Download Hybrid representation - University of Regina Computer Science

8. Temporal Reasoning
8. Temporal Reasoning
• Motivations
• Applications
• Time Representation
• Hybrid representation : the model TemPro
Malek Mouhoub, CS 820 Fall 2005
1
Motivations
Motivations
• Most real-world problems have spatial and|or temporal setting.
• Traditional mathematical formulations often not useful for
automated reasoning
– continous rather than discrete
– require complete, precise information
• Qualitative and logical representations offer computational
advantages
Malek Mouhoub, CS 820 Fall 2005
2
Applications
Applications
• Natural Language Processing
– Joe will have finished before dinner
• Temporal Databases
– When was Alfredo last seen by Dr Schweitzer ?
– How often has Jane been hospitalized ?
• Robotics, GIS, computer games
– How long until I reach X.
– I must reach X while R2D2 is active.
• Concurrent Programming
– when can process P1 add an element to the buffer ?
– Can P1 and P2 write to File F at the same time ?
Malek Mouhoub, CS 820 Fall 2005
3
Applications (cont)
Applications (cont)
• Reactive Systems
– Is it possible for the system to reach Deadlock ?
– Could one process exclude another process from ever performing its next operation ?
• Real-time Systems
– Will the velocity calculation always meet its deadline ?
– What is the maximum data rate that can be handled ?
– It is possible to schedule a list of tasks within a given deadline ?
• Scheduling and Planning
– Can task 6 and 9 be done in any order ?
– What needs to be done before doing task 12 ?
• Archaelogy
– Create a chronology for objects found in egyptian graves.
• Molecular Biology
Malek Mouhoub, CS 820 Fall 2005
4
Applications (cont)
Temporal Reasoning
Time Representation
Symbolic
Hybrid
Reified Logic
Point Interval
Algebra Algebra
Malek Mouhoub, CS 820 Fall 2005
Numeric
Resolution Techniques
Operations
Research
Domains
Arithmetic
Equations of Variables
and Inequations Values
C.S.P
Decomposition
Local
+
Consistency
Backtrack
+
Search
Backtrack
Search
5
Time representation
Time representation
• Symbolic representation
• Numeric representation
• Hybrid representation
Malek Mouhoub, CS 820 Fall 2005
6
Time representation
Symbolic representation of time
Predicate Calculus :
–
Meet(John,Laurie,April12)
–
Problem : time is represented as any other argument.
Modal Logic :
–
An extension of predicate calculus which includes notation for arguing about “when” statements are true.
–
A logic with a notion of time included. The formulas can express facts about past, present, and future states.
–
Three operators :
∗ °F : F is true at the next time instant.
∗ 2F : F is true from now.
∗ 3F : F is eventually true.
–
Two connectives :
∗ x ∪ y :x is true until y is true.
∗ x P y : x precedes y.
–
Linear time (considers only one possible future) and branching time (several alternative futures). In branching temporal
logic we have 2 extra operators :
∗ A (for ”all futures”),
∗ and E (for ”some future”).
∗ Examples :
· A(work U go home) means : I will work until I go home.
· E (work U go home) means : I may work until I go home.
Malek Mouhoub, CS 820 Fall 2005
7
Time representation
Symbolic representation of time
Reified Logic :
– TRUE(preposition,temporal qualification) or <
φ, t >
– Allen Algebra (Interval Algebra)
– Point Algebra
Malek Mouhoub, CS 820 Fall 2005
8
Time representation
Symbolic representation of time
Allen Algebra :
– Event : Couple (p.I) where p is a proposition and I the interval where p is
true.
– Given two events ev1
= (p1 , I1 ), ev2 = (p2 , I2 ), the qualitative
relation between ev1 and ev2 can be represented as follows :
R(ev1 , ev2 ) = I1 r1 ∨ · · · ∨ rn I2
The ri ’s are basic Allen relations.
Point Algebra :
– Think in terms of time points instead of intervals.
– Basic relations :
Malek Mouhoub, CS 820 Fall 2005
<, >, =
9
Time representation
Relation
Symbol Inverse
X precedes Y
P
P-
X equals Y
E
E
X meets Y
M
M-
X overlaps Y
O
O-
X during Y
D
D-
X starts Y
S
S-
X finishes Y
F
F-
Meaning
X
Y
X
Y
X
Y
X
Y
X
X
Y
Y
Y
X
Figure 1: Allen Primitives.
Malek Mouhoub, CS 820 Fall 2005
10
Time representation
Symbolic representation of time
Nonlinear Time :
– Most temporal models (points or interval) represent Linear Time :
∗ For all t1 , t2 : t1 < t2 or t1 > t2 or t1 = t2
– For some applications Partially Ordered Time Models is more
appropriate :
∗ Two time points can be Unrelated : t1 ||t2
∗ For interval models, number of relations increases from 13 (in
linear time) to 29.
Malek Mouhoub, CS 820 Fall 2005
11
Time representation
Nonlinear Time
Applications :
• Events in a distributed system
• Military intelligence information gathered by unsynchronized agents
• Relative time : astronomical events
Malek Mouhoub, CS 820 Fall 2005
12
Time representation
Nonlinear Time
Partially Followed
Partially Starts
Partially Precedes
Unrelated
Figure 2: Partially Ordered Temporal Interval Relations
Malek Mouhoub, CS 820 Fall 2005
13
Time representation
Numeric representation of time
Arithmetic Inequations :
A−B ≥d
Temporal Windows :
[begintime, endtime, step, duration]
Begintime
SOPO
Endtime
Tr
Ui
Interval
Malek Mouhoub, CS 820 Fall 2005
Step
14
Time representation
Temporal Constraint Networks
• Nodes can represent points. Arcs represent relation between
points (before,after,equal).
• Nodes can be intervals. Arcs are temporal interval relations.
• Nodes can represent either poins or intervals. Arcs are point,
interval, point-interval or interval-point relations.
• Relations can be from any appropriate model (linear/non-linear,
discrete/dense, . . . etc).
Malek Mouhoub, CS 820 Fall 2005
15
Time representation
Qualitative Network
Fred was reading the paper while eating his breakfast. He put the paper down and
drank the last of his coffee. After breakfast he went for a walk.
Breakfast
P
EDD-OO-SS-FFD
Paper
DOS
Coffee
Walk
Paper
Coffee
Breakfast
Walk
Solution
Initial problem
Figure 3: Interval Algebra Constraint Network
Malek Mouhoub, CS 820 Fall 2005
16
Time representation
Quantitative Network : the TCSP model
John goes to work either by car (30-40 minutes), or by bus (at least 60 minutes). Fred goes to
work either by car (20-30 minutes), or in a carpool (40-50 minutes) Today John left home between
7:10 and 7:20, and Fred arrived at work between 8:00 and 8:10. We also know that John arrived
at work about 10-20 minutes after Fred left home. We wish to answer queries such as: “Is the
information in the story consistent?”, “Is it possible that John took the bus, and Fred used the
carpool?”, “What are the possible times at which Fred left home?”, and so on.
[30,40]
[60, ∝ ]
1
[10,20]
2
[10,20]
0
3
[20,30]
[40,50]
4
Nodes:
0 = start time of the story
1 = John left home
2 = John arrived at work
3 = Fred left home
4 = Fred arrived at work
[60,70]
Malek Mouhoub, CS 820 Fall 2005
17
Time representation
Quantitative Network : the Disjunctive Temporal Problem (DTP)
A DTP is defined by a pair (X, C), where :
• each xi ∈ X is a time point,
• each ci (∈ C) = ci1 ∨ ci2 . . . cin ,
• and each cij = x − y ≤ b where x, y ∈ X and b ∈ R.
Example
C1 : c11 : a − b ≤ 10
C2 : c21 : b − a ≤ −15 ∨ c22 : c − a ≤ −25
C3 : c31 : b − c ≤ 10
C4 : c41 : a − c ≤ 20
Malek Mouhoub, CS 820 Fall 2005
18
Time representation
Hybrid Network : Meiri’s TCSP network
John and Fred work for a company that has local and main offices in Los Angeles, They usually
work at the local office, in which case it takes John less than 20 minutes and Fred 15-20 minutes
to get to work. Twice a week John works at the main office, in which case his commute to work
takes at least 60 minutes. Today John left home between 7:05-7:10 a.m., and Fred arrived at
work between 7:50-7:55 a.m. We also know that Fred and John met at a traffic light on their way
to work.
P
2
P
3
F
S
J
{(0,20),(60,+00)}
S Si D Di F Fi O Oi E
S
P
1
Malek Mouhoub, CS 820 Fall 2005
{(15,20)}
F
F
{(0,5)}
P
0
{(50,55)}
P
4
Nodes:
Time Intervals
J = John is going to work
F = Fred is going to work
Time Points
P0 = Begining story
P1 = John leaves home
P2 = John arrives at work
P3 = Fred leaves home
P4 = Fred arrivesat work
19
Hybrid representation of time
Hybrid representation of time
1. John, Mary and Wendy separately rode to the soccer game.
2. It takes John 30 minutes, Mary 20 minutes and Wendy 50 minutes to get to the soccer
game.
3. John either started or arrived just as Mary started.
4. John left home between 7:00 and 7:10.
5. Mary arrived at work between 7:55 and 8:00.
6. Wendy left home between 7:00 and 7:10.
7. John’s trip overlapped the soccer game.
8. Mary’s trip took place during the game or else the game took place during her trip.
9. The soccer game starts at 7:30 and lasts 105 minutes.
10. John either started or arrived just as Wendy started.
11. Mary and Wendy arrived together but started at different times.
Malek Mouhoub, CS 820 Fall 2005
20
Hybrid representation of time
Hybrid representation : the model TemPro
Problem involving
numeric and symbolic
−→
CSP
time information
using :
• the interval algebra of Allen
• and a discrete representation of time
Malek Mouhoub, CS 820 Fall 2005
21
Hybrid representation of time
Hybrid representation : the model TemPro
A Temporal Constraint Satisfaction Problem involves :
• N = {EV1 , . . . , EVn }.
• D = {D1 , . . . , Dn }.
• R.
Solving a temporal constraint satisfaction problem consists in
finding all sets of values {occ1j1 , . . . ,
occnjn } for
{EV1 , . . . , EVn } satisfying all relations belonging to R.
Malek Mouhoub, CS 820 Fall 2005
22
Hybrid representation : the model TemPro
Hybrid representation : the model TemPro
Basic Notions : Time line, Interval.
Event : couple (p, I) where p is a proposition and I the interval under
which p holds.
Quantitative Constraints : SOPO, defined by the fourfold :
[begintime, endtime, duration, step]
SOPO
Time line
Interval
Step
Qualitative Constraints : disjunctions of Allen primitives.
Malek Mouhoub, CS 820 Fall 2005
23
Hybrid representation : the model TemPro
John
Mary
Wendy
Soccer
Figure 4: John, Mary and Wendy separately rode to the soccer game.
Malek Mouhoub, CS 820 Fall 2005
24
Hybrid representation : the model TemPro
[? , ?, 30 , 1]
John
[? , ? , 50 , 1]
[? , ? , 20 , 1]
Mary
Wendy
Soccer
Figure 5: It takes John 30 minutes, Mary 20 minutes and Wendy 50
minutes to get to the soccer game.
Malek Mouhoub, CS 820 Fall 2005
25
Hybrid representation : the model TemPro
[? , ?, 30 , 1]
E S S- M
John
[? , ? , 50 , 1]
[? , ? , 20 , 1]
Mary
Wendy
Soccer
Figure 6: John either started or arrived just as Mary started.
Malek Mouhoub, CS 820 Fall 2005
26
Hybrid representation : the model TemPro
{(0 30) .. (10 40)}
E S S- M
John
{(0 50) .. (10 60)}
{(35 55) .. (40 60)}
Mary
Wendy
Soccer
Figure 7:
John left home between 7:00 and 7:10. Mary arrived at work between
7:55 and 8:00. Wendy left home between 7:00 and 7:10.
Malek Mouhoub, CS 820 Fall 2005
27
Hybrid representation : the model TemPro
{(0 30) .. (10 40)}
E S S- M
John
{(0 50) .. (10 60)}
{(35 55) .. (40 60)}
Mary
Wendy
O
Soccer
Figure 8: John’s trip overlapped the soccer game.
Malek Mouhoub, CS 820 Fall 2005
28
Hybrid representation : the model TemPro
{(0 30) .. (10 40)}
E S S- M
John
{(0 50) .. (10 60)}
{(35 55) .. (40 60)}
Mary
Wendy
O
D DSoccer
Figure 9: Mary’s trip took place during the game or else the game
took place during her trip.
Malek Mouhoub, CS 820 Fall 2005
29
Hybrid representation : the model TemPro
{(0 30) .. (10 40)}
E S S- M
John
{(0 50) .. (10 60)}
{(35 55) .. (40 60)}
Mary
Wendy
O
D DSoccer
{(30 135)}
Figure 10: The soccer game starts at 7:30 and lasts 105 minutes.
Malek Mouhoub, CS 820 Fall 2005
30
Hybrid representation : the model TemPro
{(0 30) .. (10 40)}
E S S- M
John
ESS-M
{(0 50) .. (10 60)}
{(35 55) .. (40 60)}
Mary
Wendy
O
D DSoccer
{(30 135)}
Figure 11: John either started or arrived just as Wendy started.
Malek Mouhoub, CS 820 Fall 2005
31
Hybrid representation : the model TemPro
{(0 30) .. (10 40)}
John
E S S- M
{(35 55) .. (40 60)}
Mary
ESS-M
{(0 50) .. (10 60)}
FF-
Wendy
O
D DSoccer
{(30 135)}
Figure 12: Mary and Wendy arrived together but started at different
times.
Malek Mouhoub, CS 820 Fall 2005
32
Hybrid representation : the model TemPro
Hybrid representation : the model TemPro
Example :
3 tasks T1 , T2 and T3 are processed by a mono processor
machine M. A task T4 must be processed before T1 and T2 .
T1 : 3h,10:00,15:00.
T2 : 3h,20:00,24:00.
T3 : 4h,7:00,12:00.
T4 : 1h,9:00,11:00.
Malek Mouhoub, CS 820 Fall 2005
33
Hybrid representation : the model TemPro
[20,24,1,3]={(20 23),(21 24)}
[7,12,1,4]={(7 11),(8 12)}
P v PT
T
2
P v P-
3
P v P-
I
P-
PT1
T4
[10,15,1,3]={(10 13),(11 14),(12 15)} [9,11,1,1]={(9 10),(10 11)}
I : The universal relation(disjunction of the 13 basic Allen relations).
P : Precedes, P- : precedes inverse.
Figure 13: A qualitative and quantitative constraint graph.
Malek Mouhoub, CS 820 Fall 2005
34
Hybrid representation : the model TemPro
Hybrid representation : the model TemPro
Resolution method based on constraint propagation :
• Numeric → Symbolic Conversion.
• Perform Local Consistency Algorithms.
– Arc and Path Consistencies.
• Backtrack Search.
Malek Mouhoub, CS 820 Fall 2005
35
Hybrid representation : the model TemPro
{(0 30) .. (10 40)}
John
E S S- M
{(35 55) .. (40 60)}
Mary
E S S-M
{(0 50) .. (10 60)}
F F-
Wendy
O
D DSoccer
{(30 135)}
Figure 14: Numeric → Symbolic Conversion.
Malek Mouhoub, CS 820 Fall 2005
36
Hybrid representation : the model TemPro
{(0 30)..(4 34),(5 35) .. (10 40)}
John
M
{(35 55) .. (40 60)}
Mary
S
F
{(0 50) .. (4 54),(5 55) .. (10 60)}
Wendy
O
D
Soccer
{(30 135)}
Figure 15: Arc Consistency.
Malek Mouhoub, CS 820 Fall 2005
37
Hybrid representation : the model TemPro
{(5 35) .. (10 40)}
John
M
{(35 55) .. (40 60)}
Mary
S
{(5 55) .. (10 60)}
F
Wendy
O
D
O
Soccer
{(30 135)}
Figure 16: Path Consistency.
Malek Mouhoub, CS 820 Fall 2005
38
Hybrid representation : the model TemPro
GAs for TemPro
Basic notions :
• Individual and random Individual.
• Population.
• Fitness (evaluation) function.
• Mutation.
• Crossover.
Malek Mouhoub, CS 820 Fall 2005
39
Hybrid representation : the model TemPro
{(5 35), (6 36) .. (10 40)}
John
M
S
{(35 55) .. (39 59) ,(40 60)}
Mary
{(5 55) .. (10 60) }
F
Wendy
O
D
O
Soccer
{ (30 135) }
random
individual
Population
Figure 17: GA representation of TCSPs.
Malek Mouhoub, CS 820 Fall 2005
40
Hybrid representation : the model TemPro
{(5 35), (6 36) .. (10 40)}
conflict
conflict
John
M
S
{(35 55) .. (39 59) ,(40 60)}
Mary
{(5 55) .. (10 60) }
F
Wendy
O
D
O
conflict
Soccer
{ (30 135) }
random
individual
(fitness=3)
Population
Figure 18: GA representation of TCSPs.
Malek Mouhoub, CS 820 Fall 2005
41
Hybrid representation : the model TemPro
Fitness = 3
(6 36) (39 59) (30 135) (10 60)
Fitness = 4
(6 36) (40 60) (30 135) (10 60)
Figure 19: Mutation Operator.
Malek Mouhoub, CS 820 Fall 2005
42
Hybrid representation : the model TemPro
Fitness = 3
Fitness = 3
(6 36) (39 59) (30 135) (10 60)
(5 35) (40 60) (30 135) (9 59)
(6 36) (40 60) (30 135) (10 60)
Fitness = 4
Figure 20: Crossover Operator.
Malek Mouhoub, CS 820 Fall 2005
43
Hybrid representation : the model TemPro
1.
2.
3.
4.
begin
t←1
// P (t) denotes the population containing the current solution
eval ← evaluate P (t)
5.
6.
7.
8.
9.
while termination condition is not satisfied do
t←t+1
select P (t) from P (t − 1)
alter P (t)
evaluate P (t)
10.
endwhile
11.
if solution found then
12.
return P(t)
13. end
Malek Mouhoub, CS 820 Fall 2005
44