11,833 research outputs found
Partial Quantifier Elimination By Certificate Clauses
We study partial quantifier elimination (PQE) for propositional CNF formulas.
In contrast to full quantifier elimination, in PQE, one can limit the set of
clauses taken out of the scope of quantifiers to a small subset of target
clauses. The appeal of PQE is twofold. First, PQE can be dramatically simpler
than full quantifier elimination. Second, it provides a language for performing
incremental computations. Many verification problems (e.g. equivalence checking
and model checking) are inherently incremental and so can be solved in terms of
PQE. Our approach is based on deriving clauses depending only on unquantified
variables that make the target clauses . Proving redundancy
of a target clause is done by construction of a ``certificate'' clause implying
the former. We describe a PQE algorithm called that employs
the approach above. We apply to generating properties of a
design implementation that are not implied by specification. The existence of
an property means that this implementation is buggy. Our
experiments with HWMCC-13 benchmarks suggest that can be used
for generating properties of real-life designs
Exponential separations using guarded extension variables
We study the complexity of proof systems augmenting resolution with inference
rules that allow, given a formula in conjunctive normal form, deriving
clauses that are not necessarily logically implied by but whose
addition to preserves satisfiability. When the derived clauses are
allowed to introduce variables not occurring in , the systems we
consider become equivalent to extended resolution. We are concerned with the
versions of these systems without new variables. They are called BC,
RAT, SBC, and GER, denoting respectively blocked clauses,
resolution asymmetric tautologies, set-blocked clauses, and generalized
extended resolution. Each of these systems formalizes some restricted version
of the ability to make assumptions that hold "without loss of generality,"
which is commonly used informally to simplify or shorten proofs.
Except for SBC, these systems are known to be exponentially weaker than
extended resolution. They are, however, all equivalent to it under a relaxed
notion of simulation that allows the translation of the formula along with the
proof when moving between proof systems. By taking advantage of this fact, we
construct formulas that separate RAT from GER and vice versa. With
the same strategy, we also separate SBC from RAT. Additionally, we
give polynomial-size SBC proofs of the pigeonhole principle, which
separates SBC from GER by a previously known lower bound. These
results also separate the three systems from BC since they all simulate
it. We thus give an almost complete picture of their relative strengths
New methods for 3-SAT decision and worst-case analysis
We prove the worst-case upper bound 1:5045 n for the time complexity of 3-SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2- and 3-clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3-clause-sets and can be applied also to arbitrary trees. Keywords: 3-SAT, worst-case upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3-SAT decision and prove the worst-case upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results..
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