Answer Set Programming with Quantifiers (ASP(Q)) extends Answer Set Programming (ASP) by allowing quantification over answer sets. Although probabilistic extensions to ASP exist, there is no such counterpart for ASP(Q). In this paper, we close this gap by introducing Inferential Quantified Answer Set Programming (ASP(Q)Inf), an extension of ASP(Q) that supports probabilistic inference over programs with alternating quantifiers, allowing uncertainty at the innermost level. We demonstrate the modeling capabilities of ASP(Q)Inf, analyze its computational complexity, and present an implementation based on Algebraic Model Counting. An experimental evaluation confirms its effectiveness and practical applicability.

Maximum-a-Posteriori (MAP) inference is a crucial problem in Artificial Intelligence, which requires both marginalization and maximization, and asks for the most probable value for a given set of variables such that an evidence holds. Several languages within the Statistical Relational Artificial Intelligence landscape support the encoding of MAP. Here, we focus on Probabilistic Answer Set Programming, consider the credal and smProbLog semantics, and introduce a three-level algebraic model counting representation for MAP. We implemented our approach on top of a state-of-the-art solver and compared it with existing solutions, showing the competitive performance of our proposal, even against less general tools.

Most Probable Explanation (MPE) is a fundamental problem in statistical relational artificial intelligence. In the context of Probabilistic Answer Set Programming (PASP), solving MPE is still an open research problem. In this paper, we present three novel approaches for solving the MPE task in PASP that are based on: i) Algebraic Model Counting, ii) Answer Set Programming (ASP), and iii) ASP with quantifiers (ASP(Q)). These approaches are implemented and evaluated against existing solvers across different datasets and configurations. Empirical results demonstrate that the novel solutions consistently outperform existing alternatives for non-stratified programs.

Statistical statements are an expressive tool for representing statistical information of a domain of interest. Recently, these statements were given a meaning in the context of Probabilistic Answer Set Programming (PASP), allowing one to encode properties like "x% of elements of a domain have the feature y". Although the computational complexity of different tasks in PASP is well known, the complexity of restricted programs composed only of statistical statements and probabilistic facts has not been studied. As a first contribution, we address this problem, confirming that even in seemingly restricted cases the complexity is high. Indeed, even with this restriction we do not lose expressiveness, reaching higher levels of the polynomial hierarchy. To mitigate these high complexities, we focus on the structure of the programs. Thereby, we design novel structure-guided reductions, demonstrating how one can efficiently answer queries along treewidth decompositions. We obtain precise upper bounds and we show that under reasonable assumptions in complexity theory we cannot significantly improve, as we give matching lower bounds.

Probabilistic logic-based languages offer an expressive framework for encoding uncertain information in a human-interpretable way. Among existing formalisms, Probabilistic Answer Set Programming (PASP) stands out for its ease of modeling complex scenarios. The current definition of PASP is limited to programs consisting of disjunctive rules and probabilistic facts only. To enhance the expressivity of the framework, we introduce Optimal Probabilistic Answer Set Programming, which extends the language by allowing the inclusion of weak constraints within PASP specifications. We motivate this extension through some real-world application scenarios and present a detailed computational complexity analysis for both the inference and Most Probable Explanation (MPE) tasks.

Structure learning (SL) is a fundamental task in Statistical Relational Artificial Intelligence, where the goal is to learn a program from data. Among the possible target languages, there is Probabilistic Logic Programming. Mixture models have recently gained attention thanks to their effectiveness in modeling complex distributions by combining simpler ones. In this paper, we propose learning a mixture of probabilistic logic programs to handle SL. Our method consists of three steps: 1) generating mixture components with a specific structure, 2) applying parameter learning to each component, and 3) optimizing the weights associated with each component. Furthermore, to possibly reduce the number of components and mitigate overfitting, we also explore the use of L1 and L2 regularization. Empirical results obtained by considering both the full set of components and only a fraction of them demonstrate that our approach, despite being seemingly simple, is competitive with state-of-the-art solvers.

Learning a logic program that effectively describes input data has been a long-standing goal in Artificial Intelligence , particularly within the field of Inductive Logic Programming . Learning it is even more challenging when information is uncertain, due to the inherent complexity of probabilistic reasoning . In this paper, we propose an approach based on an evolutionary algorithm to learn probabilistic logic programs. Our empirical evaluation shows that the proposed method outperforms existing tools in terms of log-likelihood, AUCROC, and AUCPR, while also providing more compact and interpretable theories.

Abstract The goal of inductive logic programming is to learn a logic program that models the examples provided as input. The search space of the possible programs is constrained by a language bias, which defines the atoms and literals allowed in rules. Answer set programming is a powerful formalism to represent complex combinatorial domains, also thanks to syntactic constructs such as aggregates. However, learning answer set programs from data is challenging, and often existing tools do not support the specification of aggregates in the language bias. In this paper, we introduce GENTIANS, a tool based on a genetic algorithm to learn answer set programs possibly with aggregates, arithmetic, and comparison operators, from examples. Empirical results, also against an existing solver, show that GENTIANS is able to provide accurate solutions even when the search space contains millions of clauses. Additionally, experiments on noisy datasets show the effectiveness of our approach.

Abstract Solving a decision theory problem usually involves finding the actions, among a set of possible ones, which optimize the expected reward, while possibly accounting for the uncertainty of the environment. In this paper, we introduce the possibility to encode decision theory problems with Probabilistic Answer Set Programming under the credal semantics via decision atoms and utility attributes. To solve the task, we propose an algorithm based on three layers of Algebraic Model Counting, that we test on several synthetic datasets against an algorithm that adopts answer set enumeration. Empirical results show that our algorithm can manage non-trivial instances of programs in a reasonable amount of time.

Abstract Novel utility computing paradigms rely upon the deployment of multi-service applications to pervasive and highly distributed cloud-edge infrastructure resources. Deciding onto which computational nodes to place services in cloud-edge networks, as per their functional and non-functional constraints, can be formulated as a combinatorial optimisation problem. Most existing solutions in this space are not able to deal with unsatisfiable problem instances, nor preferences, i.e., requirements that DevOps may agree to relax to obtain a solution. In this article, we exploit Answer Set Programming optimisation capabilities to tackle this problem. Experimental results in simulated settings show that our approach is effective on lifelike networks and applications.

Abstract Probabilistic Answer Set Programming under the credal semantics extends Answer Set Programming with probabilistic facts that represent uncertain information. The probabilistic facts are discrete with Bernoulli distributions. However, several real-world scenarios require a combination of both discrete and continuous random variables. In this paper, we extend the PASP framework to support continuous random variables and propose Hybrid Probabilistic Answer Set Programming. Moreover, we discuss, implement, and assess the performance of two exact algorithms based on projected answer set enumeration and knowledge compilation and two approximate algorithms based on sampling. Empirical results, also in line with known theoretical results, show that exact inference is feasible only for small instances, but knowledge compilation has a huge positive impact on performance. Sampling allows handling larger instances but sometimes requires an increasing amount of memory.

Abstract Probabilistic Logic Programming (PLP) under the distribution semantics is a leading approach to practical reasoning under uncertainty. An advantage of the distribution semantics is its suitability for implementation as a Prolog or Python library, available through two well-maintained implementations, namely ProbLog and cplint/PITA. However, current formulations of the distribution semantics use point-probabilities, making it difficult to express epistemic uncertainty, such as arises from, for example, hierarchical classifications from computer vision models. Belief functions generalize probability measures as non-additive capacities and address epistemic uncertainty via interval probabilities. This paper introduces interval-based Capacity Logic Programs based on an extension of the distribution semantics to include belief functions and describes properties of the new framework that make it amenable to practical applications.

Probabilistic answer set programming has recently been extended to manage imprecise probabilities by means of credal probabilistic facts and credal annotated disjunctions. This increases the expressivity of the language but, at the same time, the cost of inference. In this paper, we cast inference in probabilistic answer set programs with credal probabilistic facts and credal annotated disjunctions as a constrained nonlinear optimization problem where the function to optimize is obtained via knowledge compilation. Empirical results on different datasets with multiple configurations shows the effectiveness of our approach.

Abstract Parameter learning is a crucial task in the field of Statistical Relational Artificial Intelligence: given a probabilistic logic program and a set of observations in the form of interpretations, the goal is to learn the probabilities of the facts in the program such that the probabilities of the interpretations are maximized. In this paper, we propose two algorithms to solve such a task within the formalism of Probabilistic Answer Set Programming, both based on the extraction of symbolic equations representing the probabilities of the interpretations. The first solves the task using an off-the-shelf constrained optimization solver while the second is based on an implementation of the Expectation Maximization algorithm. Empirical results show that our proposals often outperform existing approaches based on projected answer set enumeration in terms of quality of the solution and in terms of execution time.

Abstract When we want to compute the probability of a query from a probabilistic answer set program, some parts of a program may not influence the probability of a query, but they impact on the size of the grounding. Identifying and removing them is crucial to speed up the computation. Algorithms for SLG resolution offer the possibility of returning the residual program which can be used for computing answer sets for normal programs that do have a total well-founded model. The residual program does not contain the parts of the program that do not influence the probability. In this paper, we propose to exploit the residual program for performing inference. Empirical results on graph datasets show that the approach leads to significantly faster inference. The paper has been accepted at the ICLP2024 conference and under consideration in Theory and Practice of Logic Programming (TPLP).

Abstract Probabilistic Logic Programming combines uncertainty and logic-based languages. Liftable Probabilistic Logic Programs have been recently proposed to perform inference in a lifted way. LIFTCOVER is an algorithm used to perform parameter and structure learning of liftable probabilistic logic programs. In particular, it performs parameter learning via Expectation Maximization and LBFGS. In this paper, we present an updated version of LIFTCOVER, called LIFTCOVER+, in which regularization was added to improve the quality of the solutions and LBFGS was replaced by gradient descent. We tested LIFTCOVER+ on the same 12 datasets on which LIFTCOVER was tested and compared the performances in terms of AUC-ROC, AUC-PR, and execution times. Results show that in most cases Expectation Maximization with regularization improves the quality of the solutions.

In 1990, Halpern proposed the distinction between Type 1 and Type 2 statements: the former express statistical information about a domain of interest while the latter define a degree of belief. An example of Type 1 statement is “30% of the elements of a domain share the same property” while an example of Type 2 statement is “the element x has the property y with probability p”. Recently, Type 1 statements were given an interpretation in terms of probabilistic answer set programs under the credal semantics in the PASTA framework. The algorithm proposed for inference requires the enumeration of all the answer sets of a given program, and so it is impractical for domains of not trivial size. The field of lifted inference aims to identify programs where inference can be computed without grounding the program. In this paper, we identify some classes of PASTA programs for which we apply lifted inference and develop compact formulas to compute the probability bounds of a query without the need to generate all the possible answer sets.

Abstract Probabilistic logic programming is an effective formalism for encoding problems characterized by uncertainty. Some of these problems may require the optimization of probability values subject to constraints among probability distributions of random variables. Here, we introduce a new class of probabilistic logic programs, namely probabilisticoptimizablelogic programs, and we provide an effective algorithm to find the best assignment to probabilities of random variables, such that a set of constraints is satisfied and an objective function is optimized.