe the Tsetlin Machine will kick-start completely new paths of research, with a potentially significant impact on the AI field and the applications of AI.