The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [arXiv:1312.4524]. Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: on one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement

Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting

In this paper we study lifted inference for the Weighted First-Order Model Counting problem (WFOMC), which counts the assignments that satisfy a given sentence in first-order logic (FOL); it has applications in Statistical Relational Learning (SRL) and Probabilistic Databases (PDB). We present several results. First, we describe a lifted inference algorithm that generalizes prior approaches in SRL and PDB. Second, we provide a novel dichotomy result for a non-trivial fragment of FO CNF sentences, showing that for each sentence the WFOMC problem is either in PTIME or #P-hard in the size of the input domain; we prove that, in the first case our algorithm solves the WFOMC problem in PTIME, and in the second case it fails. Third, we present several properties of the algorithm. Finally, we discuss limitations of lifted inference for symmetric probabilistic databases (where the weights of ground literals depend only on the relation name, and not on the constants of the domain), and prove the impossibility of a dichotomy result for the complexity of probabilistic inference for the entire language FOL

Connectivity of Boolean Satisfiability

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. For this implicitly defined graph, we here study the st-connectivity and connectivity problems. Building on the work of Gopalan et al. ("The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies", 2006/2009), we first investigate satisfiability problems given by CSPs, more exactly CNF(S)-formulas with constants (as considered in Schaefer's famous 1978 dichotomy theorem); we prove a computational dichotomy for the st-connectivity problem, asserting that it is either solvable in polynomial time or PSPACE-complete, and an aligned structural dichotomy, asserting that the maximal diameter of connected components is either linear in the number of variables, or can be exponential; further, we show a trichotomy for the connectivity problem, asserting that it is either in P, coNP-complete, or PSPACE-complete. Next we investigate two important variants: CNF(S)-formulas without constants, and partially quantified formulas; in both cases, we prove analogous dichotomies for st-connectivity and the diameter; for for the connectivity problem, we show a trichotomy in the case of quantified formulas, while in the case of formulas without constants, we identify fragments of a possible trichotomy. Finally, we consider the connectivity issues for B-formulas, which are arbitrarily nested formulas built from some fixed set B of connectives, and for B-circuits, which are Boolean circuits where the gates are from some finite set B; we prove a common dichotomy for both connectivity problems and the diameter; for partially quantified B-formulas, we show an analogous dichotomy.Comment: PhD thesis, 82 pages, contains all results from the previous papers arXiv:1312.4524, arXiv:1312.6679, and arXiv:1403.6165, plus additional findings. arXiv admin note: text overlap with arXiv:cs/0609072 by other author

Query Answering in Probabilistic Data and Knowledge Bases

Probabilistic data and knowledge bases are becoming increasingly important in academia and industry. They are continuously extended with new data, powered by modern information extraction tools that associate probabilities with knowledge base facts. The state of the art to store and process such data is founded on probabilistic database systems, which are widely and successfully employed. Beyond all the success stories, however, such systems still lack the fundamental machinery to convey some of the valuable knowledge hidden in them to the end user, which limits their potential applications in practice. In particular, in their classical form, such systems are typically based on strong, unrealistic limitations, such as the closed-world assumption, the closed-domain assumption, the tuple-independence assumption, and the lack of commonsense knowledge. These limitations do not only lead to unwanted consequences, but also put such systems on weak footing in important tasks, querying answering being a very central one. In this thesis, we enhance probabilistic data and knowledge bases with more realistic data models, thereby allowing for better means for querying them. Building on the long endeavor of unifying logic and probability, we develop different rigorous semantics for probabilistic data and knowledge bases, analyze their computational properties and identify sources of (in)tractability and design practical scalable query answering algorithms whenever possible. To achieve this, the current work brings together some recent paradigms from logics, probabilistic inference, and database theory

Beyond Q-Resolution and Prenex Form: A Proof System for Quantified Constraint Satisfaction

We consider the quantified constraint satisfaction problem (QCSP) which is to decide, given a structure and a first-order sentence (not assumed here to be in prenex form) built from conjunction and quantification, whether or not the sentence is true on the structure. We present a proof system for certifying the falsity of QCSP instances and develop its basic theory; for instance, we provide an algorithmic interpretation of its behavior. Our proof system places the established Q-resolution proof system in a broader context, and also allows us to derive QCSP tractability results

Reasons for Hardness in QBF Proof Complexity

Quantified Boolean Formulas (QBF) extend the canonical NP-complete satisfiability problem by including Boolean quantifiers. Determining the truth of a QBF is PSPACE-complete; this is expected to be a harder problem than satisfiability, and hence QBF solving has much wider applications in practice. QBF proof complexity forms the theoretical basis for understanding QBF solving, as well as providing insights into more general complexity theory, but is less well understood than propositional proof complexity. We begin this thesis by looking at the reasons underlying QBF hardness, and in particular when the hardness is propositional in nature, rather than arising due to the quantifiers. We introduce relaxing QU-Res, a previous model for identifying such propositional hardness, and construct an example where relaxing QU-Res is unsuccessful in this regard. We then provide a new model for identifying such hardness which we prove captures this concept. Now equipped with a means of identifying ‘genuine’ QBF hardness, we prove a new lower bound technique for tree-like QBF proof systems. Lower bounds using this technique allows us to show a new separation between tree-like and dag-like systems. We give a characterisation of lower bounds for a large class of tree-like proof systems, in which such lower bounds play a prominent role. Further to the tree-like bound, we provide a new lower bound technique for QBF proof systems in general. This technique has some similarities to the above technique for tree-like systems, but requires some refinement to provide bounds for dag-like systems. We give applications of this new technique by proving lower bounds across several systems. The first such lower bounds are for a very simple family of QBFs. We then provide a construction to combine false QBFs to give formulas for which we can show lower bounds in this way, allowing the generation of the first random QBF proof complexity lower bounds