709 research outputs found
On the Complexity of Computing Minimal Unsatisfiable LTL formulas
We show that (1) the Minimal False QCNF search-problem (MF-search) and the
Minimal Unsatisfiable LTL formula search problem (MU-search) are FPSPACE
complete because of the very expressive power of QBF/LTL, (2) we extend the
PSPACE-hardness of the MF decision problem to the MU decision problem. As a
consequence, we deduce a positive answer to the open question of PSPACE
hardness of the inherent Vacuity Checking problem. We even show that the
Inherent Non Vacuous formula search problem is also FPSPACE-complete.Comment: Minimal unsatisfiable cores For LTL causes inherent vacuity checking
redundancy coverag
Optimization Modulo Theories with Linear Rational Costs
In the contexts of automated reasoning (AR) and formal verification (FV),
important decision problems are effectively encoded into Satisfiability Modulo
Theories (SMT). In the last decade efficient SMT solvers have been developed
for several theories of practical interest (e.g., linear arithmetic, arrays,
bit-vectors). Surprisingly, little work has been done to extend SMT to deal
with optimization problems; in particular, we are not aware of any previous
work on SMT solvers able to produce solutions which minimize cost functions
over arithmetical variables. This is unfortunate, since some problems of
interest require this functionality.
In the work described in this paper we start filling this gap. We present and
discuss two general procedures for leveraging SMT to handle the minimization of
linear rational cost functions, combining SMT with standard minimization
techniques. We have implemented the procedures within the MathSAT SMT solver.
Due to the absence of competitors in the AR, FV and SMT domains, we have
experimentally evaluated our implementation against state-of-the-art tools for
the domain of linear generalized disjunctive programming (LGDP), which is
closest in spirit to our domain, on sets of problems which have been previously
proposed as benchmarks for the latter tools. The results show that our tool is
very competitive with, and often outperforms, these tools on these problems,
clearly demonstrating the potential of the approach.Comment: Submitted on january 2014 to ACM Transactions on Computational Logic,
currently under revision. arXiv admin note: text overlap with arXiv:1202.140
Unsatisfiable Linear CNF Formulas Are Large and Complex
We call a CNF formula linear if any two clauses have at most one variable in
common. We show that there exist unsatisfiable linear k-CNF formulas with at
most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at
most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic
means, and we have no explicit construction coming even close to it. One reason
for this is that unsatisfiable linear formulas exhibit a more complex structure
than general (non-linear) formulas: First, any treelike resolution refutation
of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$.
This implies that small unsatisfiable linear k-CNF formulas are hard instances
for Davis-Putnam style splitting algorithms. Second, if we require that the
formula F have a strict resolution tree, i.e. every clause of F is used only
once in the resolution tree, then we need at least a^a^...^a clauses, where a
is approximately 2 and the height of this tower is roughly k.Comment: 12 pages plus a two-page appendix; corrected an inconsistency between
title of the paper and title of the arxiv submissio
On the van der Waerden numbers w(2;3,t)
We present results and conjectures on the van der Waerden numbers w(2;3,t)
and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed
the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39,
where for t <= 30 we conjecture these lower bounds to be exact. The lower
bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we
present an improved conjecture. We also investigate regularities in the good
partitions (certificates) to better understand the lower bounds.
Motivated by such reglarities, we introduce *palindromic van der Waerden
numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers
w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good
partitions), defined as reading the same from both ends. Different from the
situation for ordinary van der Waerden numbers, these "numbers" need actually
to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide
lower bounds, which we conjecture to be exact, for t <= 35.
All computations are based on SAT solving, and we discuss the various
relations between SAT solving and Ramsey theory. Especially we introduce a
novel (open-source) SAT solver, the tawSolver, which performs best on the SAT
instances studied here, and which is actually the original DLL-solver, but with
an efficient implementation and a modern heuristic typical for look-ahead
solvers (applying the theory developed in the SAT handbook article of the
second author).Comment: Second version 25 pages, updates of numerical data, improved
formulations, and extended discussions on SAT. Third version 42 pages, with
SAT solver data (especially for new SAT solver) and improved representation.
Fourth version 47 pages, with updates and added explanation
Strongly Refuting Random CSPs Below the Spectral Threshold
Random constraint satisfaction problems (CSPs) are known to exhibit threshold
phenomena: given a uniformly random instance of a CSP with variables and
clauses, there is a value of beyond which the CSP will be
unsatisfiable with high probability. Strong refutation is the problem of
certifying that no variable assignment satisfies more than a constant fraction
of clauses; this is the natural algorithmic problem in the unsatisfiable regime
(when ).
Intuitively, strong refutation should become easier as the clause density
grows, because the contradictions introduced by the random clauses become
more locally apparent. For CSPs such as -SAT and -XOR, there is a
long-standing gap between the clause density at which efficient strong
refutation algorithms are known, , and the
clause density at which instances become unsatisfiable with high probability,
.
In this paper, we give spectral and sum-of-squares algorithms for strongly
refuting random -XOR instances with clause density in time or in
rounds of the sum-of-squares hierarchy, for any
and any integer . Our algorithms provide a smooth
transition between the clause density at which polynomial-time algorithms are
known at , and brute-force refutation at the satisfiability
threshold when . We also leverage our -XOR results to obtain
strong refutation algorithms for SAT (or any other Boolean CSP) at similar
clause densities. Our algorithms match the known sum-of-squares lower bounds
due to Grigoriev and Schonebeck, up to logarithmic factors.
Additionally, we extend our techniques to give new results for certifying
upper bounds on the injective tensor norm of random tensors
On Matrices, Automata, and Double Counting
Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables M, with the same constraint defined by a finite-state automaton A on each row of M and a global cardinality constraint gcc on each column of M. We give two methods for deriving, by double counting,
necessary conditions on the cardinality variables of the gcc constraints from the automaton A. The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We evaluate the impact of our methods on a large set of nurse rostering problem instances
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