Due to the wide employment of automated reasoning in the analysis and construction of correct systems, the results reported by automated reasoning engines must be trustworthy. For Boolean satisfiability (SAT) solvers - and more recently SAT-based maximum satisfiability (MaxSAT) solvers - trustworthiness is obtained by integrating proof logging into solvers, making solvers capable of emitting machine-verifiable proofs to certify correctness of the reasoning steps performed. In this work, we enable for the first time proof logging based on the VeriPB proof format for multi-objective MaxSAT (MO-MaxSAT) optimization techniques. Although VeriPB does not offer direct support for multi-objective problems, we detail how preorders in VeriPB can be used to provide certificates for MO-MaxSAT algorithms computing a representative solution for each element in the non-dominated set of the search space under Pareto-optimality, without extending the VeriPB format or the proof checker. By implementing VeriPB proof logging into a state-of-the-art multi-objective MaxSAT solver, we show empirically that proof logging can be made scalable for MO-MaxSAT with reasonable overhead.

In eXplainable Constraint Solving (XCS), it is common to extract a Minimal Unsatisfiable Subset (MUS) from a set of unsatisfiable constraints. This helps explain to a user why a constraint specification does not admit a solution. Finding MUSes can be computationally expensive for highly symmetric problems, as many combinations of constraints need to be considered. In the traditional context of solving satisfaction problems, symmetry has been well studied, and effective ways to detect and exploit symmetries during the search exist. However, in the setting of finding MUSes of unsatisfiable constraint programs, symmetries are understudied. In this paper, we take inspiration from existing symmetry-handling techniques and adapt well-known MUS-computation methods to exploit symmetries in the specification, speeding-up overall computation time. Our results display a significant reduction of runtime for our adapted algorithms compared to the baseline on symmetric problems.

We analyze how symmetries can be used to compress structures (also known as interpretations) onto a smaller domain without loss of information. This analysis suggests the possibility to solve satisfiability problems in the compressed domain for better performance. Thus, we propose a 2-step novel method: (i) the sentence to be satisfied is automatically translated into an equisatisfiable sentence over a ``lifted'' vocabulary that allows domain compression; (ii) satisfiability of the lifted sentence is checked by growing the (initially unknown) compressed domain until a satisfying structure is found. The key issue is to ensure that this satisfying structure can always be expanded into an uncompressed structure that satisfies the original sentence to be satisfied. We present an adequate translation for sentences in typed first-order logic extended with aggregates. Our experimental evaluation shows large speedups for generative configuration problems. The method also has applications in the verification of software operating on complex data structures. Our results justify further research in automatic translation of sentences for symmetry reduction.

We build on a recently proposed method for stepwise explaining the solutions to Constraint Satisfaction Problems (CSPs) in a human understandable way. An explanation here is a sequence of simple inference steps where simplicity is quantified by a cost function. Explanation generation algorithms rely on extracting Minimal Unsatisfiable Subsets (MUSs) of a derived unsatisfiable formula, exploiting a one-to-one correspondence between so-called non-redundant explanations and MUSs. However, MUS extraction algorithms do not guarantee subset minimality or optimality with respect to a given cost function. Therefore, we build on these formal foundations and address the main points of improvement, namely how to generate explanations efficiently that are provably optimal (with respect to the given cost metric). To this end, we developed (1) a hitting set-based algorithm for finding the optimal constrained unsatisfiable subsets; (2) a method for reusing relevant information across multiple algorithm calls; and (3) methods for exploiting domain-specific information to speed up the generation of explanation sequences. We have experimentally validated our algorithms on a large number of CSP problems. We found that our algorithms outperform the MUS approach in terms of explanation quality and computational time (on average up to 56 % faster than a standard MUS approach).

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

Many practical problems can be understood as the search for a state of affairs that extends a fixed partial state of affairs, the \emph{environment}, while satisfying certain conditions that are formally specified. Such problems are found in, e.g., engineering, law or economics. We study this class of problems in a context where some of the relevant information about the environment is not known by the user at the start of the search. During the search, the user may consider tentative solutions that make implicit hypotheses about these unknowns. To ensure that the solution is appropriate, these hypotheses must be verified by observing the environment. Furthermore, we assume that, in addition to knowledge of what constitutes a solution, knowledge of general laws of the environment is also present. We formally define partial solutions with enough verified facts to guarantee the existence of complete and appropriate solutions. Additionally, we propose an interactive system to assist the user in their search by determining 1) which hypotheses implicit in a tentative solution must be verified in the environment, and 2) which observations can bring useful information for the search. We present an efficient method to over-approximate the set of relevant information, and evaluate our implementation.

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, i.e. operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of nondeterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola, et al., and (3) we generalize the characterisations of disjunctive logic programs to disjunctive logic programs with aggregates. This is an extended version of our paper that will be presented at ICLP 2023 and will appear in the special issue of TPLP with the ICLP proceedings.

Link Traversal-based Query Processing (ltqp), in which a sparql query is evaluated over a web of documents rather than a single dataset, is often seen as a theoretically interesting yet impractical technique. However, in a time where the hypercentralization of data has increasingly come under scrutiny, a decentralized Web of Data with a simple document-based interface is appealing, as it enables data publishers to control their data and access rights. While ltqp allows evaluating complex queries over such webs, it suffers from performance issues (due to the high number of documents containing data) as well as information quality concerns (due to the many sources providing such documents). In existing ltqp approaches, the burden of finding sources to query is entirely in the hands of the data consumer. In this paper, we argue that to solve these issues, data publishers should also be able to suggest sources of interest and guide the data consumer towards relevant and trustworthy data. We introduce a theoretical framework that enables such guided link traversal and study its properties. We illustrate with a theoretic example that this can improve query results and reduce the number of network requests. We evaluate our proposal experimentally on a virtual linked web with specifications and indeed observe that not just the data quality but also the efficiency of querying improves.

Semantics of various formalisms for knowledge representation can often be described by fixpoints of corresponding operators. For example, in many logics theories of a set of formulas can be seen as fixpoints of the underlying consequence operator [52]. Likewise, in logic programming, default logic or formal argumentation, all the major semantics can be formulated as different types of fixpoints of the same operator (see [22]). Such operators are usually non-monotonic, and so one cannot always be sure whether their fixpoints exist, and how they can be constructed. In order to deal with this'illusive nature' of the fixpoints, Denecker, Marek and Truszczyński [22] introduced a method for approximating each value z of the underlying operator by a pair of elements (x, y). These elements intuitively represent lower and upper bounds on z, and so a corresponding approximation operator for the original, non-monotonic operator, is constructed. If the approximating operator that is obtained is precision-monotonic, intuitively meaning that more precise inputs of the operator give rise to more precise outputs, then by Tarski and Knaster's Fixpoint Theorem the approximating operator has fixpoints that can be constructively computed, and which in turn approximate the fixpoints of the approximated operator, if such fixpoints exist. The usefulness of the algebraic theory that underlies the computation process described above was demonstrated on several knowledge representation formalisms, such as propositional logic programming [20], default logic [23], autoepistemic logic [23], abstract argumentation and abstract dialectical frameworks [50], hybrid MKNF [39], the graph description language SCHACL [11], and active integrity constraints [10], each one of which was shown to be an instantiation of this abstract theory of approximation.