stabilizing orbits separated by delay

Separating Orbits by Local Delay:Examples of Emergent Behavior

presented by: Ted Herman, University of Iowa

April 2010

motivationm tokens on a ring

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daring thesis

controversial thesis statements

• there is more to computing than laptops, smartphones, gameconsoles (Xbox, Wii, PS3), Enterprise/Cloud, . . .

• there is more to software than Internet, Game/Graphics, HCI,Database, Mathematica, Matlab, . . .

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pervasive computing

enter computer control

computing is taking control . . .

• Runaway Prius

• Algorithmic Trading ⇒ Economic Meltdown

• coming: computer-controlled arms, legs, hands . . .

question to you

are you the windshield or are you the bug?

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pervasive computing

pervasively computers controlling things

likely future

1 many independently created (computing) systems will controldifferent things

2 these will be combined (in new, interesting, and unintendedways) in the physical world

3 cascades of failure, cross-layer interactions, timing issues – willhighlight fragile & robust properties

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pervasive complexity

computing in devices is already complex

1 by Lines of Code (LoC)• Air Force F-22 Raptor: 1.7 million• F-35 Joint Strike Fighter: 5.7 million• Boeing 787 Dreamliner: 6.5 million• Typical Mercedes/BMW/etc: 100 million

2 by interfaces

separate computing/control systems may havephysical interfaces with each other

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example for this talk

industrial process control

the practical story of robotics

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example for this talk

in the home?

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aspects of process control software

many aspects to process controlfault tolerance, parallel programming, specification languages, debugging, modeling, architecture, visualization,management, standardization, security, validation, resource optimization, . . .

But for this talk, just one, very restricted topic:

tolerance to misplacing objects in a closed system, forparticular kinds of process control networks

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aspects of timing control

even further, problem is restricted to synchronous timing (everyonerunning on same clock), with goal of “keeping things apart”(which might not be apart sometimes, due to misplacement)

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modelisation

simplified model

(overly) simplified model: Petri NetW

• unrelated to Petri Dish, Rob Petrie, . . .

• historically early attempt to representconcurrency (how to reason about parallelprograms)

• still used for control theoryW & processcontrol studies

• graphical representation, but with algebraictools

• tokens, places, transitions

W = lookup in Wikipedia

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abstract problem

emergent, distributed token behavior

1 physical process control• tokens symbolize physical entities• token transfer ⇔ physical movement• separation useful for physical reasons (timing)

2 sensor networkW

• token represents sensor power on• limiting tokens reduces power usage• separation good for sensing diversity

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problem statement as exercise

correct token behavior

1 problem setup

m tokens, initially in arbitrary locations

at most one token moves per step on each link

synchronous steps (full parallelism)

2 goals

minimum distance d between tokens

all tokens move in each step

fair visitation of tokens to nodes

problem: design a simple algorithm that would work atmicroprocessor scale; all decisions are local, without networkcommunication

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motivating technique

diversion: an open system

tokens arrive from outside, to p1

each token travels from pi to pi+1

Goal:

separation distance at least d for { pi | i > 1 }

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motivating technique

Petri net for open system

auxilliary ring/token at p1

joint transition for p1 → p2 transfer

can this convert to closed system?e.g., create a link from pn to p1, aring topology

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relay

relay process (node pi )

relay ::do forever

ri ← ri + qi−1 ; // receive queued tokens from pi−1

qi−1 ← 0 ;

if ri > 0 then

ri ← ri − 1 ;

qi ← qi + 1 // put one token in output queue

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delay

delay Process (node pi )

clocked release of tokens

ci represents clock

delay ::do forever

ri ← ri + qi−1 ; // this part is like relayqi−1 ← 0 ;

if ci > 0 then

ci ← ci − 1 // waiting for clock to reach zeroelse if ci = 0 ∧ ri > 0 then

ci ← C ; // value of C determines token separationri ← ri − 1 ;

qi ← qi + 1

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easy protocol

one delay + others relay

• unidirectional ring p1 . . . pn (pn → p1)

• p1 runs delay

• others p2–pn run relay

• m tokens, m · (C + 1) ≤ n

stabilization

the m tokens initially in arbitrary places

clock c1 arbitrarily in [0,C ]

desired: minimum intertoken distance ≥ C + 1

hoped-for invariantW: c1 = 0 whenever token arrives to p1

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more ambitious protocol

mix of delays and relays

• at least one pi runs delay

• others run relay

• could even be uniform (all can run delay)

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mixed (or uniform) protocol

result (theorem)

the delay/relay protocol, with at least one delay process, and withn > 1, m > 1, m · (C + 1) ≤ n, self-stabilizes to minimumintertoken distance at least C + 1.

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uniformity, good or bad?

does having more than one delay help?

simulation results suggest that moving to uniformity may improveconvergence rate, on average

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beyond rings?

what about other topologies?

we can try to map walks over a network to virtual rings

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line of rings

looking at another situation

• suppose there are n rings

• each has just one token, four nodes

• consistent node labels, token path is clockwise

• rings are in a line• node 0 is at left, node 2 is at right• in the line, node 2 of ring j sits next to node 0

of ring j + 1

• counters are shared by adjacent nodes

Goal: keep each tokenat least distance 2 fromany other token

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protocol for the line

relay (nodes 1 and 3)

do forever

qi , qi−1 ← qi−1, 0

delay (nodes 0 and 2)

do forever // node 0if c = 0 ∧ r0 = 1 then

c, q0, r0 ← 1, 1, 0elif q3 = 1 ∧

(c = 1 ∨ r2 = 1 ∨ q1 = 1) then

r0, q3 ← 1, 0elif q3 = 1 ∧ c = 0 ∧ r2 = 0 then

c, q0, q3 ← 1, 1, 0elif c = 1 then

c ← c − 1

do forever // node 2if c = 0 ∧ r2 = 1 then

c, q2, r2 ← 1, 1, 0elif q1 = 1 ∧ (c = 1 ∨ r0 = 1) then

r2, q1 ← q1, 0elif q1 = 1 ∧ c = 0 ∧ r0 = 0 then

c, q2, q1 ← 1, 1, 0elif c = 1 then

c ← c − 1

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protocol - example run

let’s see it start from an improper initial state

(example with replay)

(same example, step by step)

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does it always work?

Program VerificationW

intuitively, seems “obvious” that all rings will eventuallysynchronize

but how to show this?

some careful observations (about the program and the evolution ofstates in an execution) may “build up” to verification

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easy observation

Implausible States

1 some states cannot occur as result of any state step(wp(σ) = false)

2 similarly, other states cannot occur after two steps(wp2(σ) = false)

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state properties

plausible, but incorrect states

• the notion of gap is useful

• definition: ring j has gap k if the position of the token inring j + 1 is k nodes “ahead”

example: gap = 1

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sum of gaps

Loop VariantW

1 S(σ) =∑

j gapj(σ) (sum of gaps for state σ)

2 lemma: if σ0, σ1, . . . represents system execution of plausiblestates, then S(σ0), S(σ1), . . . is a “descending” sequence ofnumbers

3 proof: any resting token decrements the gap to the left whilstincrementing the gap to the right

4 observation: S(σ) = 0 implies σ is a correct state

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problem

what if S(σ0), S(σ1), . . . never decreases, but remains at somepositive value forever? (the definition of “descending” does allowthis loophole)

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back to the blackboard

more observations

• g j = 1 “moves right” to g j+1 = 1

• g j = 2 “moves right” to g j+1 = 2

• g j = 3 “moves left” to g j−1 = 3

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again at the blackboard

how to gaps interact?

1 gaps may merge:• +1 & +1 → +2• +1 & +2 → +3• +1 & +3 → +4 = 0 ⇒ decrease S

2 gaps may “run off the end of the line” ⇒ decrease S

3 gaps do not “split” (except, momentarily, for +2)

4 if gaps do not merge, they continue to move in same direction

lemma: positive S eventually decreases

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. . . in conclusion

1 emergent behavior (network of circulatingobjects, separated) achievable just bylocal delay

2 open theory question: just what ispossible, what is impossible when usinglocal delay mechanism?

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stabilizing orbits separated by delay

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