10,441 research outputs found
Pairs of SAT Assignment in Random Boolean Formulae
We investigate geometrical properties of the random K-satisfiability problem
using the notion of x-satisfiability: a formula is x-satisfiable if there exist
two SAT assignments differing in Nx variables. We show the existence of a sharp
threshold for this property as a function of the clause density. For large
enough K, we prove that there exists a region of clause density, below the
satisfiability threshold, where the landscape of Hamming distances between SAT
assignments experiences a gap: pairs of SAT-assignments exist at small x, and
around x=1/2, but they donot exist at intermediate values of x. This result is
consistent with the clustering scenario which is at the heart of the recent
heuristic analysis of satisfiability using statistical physics analysis (the
cavity method), and its algorithmic counterpart (the survey propagation
algorithm). The method uses elementary probabilistic arguments (first and
second moment methods), and might be useful in other problems of computational
and physical interest where similar phenomena appear
The quantum adversary method and classical formula size lower bounds
We introduce two new complexity measures for Boolean functions, or more
generally for functions of the form f:S->T. We call these measures sumPI and
maxPI. The quantity sumPI has been emerging through a line of research on
quantum query complexity lower bounds via the so-called quantum adversary
method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the
realization that these many different formulations are in fact equivalent.
Given that sumPI turns out to be such a robust invariant of a function, we
begin to investigate this quantity in its own right and see that it also has
applications to classical complexity theory.
As a surprising application we show that sumPI^2(f) is a lower bound on the
formula size, and even, up to a constant multiplicative factor, the
probabilistic formula size of f. We show that several formula size lower bounds
in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93],
including a key lemma of [Has98], are in fact special cases of our method.
The second quantity we introduce, maxPI(f), is always at least as large as
sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a
lower bound on formula size. While sumPI(f) is always a lower bound on the
quantum query complexity of f, this is not the case in general for maxPI(f). A
strong advantage of sumPI(f) is that it has both primal and dual
characterizations, and thus it is relatively easy to give both upper and lower
bounds on the sumPI complexity of functions. To demonstrate this, we look at a
few concrete examples, for three functions: recursive majority of three, a
function defined by Ambainis, and the collision problem.Comment: Appears in Conference on Computational Complexity 200
On Characterizing the Data Access Complexity of Programs
Technology trends will cause data movement to account for the majority of
energy expenditure and execution time on emerging computers. Therefore,
computational complexity will no longer be a sufficient metric for comparing
algorithms, and a fundamental characterization of data access complexity will
be increasingly important. The problem of developing lower bounds for data
access complexity has been modeled using the formalism of Hong & Kung's
red/blue pebble game for computational directed acyclic graphs (CDAGs).
However, previously developed approaches to lower bounds analysis for the
red/blue pebble game are very limited in effectiveness when applied to CDAGs of
real programs, with computations comprised of multiple sub-computations with
differing DAG structure. We address this problem by developing an approach for
effectively composing lower bounds based on graph decomposition. We also
develop a static analysis algorithm to derive the asymptotic data-access lower
bounds of programs, as a function of the problem size and cache size
Towards a Formal Verification Methodology for Collective Robotic Systems
We introduce a UML-based notation for graphically modeling
systems’ security aspects in a simple and intuitive
way and a model-driven process that transforms graphical
specifications of access control policies in XACML. These
XACML policies are then translated in FACPL, a policy
language with a formal semantics, and the resulting policies
are evaluated by means of a Java-based software tool
- …