Mateusz Rychlicki

Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki et al.

This paper presents a comprehensive theoretical investigation into the parameterized complexity of explanation problems in various machine learning (ML) models. Contrary to the prevalent black-box perception, our study focuses on models with transparent internal mechanisms. We address two principal types of explanation problems: abductive and contrastive, both in their local and global variants. Our analysis encompasses diverse ML models, including Decision Trees, Decision Sets, Decision Lists, Ordered Binary Decision Diagrams, Random Forests, and Boolean Circuits, and ensembles thereof, each offering unique explanatory challenges. This research fills a significant gap in explainable AI (XAI) by providing a foundational understanding of the complexities of generating explanations for these models. This work provides insights vital for further research in the domain of XAI, contributing to the broader discourse on the necessity of transparency and accountability in AI systems.

Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki et al.

This paper presents a comprehensive theoretical investigation into the parameterized complexity of explanation problems in various machine learning (ML) models. Contrary to the prevalent black-box perception, our study focuses on models with transparent internal mechanisms. We address two principal types of explanation problems: abductive and contrastive, both in their local and global variants. Our analysis encompasses diverse ML models, including Decision Trees, Decision Sets, Decision Lists, Boolean Circuits, and ensembles thereof, each offering unique explanatory challenges. This research fills a significant gap in explainable AI (XAI) by providing a foundational understanding of the complexities of generating explanations for these models. This work provides insights vital for further research in the domain of XAI, contributing to the broader discourse on the necessity of transparency and accountability in AI systems.

Leif Eriksson, Victor Lagerkvist, Sebastian Ordyniak et al.

The quantified Boolean formula problem (QBF) is a well-known PSpace-complete problem with rich expressive power, and is generally viewed as the SAT analogue for PSpace. Given that many problems today are solved in practice by reducing to SAT, and then using highly optimized SAT solvers, it is natural to ask whether problems in PSpace are amenable to this approach. While SAT solvers exploit hidden structural properties, such as backdoors to tractability, backdoor analysis for QBF is comparatively very limited. We present a comprehensive study of the (parameterized) complexity of QBF parameterized by backdoor size to the largest tractable syntactic classes: HORN, 2-SAT, and AFFINE. While SAT is in FPT under this parameterization, we prove that QBF remains PSpace-hard even on formulas with backdoors of constant size. Parameterizing additionally by the quantifier depth, we design FPT-algorithms for the classes 2-SAT and AFFINE, and show that 3-HORN is W[1]-hard. As our next contribution, we vastly extend the applicability of QBF backdoors not only for the syntactic classes defined above but also for tractable classes defined via structural restrictions, such as formulas with bounded incidence treewidth and quantifier depth. To this end, we introduce enhanced backdoors: these are separators S of size at most k in the primal graph such that S together with all variables contained in any purely universal component of the primal graph minus S is a backdoor. We design FPT-algorithms with respect to k for both evaluation and detection of enhanced backdoors to all tractable classes of QBF listed above and more.

Leif Eriksson, Victor Lagerkvist, Sebastian Ordyniak et al.

Determining the validity of a quantified Boolean formula (QBF) is a PSPACE-complete problem with rich expressive power. Despite interest in efficient solvers, there is, compared to problems in NP, a lack of positive theoretical results, and in the parameterized complexity setting one often has to restrict the quantifier prefix (e.g., bounding alternations) to obtain fixed parameter tractability (FPT). We propose a new parameter: the number of variables in clauses that has to be removed before reaching a tractable class (a clause covering (CC) backdoor). We are then interested in solving QBF in FPT time given a CC-backdoor of size $k$. We consider the three classical, tractable cases of QBF as base classes: Horn, 2-CNF, and linear equations. We establish W[1]-hardness for Horn but prove FPT for the others, and prove that in a precise, algebraic sense, we are only missing one important case for a full dichotomy. Our algorithms are non-trivial and depend on propagation, and Gaussian elimination, respectively, and are comparably unexplored for QBF.

Leif Eriksson, Victor Lagerkvist, George Osipov et al.

The quantified Boolean formula (QBF) problem is an important decision problem generally viewed as the archetype for PSPACE-completeness. Many problems of central interest in AI are in general not included in NP, e.g., planning, model checking, and non-monotonic reasoning, and for such problems QBF has successfully been used as a modelling tool. However, solvers for QBF are not as advanced as state of the art SAT solvers, which has prevented QBF from becoming a universal modelling language for PSPACE-complete problems. A theoretical explanation is that QBF (as well as many other PSPACE-complete problems) lacks natural parameters} guaranteeing fixed-parameter tractability (FPT). In this paper we tackle this problem and consider a simple but overlooked parameter: the number of existentially quantified variables. This natural parameter is virtually unexplored in the literature which one might find surprising given the general scarcity of FPT algorithms for QBF. Via this parameterization we then develop a novel FPT algorithm applicable to QBF instances in conjunctive normal form (CNF) of bounded clause length. We complement this by a W[1]-hardness result for QBF in CNF of unbounded clause length as well as sharper lower bounds for the bounded arity case under the (strong) exponential-time hypothesis.