Towards Uniform Certification in QBF

We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution. These results are obtained by taking a technique of Beyersdorff et al. (JACM 2020) that turns strategy extraction into simulation and combining it with new local strategy extraction arguments. This approach leads to simulations that are carried out mainly in propositional logic, with minimal use of the QBF rules. Our proofs therefore provide a new, largely propositional interpretation of the simulated systems. We argue that these results strengthen the case for uniform certification in QBF solving, since many QBF proof systems now fall into place underneath extended QBF Frege

QBF Proof Complexity

Quantified Boolean Formulas (QBF) and their proof complexity are not as well understood as propositional formulas, yet remain an area of interest due to their relation to QBF solving. Proof systems for QBF provide a theoretical underpinning for the performance of these solvers. We define a novel calculus IR-calc, which enables unification of the principal existing resolution-based QBF calculi and applies to the more powerful Dependency QBF (DQBF). We completely reveal the relative power of important QBF resolution systems, settling in particular the relationship between the two different types of resolution-based QBF calculi. The most challenging part of this comparison is to exhibit hard formulas that underlie the exponential separations of the proof systems. In contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. To this end we exhibit a new proof technique for showing lower bounds in QBF proof systems based on strategy extraction. We also find that the classical lower bound techniques of the prover-delayer game and feasible interpolation can be lifted to a QBF setting and provide new lower bounds. We investigate more powerful proof systems such as extended resolution and Frege systems. We define and investigate new QBF proof systems that mix propositional rules with a reduction rule, we find the strategy extraction technique also works and directly lifts lower bounds from circuit complexity. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated, but had not been formally established for propositional proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem

Frege systems for quantified Boolean logic

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.This research was supported by grant nos. 48138 and 60842 from the John Templeton Foundation, EPSRC grant EP/L024233/1, and a Doctoral Prize Fellowship from EPSRC (third author). The second author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 279611 and under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no. 648276 AUTAR. The fourth author was supported by the Austrian Science Fund (FWF) under project number P28699 and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 61507. Part of this work was done when Beyersdorff and Pich were at the University of Leeds and Bonacina at Sapienza University Rome.Peer ReviewedPostprint (published version

A game characterisation of tree-like Q-Resolution size

We provide a characterisation for the size of proofs in tree-like Q-Resolution and tree-like QU-Resolution by a Prover–Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives one of the first successful transfers of one of the lower bound techniques for classical proof systems to QBF proof systems. We apply our technique to show the hardness of three classes of formulas for tree-like Q-Resolution. In particular, we give a proof of the hardness of the parity formulas from Beyersdorff et al. (2015) for tree-like Q-Resolution and of the formulas of Kleine Büning et al. (1995) for tree-like QU-Resolution

Reinterpreting Dependency Schemes: Soundness Meets Incompleteness in DQBF

Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called ‘full exhibition’ is central. We apply our approach to QBF proof systems from two distinct paradigms, termed ‘universal reduction’ and ‘universal expansion’. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky

Nothing Lasts Forever: Environmental Discourses on the Collapse of Past Societies

The study of the collapse of past societies raises many questions for the theory and practice of archaeology. Interest in collapse extends as well into the natural sciences and environmental and sustainability policy. Despite a range of approaches to collapse, the predominant paradigm is environmental collapse, which I argue obscures recognition of the dynamic role of social processes that lie at the heart of human communities. These environmental discourses, together with confusion over terminology and the concepts of collapse, have created widespread aporia about collapse and resulted in the creation of mixed messages about complex historical and social processes

Host response dynamics following lethal infection of rhesus macaques with zaire ebolavirus

To gain further insight into the interdependent pathogenic processes in Ebola hemorrhagic fever (EHF), we have examined the dynamics of host responses in individual rhesus macaques infected with Zaire ebolavirus over the entire disease course. Examination of coagulation parameters revealed that decreased coagulation inhibitor activity triggered severe coagulopathy as indicated by prolonged coagulation times and decreased fibrinogen levels. This has been proposed as one of the significant mechanisms underlying disseminated intravascular coagulation in EHF patients. Furthermore, monitoring of expression levels for cytokines/chemokines suggested a mixed anti-inflammatory response syndrome (MARS), which indicates that a catastrophic uncontrolled immunological status contributes to the development of fatal hemorrhagic fever. These results highlight the pathological analogies between EHF and severe sepsis and not only contribute to our understanding of the pathogenic process, but will also help to establish novel postexposure treatment modalities