Proof Complexity and Beyond

Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples

Stochastic Invariants for Probabilistic Termination

Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability~1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behavior of the programs, the invariants are obtained completely ignoring the probabilistic aspect. In this work we address the probabilistic termination problem for linear-arithmetic probabilistic programs with nondeterminism. We define the notion of {\em stochastic invariants}, which are constraints along with a probability bound that the constraints hold. We introduce a concept of {\em repulsing supermartingales}. First, we show that repulsing supermartingales can be used to obtain bounds on the probability of the stochastic invariants. Second, we show the effectiveness of repulsing supermartingales in the following three ways: (1)~With a combination of ranking and repulsing supermartingales we can compute lower bounds on the probability of termination; (2)~repulsing supermartingales provide witnesses for refutation of almost-sure termination; and (3)~with a combination of ranking and repulsing supermartingales we can establish persistence properties of probabilistic programs. We also present results on related computational problems and an experimental evaluation of our approach on academic examples.Comment: Full version of a paper published at POPL 2017. 20 page

Solving Linear Constraints in Elementary Abelian p-Groups of Symmetries

Symmetries occur naturally in CSP or SAT problems and are not very difficult to discover, but using them to prune the search space tends to be very challenging. Indeed, this usually requires finding specific elements in a group of symmetries that can be huge, and the problem of their very existence is NP-hard. We formulate such an existence problem as a constraint problem on one variable (the symmetry to be used) ranging over a group, and try to find restrictions that may be solved in polynomial time. By considering a simple form of constraints (restricted by a cardinality k) and the class of groups that have the structure of Fp-vector spaces, we propose a partial algorithm based on linear algebra. This polynomial algorithm always applies when k=p=2, but may fail otherwise as we prove the problem to be NP-hard for all other values of k and p. Experiments show that this approach though restricted should allow for an efficient use of at least some groups of symmetries. We conclude with a few directions to be explored to efficiently solve this problem on the general case.Comment: 18 page

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

Quasi-Equivalence of Width and Depth of Neural Networks

While classic studies proved that wide networks allow universal approximation, recent research and successes of deep learning demonstrate the power of the network depth. Based on a symmetric consideration, we investigate if the design of artificial neural networks should have a directional preference, and what the mechanism of interaction is between the width and depth of a network. We address this fundamental question by establishing a quasi-equivalence between the width and depth of ReLU networks. Specifically, we formulate a transformation from an arbitrary ReLU network to a wide network and a deep network for either regression or classification so that an essentially same capability of the original network can be implemented. That is, a deep regression/classification ReLU network has a wide equivalent, and vice versa, subject to an arbitrarily small error. Interestingly, the quasi-equivalence between wide and deep classification ReLU networks is a data-driven version of the De Morgan law