In this paper, we develop a quantified propositional proof systems that corresponds to logarithmic-space reasoning. We begin by defining a class SigmaCNF(2) of quantified formulas that can be evaluated in log space. Then our new proof system GL^* is defined as G_1^* with cuts restricted to SigmaCNF(2) formulas and no cut formula that is not quantifier free contains a free variable that does not appear in the final formula.
To show that GL^* is strong enough to capture log space reasoning, we translate theorems of VL into a family of tautologies that have polynomial-size GL^* proofs. VL is a theory of bounded arithmetic that is known to correspond to logarithmic-space reasoning. To do the translation, we find an appropriate axiomatization of VL, and put VL proofs into a new normal form.
To show that GL^* is not too strong, we prove the soundness of GL^* in such a way that it can be formalized in VL. This is done by giving a logarithmic-space algorithm that witnesses GL^* proofs.
We study quantified propositional logics from the complexity theoretic point of view. First we introduce alternating dependency quantified boolean formulae (ADQBF) which generalize both quantified and dependency quantified boolean formulae. We show that the truth evaluation for ADQBF is AEXPTIME(poly)-complete. We also identify fragments for which the problem is complete for the levels of the exponential hierarchy. Second we study propositional team-based logics. We show that DQBF formulae correspond naturally to quantified propositional dependence logic and present a general NEXPTIME upper bo…
Modal logics allow reasoning about various modes of truth: for example, what it means for something to be possibly true, or to know that something is true as opposed to merely believing it. This report describes embeddings of propositional and quantified modal logic in the PVS verification system. The resources of PVS allow this to be done in an attractive way that supports much of the standard syntax of modal logic, while providing effective automation. The report introduces and formally specifies and verifies several standard topics in modal logic such as relationships between the standard m…
While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynom…
Quantified Boolean formulas (QBFs) generalize propositional formulas by admitting quantifications over propositional variables. QBFs can be viewed as (restricted) formulas of first-order predicate logic and easy translations of QBFs into first-order formulas exist. We analyze different translations and show that first-order resolution combined with such translations can polynomially simulate well-known deduction concepts for QBFs. Furthermore, we extend QBF calculi by the possibility to instantiate a universal variable by an existential variable of smaller level. Combining such an enhanced cal…
In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(~) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion. As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-…
While symmetries are well understood for Boolean formulas and successfully exploited in practical SAT solving, less is known about symmetries in quantified Boolean formulas (QBF). There are some works introducing adaptions of propositional symmetry breaking techniques, with a theory covering only very specific parts of QBF symmetries. We present a general framework that gives a concise characterization of symmetries of QBF. Our framework naturally incorporates the duality of universal and existential symmetries resulting in a general basis for QBF symmetry breaking.
In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems, showed that all these problems are either in P or NP-complete, and gave a simple criterion to determine which of the two cases holds. This result is surprising in light of Ladner's theorem, which implies that there are an infinite number of complexity classes between P and NP-complete (under the assumption that P is not equal to NP). Schaefer also stated a dichotomy theorem for quantified generalized Boolean formulas, but this th…
Plausible reasoning concerns situations whose inherent lack of precision is not quantified; that is, there are no degrees or levels of precision, and hence no use of numbers like probabilities. A hopefully comprehensive set of principles that clarifies what it means for a formal logic to do plausible reasoning is presented. A new propositional logic, called Propositional Plausible Logic (PPL), is defined and applied to some important examples. PPL is the only non-numeric non-monotonic logic we know of that satisfies all the principles and correctly reasons with all the examples. Some important…
Dependency quantified Boolean formulas (DQBFs) are a powerful formalism, which subsumes quantified Boolean formulas (QBFs) and allows an explicit specification of dependencies of existential variables on universal variables. Driven by the needs of various applications which can be encoded by DQBFs in a natural, compact, and elegant way, research on DQBF solving has emerged in the past few years. However, research focused on closed DQBFs in prenex form (where all quantifiers are placed in front of a propositional formula), while non-prenex DQBFs have almost not been studied in the literature. I…
The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction op…
Prolog is a well known declarative programming language based on propositional Horn formulas. It is useful in various areas, including artificial intelligence, automated theorem proving, mathematical logic and so on. An active research area for many years is to extend Prolog to larger classes of logic. Some important extensions of it includes the constraint logic programming, and the object oriented logic programming. However, it cannot solve problems having arbitrary quantified Horn formulas. To be precise, the facts, rules and queries in Prolog are not allowed to have arbitrary quantified va…
This report presents an elementary theory of unification for positive conjunctive queries. A positive conjunctive query is a formula constructed from propositional constants, equations and atoms using the conjunction $\wedge$ and the existential quantifier $\exists$. In particular, empty queries correspond to existentially quantified systems of equations -- called $\cal E$-formulas. We provide an algorithm which transforms any conjunctive query into a solved form. We prove some lattice-theoretic properties of queries. In particular, the quotient set of $\cal E$-formulas under an equivalence re…
Existing approaches to synthesize reactive systems from declarative specifications mostly rely on Binary Decision Diagrams (BDDs), inheriting their scalability issues. We present novel algorithms for safety specifications that use decision procedures for propositional formulas (SAT solvers), Quantified Boolean Formulas (QBF solvers), or Effectively Propositional Logic (EPR). Our algorithms are based on query learning, templates, reduction to EPR, QBF certification, and interpolation. A parallelization combines multiple algorithms. Our optimizations expand quantifiers and utilize unreachable st…
We consider the Quantifier Elimination (QE) problem for propositional CNF formulas with existential quantifiers. QE plays a key role in formal verification. Earlier, we presented an approach based on the following observation. To perform QE, one just needs to add a set of clauses depending on free variables that makes the quantified clauses (i.e. clauses with quantified variables) redundant. To implement this approach, we introduced a branching algorithm making quantified clauses redundant in subspaces and merging the results of branches. To implement this algorithm we developed the machinery …
In spite of the close connection between the evaluation of quantified Boolean formulas (QBF) and propositional satisfiability (SAT), tools and techniques which exploit structural properties of SAT instances are known to fail for QBF. This is especially true for the structural parameter treewidth, which has allowed the design of successful algorithms for SAT but cannot be straightforwardly applied to QBF since it does not take into account the interdependencies between quantified variables. In this work we introduce and develop dependency treewidth, a new structural parameter based on treewidth…
Many verification and synthesis approaches rely on solving techniques for quantified Boolean formulas (QBF). Consequently, solution witnesses, in the form of Boolean functions, become more and more important as they represent implementations or counterexamples. We present a recursive counterexample guided abstraction and refinement algorithm (CEGAR) for solving and certifying QBFs that exploits structural reasoning on the formula level. The algorithm decomposes the given QBF into one propositional formula for every block of quantifiers that abstracts from assignments of variables not bound by …
A quantified Boolean formula (QBF) is a propositional formula extended with universal and existential quantification over propositions. There are two methodologies in CEGAR based QBF solving techniques, one that is based on a refinement loop that builds partial expansions and a more recent one that is based on the communication of satisfied clauses. Despite their algorithmic similarity, their performance characteristics in experimental evaluations are very different and in many cases orthogonal. We compare those CEGAR approaches using proof theory developed around QBF solving and present a uni…
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-…
The Institute of Mathematical Sciences (A CI of Homi Bhabha National Institute)（数学科学研究所（霍米·巴巴国立学院成员机构））；Czech Technical University in Prague（布拉格捷克理工大学）；Indian Institute of Technology Ropar（印度理工学院罗帕尔分校）
