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A Core Course on Modeling

A Core Course on Modeling. Week 3- Time for Change.      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

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A Core Course on Modeling

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  1. A Core Course on Modeling Week 3- Time for Change    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)

  2. A Core Course on Modeling Week 3- Time for Change    Contents     2 Question: • how to model systems … • … that vary over time … • … such that their behavior • can be predicted, • can be specified, • can be verified, • can be … ?

  3. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     3 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

  4. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     4 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

  5. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     5 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 write with red pen red pen goes out

  6. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     6 • 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

  7. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     7 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 Green pen is out; other three pens are in; green pen contains some ink All four pens are in and contains some ink Blue pen is out; other three pens are in; blue pen contains some ink Red pen is out; other three pens are in; red pen contains less ink All four pens are in and contain some ink Red pen is out; other three pens are in; red pen contains some ink All four pens are in and contain some ink Red pen is out; other three pens are in; red pen contains no ink

  8. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     8 A real-life example: controlling a parking garage

  9. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     9 • 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

  10. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     10 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!)

  11. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     11 Idea: hiding and exposing quantities or values

  12. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     12 Individualproperties: use is visible level is notvisible inOut … dependsonpurpose make use an exposed quantity make level a hidden quantity make inOutexposed if color of a pen matters; otherwise hidden

  13. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     13 • Decrease amount of states: • distinguish exposed and hidden quantities or values • focus on the exposed ones

  14. writing A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     14 Example: 4-color pen Simples process description: just two states myPen.use=TRUE myPen.use=FALSE

  15. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     15 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

  16. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     16 Denote state and state transitions: statechart red.inOut=out red.level>0 red.inOut=out red.level=0 black.inOut=out black.level=0 green.inOut=out green.level>0 red.inOut=in green.inOut=in blue.inOut=in black.inOut=in green.inOut=out green.level=0 black.inOut=out black.level>0 blue.inOut=out blue.level>0 blue.inOut=out blue.level=0

  17. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     17 Denote state and state transitions: statechart red.inOut=out black.inOut= out red.inOut=in green.inOut=in blue.inOut=in black.inOut=in green.inOut=out blue.inOut=out

  18. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     18 Denote state and state transitions: statechart red.level=0 red.level>0 green.level=0 green.level>0 blue.level>0 blue.level=0 black.level>0 black.level=0

  19. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     19 Now we have a state chart model. So what? garage model was reduced to 3.3 x 106states … and couldbeanalysedby computer

  20. A Core Course on Modeling Week 3- Time for Change    Introduction to Processes     20 Now we have a state chart model. So what? • 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?

  21. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    21 Let’s start talking about time.

  22. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    22 state S0 transition T1 state S1 transition T2 state S2 transition T3 state S3

  23. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    23 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)

  24. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    24 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) ?

  25. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    25 Flavours of time (applications) partial order

  26. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    26 Flavours of time (applications) total order for all pairs of T1, T2, earlier (T1,T2) or later(T1,T2)

  27. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    27 Flavours of time (applications) total order + equal intervals

  28. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    28 Flavours of time (applications) total order + infinitesimal intervals

  29. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    29 Causality: Qcurr=F(Qprev,Pprev) Forbidden: Qcurr=F(Qcurr,…)

  30. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    30 Causality: Qcurr=F(Qprev,Pprev) Forbidden: Qcurr=F(Qcurr,…) F is a recursive function

  31. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    31 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)?

  32. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    32 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)

  33. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    33 Equal intervals a mass-spring system.

  34. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    34 Equal intervals a mass-spring system. K = ma K = C(urest-u) a = v’ v = u’ t = i

  35. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    35 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

  36. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    36 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’ ui+1 = ui + vi t = i vi+1 = vi + ai = vi + K/m = vi +  C(urest-ui) /m

  37. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    37 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’ ui+1 = ui + vi t = i vi+1 = vi + ai = 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

  38. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    38 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 t = i vi+1 = vi + ai = 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

  39. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    39 Equal intervals a mass-spring system with damping. 39

  40. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    40 Infinitesimal intervals a mass-spring system with damping.

  41. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    41 Infinitesimal intervals a mass-spring system with damping. 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)

  42. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    42 Infinitesimal intervals a mass-spring system with damping. K = K(t), u = u(t), K = C(urest-u) - v. Next try for 0: u = urest + A et C(urest-u) - u’ = mu’’ and use u’ = Aet; u’’ = A2et. C(urest-urest-Aet) - Aet = mA2et .Must hold for any t, so C +  + m2 = 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

  43. A Core Course on Modeling Week 3- Time for Change    Summary     43 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

  44. A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    44 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)

  45. the reconstructed signal samples taken every 1/44100 sec original sound A Core Course on Modeling Week 3- Time for Change    Time and State Transitions    45 Equal intervals Playing a CD: a series of 44100 samples/sec.

  46. A Core Course on Modeling Week 3- Time for Change    Summary     46 • State= snapshot of a conceptual model at some time point; • State space= collection of all states; • Change = transitionsbetween 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 exposedand hiddenproperties 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 recursivefunction Qi+1 = F(Qi , Qi-1 , Qi-2, …. , Pi , Pi-1 , Pi-2 , …) to evaluate or unrolla behavior; • equal intervals: closed form evaluation (e.g.,: periodic financial transactions; sampling); • equal, small intervals: approximation, sampling error (examples: moving point mass, … ); • infinitesimal intervals: continuoustime, differential equations (examples: mass-spring system);

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