On the Complexity of Optimization Problems based on Compiled NNF Representations

Optimization is a key task in a number of applications. When the set of feasible solutions under consideration is of combinatorial nature and described in an implicit way as a set of constraints, optimization is typically NP-hard. Fortunately, in many problems, the set of feasible solutions does not often change and is independent from the user's request. In such cases, compiling the set of constraints describing the set of feasible solutions during an off-line phase makes sense, if this compilation step renders computationally easier the generation of a non-dominated, yet feasible solution matching the user's requirements and preferences (which are only known at the on-line step). In this article, we focus on propositional constraints. The subsets L of the NNF language analyzed in Darwiche and Marquis' knowledge compilation map are considered. A number of families F of representations of objective functions over propositional variables, including linear pseudo-Boolean functions and more sophisticated ones, are considered. For each language L and each family F, the complexity of generating an optimal solution when the constraints are compiled into L and optimality is to be considered w.r.t. a function from F is identified

Algebraic model counting

Weighted model counting (WMC) is a well-known inference task on knowledge bases, and the basis for some of the most efficient techniques for probabilistic inference in graphical models. We introduce algebraic model counting (AMC), a generalization of WMC to a semiring structure that provides a unified view on a range of tasks and existing results. We show that AMC generalizes many well-known tasks in a variety of domains such as probabilistic inference, soft constraints and network and database analysis. Furthermore, we investigate AMC from a knowledge compilation perspective and show that all AMC tasks can be evaluated using sd-DNNF circuits, which are strictly more succinct, and thus more efficient to evaluate, than direct representations of sets of models. We identify further characteristics of AMC instances that allow for evaluation on even more succinct circuits

Optimized Framework based on Rough Set Theory for Big Data Pre-processing in Certain and Imprecise Contexts” -- Marie Sklodowska-Curie Project: Open Problems’

Peer reviewe

Type-elimination-based reasoning for the description logic SHIQbs using decision diagrams and disjunctive datalog

We propose a novel, type-elimination-based method for reasoning in the description logic SHIQbs including DL-safe rules. To this end, we first establish a knowledge compilation method converting the terminological part of an ALCIb knowledge base into an ordered binary decision diagram (OBDD) which represents a canonical model. This OBDD can in turn be transformed into disjunctive Datalog and merged with the assertional part of the knowledge base in order to perform combined reasoning. In order to leverage our technique for full SHIQbs, we provide a stepwise reduction from SHIQbs to ALCIb that preserves satisfiability and entailment of positive and negative ground facts. The proposed technique is shown to be worst case optimal w.r.t. combined and data complexity and easily admits extensions with ground conjunctive queries.Comment: 38 pages, 3 figures, camera ready version of paper accepted for publication in Logical Methods in Computer Scienc

IASCAR: Incremental Answer Set Counting by Anytime Refinement

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 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.Comment: Under consideration in Theory and Practice of Logic Programming (TPLP

Implementing semantic tableaux

This report describes implementions of the tableau calculus for first-order logic. First an extremely simple implementation, called leanTAP, is presented, which nonetheless covers the full functionality of the calculus and is also competitive with respect to performance. A second approach uses compilation techniques for proof search. Improvements inculding universal variables and lemmata are considered as well as more efficient data structures using reduced ordered binary decision diagrams. The implementation language is PROLOG. In all cases fully operational PROLOG code is given. For leanTAP a formal proof of the correctness of the implementation is given relying on the operational semantics of PROLOG as given by the SLD-tree model. This report will appear as a chapter in the Handbook of Tableau-based Methods in Automated Deduction edited by: D. Gabbay, M. D\u27Agostino, R. H\"{a}hnle, and J.Posegga published by: KLUWER ACADEMIC PUBLISHERS Electronic availability will be discontinued after final acceptance for publication is obtained

Interactive Cost Configuration Over Decision Diagrams

Abstract In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrack-free user interaction online. In particular, binary decision diagrams (BDDs) have been successfully used as a compilation target for product and service configuration. In this paper we discuss how to extend BDD-based configuration to scenarios involving cost functions which express user preferences. We first show that an efficient, robust and easy to implement extension is possible if the cost function is additive, and feasible solutions are represented using multi-valued decision diagrams (MDDs). We also discuss the effect on MDD size if the cost function is non-additive or if it is encoded explicitly into MDD. We then discuss interactive configuration in the presence of multiple cost functions. We prove that even in its simplest form, multiple-cost configuration is NP-hard in the input MDD. However, for solving two-cost configuration we develop a pseudo-polynomial scheme and a fully polynomial approximation scheme. The applicability of our approach is demonstrated through experiments over real-world configuration models and product-catalogue datasets. Response times are generally within a fraction of a second even for very large instances

On Formal Methods for Large-Scale Product Configuration

<p>In product development companies mass customization is widely used to achieve better customer satisfaction while keeping costs down. To efficiently implement mass customization, product platforms are often used. A product platform allows building a wide range of products from a set of predefined components. The process of matching these components to customers' needs is called product configuration. Not all components can be combined with each other due to restrictions of various kinds, for example, geometrical, marketing and legal reasons. Product design engineers develop configuration constraints to describe such restrictions. The number of constraints and the complexity of the relations between them are immense for complex product like a vehicle. Thus, it is both error-prone and time consuming to analyze, author and verify the constraints manually. Software tools based on formal methods can help engineers to avoid making errors when working with configuration constraints, thus design a correct product faster.</p> <p>This thesis introduces a number of formal methods to help engineers maintain, verify and analyze product configuration constraints. These methods provide automatic verification of constraints and computational support for analyzing and refactoring constraints. The methods also allow verifying the correctness of one specific type of constraints, item usage rules, for sets of mutually-exclusive required items, and automatic verification of equivalence of different formulations of the constraints. The thesis also introduces three methods for efficient enumeration of valid partial configurations, with benchmarking of the methods on an industrial dataset.</p> <p>Handling large-scale industrial product configuration problems demands high efficiency from the software methods. This thesis investigates a number of search-based and knowledge-compilation-based methods for working with large product configuration instances, including Boolean satisfiability solvers, binary decision diagrams and decomposable negation normal form. This thesis also proposes a novel method based on supervisory control theory for efficient reasoning about product configuration data. The methods were implemented in a tool, to investigate the applicability of the methods for handling large product configuration problems. It was found that search-based Boolean satisfiability solvers with incremental capabilities are well suited for industrial configuration problems.</p> <p>The methods proposed in this thesis exhibit good performance on practical configuration problems, and have a potential to be implemented in industry to support product design engineers in creating and maintaining configuration constraints, and speed up the development of product platforms and new products.</p