Steve Levine. Programs with Flexible Time. When? Tuesday, Feb 16 th. courtesy of JPL

Transcription

1 Programs with Flexible Time When? Contributions: Brian Williams Patrick Conrad Simon Fang Paul Morris Nicola Muscettola Pedro Santana Julie Shah John Stedl Andrew Wang Steve Levine Tuesday, Feb 16 th courtesy of JPL

2 Assignments Problems Sets: Pset 1 due tomorrow (Wednesday) at 11:59pm Pset 2 released tomorrow Interesting references: Dechter, R., I. Meiri, J. Pearl, Temporal Constraint Networks, Artificial Intelligence, 49, pp ,1991. Muscettola, N., P. Morris and I. Tsamardinos, Reformulating Temporal Plans for Efficient Execution. Intl Conf. on Knowledge Representation and Reasoning (KRR), 1998.

3 Outline Programs with Flexible Time Intro Describing temporal plans Exposing implicit constraints Consistency checking Offline scheduling Online execution Reformulating for faster online execution

4 A single cognitive system language and executive. Uhura Kirk Burton Pike Sulu Bones source unknown. All rights reserved. This content is excluded from our Creative Commons license. For moreinformation, see

5 Flexible Time Flexible time = more robustness We tell cognitive robot: Timing requirements ( engage boosters 2-4 minutes after launch but before reaching orbit ) Cognitive robot schedules autonomously. This image is in the public domain.

6 Execution of Timed Model-based Programs 1 2 in RMPL [williams et al 01] [5,10] [5,10] [2,5] [5,10] [5,10] [5,10] [2,5] [5,10] Decisions include what, how, who and when. Focus of today [Muscettola, Morris, Tsamardinos,KR 98]

7 To Execute a Temporal Plan 1. Describe Temporal Plan 1. Describe Temporal Plan 2. Test Consistency 2. Test Consistency 3. Schedule Plan 3. Reformulate Plan 4. Execute Plan 4. Dynamically Schedule Plan

8 To Execute a Temporal Plan 1. Describe Temporal Plan 2. Test Consistency 3. Schedule Plan 4. Execute Plan

9 Describe Temporal Plan Activities to perform Relationships among activities This image is in the public domain.

10 Metric Temporal Relations [10min, 30min] Activity: Going to the store Activity: Bake Bread [0d, 1d] Activity: Eat Bread

11 Simplify by reducing interval relations to relations on timepoints. Activity A A - A + Start Activity A End Activity A

12 Metric Temporal Relations [10m,30m] G - G + 10 < [G + -G - ] < 30 Start Going to Store End Going to Store [0d,1d] B + E - 0 < [E - -B + ] < 1 End Bake Bread Start Eat Bread

13 Qualitative Temporal Relations [Allen 83] X Y X Y X Y XY X Y Y X Y X

14 Qualitative Temporal Relations Expressed as timepoint inequalities: X Y X Y X + X + [0,inf] [0,0] Y - Y - and X - < Y + X X XY Y Y X Y and X + < Y + and X + < Y + and X + = Y + Y X and X + = Y + or Y + < X - [Villain & Kautz; Simmons]

15 Temporal Relations Described by a Simple Temporal Network (STN) Simple Temporal Network Tuple <X, C> where: variables X 1, X n, represent time points (real-valued domains) binary constraints C of the form: ( X X ) [ a b ]. k i ik, ik [l 1, u 1 ] X 1 X 3 [l 2, u 2 ] [l 3, u 3 ] X 2 [Dechter, Meiri, Pearl 91] called links. Sufficient to represent: simple metric constraints all Allen relations but 1 Can t represent: Allen s disjoint relation

16 Modeling Visualization

17 To Execute a Temporal Plan 1. Describe Temporal Plan A [1,10] B [1,1] D [0,9] C [2,2] 2. Test Consistency 3. Schedule Plan 4. Execute Plan

18 Consistency of an STN Input: STN <X, C> where C j = < <X k, X i >, <a j, b j > > A [1,10] [1,1] B D [0,9] C [2,2] STN is consistent iff there exists an assignment to times X satisfying C.

19 Schedule of an STN Input: STN <X, C> where C j = < <X k, X i >, <a j, b j > > A [1,10] [0,9] B C [1,1] D [2,2] Schedule is assignment to all timepoints X consistent with constraints.

20 How to find schedule? Idea: Transform STN Transform to distance-graph Common graph algorithms (i.e., shortest path) will apply

21 Transformation to distance graph (Board)

22 Simple Temporal Network Distance Graph u A [l, u] B A B -l l B A u B A u l B A A B -l [Dechter, Meiri, Pearl 91]

23 Algorithm: offline scheduling Initialize execution window to [-, ] for each event while unexecuted events: x i = pick any unexecuted event t i = pick any time in x i s execution window Propagate to all x i s neighbors & update their windows

24 Naïve (and wrong) scheduling (Board)

25 Propagating to neighbors Tighten neighbor s execution windows: - outgoing edges to neighbor: u = min(u, t i + w u ) - incoming edges from neighbor: l = max(l, t i -w) l [u, l] tightened [u, l ] w u x i = t i w l

26 Exposing Implicit Constraints (Board)

27 [10,20] [30,40] [10,20] [60,70] [40,50]

28 APSP Graph: Windows of Feasible Values

29 Dispatchable form An STN or distance graph is dispatchable if: Can be properly scheduled via local propagations to neighbors only Requires all implicit constraints be explicit

30 Checking Consistency (Board)

31 1 A [2, 1] B A B Simple Temporal Network Distance Graph

32 Summary To schedule, want a simple, local-propagation algorithm Requires exposing implicit constraints All-pairs shortest path (APSP) exposes all implicit constraints Puts network in dispatchable form Negative cycle in APSP: inconsistent.

33 To Execute a Temporal Plan 1. Describe Temporal Plan 1. Describe Temporal Plan 2. Test Consistency 2. Test Consistency 3. Schedule Plan 3. Reformulate Plan 4. Execute Plan 4. Dynamically Schedule Plan

34 Algorithm: offline scheduling Compute dispatchable form (i.e., APSP) Initialize execution window to [-, ] for each event while unexecuted events: x i = pick any unexecuted event t i = pick any time in x i s execution window Propagate to all x i s neighbors & update their windows

35 The original STN [0,10] [1, 1] [0,10] [2, 2]

36 Distance graph transformation

37 All pairs shortest path A B C D A B C D

38 Computing a schedule

39 Computing a schedule [-, ] [-, ] 1-1 [-, ] 2 0 [-, ] 11 Initialize execution windows for each event in the plan

40 Computing a schedule [-, ] t = [-, ] 2 0 [-, ] 11 Assign the first event

41 outgoing edges to neighbor: u = min(u, t i + w u ) incoming edges from neighbor: l = max(l, t i Computing a schedule w) l [-, 10] t = [-, ] 2 0 [-, ] 11 Propagate updated time bounds to neighbors

42 Computing a schedule outgoing edges to neighbor: u = min(u, t i + w u ) incoming edges from neighbor: l = max(l, t i 10 -w) l [1, 10] 1-1 t = [-, ] [-, ] 11 Propagate updated time bounds to neighbors

43 Computing a schedule [1, 10] t = [-, ] [0, 9] 11 Propagate updated time bounds to neighbors

44 Computing a schedule [1, 10] t = [2, 11] [0, 9] 11 Propagate updated time bounds to neighbors

45 Computing a schedule t = 3 t = [2, 11] 2 0 [0, 9] 11 Arbitrarily pick another time point and assign it...

46 Computing a schedule 10 t = t = [2, 11] [2, 2] 11 Propagate updated time bounds to neighbors

47 Computing a schedule 10 t = t = [4, 4] [2, 2] 11 Propagate updated time bounds to neighbors

48 Computing a schedule t = 3 t = [4, 4] 2 0 t = 2 11 Pick another event and assign it

49 Computing a schedule t = 3 t = [4, 4] 2 0 t = 2 11 Propagate to neighbors

50 Computing a schedule t = 3 t = t = t = 2 11 Assign the final event

51 Pre-computed schedules not robust against fluctuations We ve just computed a schedule: ta = 0, tb = 3, tc = 2, td = 4 But what if there s a disturbance? i.e., what if tb = 3.1? i.e., what if tb = 4? i.e., what if tb = 100? Pre-computed schedules not robust against fluctuations! Solution:Dispatch dynamically online. Schedule events on the fly, after observing past event times. Increases robustness to many unanticipated fluctuations. Flexible temporal constraints allow this!

52 To Execute a Temporal Plan 1. Describe Temporal Plan 1. Describe Temporal Plan 2. Test Consistency 2. Test Consistency 3. Schedule Plan 3. Reformulate Plan 4. Execute Plan 4. Dynamically Schedule Plan

53 How do we schedule online? First, consider naive (incorrect!) approach. Similar to offline schedule algorithm, but now online: Wait until current time in execution window ( active ) (Still a problem though as we ll see shortly)

54 Naïve (wrong!) online dispatcher Compute dispatchable form (i.e., APSP) Initialize execution window to [-, ] for each event while unexecuted events: x i = pick any unexecuted event if current time in window t i = now Propagate to all x i s neighbors & update their windows

55 Naïve (wrong!) online scheduling [-, ] t = [-, ] [-, ] 11 Arbitrarily picking next event

56 Naïve (wrong!) online scheduling 10 t = t = [2, 11] [0, 9] 11 Arbitrarily picking next event

57 Naïve (wrong!) online scheduling 10 t = t = [4, 4] t = but wait! We just assigned a past time!

58 Enablement conditions dictate the ordering of dispatched events Online: must assign monotonically increasing times whereas offline algorithms may assign in any order. How can we constrain dispatcher to do this? Solution:determine enablement conditions by analyzing negative edges. Allows us to infer if some edges must precede other edges

59 Enablement conditions dictate the ordering of dispatched events Negative edges from APSP dictate ordering constraints

60 Enablement conditions dictate the ordering of dispatched events Negative edges from APSP dictate ordering constraints B must occur after both A and C!

61 Enablement conditions dictate the ordering of dispatched events An event is enabled if all its neighbors over negative edges have already been dispatched. All predecessors have been dispatched. Modify online dispatching algorithm to only dispatch events if they are enabled. An event is active is the current time is within that event s execution window

62 Corrected online dispatcher Compute dispatchable form (i.e., APSP) Initialize execution window to [-, ] for each event E {events with no predecessors} # set of enabled events S {} # set of executed events while unexecuted events: Wait until some event x i in E is active t i = now # dispatch x i now at t i Propagate to all x i s neighbors & update their windows Add x i to S Add to E any now-enabled events

63 Running online dispatcher Compute dispatchable form (i.e., APSP) E = {A, C} Initialize execution windows to [-, ] S = {} E {events with no predecessors} S {} [-, ] while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 10 [-, ] [-, ] A, C initially in E have no [-, ] negative, outgoing edges

64 Running online dispatcher Compute dispatchable form (i.e., APSP) E = {A, C} Initialize execution windows to [-, ] S = {} E {events with no predecessors} S {} [-, ] while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 10 [-, ] [-, ] A is enabled and in E. [-, ] (could have chosen C too)

65 Running online dispatcher Compute dispatchable form (i.e., APSP) Initialize execution windows to [-, ] E {events with no predecessors} S {} [1, 10] while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events E = {C} S = {A} t = 0 [2, 11] Dispatch A and propagate [0, 9]

66 Running online dispatcher Compute dispatchable form (i.e., APSP) Initialize execution windows to [-, ] E {events with no predecessors} S {} [1, 10] while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events E = {C} S = {A} t = 0 [2, 11] B, D not enabled! But C still is. [0, 9]

67 Running online dispatcher Compute dispatchable form (i.e., APSP) E = {} Initialize execution windows to [-, ] S = {A, C} E {events with no predecessors} S {} [3, 3] while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 10 t = 0 [4, 4] Dispatch & propagate C. t =

68 Running online dispatcher Compute dispatchable form (i.e., APSP) E = {B} Initialize execution windows to [-, ] S = {A, C} E {events with no predecessors} S {} [3, 3] while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 10 t = 0 [4, 4] B is now enabled (but still not D). t =

69 Running online dispatcher Compute dispatchable form (i.e., APSP) E = {} Initialize execution windows to [-, ] S = {A, C, B} E {events with no predecessors} S {} t = 3 while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 10 t = 0 [4, 4] Dispatch & propagate B. t =

70 Running online dispatcher Compute dispatchable form (i.e., APSP) E = {D} Initialize execution windows to [-, ] S = {A, C, B} E {events with no predecessors} S {} t = 3 while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 10 t = 0 [4, 4] D is finally enabled. t =

71 Running online dispatcher Compute dispatchable form (i.e., APSP) E = {} Initialize execution windows to [-, ] S = {A, C, B, D} E {events with no predecessors} S {} t = 3 while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 10 t = 0 t = 4 Finish up by dispatching D! t =

72 Online dispatching algorithm remarks By considering predecessors, we guarantee that events assigned monotonically increasing times online. Capable of responding to fluctuations that do not affect overall temporal feasibility. (Note: must be run on an dispatchable / APSP graph!) 2

73 Online dispatcher efficiency Consider an STN with n edges. How many edges in APSP distance graph? Compute dispatchable form (i.e., APSP) Initialize execution windows to [-, ] E {events with no predecessors} S {} while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 3

74 Online dispatcher efficiency Consider an STN with n edges. How many edges in APSP distance graph? n 2. Compute dispatchable form (i.e., APSP) Initialize execution windows to [-, ] E {events with no predecessors} S {} while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 4

75 Online dispatcher efficiency Consider an STN with n edges. How many edges in APSP distance graph? n 2. How many neighbors to propagate to each step? Compute dispatchable form (i.e., APSP) Initialize execution windows to [-, ] E {events with no predecessors} S {} while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 5

76 Online dispatcher efficiency Consider an STN with n edges. How many edges in APSP distance graph? n 2. How many neighbors to propagate to each step? n. Compute dispatchable form (i.e., APSP) Initialize execution windows to [-, ] E {events with no predecessors} S {} while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 6

77 Online dispatcher efficiency Consider an STN with n edges. How many edges in APSP distance graph? n 2. How many neighbors to propagate to each step? n. Large STNs: propagation slow. Want to reduce this. Compute dispatchable form (i.e., APSP) Initialize execution windows to [-, ] E {events with no predecessors} S {} while unexecuted events: Wait until some event x i in E is active t i = now Propagate to x i s neighbors Add x i to S Add to E any now-enabled events 7

78 To Execute a Temporal Plan 1. Describe Temporal Plan 1. Describe Temporal Plan 2. Test Consistency 2. Test Consistency 3. Schedule Plan 3. Reformulate Plan 4. Execute Plan 4. Dynamically Schedule Plan 7

79 79

80 You don t need all those edges!

81 You don t need all those edges! Equivalent minimal dispatchable network 1

82 You don t need all those edges! Let s consider a specific triangle of edges

83 You don t need all those edges! 10 1 Let s consider a specific triangle of edges. Do we really need the bottom edge?

84 You don t need all those edges! 10 1 Let s consider a different triangle of edges. Do we really need the bottom edge?

85 A ropes analogy Imagine a unidirectional rope of length 10 constraining sliders on a track. 5

86 Rope analogy Imagine a unidirectional rope of length 10 constraining sliders on a track. 6

87 Rope analogy Now add in ropes for other constraints 7

88 Rope analogy Imagine pulling A and D as tightly as possible

89 Rope analogy Can we remove rope AD without changing behavior? Yes! Same possible positions for A, B, D. 8

90 Rope analogy Can we remove ropes AB, BD without changing behavior? No. AD still constrained, but B could slide freely! Not the same behavior. Collectively, AB and BD entail AD (but AD does not entail both AB and AD). 0

91 Upper dominating edges - detection from APSP d BC If d AC, d BC 0 and d AB + d BC =d AC then BC dominates AC d AC (Proof omitted - based on triangle rule property of APSP. Please see notes / reading for more info) 1

92 Lower dominating edges - detection from APSP d AB If d AB, d AC <0and d AB + d BC =d AC then AB dominates AC d AC (Proof omitted - based on triangle rule property of APSP. Please see notes / reading for more info) 2

93 Dominance example

94 Dominance example Upper dominated! 4

95 Dominance example Upper dominated! 5

96 Dominance example Upper dominated!

97 Dominance example Upper dominated!

98 Dominance example Upper dominated! 98

99 Dominance example Upper dominated! 99

100 Dominance example Lower dominated!

101 Dominance example Lower dominated!

102 Dominance example Lower dominated! 2

103 Dominance example Lower dominated! 3

104 Dominance example Lower dominated! 4

105 Dominance example Lower dominated! 5

106 Dominance example

107 10 1 Dominance example Original APSP distance graph now in minimal dispatchable form!

108 Dominance example [0,10] [1, 1] Original STN [0,10] [2, 2] 1... now in minimal dispatchable form!

109 FilteringAlgorithm(G) Input: A dispatchable APSP-graph G Output: A minimal dispatchable graph 1 for each pair of intersecting edges in G 2 if both dominate each other 3 if neither is marked 4 arbitrarily mark one for elimination 5 end if 6 else if one dominates the other 7 mark dominated edge for elimination 8 end if 9 end for 10 remove all marked edges from graph 11 return G 09

110 Problem: Avoiding Intermediate Graph Explosion All pairs shortest path table computation consumed O(n 2 ) space Only used as an intermediate - not needed after minimal dispatchable graph obtained. Solution: Interleave process of APSP construction with edge elimination. Never have to build whole APSP graph. [Tsarmardinos 1998] 0

111 Recap Recap To schedule online, times must monotonically increase - use enablement conditions Running online allows greater flexibility to fluctuations However, propagation costs can be large for large graphs Can reduce edges by using domination to make graph smaller 1

112 To Execute a Temporal Plan STN D Graph 1. Describe Temporal Plan A [1,10] [0,9] B C [1,1] [2,2] D A 10 B C D 2. Test Consistency Detect negative loops (SSSP). 3. Schedule Plan APSP + Decomposition. 4. Execute Plan 2 [Dechter, Meiri, Pearl 91]

113 To Execute a Temporal Plan 1. Describe Temporal Plan 2. Test Consistency 3. Schedule Plan 4. Execute Plan Problem: delays and fluctuations in task duration can cause plan failure. Observation: temporal constraints leave room to adapt. Flexible Execution adapts through dynamic scheduling: Assign time to event when executed. Guarantee that all constraints will be satisfied. Schedule with low latency through pre-compilation. [Muscettola, Morris, Tsmardinos KR98] 3

114 To Execute a Temporal Plan 1. Describe Temporal Plan 1. Describe Temporal Plan 2. Test Consistency 2. Test Consistency 3. Schedule Plan 3. Reformulate Plan ~ Decomposable STN 4. Execute Plan 4. Dynamically Execute Plan 4 How do we schedule on line?

115 Outline: To Execute a Temporal Plan 1. Describe Temporal Plan [1,10] B [1,1] A [0,9] C [2,2] D 2. Test Consistency 3. Reformulate Plan 4. Dynamically Execute Plan 5 [0,9] B [1,1] A [1,1] D [0,9] C t=3 t=0 B [1,1] t=4 A [1,1] D C t=2

116 MIT OpenCourseWare J / 6.834J Cognitive Robotics Spring 2016 For information about citing these materials or our Terms of Use, visit: