1,747 research outputs found
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
Efficient Generation of Craig Interpolants in Satisfiability Modulo Theories
The problem of computing Craig Interpolants has recently received a lot of
interest. In this paper, we address the problem of efficient generation of
interpolants for some important fragments of first order logic, which are
amenable for effective decision procedures, called Satisfiability Modulo Theory
solvers.
We make the following contributions.
First, we provide interpolation procedures for several basic theories of
interest: the theories of linear arithmetic over the rationals, difference
logic over rationals and integers, and UTVPI over rationals and integers.
Second, we define a novel approach to interpolate combinations of theories,
that applies to the Delayed Theory Combination approach.
Efficiency is ensured by the fact that the proposed interpolation algorithms
extend state of the art algorithms for Satisfiability Modulo Theories. Our
experimental evaluation shows that the MathSAT SMT solver can produce
interpolants with minor overhead in search, and much more efficiently than
other competitor solvers.Comment: submitted to ACM Transactions on Computational Logic (TOCL
MaxSAT Resolution and Subcube Sums
We study the MaxRes rule in the context of certifying unsatisfiability. We
show that it can be exponentially more powerful than tree-like resolution, and
when augmented with weakening (the system MaxResW), p-simulates tree-like
resolution. In devising a lower bound technique specific to MaxRes (and not
merely inheriting lower bounds from Res), we define a new proof system called
the SubCubeSums proof system. This system, which p-simulates MaxResW, can be
viewed as a special case of the semialgebraic Sherali-Adams proof system. In
expressivity, it is the integral restriction of conical juntas studied in the
contexts of communication complexity and extension complexity. We show that it
is not simulated by Res. Using a proof technique qualitatively different from
the lower bounds that MaxResW inherits from Res, we show that Tseitin
contradictions on expander graphs are hard to refute in SubCubeSums. We also
establish a lower bound technique via lifting: for formulas requiring large
degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums
Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer
We solved a long-outstanding open problem in Ramsey theory, using SAT solving
Even shorter proofs without new variables
Proof formats for SAT solvers have diversified over the last decade, enabling
new features such as extended resolution-like capabilities, very general
extension-free rules, inclusion of proof hints, and pseudo-boolean reasoning.
Interference-based methods have been proven effective, and some theoretical
work has been undertaken to better explain their limits and semantics. In this
work, we combine the subsumption redundancy notion from (Buss, Thapen 2019) and
the overwrite logic framework from (Rebola-Pardo, Suda 2018). Natural
generalizations then become apparent, enabling even shorter proofs of the
pigeonhole principle (compared to those from (Heule, Kiesl, Biere 2017)) and
smaller unsatisfiable core generation.Comment: 21 page
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
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