QMusExt: A Minimal (Un)satisfiable Core Extractor for Quantified Boolean Formulas

In this paper, we present QMusExt, a tool for the extraction of minimal unsatisfiable sets (MUS) from quantified Boolean formulas (QBFs) in prenex conjunctive normal form (PCNF). Our tool generalizes an efficient algorithm for MUS extraction from propositional formulas that analyses and rewrites resolution proofs generated by SAT solvers. In addition to extracting unsatisfiable cores from false formulas in PCNF, we apply QMusExt also to obtain satisfiable cores from Q-resolution proofs of true formulas in prenex disjunctive normal form (PDNF)

Logic-Based Explainability in Machine Learning

The last decade witnessed an ever-increasing stream of successes in Machine Learning (ML). These successes offer clear evidence that ML is bound to become pervasive in a wide range of practical uses, including many that directly affect humans. Unfortunately, the operation of the most successful ML models is incomprehensible for human decision makers. As a result, the use of ML models, especially in high-risk and safety-critical settings is not without concern. In recent years, there have been efforts on devising approaches for explaining ML models. Most of these efforts have focused on so-called model-agnostic approaches. However, all model-agnostic and related approaches offer no guarantees of rigor, hence being referred to as non-formal. For example, such non-formal explanations can be consistent with different predictions, which renders them useless in practice. This paper overviews the ongoing research efforts on computing rigorous model-based explanations of ML models; these being referred to as formal explanations. These efforts encompass a variety of topics, that include the actual definitions of explanations, the characterization of the complexity of computing explanations, the currently best logical encodings for reasoning about different ML models, and also how to make explanations interpretable for human decision makers, among others

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

Efficient local search for Pseudo Boolean Optimization

Algorithms and the Foundations of Software technolog

Complexity Results for Reachability in Cooperating Systems and Approximated Reachability by Abstract Over-Approximations

This work deals with theoretic aspects of cooperating systems, i.e., systems that consists of cooperating subsystems. Our main focus lies on the complexity theoretic classification of deciding the reachability problem and on efficiently establishing deadlock-freedom in models of cooperating systems. The formal verification of system properties is an active field of research, first attempts of which go back to the late 60's. The behavior of cooperating systems suffers from the state space explosion problem and can become very large. This is, techniques that are based on an analysis of the reachable state space have a runtime exponential in the number of subsystems. The consequence is that even modern techniques that decide whether or not a system property holds in a system can become unfeasible. We use interaction systems, introduced by Sifakis et al. in 2003, as a formalism to model cooperating systems. The reachability problem and deciding deadlock-freedom in interaction systems was proved to be PSPACE-complete. An approach to deal with this issue is to investigate subclasses of systems in which these problems can be treated efficiently. We show here that the reachability problem remains PSPACE-complete in subclasses of interaction systems with a restricted communication structure. We consider structures that from trees, stars and linear arrangements of subsystems. Our result motivates the research of techniques that treat the reachability problem in these subclasses based on sufficient conditions which exploit characteristics of the structural restrictions. In a second part of this work we investigate an approach to efficiently establish the reachability of states and deadlock-freedom in general interaction systems. We introduce abstract over-approximations -- a concept of compact representations of over-approximations of the reachable behavior of interaction systems. Families of abstract over-approximations are the basis for our approach to establish deadlock-freedom in interaction systems in polynomial time in the size of the underlying interaction system. We introduce an operator called Edge-Match for refining abstract over-approximations. The strength of our approach is illustrated on various parametrized instances of interaction systems. Furthermore, we establish a link between our refinement approach and the field of relational database theory and use this link in order to make a preciseness statement about our refinement approach

Trading inference effort versus size in CNF Knowledge Compilation

Knowledge Compilation (KC) studies compilation of boolean functions f into some formalism F, which allows to answer all queries of a certain kind in polynomial time. Due to its relevance for SAT solving, we concentrate on the query type "clausal entailment" (CE), i.e., whether a clause C follows from f or not, and we consider subclasses of CNF, i.e., clause-sets F with special properties. In this report we do not allow auxiliary variables (except of the Outlook), and thus F needs to be equivalent to f. We consider the hierarchies UC_k <= WC_k, which were introduced by the authors in 2012. Each level allows CE queries. The first two levels are well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in KC, that is, f is represented by the set of all prime implicates, while UC_1 = WC_1 is the same as UC, the class of unit-refutation complete clause-sets introduced by del Val 1994. We show that for each k there are (sequences of) boolean functions with polysize representations in UC_{k+1}, but with an exponential lower bound on representations in WC_k. Such a separation was previously only know for k=0. We also consider PC < UC, the class of propagation-complete clause-sets. We show that there are (sequences of) boolean functions with polysize representations in UC, while there is an exponential lower bound for representations in PC. These separations are steps towards a general conjecture determining the representation power of the hierarchies PC_k < UC_k <= WC_k. The strong form of this conjecture also allows auxiliary variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the separation results from the discontinued arXiv:1302.442

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers