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Temporal Constraint Management in Artificial Intelligence
- Introduction: time & temporal constraints
- The problem
- Survey of AI approaches to temporal constraints
Paolo TerenzianiDipartimento di Informatica
Universita’ del Piemonte Orientale, Alessandria, Italy
TEMPORAL CONSTRAINT
Introduction (1/3)
Time has a “peculiar” semantics, so that it deserves a specific attention
The world evolves in time: time is an intrinsic part of human way of approaching reality
Time has to be taken into account in each approach modeling (evolving) parts of the world
Introduction (2/3)
Many different approaches in the literature, e.g.,- simulation-based approaches (Petri Nets, Markov Models,
Workflows, ...)- ….- logical approaches (dynamic l., temporal l., nonmonotonic l.,
semantic nets, ….)
A MAIN DISTINCTION:“general purpose”: modeling both (part of) the world and its temporal phenomena
+ generality, homogeneous framework to deal with phenomena- computationally not efficient
VS.“specialised”: dealing only with some temporal phenomena
- generality+ computationally efficient
Introduction (3/3)“Specialised” approaches
IDEA: modularity:Building efficient solutions to well-defined parts of the whole problem
IN AI:“Knowledge Servers” [Brachman & Levesque] to be paired with other systems/problem solvers
Trade-off between expressiveness and computational complexity of (correct & complete) inferential mechanisms
NOTICE: general (not ad-hoc) solutions to a slice of temporal phenomena
Temporal Constraint Managers: the Problem (1/5)
Temporal Constraint (TC): a part of the problem that can be isolatede.g., A before B & B before C A before CREGARDLESS of the description of the events A, B, C
(1) Which constraints (representation language)?
(2) Which inferences?
Trade-off!!!
Temporal Constraint Managers: the Problem (2/5)Digression
Intended vs. supported SEMANTICS
Temporal Constraints without Temporal Reasoning (constraint propagation)- are useless- clash with users’ intuitions/expectations
Temporal Constraint Managers: the Problem (3/5)
Implied constraint (temporal reasoning):(1.6) C ends between 30 and 60 m after the start of A
(1.1) the end of A is equal to the start of B(1.2) the end of B is equal to the start of C(1.3) the duration of A is between 10 and 20 m(1.4) the duration of B is between 10 and 20 m(1.5) the duration of C is between 10 and 20 m
A B C10-20 10-2010-20
Correct (consistent) assertion:(1.7) C ends between 30 and 50 m after the start of A
Not correct (inconsistent) assertion:(1.8) C ends more than 70 m. after the start of A
However: Temporal Reasoning is NEEDED in order to support such an intended semantics!
Temporal Constraint Managers: the Problem (4/5)
DESIDERATA for Temporal Reasoning Algorithms
- tractability “reasonable” response time (important for Knowledge servers!)
- correctness no wrong inferences
- completeness reliable answers
Temporal Constraint Managers: the Problem (5/5)
Implied constraint (temporal reasoning):(1.6) C ends between 30 and 60 m after the start of A
(1.1) the end of A is equal to the start of B(1.2) the end of B is equal to the start of C(1.3) the duration of A is between 10 and 20 m(1.4) the duration of B is between 10 and 20 m(1.5) the duration of C is between 10 and 20 m
A B C10-20 10-2010-20
Suppose that temporal reasoning is NOT complete, so that (1.6) is not inferredThe answer to query (Q1) might be: YES(Q1) Is it possible that C ends more than 70 m. after the start of A?
Complete Temporal Reasoning is NEEDED in order to grant correct answers to queries!
Survey (1/18)Types of temporal entities
- Time Points
- Time Intervals
- Sets of Time Points/Intervals (repeated/periodic events)
Survey (2/18)Types of temporal constraints (1/4)
- Qualitative: relative positions of entities (e.g., A during B)
- Quantitative: metric time - dates (A on 1/1/2003 from 9:00 to 11:33)- duration (A lasted between 3 and 4 hours)- delays (B started between 5 and 10 minutes after A)
- Periodicity/repetition -based (qualitative and/or quantitative)
Survey (3/18)Types of temporal constraints (2/4)
QUALITATIVE CONSTRAINTS on TIME POINTS
Point Algebra [Vilain & Kautz, 87]
- base relations: <, =, >
- composite relations: (<,=), (<,>), (=,>), (<,=,>)
Notice: P1(r1,r2,…rk)P2 means r1(P1,P2) r2(P1,P2) … rk(P1,P2)
Continuous Pointizable Algebra [Vilain, Kautz, VanBeek]
- base relations: <, =, >
- composite relations: (<,=), (=,>), (<,=,>)
Survey (4/18)Types of temporal constraints (3/4)
QUALITATIVE CONSTRAINTS on TIME INTERVALS
Interval Algebra [Allen, 83]
- 13 base relations, 213 relations
I before J (J after I)
I meets J (J met-by I)
I overlaps J (J overlapped-by I)I finished-by J (J finishes I)
I equal J
I contains J (J during I)
I started-by J (J starts I)
Survey (5/18)Types of temporal constraints (4/4)
CONSTRAINTS on SETS OF INTERVALS(repeated/periodic events)
Periodicity-dependent durations [Loganantharaj & Gimbrone, 95]e.g. On Mondays John goes to work in 40-45 minutes
On Tuesdays John goes to work in 30-55 minutes
“Absolute” qualitative constraints on repeated events [Morris et al., 93]
e.g. Meetings always precede Lunches
Periodicity-dependent qualitative constraints on repeated events [Terenziani, 95]
e.g. From 10/1/2003 to 31/3/2003, twice each Monday, two units of Math precede one unit of Physics
>>>> QUANTITATIVE CONSTRAINTS: see below
Survey (6/18)Temporal Reasoning (1/5)
Mostly: PATH-CONSISTENCY-based TR
C3NEW C3OLD (C1 @ C2)
Different instantiations, depending on the types of constraints (and on the definitions of intersection and composition)
I J K
C3?
C2C1
Survey (7/18)Temporal Reasoning (2/5)
E.g., path-consistency on quantitative constraints between time points (STP framework [Dechter et al., 91])
I
J
K
[0,10]
H
[10,20]
[10,30]
[10,20]
[10,10]
Survey (7/18)Temporal Reasoning (2/5)
E.g., path-consistency on quantitative constraints between time points (STP framework [Dechter et al., 91])
I
J
K
[0,10]
H
[10,20]
[10,20]
[10,20]
[10,10]
IHNEW= [10,30] ([0,10][10,10]) = [10,20]
Survey (8/18)Temporal Reasoning (3/5)
STP (Simple Temporal Problem) framework [Dechter et al., 91])
Conjunction of Bounds on Difference (b.o.d.) constraints
0 J-I 1010 H-I 30
10 K-I 20
10 H-K 2010 H-J 10
- < K-J < +
I
I
J
J K
K
H
H
0
0
0
0
10
0
-10 20+
-10 -10-10
10
3020
+
i j[c,d]
i jd
-c c j-i d
Survey (9/18)Temporal Reasoning (4/5)
All-to-all shortest path algorithm [Floyd-Warshall]
For k:=1 to N doFor i:=1 to N do
For j:=1 to N doM[i,j]=Min(M[i,j],M[i,k]+M[k,j])
Property: Consistent iff no negative cycle
Complexity: O(N3)
Property: Correct & complete for b.o.d.
Survey (10/18)Temporal Reasoning (5/5)
I
J
K
[10,10]
H
[10,10]
[20,20]
[10,10]
[10,10]
[0,0]
Minimal Network (shortest path between each pair of nodes)
I
I
J
J K
K
H
H
0
0
0
0
10
-10
-10 100
-10 -10-20
10
2010
0
Survey (11/18)Approaches & Complexity (1/5)
QUALITATIVE CONSTRAINTS
Continuous Pointizable Algebra [Vilain, Kautz, VanBeek, 89]O(N3)
Point Algebra [Vilain & Kautz, 87]O(N4)
Interval Algebra [Allen, 83]ExponentialMaximal tractable fragments [Nebel & Buckert, 95], [Drakengren & Jonsson, 97]
Survey (12/18)Approaches & Complexity (2/5)
QUANTITATIVE CONSTRAINTS
STP [Dechter et al., 91]O(N3)
TCSP [Dechter et al., 91]Exponential (many optimizations)
I J[10,20][30,35]
Survey (13/18)Approaches & Complexity (3/5)
QUALITATIVE+QUANTITATIVE CONSTRAINTS
[Vilain & Kautz, 91]Combining two TRsDoes the exchange of constraints between TRs end?
[Meiri, 91] “two sorted” formalism + mapping operators
[Brusoni, Terenziani et al., 95]mapping onto STP
Survey (14/18)Approaches & Complexity (4/5)
STP (and TCSP) and QUALITATIVE CONSTRAINTS
STP (and TCSP) can also represent (a subset of) qualitative constraints
Continuous Poitizable relationse.g., P1<P2 0<P2-P1
Some Interval Algebra relatione.g., I (started-by,contains, finished-by,equal) J 0 Start(J)-Start(I) 0 < End(I)-End(J)
BUT NOT ALL RELATIONSe.g., P1(<,>)P2 0 < P1-P2 0 < P2-P1 (in TCST but not in STP)e.g., I (before,after) J 0 < End(I)-Start(J) 0 < End(J)-Start(I) (neither in STP nor in TCSP)
Survey (15/18)Approaches & Complexity (5/5)
SURVEY NOT EXHAUSTIVE !!!
E.g., relative duration
E.g., “A lasted more than B”
[Pujary & Sattar, 99]
[Jonsson & Backstrom, 98] homogeneous approach based on linear programming
Survey (16/18)TRs & Applications
MANY TRs (knowledge servers) in AI
TMM [Dean & McDermott, 87] Timelogic [Koomen, 89] MATS [Kautz & Ladkin, 91] Timegraph Gerevini & Schubert, 95] ….. Later [Brusoni, Terenziani et al., 95]
Comparison of several systems in [Allen & Yampratoom, 93]
Survey (17/18)TRs & Applications
MANY APPLICATIONS
Scheduling Planning Natural Language Understanding Diagnosis….. Multimedia Presentations Clinical Guidelines
Survey (18/18)TRs & Applications
REFERENCES TO SURVEYS
M. Vilain, H. Kautz, and P. VanBeek. "Constraint Propagation Algorithms for temporal reasoning: a Revised Report", D.S. Weld, J. deKleer, eds., Readings in Qualitative Reasoning about Physical Systems. Morgan Kaufmann, 373-381, 1990.
J. Allen, “Time and Time Again: The Many Ways to Represent Time”, Int’l Journal of Intelligent Systems 6(4), 341-355, 1991.
E. Yampratoom, J. Allen, “Performance of Temporal reasoning Systems”, Sigart Bull. 4(3), 26-29, 1993.
L. Vila. 1994, "A Survey on Temporal Reasoning in Artificial Intelligence", AI Communications 7(1):4-28, 1994.
….. P. Terenziani, “Reasoning about time”, Encyclopedia of
Cognitive Science, Macmillan Reference Ltd, Vo.3, 869-874, 2003.
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