Quantified Boolean Formulas: Proof Complexity and Models of Solving

Quantified Boolean formulas (QBF), which form the canonical PSPACE-complete decision problem, are a decidable fragment of first-order logic. Any problem that can be solved within a polynomial-size space can be encoded succinctly as a QBF, including many concrete problems in computer science from domains such as verification, synthesis and planning. Automated solvers for QBF are now reaching the point of industrial applicability. In this thesis, we focus on dependency awareness, a dedicated solving paradigm for QBF. We show that dependency schemes can be envisaged in terms of dependency quantified Boolean formulas (DQBF), exposing strong connections between these two previously disparate entities. By introducing new lower-bound techniques for QBF proof systems, we study the relative strengths of models of dependency-aware solving, including the proposal of new, stronger models. Proof Complexity: Using the strategy extraction paradigm, we introduce new lower-bound techniques that apply to resolution-based QBF proof systems. In particular, we use the technique to prove exponential lower bounds for a new family of QBFs called the equality formulas. Our technique also affords considerably simpler, more intuitive proofs of some existing QBF proof-size lower bounds. Models of Solving: We apply our lower bound techniques to show new separations for QBF proof systems parametrised by dependency schemes. We also propose new models of dynamic dependency-aware solving and prove that they are exponentially stronger than the existing static models. Finally, we introduce Merge Resolution, a proof system modelling CDCL-style solving for DQBF, which is the first of its kind

A prototype implementation of the AUnit test automation framework for alloy

Alloy is a declarative language based on relational first-order logic. Unlike commonly used procedural languages, the testing criteria of declarative languages like Alloy has remained largely ad hoc. Recent work on the AUnit test automation framework introduced a foundation for testing Alloy models. This report presents our effort on developing a prototype implementation of AUnit based on the standard Alloy distribution. Our implementation of AUnit has all core functionalities for writing unit tests, running all tests, showing the test execution results including the number of tests ran, the number of tests failed, coverage obtained (which is highlighted using coloring), all test requirements, and all uncovered requirements. We compute coverage for signatures, fields, predicates and specifically for primitive Booleans and quantified formulas. Our implementation can allow users to check the quality of their models in the spirit of traditional unit testing.Electrical and Computer Engineerin

Tackling Universal Properties of Minimal Trap Spaces of Boolean Networks

Minimal trap spaces (MTSs) capture subspaces in which the Boolean dynamics is trapped, whatever the update mode. They correspond to the attractors of the most permissive mode. Due to their versatility, the computation of MTSs has recently gained traction, essentially by focusing on their enumeration. In this paper, we address the logical reasoning on universal properties of MTSs in the scope of two problems: the reprogramming of Boolean networks for identifying the permanent freeze of Boolean variables that enforce a given property on all the MTSs, and the synthesis of Boolean networks from universal properties on their MTSs. Both problems reduce to solving the satisfiability of quantified propositional logic formula with 3 levels of quantifiers ($\exists\forall\exists$). In this paper, we introduce a Counter-Example Guided Refinement Abstraction (CEGAR) to efficiently solve these problems by coupling the resolution of two simpler formulas. We provide a prototype relying on Answer-Set Programming for each formula and show its tractability on a wide range of Boolean models of biological networks.Comment: Accepted at 21st International Conference on Computational Methods in Systems Biology (CMSB 2023

On QBF Proofs and Preprocessing

QBFs (quantified boolean formulas), which are a superset of propositional formulas, provide a canonical representation for PSPACE problems. To overcome the inherent complexity of QBF, significant effort has been invested in developing QBF solvers as well as the underlying proof systems. At the same time, formula preprocessing is crucial for the application of QBF solvers. This paper focuses on a missing link in currently-available technology: How to obtain a certificate (e.g. proof) for a formula that had been preprocessed before it was given to a solver? The paper targets a suite of commonly-used preprocessing techniques and shows how to reconstruct certificates for them. On the negative side, the paper discusses certain limitations of the currently-used proof systems in the light of preprocessing. The presented techniques were implemented and evaluated in the state-of-the-art QBF preprocessor bloqqer.Comment: LPAR 201