Structural measures of graphs, such as treewidth, are central tools in computational complexity resulting in efficient algorithms when exploiting the parameter. It is even known that modern SAT solvers work efficiently on instances of small treewidth. Since these solvers are widely applied, research interests in compact encodings into (Q)SAT for solving and to understand encoding limitations. Even more general is the graph parameter clique-width, which unlike treewidth can be small for dense graphs. Although algorithms are available for clique-width, little is known about encodings. We initiate the quest to understand encoding capabilities with clique-width by considering abstract argumentation, which is a robust framework for reasoning with conflicting arguments. It is based on directed graphs and asks for computationally challenging properties, making it a natural candidate to study computational properties. We design novel reductions from argumentation problems to (Q)SAT. Our reductions linearly preserve the clique-width, resulting in directed decomposition-guided (DDG) reductions. We establish novel results for all argumentation semantics, including counting. Notably, the overhead caused by our DDG reductions cannot be significantly improved under reasonable assumptions.

Answer Set Programming (ASP) is a mature and widely used framework for modeling and solving problems in AI, knowledge representation and reasoning, and combinatorial search. Counting answer sets is of growing importance for analyzing search spaces, navigating ASP programs, and enabling probabilistic reasoning. While Truszczynski established a complete hierarchy for the computational complexity of ASP decision and reasoning problems (skeptical and credulous), a corresponding systematic treatment of counting problems has been missing so far. We close this gap by providing an almost complete characterisation of the counting complexity landscape for ASP. A remaining gap arises between Krom and Horn programs, caused by the minimality of disjunctions in Krom rule heads for guessing. To address this issue, we replace disjunctions with choice rules and introduce a controlled fragment in which choices are allowed and every rule is simultaneously Horn and Krom (Choice-Horn-Krom). We show that this fragment does not admit an polynomial-time approximation scheme (FPRAS) under standard complexity-theoretic assumptions. However, we prove that counting answer sets of an arbitrary ASP program can already be done by counting answer sets of two Choice-Horn-Krom programs. This result demonstrates the expressive power of ASP and yields a conceptually simpler alternative to Valiant's classical reduction from #SAT to #Krom-SAT, which a very well-known result in propositional logic.

Abstract The propositional model counting problem #SAT asks to compute the number of satisfying assignments for a given propositional formula. Recently, three #SAT proof systems $$\textsf{kcps}$$ kcps (knowledge compilation proof system), $$\textsf{MICE}$$ MICE (model counting induction by claim extension), and $$\textsf{CPOG}$$ CPOG (certified partitioned-operation graphs) have been introduced with the aim to model #SAT solving and enable proof logging for solvers. A fourth system, $$\textsf{CLIP}$$ CLIP (circuit linear introduction proposition), is a very powerful proof system of theoretical interest. Prior to this paper, it was only known that $$\textsf{CLIP}$$ CLIP simulates the three other systems. All the remaining relations between the systems have been unclear and very few proof complexity results are known. We completely determine the simulation order of the four systems, establishing that $$\textsf{CPOG}$$ CPOG simulates both $$\textsf{MICE}$$ MICE and $$\textsf{kcps}$$ kcps , while $$\textsf{MICE}$$ MICE and $$\textsf{kcps}$$ kcps are exponentially incomparable. This implies that $$\textsf{CPOG}$$ CPOG is strictly stronger than the other two systems.

Classical planning asks for a sequence of operators reaching a given goal. While the most common case is to compute a plan, many scenarios require more than that. However, quantitative reasoning on the plan space remains mostly unexplored. A fundamental problem is to count plans, which relates to the conditional probability on the plan space. Indeed, qualitative and quantitative approaches are well-established in various other areas of automated reasoning. We present the first study to quantitative and qualitative reasoning on the plan space. In particular, we focus on polynomially bounded plans. On the theoretical side, we study its complexity, which gives rise to rich reasoning modes. Since counting is hard in general, we introduce the easier notion of facets, which enables understanding the significance of operators. On the practical side, we implement quantitative reasoning for planning. Thereby, we transform a planning task into a propositional formula and use knowledge compilation to count different plans. This framework scales well to large plan spaces, while enabling rich reasoning capabilities such as learning pruning functions and explainable planning.

Argumentation is a central subarea of Artificial Intelligence (AI) for modeling and reasoning about arguments. The semantics of abstract argumentation frameworks (AFs) is given by sets of arguments (extensions) and conditions on the relationship between arguments, such as stable or admissible. Today's solvers implement tasks such as finding extensions, deciding credulously or skeptically acceptance, counting, or enumerating extensions. While these tasks are well charted, the area between decision and counting/enumeration and fine-grained reasoning requires expensive reasoning so far. We introduce a novel concept (facets) for reasoning between decision and enumeration. Facets are arguments that belong to some extensions (credulous) but not to all extensions (skeptical). They are most natural when a user aims to navigate, filter, or comprehend specific arguments, according to their needs. We study the complexity and show that tasks involving facets are much easier than counting extensions. Finally, we provide an implementation, and conduct experiments to demonstrate feasibility.

Many planning applications require not only a single solution but benefit substantially from having a set of possible plans from which users can select, for example, when explaining plans. For decades, research in classical AI planning has primarily focused on quickly finding single plans. Only recently researchers have started to investigate preferences, enumerate plans by top-k planning, or count plans to reason about the plan space. Unfortunately, reasoning about the plan space is computationally extremely hard and feeding many similar plans to the user is hardly practical. To circumvent computational shortcomings while still being able to reason about variability in plans, faceted actions have been introduced very recently. These are meaningful actions that can be used by some plan but are not required by all plans. Enforcing or forbidding such facets allows for navigating even large plan spaces while ensuring desired properties quickly and step by step. In this paper, we illustrate an industrial challenge, the Beluga logistics problem of Airbus, where reasoning with facets enables targeted plan space navigation. We present an approach to handle large plan spaces iteratively and interactively and present a tool that we call PlanPilot.

Abstract Abductive reasoning is a popular non-monotonic paradigm that aims to explain observed symptoms and manifestations. It has many applications, such as diagnosis and planning in artificial intelligence and database updates. In propositional abduction, we focus on specifying knowledge by a propositional formula. The computational complexity of tasks in propositional abduction has been systematically characterized – even with detailed classifications for Boolean fragments. Unsurprisingly, the most insightful reasoning problems (counting and enumeration) are computationally highly challenging. Therefore, we consider reasoning between decisions and counting, allowing us to understand explanations better while maintaining favorable complexity. We introduce facets to propositional abductions, which are literals that occur in some explanation (relevant) but not all explanations (dispensable). Reasoning with facets provides a more fine-grained understanding of variability in explanations (heterogeneous). In addition, we consider the distance between two explanations, enabling a better understanding of heterogeneity/homogeneity. We comprehensively analyze facets of propositional abduction in various settings, including an almost complete characterization in Post’s framework.

Computational evaluations are crucial in modern problem-solving when we surpass theoretical algorithms or bounds. These experiments frequently take much work, and the sheer amount of needed resources makes it impossible to execute them on a single personal computer or laptop. Cluster schedulers allow for automatizing these tasks and scale to many computers. But, when we evaluate implementations of combinatorial algorithms, we depend on stable runtime results. Common approaches either limit parallelism or suffer from unstable runtime measurements due to interference among jobs on modern hardware. The former is inefficient and not sustainable. The latter results in unreplicable experiments. In this work, we address this issue and offer an acceptable balance between efficiency, software, hardware complexity, reliability, and replicability. We investigate effects towards replicability stability and illustrate how to efficiently use widely employed cluster resources for parallel evaluations. Furthermore, we present solutions which mitigate issues that emerge from the concurrent execution of benchmark jobs. Our experimental evaluation shows that – despite parallel execution – our approach reduces the runtime instability on the majority of instances to one second.

Abstract argumentation is a popular toolkit for modeling, evaluating, and comparing arguments. Relationships between arguments are specified in argumentation frameworks (AFs), and conditions are placed on sets (extensions) of arguments that allow AFs to be evaluated. For more expressiveness, AFs are augmented with acceptance conditions on directly interacting arguments or a constraint on the admissible sets of arguments, resulting in dialectic frameworks or constrained argumentation frameworks. In this paper, we consider flexible conditions for rejecting an argument from an extension, which we call rejection conditions (RCs). On the technical level, we associate each argument with a specific logic program. We analyze the resulting complexity, including the structural parameter treewidth. Rejection AFs are highly expressive, giving rise to natural problems on higher levels of the polynomial hierarchy.

Counting solutions is a central task in mathematics and computer science. By counting, we can directly compute the probability of an event, making it fundamental to many application domains. However, often, we are interested in counting solutions only with respect to a “shown part” of the instance while still satisfying the “forgotten (hidden) part” called projected solution counting (PSC). In this paper, we introduce algorithms for PSC and a large variety of knowledge representation and reasoning (KRR) formalisms. Our algorithms employ small treewidth of the input instance, which yields polynomial-time solvability in the input size for instances of bounded treewidth. We obtain a generic result that allows lifting many existing given dynamic programming (DP) algorithms from decisions to PSC. Exemplarily, we present PSC for quantified Boolean formulas (QBFs), which is a canonical reasoning problem. Finally, we present computational lower bounds for various KRR formalisms where we cannot expect significant improvement for PSC assuming that the exponential-time hypothesis (ETH) holds.

Answer set programming is a popular declarative paradigm with countless applications for modeling and solving combinatorial problems. We can view a program as a knowledge database compactly representing conditions for solutions. Often we are interested in reasoning about solutions of filtering answer sets. At the heart of these questions is brave and cautious reasoning. For browsing answer sets, we combine both as restricting atoms of answer sets is only meaningful for atoms called facets that belong to some (brave) but not to all answer sets (cautious). Surprisingly, the precise computational complexity of facet problems remained widely open so far. In this paper, we study the complexity of answer set facets. We establish tight results for reasoning with facets, deciding upper and lower bounds as well as the exact number of facets, and comparing facets. Facet reasoning seems to be a natural problem formalism, residing in complexity families Σᴾ, Πᴾ, Dᴾ, and Θᴾ, up to the third level. Moreover, our study considers quantitative importance questions on facets and generalizing from facets to conjunctions, disjunctions, and arbitrary queries. We complete our results by an experimental evaluation.

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics, including credulous reasoning. When asking for projected counts, we are interested in counting the number of extensions of a given argumentation framework, while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by the treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming. Our algorithms run in double or triple exponential time in the treewidth, depending on the semantics under consideration. Finally, we establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension under the exponential time hypothesis (ETH).

In this article, we introduce a notion of backdoors to Reiter’s propositional default logic and study structural properties of it. Also we consider the problems of backdoor detection (parameterised by the solution size) as well as backdoor evaluation (parameterised by the size of the given backdoor) for various kinds of target classes (CNF, KROM, MONOTONE) and all SCHAEFER classes. Also, we study generalisations of HORN-formulas, namely QHORN, RHORN, as well as DUALHORN. For these classes, we also classify the computational complexity of the implication problem. We show that backdoor detection is fixed-parameter tractable for the considered target classes and prove a complete trichotomy for backdoor evaluation. The problems are either fixed-parameter tractable, para-DeltaP2-complete, or para-NP-complete, depending on the target class.

Abstract Answer set programming (ASP) is a popular declarative programming paradigm with various applications. Programs can easily have many answer sets that cannot be enumerated in practice, but counting still allows quantifying solution spaces. If one counts under assumptions on literals, one obtains a tool to comprehend parts of the solution space, so-called answer set navigation. However, navigating through parts of the solution space requires counting many times, which is expensive in theory. Knowledge compilation compiles instances into representations on which counting works in polynomial time. However, these techniques exist only for conjunctive normal form (CNF) formulas, and compiling ASP programs into CNF formulas can introduce an exponential overhead. This paper introduces a technique to iteratively count answer sets under assumptions on knowledge compilations of CNFs that encode supported models. Our anytime technique uses the inclusion–exclusion principle to improve bounds by over- and undercounting systematically. In a preliminary empirical analysis, we demonstrate promising results. After compiling the input (offline phase), our approach quickly (re)counts.

Answer Set Programming (ASP) is a prominent modeling and solving framework. An inconsistent core (IC) of an ASP program is an inconsistent subset of rules. In the case of inconsistent programs, a smallest or subset-minimal IC contains crucial rules for the inconsistency. In this work, we study fnding minimal ICs of ASP programs and key fragments from a complexity-theoretic perspective. Interestingly, due to ASP’s non-monotonic behavior, also consistent programs admit ICs. It turns out that there is an entire landscape of problems involving ICs with a diverse range of complexities up to the fourth level of the Polynomial Hierarchy. Deciding the existence of an IC is, already for tight programs, on the second level of the Polynomial Hierarchy. Furthermore, we give encodings for IC-related problems on the fragment of tight programs and illustrate feasibility on small instance sets.

Argumentation is a well-established formalism for nonmonotonic reasoning and a vibrant area of research in AI. Claim-augmented argumentation frameworks (CAFs) have been introduced to deploy a conclusion-oriented perspective. CAFs expand argumentation frameworks by an additional step which involves retaining claims for an accepted set of arguments. We introduce a novel concept of a justification status for claims, a quantitative measure of extensions supporting a particular claim. The well-studied problems of credulous and skeptical reasoning can then be seen as simply the two endpoints of the spectrum when considered as a justification level of a claim. Furthermore, we explore the parameterized complexity of various reasoning problems for CAFs, including the quantitative reasoning for claim assertions. We begin by presenting a suitable graph representation that includes arguments and their associated claims. Our analysis includes the parameter treewidth, and we present decomposition-guided reductions between reasoning problems in CAF and the validity problem for QBF.

The QSAT problem, which asks to evaluate a quantified Boolean formula (QBF), is of fundamental interest in approximation, counting, decision, and probabilistic complexity and is also considered the prototypical PSPACE-complete problem. As such, it has previously been studied under various structural restrictions (parameters), most notably parameterizations of the primal graph representation of instances. Indeed, it is known that QSAT remains PSPACE-complete even when restricted to instances with constant treewidth of the primal graph, but the problem admits a double-exponential fixed-parameter algorithm parameterized by the vertex cover number (primal graph).However, prior works have left a gap in our understanding of the complexity of QSAT when viewed from the perspective of other natural representations of instances, most notably via incidence graphs. In this paper, we develop structure-aware reductions which allow us to obtain essentially tight lower bounds for highly restricted instances of QSAT, including instances whose incidence graphs have bounded treedepth or feedback vertex number. We complement these lower bounds with novel algorithms for QSAT which establish a nearly-complete picture of the problem's complexity under standard graph-theoretic parameterizations. We also show implications for other natural graph representations, and obtain novel upper as well as lower bounds for QSAT under more fine-grained parameterizations of the primal graph.

Answer set programming (ASP) is a popular declarative programming paradigm with a wide range of applications in artificial intelligence. Oftentimes, when modeling an AI problem with ASP, and in particular when we are interested beyond simple search for optimal solutions, an actual solution, differences between solutions, or number of solutions of the ASP program matter. For example, when a user aims to identify a specific answer set according to her needs, or requires the total number of diverging solutions to comprehend probabilistic applications such as reasoning in medical domains. Then, there are only certain problem specific and handcrafted encoding techniques available to navigate the solution space of ASP programs, which is oftentimes not enough. In this paper, we propose a formal and general framework for interactive navigation toward desired subsets of answer sets analogous to faceted browsing. Our approach enables the user to explore the solution space by consciously zooming in or out of sub-spaces of solutions at a certain configurable pace. We illustrate that weighted faceted navigation is computationally hard. Finally, we provide an implementation of our approach that demonstrates the feasibility of our framework for incomprehensible solution spaces.

Answer Set Programming (ASP) is a framework in artificial intelligence and knowledge representation for declarative modeling and problem solving. Modern ASP solvers focus on the computation or enumeration of answer sets. However, a variety of probabilistic applications in reasoning or logic programming require counting answer sets. While counting can be done by enumeration, simple enumeration becomes immediately infeasible if the number of solutions is high. On the other hand, approaches to exact counting are of high worst-case complexity. In fact, in propositional model counting, exact counting becomes impractical. In this work, we present a scalable approach to approximate counting for answer set programming. Our approach is based on systematically adding XOR constraints to ASP programs, which divide the search space. We prove that adding random XOR constraints partitions the answer sets of an ASP program. In practice, we use a Gaussian elimination-based approach by lifting ideas from SAT to ASP and integrating it into a state of the art ASP solver, which we call ApproxASP. Finally, our experimental evaluation shows the scalability of our approach over the existing ASP systems.

Answer set programming is a form of declarative programming widely used to solve difficult search problems. Probabilistic applications however require to go beyond simple search for one solution and need counting. One such application is plausibility reasoning, which provides more fine-grained reasoning mode between simple brave and cautious reasoning. When modeling with ASP, we oftentimes introduce auxiliary atoms in the program. If these atoms are functionally independent of the atoms of interest, we need to hide the auxiliary atoms and project the count to the atoms of interest resulting in the problem projected answer set counting. In practice, counting becomes quickly infeasible with standard systems such as clasp. In this paper, we present a novel hybrid approach for plausibility reasoning under projections, thereby relying on projected answer set counting as basis. Our approach combines existing systems with fast dynamic programming, which in our experiments shows advantages over existing ASP systems.

Although propositional model counting (#SAT) was long considered too hard to be practical, today’s highly efficient solvers facilitate applications in probabilistic reasoning, reliability estimation, quantitative design space exploration, and more. The current trend of solvers growing more capable every year is likely to continue as a diverse range of algorithms are explored in the field. However, to establish model counters as reliable tools like SAT-solvers, correctness is as critical as speed. As in the nature of complex systems, bugs emerge as soon as the tools are widely used. To identify and avoid bugs, explain decisions, and provide trustworthy results, we need verifiable results. We propose a novel system for certifying model counts. We show how proof traces can be generated for exact model counters based on dynamic programming, counting CDCL with component caching, and knowledge compilation to Decision-DNNF, which are the predominant techniques in today’s exact implementations. We provide proof-of-concepts for emitting proofs and a parallel trace checker. Based on this, we show the feasibility of using certified model counting in an empirical experiment.

Abstract Bounded treewidth is one of the most cited combinatorial invariants in the literature. It was also applied for solving several counting problems efficiently. A canonical counting problem is #Sat, which asks to count the satisfying assignments of a Boolean formula. Recent work shows that benchmarking instances for #Sat often have reasonably small treewidth. This paper deals with counting problems for instances of small treewidth. We introduce a general framework to solve counting questions based on state-of-the-art database management systems (DBMSs). Our framework takes explicitly advantage of small treewidth by solving instances using dynamic programming (DP) on tree decompositions (TD). Therefore, we implement the concept of DP into a DBMS (PostgreSQL), since DP algorithms are already often given in terms of table manipulations in theory. This allows for elegant specifications of DP algorithms and the use of SQL to manipulate records and tables, which gives us a natural approach to bring DP algorithms into practice. To the best of our knowledge, we present the first approach to employ a DBMS for algorithms on TDs. A key advantage of our approach is that DBMSs naturally allow for dealing with huge tables with a limited amount of main memory (RAM).

Description logics (DLs) are knowledge representation languages that are used in the field of artificial intelligence (AI). A common technique is to query DL knowledge bases, e.g., by Boolean Datalog queries, and ask for entailment. But real world knowledge-bases are often obtained by combining data from various sources. This, inherently, might result in certain inconsistencies (with respect to a given query) and requires to estimate a degree of inconsistency before using a knowledge-base. In this paper, we provide a complexity analysis of fixed-domain non-entailment (NE) on Datalog programs for well-established families of knowledge bases (KBs). We exhibit a detailed complexity map for the decision cases, counting and projected counting, which may serve as a quantitative measure for inconsistency of a KB with respect to a query. Our results show that NE is natural for the second, third, and fourth level of the polynomial (counting) hierarchy depending on the type of the studied query (stratified, normal, disjunctive) and one level higher for the projected versions. Further, we show fixed-parameter tractability by bounding the treewidth, provide a constructive algorithm, and show its theoretical limitation in terms of conditional lower bounds.

In this paper, we revisit the approach to empirical experiments for combinatorial solvers. We provide a brief survey on tools that can help to make empirical work easier. We illustrate origins of uncertainty in modern hardware and show how strong the influence of certain aspects of modern hardware and its experimental setup can be in an actual experimental evaluation. More specifically, there can be situations where (i) two different researchers run a reasonable-looking experiment comparing the same solvers and come to different conclusions and (ii) one researcher runs the same experiment twice on the same hardware and reaches different conclusions based upon how the hardware is configured and used. We investigate these situations from a hardware perspective. Furthermore, we provide an overview on standard measures, detailed explanations on effects, potential errors, and biased suggestions for useful tools. Alongside the tools, we discuss their feasibility as experiments often run on clusters to which the experimentalist has only limited access. Our work sheds light on a number of benchmarking-related issues which could be considered to be folklore or even myths.

A promising new algebraic approach to weighted model counting makes use of tensor networks, following a reduction from weighted model counting to tensor-network contraction. Prior work has focused on analyzing the single-core performance of this approach, and demonstrated that it is an effective addition to the current portfolio of weighted-model-counting algorithms. In this work, we explore the impact of multi-core and GPU use on tensor-network contraction for weighted model counting. To leverage multiple cores, we implement a parallel portfolio of tree-decomposition solvers to find an order to contract tensors. To leverage a GPU, we use TensorFlow to perform the contractions. We compare the resulting weighted model counter on 1914 standard weighted model counting benchmarks and show that it significantly improves the virtual best solver.

Argumentation is a widely applied framework for modeling and evaluating arguments and its reasoning with various applications. Popular frameworks are abstract argumentation (Dung’s framework) or logic-based argumentation (Besnard-Hunter’s framework). Their computational complexity has been studied quite in-depth. Incorporating treewidth into the complexity analysis is particularly interesting, as solvers oftentimes employ SAT-based solvers, which can solve instances of low treewidth fast. In this paper, we address whether one can design reductions from argumentation problems to SAT-problems while linearly preserving the treewidth, which results in decomposition-guided (DG) reductions. It turns out that the linear treewidth overhead caused by our DG reductions, cannot be significantly improved under reasonable assumptions. Finally, we consider logic-based argumentation and establish new upper bounds using DG reductions and lower bounds.

The Steiner tree problem is a well-known problem in network design, routing, and VLSI design. Given a graph, edge costs, and a set of dedicated vertices (terminals), the Steiner tree problem asks to output a sub-graph that connects all terminals at minimum cost. A state-of-the-art algorithm to solve the Steiner tree problem by means of dynamic programming is the Dijkstra-Steiner algorithm. The algorithm builds a Steiner tree of the entire instance by systematically searching for smaller instances, based on subsets of the terminals, and combining Steiner trees for these smaller instances. The search heavily relies on a guiding heuristic function in order to prune the search space. However, to ensure correctness, this algorithm allows only for limited heuristic functions, namely, those that satisfy a so-called consistency condition. In this paper, we enhance the Dijkstra-Steiner algorithm and establish a revisited algorithm, called DS*. The DS* algorithm allows for arbitrary lower bounds as heuristics relaxing the previous condition on the heuristic function. Notably, we can now use linear programming based lower bounds. Further, we capture new requirements for a heuristic function in a condition, which we call admissibility. We show that admissibility is indeed weaker than consistency and establish correctness of the DS* algorithm when using an admissible heuristic function. We implement DS* and combine it with modern preprocessing, resulting in an open-source solver (DS*Solve). Finally, we compare its performance on standard benchmarks and observe a competitive behavior.

The problem of deciding the validity (QSat) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a successful parameter. A well-known result by Chen [9] is that QSat when parameterized by the treewidth of the primal graph and the quantifier rank of the input formula is fixed-parameter tractable. More precisely, the runtime of such an algorithm is polynomial in the formula size and exponential in the treewidth, where the exponential function in the treewidth is a tower, whose height is the quantifier rank. A natural question is whether one can significantly improve these results and decrease the tower while assuming the Exponential Time Hypothesis (ETH). In the last years, there has been a growing interest in the quest of establishing lower bounds under ETH, showing mostly problem-specific lower bounds up to the third level of the polynomial hierarchy. Still, an important question is to settle this as general as possible and to cover the whole polynomial hierarchy. In this work, we show lower bounds based on the ETH for arbitrary QBFs parameterized by treewidth and quantifier rank. More formally, we establish lower bounds for QSat and treewidth, namely, that under ETH there cannot be an algorithm that solves QSat of quantifier rank i in runtime significantly better than i-fold exponential in the treewidth and polynomial in the input size. In doing so, we provide a reduction technique to compress treewidth that encodes dynamic programming on arbitrary tree decompositions. Further, we describe a general methodology for a more finegrained analysis of problems parameterized by treewidth that are at higher levels of the polynomial hierarchy. Finally, we illustrate the usefulness of our results by discussing various applications of our results to problems that are located higher on the polynomial hierarchy, in particular, various problems from the literature such as projected model counting problems.

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics. When asking for projected counts we are interested in counting the number of extensions of a given argumentation framework while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming (DP). Our algorithms run in time double or triple exponential in the treewidth depending on the considered semantics. Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension.

The organizers of the 4th Parameterized Algorithms and Computational Experiments challenge (PACE 2019) report on the 4th iteration of the PACE challenge. This year, the first track featured the MinVertexCover problem, which asks given an undirected graph G=(V,E) to output a set S subseteq V of vertices such that for every edge vw in E at least one endpoint belongs to S. The exact decision version of this problem is one of the most discussed problem if not even the prototypical problem in parameterized complexity theory. Another two tracks were dedicated to computing the hypertree width of a given hypergraph, which is a certain generalization of tree decompositions to hypergraphs that has widely been applied to problems in databases, constraint programming, and artificial intelligence. On one track we asked for submissions that compute hypertree decompositions of minimum width (MinHypertreeWidth) and on the other track we asked to heuristically compute hypertree decompositions of small width quickly (HeurHypertreeWidth). We received 28 implementations from 26 teams. This year we asked participants to submit solver descriptions in order to count as a submission for the challenge. We received those from 16 teams with overall 33 participants from 10 countries. One team submitted successful solutions to all three tracks.

Abstract Answer Set Programming (ASP) solvers are highly-tuned and complex procedures that implicitly solve the consistency problem, i.e., deciding whether a logic program admits an answer set. Verifying whether a claimed answer set is formally a correct answer set of the program can be decided in polynomial time for (normal) programs. However, it is far from immediate to verify whether a program that is claimed to be inconsistent, indeed does not admit any answer sets. In this paper, we address this problem and develop the new proof format ASP-DRUPE for propositional, disjunctive logic programs, including weight and choice rules. ASP-DRUPE is based on the Reverse Unit Propagation (RUP) format designed for Boolean satisfiability. We establish correctness of ASP-DRUPE and discuss how to integrate it into modern ASP solvers. Later, we provide an implementation of ASP-DRUPE into the wasp solver for normal logic programs.

We propose a novel solver that efficiently finds almost the exact number of solutions of a Boolean formula (#Sat) and the weighted model count of a weighted Boolean formula (WMC) if the treewidth of the given formula is sufficiently small. The basis of our approach are dynamic programming algorithms on tree decompositions, which we engineered towards efficient parallel execution on the GPU. We provide thorough experiments and compare the runtime of our system with state-of-the-art #Sat and WMC solvers. Our results are encouraging in the sense that also complex reasoning problems can be tackled by parameterized algorithms executed on the GPU if instances have treewidth at most 30, which is the case for more than half of counting and weighted counting benchmark instances.

Today, health research and health care generate a steadily increasing amount of data. Making these available for secondary use cases is essential for efficiency gains in health research, e.g. by reducing time-and costs-intensive acquisition of data. In this contribution, we introduce our SAHRA software platform enabling reproducible research, e.g. by combining multiple data sources, performing data de-identification, and content filtering. We define an innovative research process combining retrospective and prospective research for the first time. Thus, authorized users, e.g. clinical researchers, are able to gain access through our system relevant research data and to perform interactive analyses. As a result, existing sensitive health data is securely transformed into de-identified research data, which can be used to improve future health research.

The main reasoning problems for disjunctive logic programs are complete for the second level of the polynomial hierarchy and hence considered harder than the same problems for normal (i.e., disjunction-free) programs, which are on the first level. We propose a new exact method for solving the disjunctive problems which exploits the small distance of a disjunctive programs from being normal. The distance is measured in terms of the size of a smallest “backdoor to normality,” which is the smallest number of atoms whose deletion makes the program normal. Our method consists of three phases. In the first phase, a smallest backdoor is computed. We show that this can be done using an efficient algorithm for computing a smallest vertex cover of a graph. In the second phase, the backdoor is used to transform the logic program into a quantified Boolean formula (QBF) where the number of universally quantified variables equals the size of the backdoor and where the total size of the quantified Boolean formula is quasilinear in the size of the given logic program. The quasilinearity is achieved by means of a characterization of the least model of a Horn program in terms of level numberings. In a third phase, the universal variables are eliminated using universal expansion yielding a propositional formula. The blowup in the last phase is confined to a factor that is exponential in the size of the backdoor but linear in the size of the quantified Boolean formula. By checking the satisfiability of the resulting formula with a S at solver (or by checking the satisfiability of the quantified Boolean formula by a Q bf -S at solver), we can decide the A sp reasoning problems on the input program. In consequence, we have a transformation from A sp problems to propositional satisfiability where the combinatorial explosion, which is expected when transforming a problem from the second level of the polynomial hierarchy to the first level, is confined to a function of the distance to normality of the input program. In terms of parameterized complexity, the transformation is fixed-parameter tractable. We complement this result by showing that (under plausible complexity-theoretic assumptions) such a fixed-parameter tractable transformation is not possible if we consider the distance to tightness instead of distance to normality.

Abstract Disjunctive Answer Set Programming is a powerful declarative programming paradigm with complexity beyond NP. Identifying classes of programs for which the consistency problem is in NP is of interest from the theoretical standpoint and can potentially lead to improvements in the design of answer set programming solvers. One of such classes consists of dual-normal programs, where the number of positive body atoms in proper rules is at most one. Unlike other classes of programs, dual-normal programs have received little attention so far. In this paper we study this class. We relate dual-normal programs to propositional theories and to normal programs by presenting several inter-translations. With the translation from dual-normal to normal programs at hand, we introduce the novel class of body-cycle free programs, which are in many respects dual to head-cycle free programs. We establish the expressive power of dual-normal programs in terms of SE- and UE-models, and compare them to normal programs. We also discuss the complexity of deciding whether dual-normal programs are strongly and uniformly equivalent.

The practical results of answer-set programming indicate that classical complexity theory is insufficient as a theoretical framework to explain why modern answer-set programming solvers work fast on industrial applications. Complexity analysis by means of parameterized complexity theory seems to be promising, because we think that the reason for the gap between theory and practice is the presence of a "hidden structure" in real-world instances. The application of parameterized complexity theory to answer-set programming would give a crucial understanding of how solver heuristics work. This profound understanding can be used to improve the decision heuristics of modern solvers and yields new efficient algorithms for decision problems in the nonmonotonic setting. My research aims to explain the gap between theoretical upper bounds and the effort to solve real-world instances. I will further develop by means of parameterized complexity exact algorithms which work efficiently for real-world instances. The approach is based on backdoors which are small sets of atoms that represent "clever reasoning shortcuts" through the search space. The concept of backdoors is widely used in the areas of propositional satisfiability and constraint satisfaction. I will show how this concept can be adapted to the nonmonotonic setting and how it can be utilized to improve common algorithms.

Over the last two decades, propositional satisfiability (SAT) has become one of the most successful and widely applied techniques for the solution of NP-complete problems. The aim of this paper is to investigate theoretically how SAT can be utilized for the efficient solution of problems that are harder than NP or co-NP. In particular, we consider the fundamental reasoning problems in propositional disjunctive answer set programming (ASP), BRAVE REASONING and SKEPTICA REASONING, which ask whether a given atom is contained in at least one or in all answer sets, respectively. Both problems are located at the second level of the Polynomial Hierarchy and thus assumed to be harder than NP or co-NP. One cannot transform these two reasoning problems into SAT in polynomial time, unless the Polynomial Hierarchy collapses. We show that certain structural aspects of disjunctive logic programs can be utilized to break through this complexity barrier, using new techniques from Parameterized Complexity. In particular, we exhibit transformations from BRAVE and SKEPTICAL REASONING to SAT that run in time $O(2^k n^2)$ where k is a structural parameter of the instance and n the input size. In other words, the reduction is fixed-parameter tractable for parameter k. As the parameter k we take the size of a smallest backdoor with respect to the class of normal (i.e., disjunction-free) programs. Such a backdoor is a set of atoms that when deleted makes the program normal. In consequence, the combinatorial explosion, which is expected when transforming a problem from the second level of the Polynomial Hierarchy to the first level, can now be confined to the parameter k, while the running time of the reduction is polynomial in the input size n, where the order of the polynomial is independent of k. We show that such a transformation is not possible if we consider backdoors with respect to tightness instead of normality. We think that our approach is applicable to many other hard combinatorial problems that lie beyond NP or co-NP, and thus significantly enlarge the applicability of SAT.

We present a unifying approach to the efficient evaluation of propositional answer-set programs. Our approach is based on backdoors which are small sets of atoms that represent clever reasoning shortcuts through the search space. The concept of backdoors is widely used in the areas of propositional satisfiability and constraint satisfaction. We show how this concept can be adapted to the nonmonotonic setting and how it allows to augment various known tractable subproblems, such as the evaluation of Horn and acyclic programs. In order to use backdoors we need to find them first. We utilize recent advances in fixed-parameter algorithmics to detect small backdoors. This implies fixed-parameter tractability of the evaluation of propositional answer-set programs, parameterized by the size of backdoors. Hence backdoor size provides a structural parameter similar to the treewidth parameter previously considered. We show that backdoor size and treewidth are incomparable, hence there are instances that are hard for one and easy for the other parameter. We complement our theoretical results with first empirical results.

We offer a new understanding of some aspects of practical SAT-solvers that are based on DPLL with unit-clause propagation, clause-learning, and restarts. We do so by analyzing a concrete algorithm which we claim is faithful to what practical solvers do. In particular, before making any new decision or restart, the solver repeatedly applies the unit-resolution rule until saturation, and leaves no component to the mercy of non-determinism except for some internal randomness. We prove the perhaps surprising fact that, although the solver is not explicitly designed for it, with high probability it ends up behaving as width-k resolution after no more than O(n^{2k+2}) conflicts and restarts, where n is the number of variables. In other words, width-k resolution can be thought of as O(n^{2k+2}) restarts of the unit-resolution rule with learning.