Admittedly, NP variants can be applied aseries of downstream tasks. Our selection of benchmark missions is based on existing literature for NP models.

Here we collect some frequently asked questions from reviewers and other literature researchers.

The variational autoencoder (VAE) is a simple and efficient generative artificial intelligence method for modeling complex probability distributions of various types of data, such as images and texts. However, it suffers some main shortcomings, such as lack of interpretability in the latent variables, difficulties in tuning hyperparameters while training, producing blurry, unrealistic downstream outputs or loss of information due to how it calculates loss functions and recovers data distributions, overfitting, and origin gravity effect for small data sets, among other issues. These and other limitations have caused unsatisfactory generation effects for the data with complex distributions. In this work, we proposed and developed a polynomial hierarchical variational autoencoder (PH-VAE), in which we used a polynomial hierarchical date format to generate or to reconstruct the data distributions. In doing so, we also proposed a novel Polynomial Divergence in the loss function to replace or generalize the Kullback-Leibler (KL) divergence, which results in systematic and drastic improvements in both accuracy and reproducibility of the re-constructed distribution function as well as the quality of re-constructed data images while keeping the dataset size the same but capturing fine resolution of the data. Moreover, we showed that the proposed PH-VAE has some form of disentangled representation learning ability.

While large language models (LLMs) have shown impressive capabilities across a wide range of domains, they still encounter significant challenges in reasoning tasks that require gathering evidence over multiple turns and drawing logical conclusions from this evidence. These challenges present significant obstacles for LLM chat user interfaces, which rely on multi-turn interactions to facilitate effective collaboration. This limitation leads to real-world issues; for example, service chatbots must gather necessary information from customers over multiple turns to diagnose and resolve problems effectively. Despite the multi-turn nature of many real-world LLM use cases, most existing benchmarks rely on carefully curated single-turn tests, which often blur the line between memorization and genuine reasoning. To address this, we introduce the Wason Inductive Logic Test (WILT), a simple yet challenging multi-turn reasoning benchmark designed to resist memorization. WILT is inspired by the Wason 2-4-6 task (Wason, 1960), where participants must infer a basic boolean function involving three variables (e.g., x < y < z) by proposing test cases (such as (2, 4, 6)). In WILT, each test starts from a clean slate, with only the initial instructions provided, preventing models from relying on pre-learned responses. Over several turns, models must interact with the environment by suggesting test cases to narrow the possible hypotheses and ultimately infer the hidden function based on the outcomes. Our findings reveal that LLMs struggle with this task, exhibiting distinct strengths and weaknesses: some are better at narrowing down the hypothesis space by proposing valuable test cases, while others are more adept at deducing the hidden function from observed cases. Despite these variations, the best-performing model achieves only 28% accuracy, highlighting a significant gap in LLM performance on complex multi-turn reasoning tasks. Large language models (LLMs) powered by the transformer architecture (Vaswani, 2017) have enabled a new computing paradigm driven by natural language. These models are increasingly integrated into day-to-day life beyond the machine learning research space, where they help many people with common tasks. These models interact with users through multi-turn conversations, a capability of next-token-prediction models bolstered via instruction-tuning (Mishra et al., 2021) and alignment post-training phases (Ouyang et al., 2022).

Exact lifted inference methods, like their propositional counterparts, work by recursively decomposing the model and the problem. In the propositional case, there exist formal structures, such as decomposition trees (dtrees), that represent such a decomposition and allow us to determine the complexity of inference a priori. However, there is currently no equivalent structure nor analogous complexity results for lifted inference. In this paper, we introduce FO-dtrees, which upgrade propositional dtrees to the first-order level. We show how these trees can characterize a lifted inference solution for a probabilistic logical model (in terms of a sequence of lifted operations), and make a theoretical analysis of the complexity of lifted inference in terms of the novel notion of lifted width for the tree.

Lifted probabilistic inference exploits symmetries in a probabilistic model to allow for tractable probabilistic inference with respect to domain sizes. To apply lifted inference, a lifted representation has to be obtained, and to do so, the so-called colour passing algorithm is the state of the art. The colour passing algorithm, however, is bound to a specific inference algorithm and we found that it ignores commutativity of factors while constructing a lifted representation. We contribute a modified version of the colour passing algorithm that uses logical variables to construct a lifted representation independent of a specific inference algorithm while at the same time exploiting commutativity of factors during an offline-step. Our proposed algorithm efficiently detects more symmetries than the state of the art and thereby drastically increases compression, yielding significantly faster online query times for probabilistic inference when the resulting model is applied.

Lifted inference allows to perform inference in polynomial time w.r.t. domain sizes. For a lifted algorithm, completeness investigates model classes for which the algorithm is guaranteed to compute a lifted solution. We contribute, to the best of our knowledge, the first completeness and complexity analysis for a temporal lifted algorithm, the so-called lifted dynamic junction tree algorithm (LDJT). To treat time as a first class citizen, LDJT introduces some constraints. Given these constraints, we analyse the classes of liftable models. Further, we show that LDJT has many advantages from a complexity point of view compared to a propositional temporal inference algorithm w.r.t. domain sizes. Therefore, LDJT advances the number of models for which inference tasks can be solved in reasonable time not only from a practically point of view, but also from a theoretical point of view.

Lifted inference has been proposed for various probabilistic logical frameworks in order to compute the probability of queries in a time that depends on the size of the domains of the random variables rather than the number of instances. Even if various authors have underlined its importance for probabilistic logic programming (PLP), lifted inference has been applied up to now only to relational languages outside of logic programming. In this paper we adapt Generalized Counting First Order Variable Elimination (GC-FOVE) to the problem of computing the probability of queries to probabilistic logic programs under the distribution semantics. In particular, we extend the Prolog Factor Language (PFL) to include two new types of factors that are needed for representing ProbLog programs. These factors take into account the existing causal independence relationships among random variables and are managed by the extension to variable elimination proposed by Zhang and Poole for dealing with convergent variables and heterogeneous factors. Two new operators are added to GC-FOVE for treating heterogeneous factors. The resulting algorithm, called LP$^2$ for Lifted Probabilistic Logic Programming, has been implemented by modifying the PFL implementation of GC-FOVE and tested on three benchmarks for lifted inference. A comparison with PITA and ProbLog2 shows the potential of the approach.

Lifting attempts to speedup probabilistic inference by exploiting symmetries in the model. Exact lifted inference methods, like their propositional counterparts, work by recursively decomposing the model and the problem. In the propositional case, there exist formal structures, such as decomposition trees (dtrees), that represent such a decomposition and allow us to determine the complexity of inference a priori. However, there is currently no equivalent structure nor analogous complexity results for lifted inference. In this paper, we introduce FO-dtrees, which upgrade propositional dtrees to the first-order level. We show how these trees can characterize a lifted inference solution for a probabilistic logical model (in terms of a sequence of lifted operations), and make a theoretical analysis of the complexity of lifted inference in terms of the novel notion of lifted width for the tree.

Lifted probabilistic inference algorithms exploit regularities in the structure of graphical models to perform inference more efficiently. More specifically, they identify groups of interchangeable variables and perform inference once per group, as opposed to once per variable. The groups are defined by means of constraints, so the flexibility of the grouping is determined by the expressivity of the constraint language. Existing approaches for exact lifted inference use specific languages for (in)equality constraints, which often have limited expressivity. In this article, we decouple lifted inference from the constraint language. We define operators for lifted inference in terms of relational algebra operators, so that they operate on the semantic level (the constraints' extension) rather than on the syntactic level, making them language-independent. As a result, lifted inference can be performed using more powerful constraint languages, which provide more opportunities for lifting. We empirically demonstrate that this can improve inference efficiency by orders of magnitude, allowing exact inference where until now only approximate inference was feasible.