Temporal Constraint Networks

Transcription

1 Ch. 5b p.1/24 Temporal Constraint Networks Addition to Chapter 5

2 Ch. 5b p.2/24 Outline Qualitative networks The interval algebra The point algebra Quantitative temporal networks Reference: Chapter 12 of the book titled "Constraint Processing," by Rina Dechter, Morgan Kaufmann Publishers (Elsevier Science), 2003.

3 Ch. 5b p.3/24 Temporal Reasoning Given temporal information I went to lunch before class Please come before or after lunch Lunch starts at 12:00 Lunch takes half an hour to an hour Derive answers to queries Is it possible that a proposition P holds at time t 1? What are the possible times at which a proposition P holds? What are the possible temporal relationships between two propositions P and Q?

4 Ch. 5b p.4/24 Example John was not in the room when I touched the switch to turn on the light, but John was in the room later when the light went out. Events: Switch: time of touching the switch Light: time the light was on Room: time that John was in the room Is this information consistent? If it is consistent what are the possible scenarios?

5 Ch. 5b p.5/24 Interval algebra (IA) Relation Symbol Inverse Example X before b bi X X equal = = X X meets m mi X X overlaps o oi X X during d di X X starts s si X X finishes f fi X

6 Ch. 5b p.6/24 Representation I r 1 rk J represents si di I r 1 J f i oi I r k J For example I s d f o J expresses the fact that intervals I and J intersect (it exludes b bi m mi). John was not in the room when I touched the switch to turn on the light, but John was in the room later when the light went out. 1. Switch o 2. Switch b 3. Light o m m s d Light mi a Room Room

7 Ch. 5b p.7/24 IA Constraint graph (network) Light Switch {o, m} {o, s, d} Light Switch {b,m,mi,a} Room Room A solution is an assignment of a pair of numbers to each variable such that no constraint is violated

8 Ch. 5b p.8/24 Constraint graph terms In a constraint graph, nodes represent variables and an edge represents a direct constraint (coming from the IA relation set) A universal constraint permits all relationships between two variables and is represented by the lack of an edge between the variables. A constraint C can be tighter than constraint C, denoted by C C, yielding a partial order between IA networks. A network N is tighter than network N if the partial order is satisfied for all the corresponding constraints. The minimal network of M is the unique equivalent network of M which is minimal with respect to.

9 Ch. 5b p.9/24 Minimal network Light Light {o, m} {o, s, d} {o, m} {o, s} Switch Room Switch Room {b,m, mi, a} {b,m}

10 Ch. 5b p.10/24 Example 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 Breakfast (eq, d, di, o, oi, s, si, f, fi} {d} {bi} (d, o, s} {d} {bi} Paper Walk Paper {b} Walk {d, o, s} {d, o, s} {b} Coffee Coffee Paper Coffee Breakfast Walk

11 Ch. 5b p.11/24 Path Consistency in CSPs x j ai xk Given a constraint network R X C, a two-variable set x i is path-consistent relative to variable x k iff for every consistent assignment x i j x a j there is a value a k ai D k such that the assignment x i a k is consistent and is consistent. x k ak D x j a j Alternatively, a binary constraint R i j is path-consistent relative to x k iff for every pair a i a j are from their respective domains, there is a value a k such that R ik and R k j. a i ak a k a j R i j where a i and a j D k

12 Ch. 5b p.12/24 Path-consistency in CSPs (cont d) A subnetwork over three variables path-consistent iff for any permutation of path-consistent relative to x k. x i x j x k i j is k, R i j is A network is path-consistent iff for every R i j (including universal binary relations) and for every k i j, R i j is path-consistent relative to x k.

13 Ch. 5b p.13/24 Path-consistency in IA An IA network is path-consistent if for every three variables x i j x x k, C i j C ik C k j. The intersection of two IA relations R and R, denoted by R R, is the set-theoretic intersection R R. The composition of two IA relations, R R, R between intervals I and K and R between intervals K and J, is a new relation between intervals I and J, induced by R and R as follows.

14 Ch. 5b p.14/24 Composition ( ) The composition of two basic relations r and r is defined by a transitivity table (see a portion of it on the next slide). The composition of two composite relations R and R, denoted by R basic relations: R, is the composition of the constituent R R r r r R r R

15 Ch. 5b p.15/24 Composition of basic relations b s d o m b b b b o m d s b b s b s d b o m b d b d d b o m d s b o b o o d s b o m b m b m o d s b b

16 Ch. 5b p.16/24 Qualitative Path Consistency (QPC) Algorithm X before, before Z X before Z X Z X before, during Z X {b, o, m, d, s} Z X X before overlaps Z Z

17 Ch. 5b p.17/24 Qualitative Path Consistency (QPC) Algorithm function QPC-1 (T ) returns a path consistent IA network input: T, an IA network with n variables repeat S for k T for i,j C i j until S = T return T 1 to n do 1 to n do C i j C ik C k j

18 Ch. 5b p.18/24 Example Apply CSR CSR CSR b CSR b CSR b m m i i a a CSL CLR b o m o m o s s d m o o o s o d m o m s m d b o o b m o o d d m s s d

19 Ch. 5b p.19/24 Minimizing networks using path-consistency In some cases, path-consistency algorithms are exact they are guaranteed to generate the minimal network and therefore decide consistency. In general, IA networks are NP-complete, backtracking search is needed to generate a solution. Even when the minimal network is available, it is not guaranteed to be globally consistent to allow backtrack-free search. Path-consistency can be used for forward checking.

20 Ch. 5b p.20/24 The point algebra (PA) It is a model alternative to IA. It is less expressive: there are three basic types of constraints between points P and Q: P Q, P Q, Q. P Reasoning tasks over PAs are polynomial

21 Ch. 5b p.21/24 Example Fred put the paper down and drank the last of his coffee. Paper < < > Coffee < Paper Coffee Paper + < Coffee +

22 z t Ch. 5b p.22/24 Examples I s d f represented with J where I x y and J z t can be x y x t x z y t y z However, I b a J where I represented with a PA network x y and J z t cannot be

23 Ch. 5b p.23/24 Composition in the PA < = > < < <? = < = > >? = >? expresses the universal relation.

24 Ch. 5b p.24/24 Path consistency It is defined using composition and the transitivity table Path consistency decides the consistency of a PA network in O(n 3 ) steps. Consistency and solution generation of PA networks can also be accomplished in O(n 2 ). The minimal network of a PA consistent network can be obtained using 4-consistency in O(n 4 ) steps. The minimal network of CPA networks can be obtained by path-consistency in O(n 3 ). Convex PA (CPA) networks have only and not.