16,546 research outputs found
Kinematic Self-Similar Solutions of Locally Rotationally Symmetric Spacetimes
This paper contains locally rotationally symmetric kinematic self-similar
perfect fluid and dust solutions. We consider three families of metrics which
admit kinematic self-similar vectors of the first, second, zeroth and infinite
kinds, not only for the tilted fluid case but also for the parallel and
orthogonal cases. It is found that the orthogonal case gives contradiction both
in perfect fluid and dust cases for all the three metrics while the tilted case
reduces to the parallel case in both perfect fluid and dust cases for the
second metric. The remaining cases give self-similar solutions of different
kinds. We obtain a total of seventeen independent solutions out of which two
are vacuum. The third metric yields contradiction in all the cases.Comment: 17 pages, accepted for publication Brazilian J. Physic
The Hardness of Finding Linear Ranking Functions for Lasso Programs
Finding whether a linear-constraint loop has a linear ranking function is an
important key to understanding the loop behavior, proving its termination and
establishing iteration bounds. If no preconditions are provided, the decision
problem is known to be in coNP when variables range over the integers and in
PTIME for the rational numbers, or real numbers. Here we show that deciding
whether a linear-constraint loop with a precondition, specifically with
partially-specified input, has a linear ranking function is EXPSPACE-hard over
the integers, and PSPACE-hard over the rationals. The precise complexity of
these decision problems is yet unknown. The EXPSPACE lower bound is derived
from the reachability problem for Petri nets (equivalently, Vector Addition
Systems), and possibly indicates an even stronger lower bound (subject to open
problems in VAS theory). The lower bound for the rationals follows from a novel
simulation of Boolean programs. Lower bounds are also given for the problem of
deciding if a linear ranking-function supported by a particular form of
inductive invariant exists. For loops over integers, the problem is PSPACE-hard
for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of
natural numbers as invariants.Comment: In Proceedings GandALF 2014, arXiv:1408.5560. I thank the organizers
of the Dagstuhl Seminar 14141, "Reachability Problems for Infinite-State
Systems", for the opportunity to present an early draft of this wor
A Comment on Budach's Mouse-in-an-Octant Problem
Budach's Mouse-in-an-Octant Problem (attributed to Lothar Budach in a 1980
article by van Emde Boas and Karpinski) concerns the behaviour of a very simple
finite-state machine ("the mouse") moving on the integer two-dimensional grid.
Its decidability is apparently still open. This note sketches a proof that an
extended version of the problem (a super-mouse) is undecidable.Comment: 3 pages, 2 bibliographic reference
- …