1,704 research outputs found
Flow Logic
Flow networks have attracted a lot of research in computer science. Indeed,
many questions in numerous application areas can be reduced to questions about
flow networks. Many of these applications would benefit from a framework in
which one can formally reason about properties of flow networks that go beyond
their maximal flow. We introduce Flow Logics: modal logics that treat flow
functions as explicit first-order objects and enable the specification of rich
properties of flow networks. The syntax of our logic BFL* (Branching Flow
Logic) is similar to the syntax of the temporal logic CTL*, except that atomic
assertions may be flow propositions, like or , for
, which refer to the value of the flow in a vertex, and
that first-order quantification can be applied both to paths and to flow
functions. We present an exhaustive study of the theoretical and practical
aspects of BFL*, as well as extensions and fragments of it. Our extensions
include flow quantifications that range over non-integral flow functions or
over maximal flow functions, path quantification that ranges over paths along
which non-zero flow travels, past operators, and first-order quantification of
flow values. We focus on the model-checking problem and show that it is
PSPACE-complete, as it is for CTL*. Handling of flow quantifiers, however,
increases the complexity in terms of the network to , even
for the LFL and BFL fragments, which are the flow-counterparts of LTL and CTL.
We are still able to point to a useful fragment of BFL* for which the
model-checking problem can be solved in polynomial time. Finally, we introduce
and study the query-checking problem for BFL*, where under-specified BFL*
formulas are used for network exploration
Coverage and Vacuity in Network Formation Games
The frameworks of coverage and vacuity in formal verification analyze the effect of mutations applied to systems or their specifications. We adopt these notions to network formation games, analyzing the effect of a change in the cost of a resource. We consider two measures to be affected: the cost of the Social Optimum and extremums of costs of Nash Equilibria. Our results offer a formal framework to the effect of mutations in network formation games and include a complexity analysis of related decision problems. They also tighten the relation between algorithmic game theory and formal verification, suggesting refined definitions of coverage and vacuity for the latter
Black hole entropy divergence and the uncertainty principle
Black hole entropy has been shown by 't Hooft to diverge at the horizon. The
region near the horizon is in a thermal state, so entropy is linear to energy
which consequently also diverges. We find a similar divergence for the energy
of the reduced density matrix of relativistic and non-relativistic field
theories, extending previous results in quantum mechanics. This divergence is
due to an infinitely sharp division between the observable and unobservable
regions of space, and it stems from the position/momentum uncertainty relation
in the same way that the momentum fluctuations of a precisely localized quantum
particle diverge. We show that when the boundary between the observable and
unobservable regions is smoothed the divergence is tamed. We argue that the
divergence of black hole entropy can also be interpreted as a consequence of
position/momentum uncertainty, and that 't Hooft's brick wall tames the
divergence in the same way, by smoothing the boundary.Comment: Added clarifications and explanation
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