We study generic integer programming (IP) encodings of acyclicity in directed graphs as a key constraint in various real-world problem domains. We analyze both classical and more recently-proposed generic acyclicity encodings, including Miller-Tucker-Zemlin (MTZ), feedback vertex set (FVS), vertex elimination (VE), and cycle elimination (CE) based encodings in terms of their linear programming (LP) relaxation tightness. We also introduce hybrid encodings combining sought-after properties of the individual encodings. For the hybrids, we establish tightness guarantees for their LP relaxations that interpolate smoothly between the individual encodings. Our results show that VE and CE yield equally strong relaxations and strictly dominate MTZ and FVS, while the hybrid encoding schemes become increasingly tight as the elimination prefix grows. Mapping theory to practice, we empirically evaluate the encodings on both direct IP encodings of problem domains, where acyclicity is a key constraint. The results both validate our theoretical findings and yield promising runtime performance.

Answer set programming (ASP) in the NP fragment can be solved by translating into propositional satisfiability (SAT). However, for non-tight programs, this requires additional encodings to enforce acyclicity of the underlying dependency graph, making acyclicity handling a key challenge in translating ASP encodings of decision and optimization problems into SAT and maximum satisfiability (MaxSAT). Various SAT encodings of acyclicity exploiting structural graph properties have recently been proposed for various settings. Focusing on ASP, we show that such encodings are captured by a generalized transitive closure framework. The framework can be instantiated for obtaining various types of refined transitive closure encodings. We consider four concrete instantiations framework, analyzing their correctness and size. Putting the framework into practice, we show through extensive empirical evaluation that current state-of-the-art SAT and MaxSAT solvers are competitive with and often even outperform state-of-the-art native ASP solvers on both decision and optimization problems.

The dominant approaches for solving NP-hard reasoning problems in computational argumentation are declarative—namely, Boolean satisfiability (SAT) in the case of abstract argumentation and answer set programming (ASP) in the case of structured formalisms such as assumption-based argumentation (ABA). ASP is particularly suited for the commonly-studied logic programming variant of ABA as acyclic derivations in ABA can be naturally modelled in ASP. In this work, we develop and evaluate various alternative approaches to realizing SAT-based reasoning for ABA, motivated by the success of SAT solvers in the realm of abstract argumentation. In contrast to ASP, non-trivial encodings or extensions to SAT solvers are needed to efficiently handle the acyclicity constraint underlying ABA reasoning. We develop and evaluate both advanced encodings and user-defined propagation mechanisms for realizing efficient SAT-based reasoning in ABA. As a result, we provide a first SAT-based ABA reasoner that can outperform the current state-of-the-art ASP approach to ABA.

We propose a maximum satisfiability (MaxSAT) based approach to cost-optimal delete-free planning, also known as optimal relaxed planning. Relaxed planning is a central subclass of classical planning, consisting of computing the h+ heuristic for classical planning. As an alternative to the existing approaches to exactly computing h+, we propose a maximum satisfiability (MaxSAT) based approach, motivated by the success of SAT-based planners and significant recent advances in MaxSAT solvers. Concretely, we both adapt a recent answer set optimization approach to computing h+ for MaxSAT, propose further MaxSAT encoding variants for both representing cost-optimal plans and plan acyclicity, and combine them for further runtime improvements. Overall, our MaxSAT approach compares favourably to the current state-of-the-art answer set optimization approach.

We conceptualize explainability in terms of logic and formula size, giving a number of related definitions of explainability in a very general setting. Our main interest is the so-called local explanation problem which aims to explain the truth value of an input formula in an input model. The explanation is a formula of minimal size that (1) obtains the same truth value as the input formula on the input model and (2) transmits that truth value to the input formula globally, i.e., on every model. As an important example case, we study propositional logic in this setting and show that the local explainability problem is complete for the second level of the polynomial hierarchy. The hardness result holds already for DNF-formulas. We also give parameterized versions of these problems leading to NP-completeness. The generality of our definitions allows us to lift complexity results also, e.g., to S5 modal logic and ensembles of decision trees. We also provide an implementation in answer set programming and investigate its capacity in relation to explaining answers to the n-queens and dominating set problems. Furthermore, we give an example of explaining the behavior of a black-box classifier.

Finding a graph with given properties occurs as a sub-problem of many important problems in A.I. and other areas of computer science. Main approaches to solving such problems include automated reasoning and constraint satisfaction methods. These can often be substantially sped up by considering only a subset of graphs for each equivalence class of isomorphic graphs, motivating the use of symmetry-breaking constraints for graphs. We present a symmetry-breaking constraint for directed graphs, generalizing earlier works that have presented such constraints for undirected graphs without loops, and experimentally demonstrate their effectiveness.

Abstract We investigate explainability via short Boolean formulas in the data model based on unary relations. As an explanation of lengthk, we take a Boolean formula of lengthkthat minimizes the error with respect to the target attribute to be explained. We first provide novel quantitative bounds for the expected error in this scenario. We then also demonstrate how the setting works in practice by studying three concrete data sets. In each case, we calculate explanation formulas of different lengths using an encoding in Answer Set Programming. The most accurate formulas we obtain achieve errors similar to other methods on the same data sets. However, due to overfitting, these formulas are not necessarily ideal explanations, so we use cross validation to identify a suitable length for explanations. By limiting to shorter formulas, we obtain explanations that avoid overfitting but are still reasonably accurate and also, importantly, human interpretable.

Abstract We establish a novel relation between delete-free planning, an important task for the AI planning community also known as relaxed planning, and logic programming. We show that given a planning problem, all subsets of actions that could be ordered to produce relaxed plans for the problem can be bijectively captured with stable models of a logic program describing the corresponding relaxed planning problem. We also consider the supported model semantics of logic programs, and introduce one causal and one diagnostic encoding of the relaxed planning problem as logic programs, both capturing relaxed plans with their supported models. Our experimental results show that these new encodings can provide major performance gain when computing optimal relaxed plans, with our diagnostic encoding outperforming state-of-the-art approaches to relaxed planning regardless of the given time limit when measured on a wide collection of STRIPS planning benchmarks.

We introduce novel methods for encoding acyclicity and s-t-reachability constraints for propositional formulas with underlying directed graphs, based on vertex elimination graphs, which makes them suitable for cases where the underlying graph has a low directed elimination width. In contrast to solvers with ad hoc constraint propagators for graph constraints such as GraphSAT, our methods encode these constraints as standard propositional clauses, making them directly applicable with any SAT solver. An empirical study demonstrates that our methods do often outperform both earlier encodings of these constraints as well as GraphSAT especially when underlying graphs have a low directed elimination width.

We introduce a novel method for encoding cost optimal delete-free STRIPS Planning as SAT. Our method is based on representing relaxed plans as partial functions from the set of propositions to the set of actions. This function can map any proposition to a unique action that adds the proposition during execution of the relaxed plan. We show that a relaxed plan can be produced by maintaining acyclicity in the graph of all causal relations among propositions, represented by the mentioned partial function. We also show that by efficient encoding of action cost propagation and enforcing a series of upper bounds on the total costs of the output plan, an optimal plan can effectively be produced for a given delete-free STRIPS problem. Our empirical results indicate that this method is quite competitive with the state of the art, demonstrating a better coverage compared to that of competing methods on standard STRIPS planning benchmark problems.

Planning as satisfiability is known as an efficient approach to deal with many types of planning problems. However, this approach has not been competitive with the state-space based methods in temporal planning. This paper describes ITSAT as an efficient SAT-based (satisfiability based) temporal planner capable of temporally expressive planning. The novelty of ITSAT lies in the way it handles temporal constraints of given problems without getting involved in the difficulties of introducing continuous variables into the corresponding satisfiability problems. We also show how, as in SAT-based classical planning, carefully devised preprocessing and encoding schemata can considerably improve the efficiency of SAT-based temporal planning. We present two preprocessing methods for mutex relation extraction and action compression. We also show that the separation of causal and temporal reasoning enables us to employ compact encodings that are based on the concept of parallel execution semantics. Although such encodings have been shown to be quite effective in classical planning, ITSAT is the first temporal planner utilizing this type of encoding. Our empirical results show that not only does ITSAT outperform the state-of-the-art temporally expressive planners, it is also competitive with the fast temporal planners that cannot handle required concurrency.

Since that heuristic state space planners have been very successful in classical planning, this approach is currently the most popular strategy in dealing with temporal planning, too. However, all current state-space temporal planners use a search method known as decision epoch planning, which is not complete for problems with required concurrency. In theory, this flaw can be overcome by employing another search method, called temporally lifted progression planning. In this paper, we show that there are two major problems which, if not tackled properly, can cause the latter method to be very inefficient in practice. The first problem is dealing with the remarkably large state space of temporally lifted progression planning. We present a pruning method for solving this problem and prove it to be both complete and optimality preserving. The next troublesome issue is solving a simple temporal problem (STP) in each state for computing g-values. We exploit the properties of such STPs and introduce a new method that solves them more efficiently than the state of the art algorithms do. Our experiments show that the new search method can add completeness to a state-of-the-art incomplete planner, TFD, without considerably worsening its performance in most standard domains.