Mapping constrained optimization problems to quantum annealing with application to fault diagnosis

Current quantum annealing (QA) hardware suffers from practical limitations such as finite temperature, sparse connectivity, small qubit numbers, and control error. We propose new algorithms for mapping boolean constraint satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In particular we develop a new embedding algorithm for mapping a CSP onto a hardware Ising model with a fixed sparse set of interactions, and propose two new decomposition algorithms for solving problems too large to map directly into hardware. The mapping technique is locally-structured, as hardware compatible Ising models are generated for each problem constraint, and variables appearing in different constraints are chained together using ferromagnetic couplings. In contrast, global embedding techniques generate a hardware independent Ising model for all the constraints, and then use a minor-embedding algorithm to generate a hardware compatible Ising model. We give an example of a class of CSPs for which the scaling performance of D-Wave's QA hardware using the local mapping technique is significantly better than global embedding. We validate the approach by applying D-Wave's hardware to circuit-based fault-diagnosis. For circuits that embed directly, we find that the hardware is typically able to find all solutions from a min-fault diagnosis set of size N using 1000N samples, using an annealing rate that is 25 times faster than a leading SAT-based sampling method. Further, we apply decomposition algorithms to find min-cardinality faults for circuits that are up to 5 times larger than can be solved directly on current hardware.Comment: 22 pages, 4 figure

Understanding Inconsistency -- A Contribution to the Field of Non-monotonic Reasoning

Conflicting information in an agent's knowledge base may lead to a semantical defect, that is, a situation where it is impossible to draw any plausible conclusion. Finding out the reasons for the observed inconsistency and restoring consistency in a certain minimal way are frequently occurring issues in the research area of knowledge representation and reasoning. In a seminal paper Raymond Reiter proves a duality between maximal consistent subsets of a propositional knowledge base and minimal hitting sets of each minimal conflict -- the famous hitting set duality. We extend Reiter's result to arbitrary non-monotonic logics. To this end, we develop a refined notion of inconsistency, called strong inconsistency. We show that minimal strongly inconsistent subsets play a similar role as minimal inconsistent subsets in propositional logic. In particular, the duality between hitting sets of minimal inconsistent subsets and maximal consistent subsets generalizes to arbitrary logics if the stronger notion of inconsistency is used. We cover various notions of repairs and characterize them using analogous hitting set dualities. Our analysis also includes an investigation of structural properties of knowledge bases with respect to our notions. Minimal inconsistent subsets of knowledge bases in monotonic logics play an important role when investigating the reasons for conflicts and trying to handle them, but also for inconsistency measurement. Our notion of strong inconsistency thus allows us to extend existing results to non-monotonic logics. While measuring inconsistency in propositional logic has been investigated for some time now, taking the non-monotony into account poses new challenges. In order to tackle them, we focus on the structure of minimal strongly inconsistent subsets of a knowledge base. We propose measures based on this notion and investigate their behavior in a non-monotonic setting by revisiting existing rationality postulates, and analyzing the compliance of the proposed measures with these postulates. We provide a series of first results in the context of inconsistency in abstract argumentation theory regarding the two most important reasoning modes, namely credulous as well as skeptical acceptance. Our analysis includes the following problems regarding minimal repairs: existence, verification, computation of one and characterization of all solutions. The latter will be tackled with our previously obtained duality results. Finally, we investigate the complexity of various related reasoning problems and compare our results to existing ones for monotonic logics

Automated model-based spreadsheet debugging

Spreadsheets are interactive data organization and calculation programs that are developed in spreadsheet environments like Microsoft Excel or LibreOffice Calc. They are probably the most successful example of end-user developed software and are utilized in almost all branches and at all levels of companies. Although spreadsheets often support important decision making processes, they are, like all software, prone to error. In several cases, faults in spreadsheets have caused severe losses of money. Spreadsheet developers are usually not educated in the practices of software development. As they are thus not familiar with quality control methods like systematic testing or debugging, they have to be supported by the spreadsheet environment itself to search for faults in their calculations in order to ensure the correctness and a better overall quality of the developed spreadsheets. This thesis by publication introduces several approaches to locate faults in spreadsheets. The presented approaches are based on the principles of Model-Based Diagnosis (MBD), which is a technique to find the possible reasons why a system does not behave as expected. Several new algorithmic enhancements of the general MBD approach are combined in this thesis to allow spreadsheet users to debug their spreadsheets and to efficiently find the reason of the observed unexpected output values. In order to assure a seamless integration into the environment that is well-known to the spreadsheet developers, the presented approaches are implemented as an extension for Microsoft Excel. The first part of the thesis outlines the different algorithmic approaches that are introduced in this thesis and summarizes the improvements that were achieved over the general MBD approach. In the second part, the appendix, a selection of the author's publications are presented. These publications comprise (a) a survey of the research in the area of spreadsheet quality assurance, (b) a work describing how to adapt the general MBD approach to spreadsheets, (c) two new algorithmic improvements of the general technique to speed up the calculation of the possible reasons of an observed fault, (d) a new concept and algorithm to efficiently determine questions that a user can be asked during debugging in order to reduce the number of possible reasons for the observed unexpected output values, and (e) a new method to find faults in a set of spreadsheets and a new corpus of real-world spreadsheets containing faults that can be used to evaluate the proposed debugging approaches