A Core Course on Modeling

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A Core Course on Modeling

Contents • Change needs Time

• Introduction to Processes

•States and State Charts

•Applying State Charts

• Time and State Transitions

•Partially Ordered Time

•Totally Ordered Time

•Totally Ordered Time; Equal Intervals

•Totally Ordered Time; Infinitesimal Intervals

• Summary

• References to lecture notes + book

• References to quiz-questions and homework assignments (lecture notes)

Week 3- Time for Change

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Contents Question:

• how to model systems …

• … that vary over time …

• … such that their behavior

•can be predicted,

•can be specified,

•can be verified,

•can be … ?


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Introduction to Processes

Running example: 4-color ballpoint pen

Purpose: the model should …

• simulate a virtual 4-color ballpoint, or

• warn when one of the pens is about to run out, or

• train users to handle a 4-color ballpoint, or

• specificatin of a Photoshop plugin


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Running example: 4-color ballpoint pen

Step 1: conceptual model with properties:

myPen: fourCPen;

fourCPen:

[red: pen, green: pen, blue: pen, black: pen,

use: {TRUE,FALSE}];

pen:[level: {0…99}%,inOut: {in,out}]

myPen.use is true if there is a pen p with p.level>0 and p.inOut=out


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Describe processes: statecharts

Process = series of states

All four pens are in

and contain some ink

Red pen is out;

other three pens

are in; red pen

contains some ink

Red pen is out;

other three pens

are in; red pen

contains less ink

red pen goes out

write with red pen



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•State = all properties in the conceptual model with their values

•State space = the collection of all states

•A behavior = a route through statespace

•A process = the collection of behaviors of a system



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Red pen is out;

other three pens

are in; red pen

contains some ink

Red pen is out;

other three pens

are in; red pen

contains less ink

Green pen is out;

other three pens

are in; green pen

contains some ink


and contains some ink

Blue pen is out;

other three pens

are in; blue pen

contains some ink

Green pen is out;

other three pens

are in; red pen

contains some ink

Red pen is out;

other three pens

are in; red pen

contains some ink



Red pen is out;

other three pens

are in; red pen

contains no ink



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A real-life example: controlling a parking garage

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•With n quantities, each mi values, nr states = i=1 … n mi

•Number of behaviors with p steps = j=1 … p i=1 … n mi

•State and process explosion

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For the 4-color ballpoint, myPen:fourCPen;

fourCPen: [red: pen,green: pen,

blue: pen,black: pen,

use: {TRUE,FALSE}];

pen: [level:{0…99}%, inOut: {in,out}]

the state space contains

100 x 100 x 100 x 100 x 2 x 2 x 2 x 2

= 1600000000 states

Parking garage: 1080 states (including folded cars!)

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Idea: hidingIdea: hiding and exposing and exposing

quantities orquantities or

valuesvalues

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make use an exposed quantity

make level a hidden quantity

make inOut exposed if color of a pen matters; otherwise hidden

make use an exposed quantity

make level a hidden quantity

make inOut exposed if color of a pen matters; otherwise hidden



Individual properties:

use is visiblelevel is not visibleinOut … depends on purpose

Individual properties:

use is visiblelevel is not visibleinOut … depends on purpose

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•Decrease amount of states:

•distinguish exposed and hidden quantities or values

•focus on the exposed ones



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Example: 4-color pen

Simples process description: just two states

myPen.use=TRUE myPen.use=FALSEwriting



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Example: 4-color pen

The 2-state model is too naive:

• pen must be out to write

• only one pen out at a time

• ink only decreases if a pen is out



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Denote state and state transitions: statechart

red.inOut=out

red.level>0

red.inOut=out

red.level=0

green.inOut=out

green.level>0

green.inOut=out

green.level=0

blue.inOut=out

blue.level>0

blue.inOut=out

blue.level=0

black.inOut=out

black.level>0

black.inOut=out

black.level=0

red.inOut=in

green.inOut=in

blue.inOut=in

black.inOut=in



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black.inOut=

out

blue.inOut=out

green.inOut=out

red.inOut=out

red.inOut=in

green.inOut=in

blue.inOut=in

black.inOut=in




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red.level>0

green.level>0

blue.level>0

black.level>0

red.level=0

green.level=0

blue.level=0

black.level=0




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Now we have a state chart model. So what?



garage model was reduced to garage model was reduced to 3.3 x 103.3 x 1066 states states

… … and could be analysed by and could be analysed by computercomputer

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•predict: how will process run?

•specify: desired state, process

•verify: undesired state won’t occur

•analyse: no deadlocks?

•analyse: reachability

•analyse: correct order?

•analyse: only permitted transitions?



Now we have a state chart model. So what?

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Time and State Transitions


Let’s start talking about time.

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state S0

state S1

state S2

state S3

transition T1

transition T2

transition T3

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Flavours of time

partial order

for 0 pairs of T1, T2, earlier(T1,T2) or later(T1,T2)

total order

for all pairs of T1, T2, earlier (T1,T2) or later(T1,T2)



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Flavours of time

specify coin operated vending machine:

ealier(insertCoin,giveProduct)

earlier(makeChoice,giveProduct)

earlier(insertCoin,giveChange)

earlier(makeChoice,giveChange)

earlier(giveProduct,giveChange) ?

earlier(giveChange,giveProduct) ?



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Flavours of time (applications)

partial order



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total order

for all pairs of T1, T2, earlier (T1,T2) or later(T1,T2)

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total order + equal intervals

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total order + infinitesimal intervals

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Causality:

Qcurr=F(Qprev,Pprev)

Forbidden: Qcurr=F(Qcurr,…)



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Causality:

Qcurr=F(Qprev,Pprev)

Forbidden: Qcurr=F(Qcurr,…)

F is a recursive function

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Arbitrary intervals

a 20 km march route;

15 control posts;

every post: a box of bananas;

a list of ‘efforts’ (kCal) between posts, {Ei}.

At which posts should you pick banana(s)?



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Arbitrary intervals

Qi=amount of kCal in stomach at post i, i > 0

Q0=amount of kCal in stomach at start.

Qi+1 = Qi - Ei (don’t eat);

Qi+1 = Qi – Ei + niB (eat n bananas at post i);

ni such that i, Qi+1 > 0, and as small as possible.

Qi+1 = Qi – Ei + niB; Qi – Ei + niB > 0,

so niB > Ei - Qi; ni = max(0,(Ei - Qi) / B), so

Qi+1 = Qi – Ei + B * max(0,(Ei - Qi) / B) or: Qi+1 = F(Qi,Pi)

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Equal intervals

a mass-spring system.




Equal intervals

a mass-spring system.

K = ma K = C(urest-u)

a = v’

v = u’

t = i

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Equal intervals

a mass-spring system.K = ma K(t) Ki

K = C(urest-u) a(t) (v(t+)-v(t))/ = (vi+1 - vi) /

a = v’ v(t) (u(t+ )-u(t))/ = (ui+1 - ui) /

v = u’

t = i

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Equal intervals


K = C(urest-u) a(t) (v(t+)-v(t))/ = (vi+1 - vi) / a = v’ v(t) (u(t+ )-u(t))/ = (ui+1 - ui) / v = u’ ui+1 = ui + vi

t = i vi+1 = vi + ai

= vi + K/m

= vi + C(urest-ui) /m

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Equal intervals


K = C(urest-u) a(t) (v(t+)-v(t))/ = (vi+1 - vi) / a = v’ v(t) (u(t+ )-u(t))/ = (ui+1 - ui) / v = u’ ui+1 = ui + vi


= vi + K/m

= vi + C(urest-ui) /m

So Qcurr = F(Qprev, Pprev):

ucurr = F1(uprev,vprev) = uprev+vprev

vcurr = F2(vprev,uprev) = vprev+ C(urest-uprev) /m, with given u0 and v0

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Equal intervals

a mass-spring system with damping.K = ma K(t) Ki

K = C(urest-u) - v a(t) (v(t+)-v(t))/ = (vi+1 - vi) / a = v’ v(t) (u(t+ )-u(t))/ = (ui+1 - ui) / v = u’ ui+1 = ui + vi


= vi + K/m

= vi + (C(urest-ui) - vi ) /m

So Qcurr = F(Qprev, Pprev):

ucurr = F1(uprev,vprev) = uprev+vprev

vcurr = F2(vprev,uprev) = vprev+(C(urest-uprev)-vprev )/m, with given u0 and v0

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Equal intervals

a mass-spring system with damping.

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Infinitesimal intervalsa mass-spring system with damping.

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Infinitesimal intervals


K = K(t), u = u(t), K = C(urest-u) - v.

First try for = 0:u = urest+A sin(t/T)

u’’ = -T-2Asin(t/T)

Substitute back:-mT-2Asin(t/T) = C(urest-urest-Asin(t/T),

mT-2 = C, or T = m/C ( =:T0)

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Infinitesimal intervals


K = K(t), u = u(t), K = C(urest-u) - v.

Next try for 0:u = urest + A et

C(urest-u) - u’ = mu’’ and use u’ = Aet; u’’ = A2et.

C(urest-urest-Aet) - Aet = mA2et . Must hold for any t, so

C + + m2 = 0, hence 1,2 = (- (2-4mC))/2m

Let 0 = (4mC).

For = 0 critical damping;

for < 0 oscillations; T=T0 /(1- (/0)2)

for > 0 super critical damping

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Summary

Purpose exposed and hidden variables

Purpose partial order or total order ?

If total order: unknown or known intervals? What determines (sampling) intervals?

If verification or specification: consider statecharts and process models

If simulation or prediction: use Qcurr=F(Qprev,Pprev)

interested in outcomes?

use simplest possible numerical techniques,

small : accuracy large : performance

interested in insight?

try symbolic methods


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Equal intervals

Playing a CD: a series of 44100 samples/sec.

Playing the sound: take a sample Pi

Process the sound (e.g., Qi+1=Qi+(1-)Pi)

Output Qi+1 to the speaker

Qi+1 = F(Qi,Pi)

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samples taken every 1/44100 sec

original sound

the reconstructed signal

Equal intervals

Playing a CD: a series of 44100 samples/sec.

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Summary

• State = snapshot of a conceptual model at some time point;

• State space = collection of all states;

• Change = transitions between states; state chart = graph; nodes (states) and arrows (transitions);

• Behavior = path through state space;

• State space explosion: number of states is huge for non trivial cases

• Projection: given a purpose, distinguish exposed and hidden properties or value sets;

• Multiple flavors of time:

• partially ordered time, e.g. specification and verification;

• totally ordered time, e.g. prediction, steering and control;

• A recursive function Qi+1 = F(Qi , Qi-1 , Qi-2, …. , Pi , Pi-1 , Pi-2 , …) to evaluate or unroll a behavior;

• equal intervals: closed form evaluation (e.g.,: periodic financial transactions; sampling);

• equal, small intervals: approximation, sampling error (examples: moving point mass, … );

• infinitesimal intervals: continuous time, differential equations (examples: mass-spring system);


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A Core Course on Modeling