517 research outputs found
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
Decision procedures for linear arithmetic
In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit präsentieren wir neue Entscheidungsprozeduren für lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollständiger Kalkül für ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der Größte-Würfeltest und der Einheitswürfeltest, zwei korrekte (wenn auch unvollständige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollständig unbegrenzten Systemen, welche für viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren präsentieren wir mehrere Anwendungsmöglichkeiten für diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfüllbarkeitsäquivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollständige und effiziente Entscheidungsprozedur für unbeschränkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver für lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf präsentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestätigen
False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time
False-name manipulation refers to the question of whether a player in a
weighted voting game can increase her power by splitting into several players
and distributing her weight among these false identities. Analogously to this
splitting problem, the beneficial merging problem asks whether a coalition of
players can increase their power in a weighted voting game by merging their
weights. Aziz et al. [ABEP11] analyze the problem of whether merging or
splitting players in weighted voting games is beneficial in terms of the
Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10]
for the probabilistic Banzhaf index. All these results provide merely
NP-hardness lower bounds for these problems, leaving the question about their
exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf
index, we raise these lower bounds to hardness for PP, "probabilistic
polynomial time", and provide matching upper bounds for beneficial merging and,
whenever the number of false identities is fixed, also for beneficial
splitting, thus resolving previous conjectures in the affirmative. It follows
from our results that beneficial merging and splitting for these two power
indices cannot be solved in NP, unless the polynomial hierarchy collapses,
which is considered highly unlikely
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Quantum Stochastic Processes and Quantum Many-Body Physics
This dissertation investigates the theory of quantum stochastic processes and its applications in quantum many-body physics.
The main goal is to analyse complexity-theoretic aspects of both static and dynamic properties of physical systems modelled by quantum stochastic processes.
The thesis consists of two parts: the first one addresses the computational complexity of certain quantum and classical divisibility questions, whereas the second one addresses the topic of Hamiltonian complexity theory.
In the divisibility part, we discuss the question whether one can efficiently sub-divide a map describing the evolution of a system in a noisy environment, i.e. a CPTP- or stochastic map for quantum and classical processes, respectively, and we prove that taking the nth root of a CPTP or stochastic map is an NP-complete problem.
Furthermore, we show that answering the question whether one can divide up a random variable into a sum of iid random variables , i.e. , is poly-time computable; relaxing the iid condition renders the problem NP-hard.
In the local Hamiltonian part, we study computation embedded into the ground state of a many-body quantum system, going beyond "history state" constructions with a linear clock.
We first develop a series of mathematical techniques which allow us to study the energy spectrum of the resulting Hamiltonian, and extend classical string rewriting to the quantum setting.
This allows us to construct the most physically-realistic QMAEXP-complete instances for the LOCAL HAMILTONIAN problem (i.e. the question of estimating the ground state energy of a quantum many-body system) known to date, both in one- and three dimensions.
Furthermore, we study weighted versions of linear history state constructions, allowing us to obtain tight lower and upper bounds on the promise gap of the LOCAL HAMILTONIAN problem in various cases.
We finally study a classical embedding of a Busy Beaver Turing Machine into a low-dimensional lattice spin model, which allows us to dictate a transition from a purely classical phase to a Toric Code phase at arbitrarily large and potentially even uncomputable system sizes
Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets
Let be a real closed field and
an ordered domain. We consider the algorithmic problem of computing the
generalized Euler-Poincar\'e characteristic of real algebraic as well as
semi-algebraic subsets of , which are defined by symmetric
polynomials with coefficients in . We give algorithms for computing
the generalized Euler-Poincar\'e characteristic of such sets, whose
complexities measured by the number the number of arithmetic operations in
, are polynomially bounded in terms of and the number of
polynomials in the input, assuming that the degrees of the input polynomials
are bounded by a constant. This is in contrast to the best complexity of the
known algorithms for the same problems in the non-symmetric situation, which
are singly exponential. This singly exponential complexity for the latter
problem is unlikely to be improved because of hardness result
(-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with
arXiv:1312.658
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