Combining Relational Learning with SMT Solvers Using CEGAR

SMT solvers combine SAT reasoning with specialized theory solvers either to find a feasible solution to a set of constraints or to prove that no such solution exists. Linear programming (LP) solvers come from the tradition of optimization, and are designed to find feasible solutions that are optimal with respect to some optimization function. Typical LP solvers are designed to solve large systems quickly using floating point arithmetic. Because floating point arithmetic is inexact, rounding errors can lead to incorrect results, making inexact solvers inappropriate for direct use in theorem proving. Previous efforts to leverage such solvers in the context of SMT have concluded that in addition to being potentially unsound, such solvers are too heavyweight to compete in the context of SMT. In this paper, we describe a technique for integrating LP solvers that improves the performance of SMT solvers without compromising correctness. These techniques have been implemented using the SMT solver CVC4 and the LP solver GLPK. Experiments show that this implementation outperforms other state-of-the-art SMT solvers on the QF_LRA SMT-LIB benchmarks and is competitive on the QF_LIA benchmarks.

People nowadays base their behavior by making choices through word of mouth, media, public opinion, surveys, etc. One of the most prominent techniques of recommender systems is Collaborative filtering (CF), which utilizes the known preferences of several users to develop recommendation for other users. CF can introduce limitations like new-item problem, new-user problem or data sparsity, which can be mitigated by employing Statistical Relational Learning (SRLs). This review chapter presents a comprehensive scientific survey from the basic and traditional techniques to the-state-of-the-art of SRL algorithms implemented for collaborative filtering issues. Authors provide a comprehensive review of SRL for CF tasks and demonstrate strong evidence that SRL can be successfully implemented in the recommender systems domain. Finally, the chapter is concluded with a summarization of the key issues that SRLs tackle in the collaborative filtering area and suggest further open issues in order to advance in this field of research.

Planning for hybrid systems is important for dealing with real-world applications, and PDDL+ supports this representation of domains with mixed discrete and continuous dynamics. In this paper we present a new approach for planning for hybrid systems, based on encoding the planning problem as a Satisfiability Modulo Theories (SMT) formula. This is the first SMT encoding that can handle the whole set of PDDL+ features (including processes and events), and is implemented in the planner SMTPlan. SMTPlan not only covers the full semantics of PDDL+, but can also deal with non-linear polynomial continuous change without discretization. This allows it to generate plans with non-linear dynamics that are correct-by-construction. The encoding is based on the notion of happenings, and can be applied on domains with nonlinear continuous change. We describe the encoding in detail and provide in-depth examples. We apply this encoding in an iterative deepening planning algorithm. Experimental results show that the approach dramatically outperforms existing work in finding plans for PDDL+ problems. We also present experiments which explore the performance of the proposed approach on temporal planning problems, showing that the scalability of the approach is limited by the size of the discrete search space. We further extend the encoding to include planning with control parameters. The extended encoding allows the definition of actions to include infinite domain parameters, called control parameters. We present experiments on a set of problems with control parameters to demonstrate the positive effect they provide to the approach of planning via SMT.

Satisfiability Modulo Theories (SMT) solvers have been widely applied in automated software analysis to reason about the queries that encode the essence of program semantics, relieving the heavy burden of manual analysis. Many SMT solving techniques rely on solving Boolean satisfiability problem (SAT), which is an NP-complete problem, so they use heuristic search strategies to seek possible solutions, especially when no known theorem can efficiently reduce the problem. An emerging challenge, named Mixed-Bitwise-Arithmetic (MBA) obfuscation, impedes SMT solving by constructing identity equations with both bitwise operations (and, or, negate) and arithmetic computation (add, minus, multiply). Common math theorems for bitwise or arithmetic computation are inapplicable to simplifying MBA equations, leading to performance bottlenecks in SMT solving. In this paper, we first scrutinize solvers' performance on solving different categories of MBA expressions: linear, polynomial, and non-polynomial. We observe that solvers can handle simple linear MBA expressions, but facing a severe performance slowdown when solving complex linear and non-linear MBA expressions. The root cause is that complex MBA expressions break the reduction laws for pure arithmetic or bitwise computation. To boost solvers' performance, we propose a semantic-preserving transformation to reduce the mixing degree of bitwise and arithmetic operations. We first calculate a signature vector based on the truth table extracted from an MBA expression, which captures the complete MBA semantics. Next, we generate a simpler MBA expression from the signature vector. Our large-scale evaluation on 3000 complex MBA equations shows that our technique significantly boost modern SMT solvers' performance on solving MBA formulas.

This paper describes a method for combining “off-the-shelf” SAT and constraint solvers for building an efficient Satisfiability Modulo Theories (SMT) solver for a wide range of theories. Our method follows the abstraction/refinement approach to simplify the implementation of custom SMT solvers. The expected performance penalty by not using an interweaved combination of SAT and theory solvers is reduced by generalising a Boolean solution of an SMT problem first via assigning don’t care to as many variables as possible. We then use the generalised solution to determine a thereby smaller constraint set to be handed over to the constraint solver for a background theory. We show that for many benchmarks and real-world problems, this optimisation results in considerably smaller and less complex constraint problems. The presented approach is particularly useful for assembling a practically viable SMT solver quickly, when neither a suitable SMT solver nor a corresponding incremental theory solver is available. We have implemented our approach in the ABsolver framework and applied the resulting solver successfully to an industrial case-study: the verification problems arising in verifying an electronic car steering control system impose non-linear arithmetic constraints, which do not fall into the domain of any other available solver.

Graph neural networks (GNNs) have emerged in recent years as a very powerful and popular modeling tool for graph and network data. Though much of the work on GNNs has focused on graphs with a single edge relation, they have also been adapted to multi-relational graphs, including knowledge graphs. In such multi-relational domains, the objectives and possible applications of GNNs become quite similar to what for many years has been investigated and developed in the field of statistical relational learning (SRL). This article first gives a brief overview of the main features of GNN and SRL approaches to learning and reasoning with graph data. It analyzes then in more detail their commonalities and differences with respect to semantics, representation, parameterization, interpretability, and flexibility. A particular focus will be on relational Bayesian networks (RBNs) as the SRL framework that is most closely related to GNNs. We show how common GNN architectures can be directly encoded as RBNs, thus enabling the direct integration of "low level" neural model components with the "high level" symbolic representation and flexible inference capabilities of SRL.

Online structure learning approaches, such as those stemming from Statistical Relational Learning, enable the discovery of complex relations in noisy data streams. However, these methods assume the existence of fully-labelled training data, which is unrealistic for most real-world applications. We present a novel approach for completing the supervision of a semi-supervised structure learning task. We incorporate graph-cut minimisation, a technique that derives labels for unlabelled data, based on their distance to their labelled counterparts. In order to adapt graph-cut minimisation to first order logic, we employ a suitable structural distance for measuring the distance between sets of logical atoms. The labelling process is achieved online (single-pass) by means of a caching mechanism and the Hoeffding bound, a statistical tool to approximate globally-optimal decisions from locally-optimal ones. We evaluate our approach on the task of composite event recognition by using a benchmark dataset for human activity recognition, as well as a real dataset for maritime monitoring. The evaluation suggests that our approach can effectively complete the missing labels and eventually, improve the accuracy of the underlying structure learning system.

One of the most appreciated functionality of computers nowadays is their being a means for communication and information sharing among people. With the spread of the internet, several complex interactions have taken place among people, giving rise to huge information networks based on these interactions. Social networks potentially represent an invaluable source of information that can be exploited for scientific and commercial purposes. On the other hand, due to their distinguishing peculiarities (huge size and inherent relational setting) with respect to all previous information extraction tasks faced in computer science, they require new techniques to gather this information. Social network mining (SNM) is the corresponding research area, aimed at extracting information about the network objects and behaviour that cannot be obtained based on the explicit/implicit description of the objects alone, ignoring their explicit/implicit relationships. Statistical relational learning (SRL) is a very promising approach to SNM, since it combines expressive representation formalisms, able to model complex relational networks, with statistical methods able to handle uncertainty about objects and relations. This paper is a survey of some SRL formalisms and techniques adopted to solve some SNM tasks.

One key challenge in statistical relational learning (SRL) is scalable inference. Unfortunately, most real-world problems in SRL have expressive models that translate into large grounded networks, representing a bottleneck for any inference method and weakening its scalability. In this paper we introduce Preference Relaxation (PR), a two-stage strategy that uses the determinism present in the underlying model to improve the scalability of relational inference. The basic idea of PR is that if the underlying model involves mandatory (i.e. hard) constraints as well as preferences (i.e. soft constraints) then it is potentially wasteful to allocate memory for all constraints in advance when performing inference. To avoid this, PR starts by relaxing preferences and performing inference with hard constraints only. It then removes variables that violate hard constraints, thereby avoiding irrelevant computations involving preferences. In addition it uses the removed variables to enlarge the evidence database. This reduces the effective size of the grounded network. Our approach is general and can be applied to various inference methods in relational domains. Experiments on real-world applications show how PR substantially scales relational inference with a minor impact on accuracy.

Recent years have seen a surge of interest in Statistical Relational Learning (SRL) models that combine logic with probabilities. One prominent and highly expressive SRL model is Markov Logic Networks (MLNs), but the expressivity comes at the cost of learning complexity. Most of the current methods for learning MLN structure follow a two-step approach where first they search through the space of possible clauses (i.e. structures) and then learn weights via gradient descent for these clauses. We present a functional-gradient boosting algorithm to learn both the weights (in closed form) and the structure of the MLN simultaneously. Moreover most of the learning approaches for SRL apply the closed-world assumption, i.e., whatever is not observed is assumed to be false in the world. We attempt to open this assumption. We extend our algorithm for MLN structure learning to handle missing data by using an EM-based approach and show this algorithm can also be used to learn Relational Dependency Networks and relational policies. Our results in many domains demonstrate that our approach can effectively learn MLNs even in the presence of missing data.

Large Language Models (LLMs) have emerged as a promising alternative to traditional static program analysis methods, such as symbolic execution, offering the ability to reason over code directly without relying on theorem provers or SMT solvers. However, LLMs are also inherently approximate by nature, and therefore face significant challenges in relation to the accuracy and scale of analysis in real-world applications. Such issues often necessitate the use of larger LLMs with higher token limits, but this requires enterprise-grade hardware (GPUs) and thus limits accessibility for many users. In this paper, we propose LLM-based symbolic execution —a novel approach that enhances LLM inference via a path-based decomposition of the program analysis tasks into smaller (more tractable) subtasks. The core idea is to generalize path constraints using a generic code-based representation that the LLM can directly reason over, and without translation into another (less-expressive) formal language. We implement our approach in the form of AutoBug, an LLM-based symbolic execution engine that is lightweight and language-agnostic, making it a practical tool for analyzing code that is challenging for traditional approaches. We show that AutoBug can improve both the accuracy and scale of LLM-based program analysis, especially for smaller LLMs that can run on consumer-grade hardware.

Semantic Image Interpretation (SII) is the task of extracting structured semantic descriptions from images. It is widely agreed that the combined use of visual data and background knowledge is of great importance for SII. Recently, Statistical Relational Learning (SRL) approaches have been developed for reasoning under uncertainty and learning in the presence of data and rich knowledge. Logic Tensor Networks (LTNs) are a SRL framework which integrates neural networks with first-order fuzzy logic to allow (i) efficient learning from noisy data in the presence of logical constraints, and (ii) reasoning with logical formulas describing general properties of the data. In this paper, we develop and apply LTNs to two of the main tasks of SII, namely, the classification of an image's bounding boxes and the detection of the relevant part-of relations between objects. To the best of our knowledge, this is the first successful application of SRL to such SII tasks. The proposed approach is evaluated on a standard image processing benchmark. Experiments show that background knowledge in the form of logical constraints can improve the performance of purely data-driven approaches, including the state-of-the-art Fast Region-based Convolutional Neural Networks (Fast R-CNN). Moreover, we show that the use of logical background knowledge adds robustness to the learning system when errors are present in the labels of the training data.

Cyber-Physical Systems (CPS) intrinsically combine hardware and physical systems with software and network, which are together creating complex and correlated interactions. CPS applications often experience uncertainty in interacting with environment through unreliable sensors. They can be faulty and exhibit runtime errors if developers have not considered environmental interaction uncertainty adequately. Existing work in verifying CPS applications ignores interaction uncertainty and thus may overlook uncertainty-related faults. To improve verification accuracy, in this article we propose a novel approach to verifying CPS applications with explicit modeling of uncertainty arisen in the interaction between them and the environment. Our approach builds an Interactive State Machine network for a CPS application and models interaction uncertainty by error ranges and distributions. Then it encodes both the application and uncertainty models to Satisfiability Modulo Theories (SMT) formula to leverage SMT solvers searching for counterexamples that represent application failures. The precision of uncertainty model can affect the verification results. However, it may be difficult to model interaction uncertainty precisely enough at the beginning, because of the uncontrollable noise of sensors and insufficient data sample size. To further improve the accuracy of the verification results, we propose an approach to identifying and calibrating imprecise uncertainty models. We exploit the inconsistency between the counterexamples’ estimate and actual occurrence probabilities to identify possible imprecision in uncertainty models, and the calibration of imprecise models is to minimize the inconsistency, which is reduced to a Search-Based Software Engineering problem. We experimentally evaluated our verification and calibration approaches with real-world CPS applications, and the experimental results confirmed their effectiveness and efficiency.

A workmanlike, but nevertheless very effective combination of statistical and relational learning uses a statistical learner to construct models with features identified (quite often, separately) by a relational learner. This form of model-building has a long history in Inductive Logic Programming (ILP), with roots in the early 1990s with the LINUS system. Additional work has also been done in the field under the categories of propositionalisation and relational subgroup discovery, where a distinction has been made between elementary and non-elementary features, and statistical models have been constructed using one or the other kind of feature. More recently, constructing relational features has become an essential step in many model-building programs in the emerging area of Statistical Relational Learning (SRL). To date, not much work—theoretical or empirical—has been done on what kinds of relational features are sufficient to build good statistical models. On the face of it, the features that are needed are those that capture diverse and complex relational structure. This suggests that the feature-constructor should examine as rich a space as possible, in terms of relational descriptions. One example is the space of all possible features in first-order logic, given constraints of the problem being addressed. Practically, it may be intractable for a relational learner to search such a space effectively for features that may be potentially useful for a statistical learner. Additionally, the statistical learner may also be able to capture some kinds of complex structure by combining simpler features together. Based on these observations, we investigate empirically whether it is acceptable for a relational learner to examine a more restricted space of features than that actually necessary for the full statistical model. Specifically, we consider five sets of features, partially ordered by the subset relation, bounded on top by the set F d , the set of features corresponding to definite clauses subject to domain-specific restrictions; and bounded at the bottom by F e , the set of “elementary” features with substantial additional constraints. Our results suggest that: (a) For relational datasets used in the ILP literature, features from F d may not be required; and (b) Models obtained with a standard statistical learner with features from subsets of features are comparable to the best obtained to date.

Statistical relational learning (SRL) frameworks are effective at defining probabilistic models over complex relational data. They often use weighted first-order logical rules where the weights of the rules govern probabilistic interactions and are usually learned from data. Existing weight learning approaches typically attempt to learn a set of weights that maximizes some function of data likelihood; however, this does not always translate to optimal performance on a desired domain metric, such as accuracy or F1 score. In this paper, we introduce a taxonomy of search-based weight learning approaches for SRL frameworks that directly optimize weights on a chosen domain performance metric. To effectively apply these search-based approaches, we introduce a novel projection, referred to as scaled space (SS), that is an accurate representation of the true weight space. We show that SS removes redundancies in the weight space and captures the semantic distance between the possible weight configurations. In order to improve the efficiency of search, we also introduce an approximation of SS which simplifies the process of sampling weight configurations. We demonstrate these approaches on two state-of-the-art SRL frameworks: Markov logic networks and probabilistic soft logic. We perform empirical evaluation on five real-world datasets and evaluate them each on two different metrics. We also compare them against four other weight learning approaches. Our experimental results show that our proposed search-based approaches outperform likelihood-based approaches and yield up to a 10% improvement across a variety of performance metrics. Further, we perform an extensive evaluation to measure the robustness of our approach to different initializations and hyperparameters. The results indicate that our approach is both accurate and robust.