677 research outputs found
Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances
An analysis of the average-case complexity of solving random 3-Satisfiability
(SAT) instances with backtrack algorithms is presented. We first interpret
previous rigorous works in a unifying framework based on the statistical
physics notions of dynamical trajectories, phase diagram and growth process. It
is argued that, under the action of the Davis--Putnam--Loveland--Logemann
(DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose
characteristic parameters (ratio alpha of clauses per variable, fraction p of
3-clauses) can be followed during the operation, and define resolution
trajectories. Depending on the location of trajectories in the phase diagram of
the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are
generated. Three regimes are identified, depending on the ratio alpha of the
3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost
surely finds a solution in a time growing linearly with the number N of
variables. Upper sat phase: for intermediate ratios, instances are almost
surely satisfiable but finding a solution requires exponential time (2 ^ (N
omega) with omega>0) with high probability. Unsat phase: for large ratios,
there is almost always no solution and proofs of refutation are exponential. An
analysis of the growth of the search tree in both upper sat and unsat regimes
is presented, and allows us to estimate omega as a function of alpha. This
analysis is based on an exact relationship between the average size of the
search tree and the powers of the evolution operator encoding the elementary
steps of the search heuristic.Comment: to appear in Theoretical Computer Scienc
Analysis of the computational complexity of solving random satisfiability problems using branch and bound search algorithms
The computational complexity of solving random 3-Satisfiability (3-SAT)
problems is investigated. 3-SAT is a representative example of hard
computational tasks; it consists in knowing whether a set of alpha N randomly
drawn logical constraints involving N Boolean variables can be satisfied
altogether or not. Widely used solving procedures, as the
Davis-Putnam-Loveland-Logeman (DPLL) algorithm, perform a systematic search for
a solution, through a sequence of trials and errors represented by a search
tree. In the present study, we identify, using theory and numerical
experiments, easy (size of the search tree scaling polynomially with N) and
hard (exponential scaling) regimes as a function of the ratio alpha of
constraints per variable. The typical complexity is explicitly calculated in
the different regimes, in very good agreement with numerical simulations. Our
theoretical approach is based on the analysis of the growth of the branches in
the search tree under the operation of DPLL. On each branch, the initial 3-SAT
problem is dynamically turned into a more generic 2+p-SAT problem, where p and
1-p are the fractions of constraints involving three and two variables
respectively. The growth of each branch is monitored by the dynamical evolution
of alpha and p and is represented by a trajectory in the static phase diagram
of the random 2+p-SAT problem. Depending on whether or not the trajectories
cross the boundary between phases, single branches or full trees are generated
by DPLL, resulting in easy or hard resolutions.Comment: 37 RevTeX pages, 15 figures; submitted to Phys.Rev.
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
High-dynamic GPS tracking
The results of comparing four different frequency estimation schemes in the presence of high dynamics and low carrier-to-noise ratios are given. The comparison is based on measured data from a hardware demonstration. The tested algorithms include a digital phase-locked loop, a cross-product automatic frequency tracking loop, and extended Kalman filter, and finally, a fast Fourier transformation-aided cross-product frequency tracking loop. The tracking algorithms are compared on their frequency error performance and their ability to maintain lock during severe maneuvers at various carrier-to-noise ratios. The measured results are shown to agree with simulation results carried out and reported previously
Digital phase-lock loop
An improved digital phase lock loop incorporates several distinctive features that attain better performance at high loop gain and better phase accuracy. These features include: phase feedback to a number-controlled oscillator in addition to phase rate; analytical tracking of phase (both integer and fractional cycles); an amplitude-insensitive phase extractor; a more accurate method for extracting measured phase; a method for changing loop gain during a track without loss of lock; and a method for avoiding loss of sampled data during computation delay, while maintaining excellent tracking performance. The advantages of using phase and phase-rate feedback are demonstrated by comparing performance with that of rate-only feedback. Extraction of phase by the method of modeling provides accurate phase measurements even when the number-controlled oscillator phase is discontinuously updated
A Simple and Flexible Way of Computing Small Unsatisfiable Cores in SAT Modulo Theories
Finding small unsatisfiable cores for SAT problems has recently received a lot of interest, mostly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be more naturally addressed in the framework of Satisfiability Modulo Theories, SMT. Surprisingly, the problem of finding unsatisfiable cores in SMT has received very little attention in the literature; in particular, we are not aware of any work aiming at producing small unsatisfiable cores in SMT. In this paper we present a novel approach to this problem. The main idea is to combine an SMT solver with an external propositional core extractor: the SMT solver produces the theory lemmas found during the search; the core extractor is then called on the boolean abstraction of the original SMT problem and of the theory lemmas. This results in an unsatisfiable core for the original SMT problem, once the remaining theory lemmas have been removed. The approach is conceptually interesting, since the SMT solver is used to dynamically lift the suitable amount of theory information to the boolean level, and it also has several advantages in practice. In fact, it is extremely simple to implement and to update, and it can be interfaced with every propositional core extractor in a plug-and-play manner, so that to benefit for free of all unsat-core reduction techniques which have been or will be made available. We have evaluated our approach by an extensive empirical test on SMT-LIB benchmarks, which confirms the validity and potential of this approach
The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic
We study the dynamics of a backtracking procedure capable of proving
uncolourability of graphs, and calculate its average running time T for sparse
random graphs, as a function of the average degree c and the number of vertices
N. The analysis is carried out by mapping the history of the search process
onto an out-of-equilibrium (multi-dimensional) surface growth problem. The
growth exponent of the average running time is quantitatively predicted, in
agreement with simulations.Comment: 5 figure
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
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