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On the Convergence of Population Protocols When Population Goes to Infinity
Population protocols have been introduced as a model of sensor networks
consisting of very limited mobile agents with no control over their own
movement. A population protocol corresponds to a collection of anonymous
agents, modeled by finite automata, that interact with one another to carry out
computations, by updating their states, using some rules. Their computational
power has been investigated under several hypotheses but always when restricted
to finite size populations. In particular, predicates stably computable in the
original model have been characterized as those definable in Presburger
arithmetic. We study mathematically the convergence of population protocols
when the size of the population goes to infinity. We do so by giving general
results, that we illustrate through the example of a particular population
protocol for which we even obtain an asymptotic development. This example shows
in particular that these protocols seem to have a rather different
computational power when a huge population hypothesis is considered.Comment: Submitted to Applied Mathematics and Computation. 200
Playing With Population Protocols
Population protocols have been introduced as a model of sensor networks
consisting of very limited mobile agents with no control over their own
movement: A collection of anonymous agents, modeled by finite automata,
interact in pairs according to some rules.
Predicates on the initial configurations that can be computed by such
protocols have been characterized under several hypotheses.
We discuss here whether and when the rules of interactions between agents can
be seen as a game from game theory. We do so by discussing several basic
protocols
Passively Mobile Communicating Logarithmic Space Machines
We propose a new theoretical model for passively mobile Wireless Sensor
Networks. We call it the PALOMA model, standing for PAssively mobile
LOgarithmic space MAchines. The main modification w.r.t. the Population
Protocol model is that agents now, instead of being automata, are Turing
Machines whose memory is logarithmic in the population size n. Note that the
new model is still easily implementable with current technology. We focus on
complete communication graphs. We define the complexity class PLM, consisting
of all symmetric predicates on input assignments that are stably computable by
the PALOMA model. We assume that the agents are initially identical.
Surprisingly, it turns out that the PALOMA model can assign unique consecutive
ids to the agents and inform them of the population size! This allows us to
give a direct simulation of a Deterministic Turing Machine of O(nlogn) space,
thus, establishing that any symmetric predicate in SPACE(nlogn) also belongs to
PLM. We next prove that the PALOMA model can simulate the Community Protocol
model, thus, improving the previous lower bound to all symmetric predicates in
NSPACE(nlogn). Going one step further, we generalize the simulation of the
deterministic TM to prove that the PALOMA model can simulate a Nondeterministic
TM of O(nlogn) space. Although providing the same lower bound, the important
remark here is that the bound is now obtained in a direct manner, in the sense
that it does not depend on the simulation of a TM by a Pointer Machine.
Finally, by showing that a Nondeterministic TM of O(nlogn) space decides any
language stably computable by the PALOMA model, we end up with an exact
characterization for PLM: it is precisely the class of all symmetric predicates
in NSPACE(nlogn).Comment: 22 page
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This guidance is for NHS and other commissioners, managers and practitioners who have a direct or indirect role in, and responsibility for, increasing the uptake of HIV testing among men who have sex with men. This includes those working in local authorities and the wider public, private, voluntary and community sectors. It will also be of interest to members of the public, in particular men who have sex with men.
The focus of the guidance is on increasing the uptake of HIV testing to reduce undiagnosed infection and prevent transmission.
The recommendations include advice on:
planning services, including assessing local need and developing a strategy
promoting HIV testing among men who have sex with men, including outreach schemes and providing rapid point-of-care tests
offering and recommending an HIV test in primary care, secondary care and specialist sexual health services
repeat testing
HIV referral pathways
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