291 research outputs found
The Challenge of Unifying Semantic and Syntactic Inference Restrictions
While syntactic inference restrictions don't play an important role for SAT, they are an essential reasoning technique for more expressive logics, such as first-order logic, or fragments thereof. In particular, they can result in short proofs or model representations. On the other hand, semantically guided inference systems enjoy important properties, such as the generation of solely non-redundant clauses. I discuss to what extend the two paradigms may be unifiable
Towards Verifying Nonlinear Integer Arithmetic
We eliminate a key roadblock to efficient verification of nonlinear integer
arithmetic using CDCL SAT solvers, by showing how to construct short resolution
proofs for many properties of the most widely used multiplier circuits. Such
short proofs were conjectured not to exist. More precisely, we give n^{O(1)}
size regular resolution proofs for arbitrary degree 2 identities on array,
diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs
for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result
Scavenger 0.1: A Theorem Prover Based on Conflict Resolution
This paper introduces Scavenger, the first theorem prover for pure
first-order logic without equality based on the new conflict resolution
calculus. Conflict resolution has a restricted resolution inference rule that
resembles (a first-order generalization of) unit propagation as well as a rule
for assuming decision literals and a rule for deriving new clauses by (a
first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201
Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution
QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited.
In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent.
This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs.
We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas.
We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution
QBF solvers implementing the QCDCL paradigm are powerful algorithms that
successfully tackle many computationally complex applications. However, our
theoretical understanding of the strength and limitations of these QCDCL
solvers is very limited.
In this paper we suggest to formally model QCDCL solvers as proof systems. We
define different policies that can be used for decision heuristics and unit
propagation and give rise to a number of sound and complete QBF proof systems
(and hence new QCDCL algorithms). With respect to the standard policies used in
practical QCDCL solving, we show that the corresponding QCDCL proof system is
incomparable (via exponential separations) to Q-resolution, the classical QBF
resolution system used in the literature. This is in stark contrast to the
propositional setting where CDCL and resolution are known to be p-equivalent.
This raises the question what formulas are hard for standard QCDCL, since
Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In
answer to this question we prove several lower bounds for QCDCL, including
exponential lower bounds for a large class of random QBFs.
We also introduce a strengthening of the decision heuristic used in classical
QCDCL, which does not necessarily decide variables in order of the prefix, but
still allows to learn asserting clauses. We show that with this decision
policy, QCDCL can be exponentially faster on some formulas.
We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution.
In comparison to classical QCDCL, this new QCDCL version adapts both decision
and unit propagation policies
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