The Boolean satisfiability problem is a well-known NP-complete problem. Due to the rapid advance of SA T solvers, many combinatorial problems are today solved by reducing to SA T, which can then be solved with off-the-shelf solvers. SA T fundamentally encodes a form of monotonic reasoning in the sense that conclusions remain valid regardless if new information is added. However, the real world is non-monotonic, meaning that one should be able to retract a statement if new data is added which violates the previous conclusion. One of the best known examples of non-monotonic reasoning is abductive reasoning where we are interested in finding an explanation, vis- ` a-vis a hypothesis, of some observed manifestation. Abduction has many practical applications, e.g., scientific discovery [10], network security [1], computational biology [22], medical diagnosis [18], knowledge base updates [23], and explainability issues in machine learning and decision support systems [7, 8, 2, 21]. This may be especially poignant in forthcoming decades due to the continued emergence of AI in new and surprising applications, which need to be made GDPR compliant [26] and explainable. The incitement for solving abduction fast, even when it is classically intractable, thus seems highly practically motivated. Can non-monotonic reasoning be performed as efficiently as monotonic reasoning, or are there fundamental differences between the two?

We explore the problem of explaining observations starting from a classically inconsistent theory by adopting a paraconsistent framework. We consider two expansions of the well-known Belnap--Dunn paraconsistent four-valued logic $\mathsf{BD}$: $\mathsf{BD}_\circ$ introduces formulas of the form $\circ\phi$ (the information on $\phi$ is reliable), while $\mathsf{BD}_\triangle$ augments the language with $\triangle\phi$'s (there is information that $\phi$ is true). We define and motivate the notions of abduction problems and explanations in $\mathsf{BD}_\circ$ and $\mathsf{BD}_\triangle$ and show that they are not reducible to one another. We analyse the complexity of standard abductive reasoning tasks (solution recognition, solution existence, and relevance / necessity of hypotheses) in both logics. Finally, we show how to reduce abduction in $\mathsf{BD}_\circ$ and $\mathsf{BD}_\triangle$ to abduction in classical propositional logic, thereby enabling the reuse of existing abductive reasoning procedures.

Given a knowledge base and an observation as a set of facts, ABox abduction aims at computing a hypothesis that, when added to the knowledge base, is sufficient to entail the observation. In signature-based ABox abduction, the hypothesis is further required to use only names from a given set. This form of abduction has applications such as diagnosis, KB repair, or explaining missing entailments. It is possible that hypotheses for a given observation only exist if we admit the use of fresh individuals and/or complex concepts built from the given signature, something most approaches for ABox abduction so far do not support or only support with restrictions. In this paper, we investigate the computational complexity of this form of abduction -- allowing either fresh individuals, complex concepts, or both -- for various description logics, and give size bounds on the hypotheses if they exist.

Signature-based abduction aims at building hypotheses over a specified set of names, the signature, that explain an observation relative to some background knowledge. This type of abduction is useful for tasks such as diagnosis, where the vocabulary used for observed symptoms differs from the vocabulary expected to explain those symptoms. We present the first complete method solving signature-based abduction for observations expressed in the expressive description logic ALC, which can include TBox and ABox axioms, thereby solving the knowledge base abduction problem. The method is guaranteed to compute a finite and complete set of hypotheses, and is evaluated on a set of realistic knowledge bases.

While looking for abductive explanations of a given set of manifestations, an ordering between possible solutions is often assumed. The complexity of finding/verifying optimal solutions is already known. In this paper we consider the computational complexity of finding second-best solutions. We consider different orderings, and consider also different possible definitions of what a second-best solution is.

With the current upward trend in semantically annotated data, ontology-based data access (OBDA) was formulated to tackle the problem of data integration and query answering, where an ontology is formalized as a description logic TBox. In order to meet usability requirements set by users, efforts have been made to equip OBDA system with explanation facilities. One important explanation tool for DL ontologies, referred to as query abduction, can be formalised as abductive reasoning. In particular, given an ontology and an observation (i.e., a query with an answer), an explanation to the observation is a set of facts that together with the ontology can entail the observation. In this paper, we develop a sound and complete algorithm of query abduction for general conjunctive queries in ELH ontologies. This is achieved through ontology approximation and query rewriting. We implemented a prototypical system using the highly optimized Prolog engine XSB. We evaluated our algorithm over university benchmark ontology and our experimental results show that the system is capable of handling query abduction problems for ontology that has approximately 10 millions ABox assertions.

Abduction is a form of nonmonotonic reasoning that looks for an explanation, built from a given set of hypotheses, for an observed manifestation according to some knowledge base. Following the concept behind the Schaefer's parametrization CSP(Gamma) of the Constraint Satisfaction Problem (CSP), we study here the complexity of the abduction problem Abduction(Gamma, Hyp, M) parametrized by certain (omega-categorical) infinite relational structures Gamma, Hyp, and M from which a knowledge base, hypotheses and a manifestation are built, respectively. We say that Gamma has local-to-global consistency if there is k such that establishing strong k-consistency on an instance of CSP(Gamma) yields a globally consistent (whose every solution may be obtained straightforwardly from partial solutions) set of constraints. In this case CSP(Gamma) is solvable in polynomial time. Our main contribution is an algorithm that under some natural conditions decides Abduction(Gamma, Hyp, M) in P when Gamma has local-to-global consistency. As we show in the number of examples, our approach offers an opportunity to consider abduction in the context of spatial and temporal reasoning (qualitative calculi such as Allen's interval algebra or RCC-5) and that our procedure solves some related abduction problems in polynomial time.

ABox abduction is an important reasoning mechanism for description logic ontologies. It computes all minimal explanations (sets of ABox assertions) whose appending to a consistent ontology enforces the entailment of an observation while keeps the ontology consistent. We focus on practical computation for a general problem of ABox abduction, called the query abduction problem, where an observation is a Boolean conjunctive query and the explanations may contain fresh individuals neither in the ontology nor in the observation. However, in this problem there can be infinitely many minimal explanations. Hence we first identify a class of TBoxes called first-order rewritable TBoxes. It guarantees the existence of finitely many minimal explanations and is sufficient for many ontology applications. To reduce the number of explanations that need to be computed, we introduce a special kind of minimal explanations called representative explanations from which all minimal explanations can be retrieved. We develop a tractable method (in data complexity) for computing all representative explanations in a consistent ontology. xperimental results demonstrate that the method is efficient and scalable for ontologies with large ABoxes.

ABox abduction is an important aspect for abductive reasoning in Description Logics (DLs). It finds all minimal sets of ABox axioms that should be added to a background ontology to enforce entailment of a specified set of ABox axioms. As far as we know, by now there is only one ABox abduction method in expressive DLs computing abductive solutions with certain minimality. However, the method targets an ABox abduction problem that may have infinitely many abductive solutions and may not output an abductive solution in finite time. Hence, in this paper we propose a new ABox abduction problem which has only finitely many abductive solutions and also propose a novel method to solve it. The method reduces the original problem to an abduction problem in logic programming and solves it with Prolog engines. Experimental results show that the method is able to compute abductive solutions in benchmark OWL DL ontologies with large ABoxes.

Abduction is a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining how the world behaves it aims at finding an explanation for some observed manifestation. In this paper we focus on propositional abduction, where the knowledge base and the manifestation are represented by propositional formulae. The problem of deciding whether there exists an explanation has been shown to be SigmaP2-complete in general. We consider variants obtained by restricting the allowed connectives in the formulae to certain sets of Boolean functions. We give a complete classification of the complexity for all considerable sets of Boolean functions. In this way, we identify easier cases, namely NP-complete and polynomial cases; and we highlight sources of intractability. Further, we address the problem of counting the explanations and draw a complete picture for the counting complexity.