76,911 research outputs found
Computing the Maximum using (min, +) Formulas
We study computation by formulas over (min,+). We consider the
computation of max{x_1,...,x_n} over N as a difference of
(min,+) formulas, and show that size n + n log n is sufficient
and necessary. Our proof also shows that any (min,+) formula
computing the minimum of all sums of n-1 out of n variables must
have n log n leaves; this too is tight. Our proofs use a
complexity measure for (min,+) functions based on minterm-like
behaviour and on the entropy of an associated graph
Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds
Given a formula in quantifier-free Presburger arithmetic, if it has a
satisfying solution, there is one whose size, measured in bits, is polynomially
bounded in the size of the formula. In this paper, we consider a special class
of quantifier-free Presburger formulas in which most linear constraints are
difference (separation) constraints, and the non-difference constraints are
sparse. This class has been observed to commonly occur in software
verification. We derive a new solution bound in terms of parameters
characterizing the sparseness of linear constraints and the number of
non-difference constraints, in addition to traditional measures of formula
size. In particular, we show that the number of bits needed per integer
variable is linear in the number of non-difference constraints and logarithmic
in the number and size of non-zero coefficients in them, but is otherwise
independent of the total number of linear constraints in the formula. The
derived bound can be used in a decision procedure based on instantiating
integer variables over a finite domain and translating the input
quantifier-free Presburger formula to an equi-satisfiable Boolean formula,
which is then checked using a Boolean satisfiability solver. In addition to our
main theoretical result, we discuss several optimizations for deriving tighter
bounds in practice. Empirical evidence indicates that our decision procedure
can greatly outperform other decision procedures.Comment: 26 page
Hamiltonian Simulation Using Linear Combinations of Unitary Operations
We present a new approach to simulating Hamiltonian dynamics based on
implementing linear combinations of unitary operations rather than products of
unitary operations. The resulting algorithm has superior performance to
existing simulation algorithms based on product formulas and, most notably,
scales better with the simulation error than any known Hamiltonian simulation
technique. Our main tool is a general method to nearly deterministically
implement linear combinations of nearby unitary operations, which we show is
optimal among a large class of methods.Comment: 18 pages, 3 figure
Robust Online Monitoring of Signal Temporal Logic
Signal Temporal Logic (STL) is a formalism used to rigorously specify
requirements of cyberphysical systems (CPS), i.e., systems mixing digital or
discrete components in interaction with a continuous environment or analog com-
ponents. STL is naturally equipped with a quantitative semantics which can be
used for various purposes: from assessing the robustness of a specification to
guiding searches over the input and parameter space with the goal of falsifying
the given property over system behaviors. Algorithms have been proposed and
implemented for offline computation of such quantitative semantics, but only
few methods exist for an online setting, where one would want to monitor the
satisfaction of a formula during simulation. In this paper, we formalize a
semantics for robust online monitoring of partial traces, i.e., traces for
which there might not be enough data to decide the Boolean satisfaction (and to
compute its quantitative counterpart). We propose an efficient algorithm to
compute it and demonstrate its usage on two large scale real-world case studies
coming from the automotive domain and from CPS education in a Massively Open
Online Course (MOOC) setting. We show that savings in computationally expensive
simulations far outweigh any overheads incurred by an online approach
Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover
Stochastic Boolean Function Evaluation is the problem of determining the
value of a given Boolean function f on an unknown input x, when each bit of x_i
of x can only be determined by paying an associated cost c_i. The assumption is
that x is drawn from a given product distribution, and the goal is to minimize
the expected cost. This problem has been studied in Operations Research, where
it is known as "sequential testing" of Boolean functions. It has also been
studied in learning theory in the context of learning with attribute costs. We
consider the general problem of developing approximation algorithms for
Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for
evaluating Boolean linear threshold formulas. We also present an approximation
algorithm for evaluating CDNF formulas (and decision trees) achieving a factor
of O(log kd), where k is the number of terms in the DNF formula, and d is the
number of clauses in the CNF formula. In addition, we present approximation
algorithms for simultaneous evaluation of linear threshold functions, and for
ranking of linear functions.
Our function evaluation algorithms are based on reductions to the Stochastic
Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and
Krause. They presented an approximation algorithm for the problem, called
Adaptive Greedy. Our main technical contribution is a new approximation
algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an
extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito,
which is a generalization of Hochbaum's algorithm for the classical Set Cover
Problem. We also give a new bound on the approximation achieved by the Adaptive
Greedy algorithm of Golovin and Krause
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