Parameterized Compilation Lower Bounds for Restricted CNF-formulas

We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that - there are CNF formulas of size $n$ and modular incidence treewidth $k$ whose smallest DNNF-encoding has size $n^{\Omega(k)}$, and - there are CNF formulas of size $n$ and incidence neighborhood diversity $k$ whose smallest DNNF-encoding has size $n^{\Omega(\sqrt{k})}$. These results complement recent upper bounds for compiling CNF into DNNF and strengthen---quantitatively and qualitatively---known conditional low\-er bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth

SDDs are Exponentially More Succinct than OBDDs

Introduced by Darwiche (2011), sentential decision diagrams (SDDs) are essentially as tractable as ordered binary decision diagrams (OBDDs), but tend to be more succinct in practice. This makes SDDs a prominent representation language, with many applications in artificial intelligence and knowledge compilation. We prove that SDDs are more succinct than OBDDs also in theory, by constructing a family of boolean functions where each member has polynomial SDD size but exponential OBDD size. This exponential separation improves a quasipolynomial separation recently established by Razgon (2013), and settles an open problem in knowledge compilation

Certified Knowledge Compilation with Application to Verified Model Counting

Computing many useful properties of Boolean formulas, such as their weighted or unweighted model count, is intractable on general representations. It can become tractable when formulas are expressed in a special form, such as the decision-decomposable, negation normal form (dec-DNNF) . Knowledge compilation is the process of converting a formula into such a form. Unfortunately existing knowledge compilers provide no guarantee that their output correctly represents the original formula, and therefore they cannot validate a model count, or any other computed value. We present Partitioned-Operation Graphs (POGs), a form that can encode all of the representations used by existing knowledge compilers. We have designed CPOG, a framework that can express proofs of equivalence between a POG and a Boolean formula in conjunctive normal form (CNF). We have developed a program that generates POG representations from dec-DNNF graphs produced by the state-of-the-art knowledge compiler D4, as well as checkable CPOG proofs certifying that the output POGs are equivalent to the input CNF formulas. Our toolchain for generating and verifying POGs scales to all but the largest graphs produced by D4 for formulas from a recent model counting competition. Additionally, we have developed a formally verified CPOG checker and model counter for POGs in the Lean 4 proof assistant. In doing so, we proved the soundness of our proof framework. These programs comprise the first formally verified toolchain for weighted and unweighted model counting

Anwendungen von #SAT Solvern für Produktlinien: Masterarbeit

Product lines are widely used for managing families of similar products. Typically, product lines are complex and infeasible to analyze manually. In the last two decades, product-line analyses have been reduced to satisfiability problems which are well understood. However, there are methods for which satisfiability is not sufficient. Recently, researchers begun to reduce other problems to #SAT. Yet, only few applications have been considered and those are fairly limited in their scope. Furthermore, the authors mainly propose ad-hoc solutions that are only applicable under certain restrictions or do not scale to large product lines. In this thesis, we aim show the benefits of applying #SAT for the analysis of product lines. To this end, we make the following contributions: First, we summarize applications dependent on #AT considered in the literature and propose new applications to motivate the usage of #SAT technology. Second, we present a variety of algorithms and optimizations for these applications including new proposals. Third, we empirically evaluate 10 proposed algorithms with 14 off-the-shelf #SAT solvers on 131 industrial feature models to identify the fastest algorithms and solvers. Our results show that for each analysis at least one algorithm and solver scale on a vast majority of the feature models, whereas Linux and an automotive model not be analyzed at all. In addition, our results further reveal the benefits of knowledge compilation to deterministic decomposable negation normal form for performing counting-based analyses. Overall, our work shows that #SAT dependent analyses for feature models open a new variety of different applications and scale to a large number of industrial feature models.Produktlinien sind weit verbreitet für die Verwaltung von Familien verwandter Pro- dukte. In der Regel sind Produktlinien komplex und manuell schwer zu analysieren. In den letzten zwei Jahrzehnten wurden Produktlinienanalysen auf Erfüllbarkeit- sprobleme reduziert, für welche es eine Vielzahl an effizienten Werkzeugen gibt. Allerdings ist Erfüllbarkeit nicht für alle Analysen hinreichend. Kürzlich haben Forscher damit begonnen, andere Probleme auf #SAT zu reduzieren. Es wur- den jedoch nur wenige Anwendungen in Betracht gezogen und auch der Anwen- dungsbereich ist begrenzt. Darüber hinaus schlagen die Autoren hauptsächlich Ad-hoc-Lösungen vor, die nur unter bestimmten Einschränkungen der Produktlin- ien anwendbar sind oder nicht für große Produktlinien skalieren. In dieser Arbeit zeigen wir die Vorteile von #SAT Anwendungen für Produtlinien auf. Unser wis- senschaftlicher Beitrag besteht aus den folgenden drei Punkten: Zuerst fassen wir die in der Literatur betrachteten #SAT-Anwendungen zusammen und schlagen neue Anwendungen vor, um den Einsatz von #SAT-Technologien zu motivieren. Zweit- ens stellen wir eine Vielzahl von Algorithmen und Optimierungen für diese Anwen- dungen vor, einschließlich neuer Vorschläge. Drittens führen wir eine empirische Evaluation von 10 der vorgeschlagenen Algorithmen mit 14 #SAT-Solvern auf 131 industriellen Feature-Modellen aus, um die schnellsten Algorithmen und Solver zu identifizieren. Die Ergebnisse der Evaluation zeigen, dass wir für jede Analyse wenig- stens einen Algorithmus und Solver identifiziert haben, die für industrielle Feature- Modelle skalieren. Dazu sind die Ergebnisse ein starker Indikator für die Vorteile des Einsatzes von d-DNNFs bei #SAT-Anwendungen. Insgesamt zeigt unsere Ar- beit, dass #SAT-abhängige Analysen für Feature-Modelle eine Vielzahl neuer un- terschiedlicher Anwendungen ermöglicht und für viele industirelle Feature-Modelle skaliert

The Language of Search

This paper is concerned with a class of algorithms that perform exhaustive search on propositional knowledge bases. We show that each of these algorithms defines and generates a propositional language. Specifically, we show that the trace of a search can be interpreted as a combinational circuit, and a search algorithm then defines a propositional language consisting of circuits that are generated across all possible executions of the algorithm. In particular, we show that several versions of exhaustive DPLL search correspond to such well-known languages as FBDD, OBDD, and a precisely-defined subset of d-DNNF. By thus mapping search algorithms to propositional languages, we provide a uniform and practical framework in which successful search techniques can be harnessed for compilation of knowledge into various languages of interest, and a new methodology whereby the power and limitations of search algorithms can be understood by looking up the tractability and succinctness of the corresponding propositional languages