63 research outputs found
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)
Lifting QBF Resolution Calculi to DQBF
We examine the existing resolution systems for quantified Boolean formulas (QBF) and answer the question which of these calculi can be lifted to the more powerful Dependency QBFs (DQBF). An interesting picture emerges: While for QBF we have the strict chain of proof systems Q-Res < IR-calc < IRM-calc, the situation is quite different in DQBF. Q-Res and likewise universal resolution are too weak: they are not complete. IR-calc has the right strength: it is sound and complete. IRM-calc is too strong: it is not sound any more, and the same applies to long-distance resolution. Conceptually, we use the relation of DQBF to EPR and explain our new DQBF calculus based on IR-calc as a subsystem of first-order resolutio
QRAT+: Generalizing QRAT by a More Powerful QBF Redundancy Property
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 operation to
eliminate universal literals from clauses. We apply QUP to an abstraction of
the QBF where certain universal quantifiers are converted into existential
ones. This way, we obtain a generalization of QRAT we call QRAT+. The
redundancy property in QRAT+ based on QUP is more powerful than the one in QRAT
based on UP. We report on proof theoretical improvements and experimental
results to illustrate the benefits of QRAT+ for QBF preprocessing.Comment: preprint of a paper to be published at IJCAR 2018, LNCS, Springer,
including appendi
Skolem Functions for Factored Formulas
Given a propositional formula F(x,y), a Skolem function for x is a function
\Psi(y), such that substituting \Psi(y) for x in F gives a formula semantically
equivalent to \exists F. Automatically generating Skolem functions is of
significant interest in several applications including certified QBF solving,
finding strategies of players in games, synthesising circuits and bit-vector
programs from specifications, disjunctive decomposition of sequential circuits
etc. In many such applications, F is given as a conjunction of factors, each of
which depends on a small subset of variables. Existing algorithms for Skolem
function generation ignore any such factored form and treat F as a monolithic
function. This presents scalability hurdles in medium to large problem
instances. In this paper, we argue that exploiting the factored form of F can
give significant performance improvements in practice when computing Skolem
functions. We present a new CEGAR style algorithm for generating Skolem
functions from factored propositional formulas. In contrast to earlier work,
our algorithm neither requires a proof of QBF satisfiability nor uses
composition of monolithic conjunctions of factors. We show experimentally that
our algorithm generates smaller Skolem functions and outperforms
state-of-the-art approaches on several large benchmarks.Comment: Full version of FMCAD 2015 conference publicatio
Building Strategies into QBF Proofs
Strategy extraction is of great importance for quantified Boolean formulas (QBF), both in solving and proof complexity. So far in the QBF literature, strategy extraction has been algorithmically performed from proofs. Here we devise the first QBF system where (partial) strategies are built into the proof and are piecewise constructed by simple operations along with the derivation. This has several advantages: (1) lines of our calculus have a clear semantic meaning as they are accompanied by semantic objects; (2) partial strategies are represented succinctly (in contrast to some previous approaches); (3) our calculus has strategy extraction by design; and (4) the partial strategies allow new sound inference steps which are disallowed in previous central QBF calculi such as Q-Resolution and long-distance Q-Resolution. The last item (4) allows us to show an exponential separation between our new system and the previously studied reductionless long-distance resolution calculus. Our approach also naturally lifts to dependency QBFs (DQBF), where it yields the first sound and complete CDCL-style calculus for DQBF, thus opening future avenues into CDCL-based DQBF solving
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