Constraint Programming on Infinite Data Streams

Transcription

1 Constraint Programming on Infinite Data Streams A. Lallouet 1, Y.C. Law 2, J.H.M. Lee 2, and C.F.K. Siu 2 1 Université de Caen, France 2 The Chinese University of Hong Kong, Hong Kong 1

2 Infinite Data Streams Traffic Light Temperature :00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:0 Time 2

3 Infinite Data Streams A piece of Music Sequence of notes Follow some rules of harmony Juggling Pattern Sequence of juggling actions Follow the laws of physics 3

4 Overview Stream Constraint Satisfaction Problem Search Strategy Consistency Notion Experimental Result 4

5 Infinite Data Streams A stream 3, 5, 0, 0, 2, 4, 2, 1, 0, Time Each value in a stream is a daton 5

6 Operators on Streams The operators defined on streams are adopted from Lucid [Wadge and Ashcroft, 1985]. Pointwise Operators Arithmetic Operators Add (+), Minus (-) Boolean Operators Equivalent (==), Logical And (and), Logical Or (or) Temporal Operators First (first) Next (next) Followed-By (fby) Conditional Operator If-then-else (if-then-else) 6

7 Pointwise Operators Applied on the stream pointwisely Example 1: Add (+) 1, 2, 3, 3, 3, 3, 3, + = 2, 3, 4, 3, 4, 3, 4, = 3, 5, 7, 6, 7, 6, 7, 7

8 Pointwise Operators Example 2: If-Then-Else (if-then-else) if 1, 0, 1, then 2, 3, 1, else 6, 8, 9, = 2, 8, 1, 8

9 Temporal Operators First (first) first 2, 3, 1, 5, 5, = 2, 2, 2, 2, 2, Next (next) next 2, 3, 1, 5, 5, = 3, 1, 5, 5, Followed-By (fby) 6, 3, 4, fby 2, 3, 1, = 6, 2, 3, 1, 9

10 Stream Constraint Satisfaction Problem Variables a set of unknown streams in the problem Domains each domain is a possibly infinite set of streams for each variables to take Constraints restrict the values taken by the variables at the same time Solutions: Tuples of streams satisfy the constraints 10

11 Streams Constraints Stream expressions are composed of stream variables and stream operators A + B X fby Y Stream constraints are relations on the stream expressions Equality (=) Disequality ( ) Greater-than-or-equal-to ( ) Less-than-or-equal-to ( ) Example A + B = X fby Y stream expression stream expression 11

12 Domain: A Set of Streams Storing a set of streams explicitly is infeasible The streams are of infinite length There can be infinite number of streams in a set There is finite representation of an infinite set of streams which is an Omega-regular language (recognizable by Büchi automaton) 12

13 Domain: A Set of Streams (2 3) ω is the set { 1, 2, 5, 2, 2, 2,, 1, 2, 5, 2, 2, 3,, } 1, 2, 5, 2, 3, 2,, Any infinite sequence containing 2 and 3 13

14 Juggling A basic juggling involving n balls. Every ball can be thrown with maximum m units of force so that the ball will be caught after m time points. There is at most one ball being caught at any time. 14

15 Juggling Suppose for 1 ball X The juggler throws the ball with 4 units of force Time X

16 Juggling Suppose there are 3 balls and max. 5 units of force Variables: X, Y, Z time to be caught for a ball A unit of force to throw the ball at hand Domains: D(X) = D(Y) = D(Z) = ( ) ω D(A) = ( ) ω X Y Z

17 Juggling Constraints Decrease the value for each ball over time points; Decide the force for throwing the ball when the ball is being caught (=1) next X = ( if X == 1 then A else X 1 ) next Y = ( if Y == 1 then A else Y 1 ) next Z = ( if Z == 1 then A else Z 1 ) 17

18 Juggling Constraints At most one ball is being caught at any time points X Y Y Z Z X 18

19 Searching Using depth first search A stream is of infinite size A variable cannot be instantiated with a stream value completely We instantiate the values in the stream in the order of time point X =?,?,?, Y =?,?,?, 19

20 Searching Using depth first search A stream is of infinite size A variable cannot be instantiated with a stream value completely We instantiate the values in the stream in the order of time point X = 1,?,?, Y =?,?,?, X =?,?,?, Y =?,?,?, 20

21 Searching Using depth first search A stream is of infinite size A variable cannot be instantiated with a stream value completely We instantiate the values in the stream in the order of time point X = 1,?,?, Y =?,?,?, X =?,?,?, Y =?,?,?, All datons at t=0 are labeled X = 1,?,?, Y = 2,?,?, 21

22 Searching Using depth first search A stream is of infinite size A variable cannot be instantiated with a stream value completely We instantiate the values in the stream in the order of time point X = 1,?,?, Y =?,?,?, X =?,?,?, Y =?,?,?, Start labeling datons at t=1 X = 1,?,?, Y = 2,?,?, X = 1, 0,?, Y = 2,?,?, 22

23 Searching Using depth first search A stream is of infinite size A variable cannot be instantiated with a stream value completely We instantiate the values in the stream in the order of time point X =?,?,?, Y =?,?,?, X = 1,?,?, Y = 2,?,?, X = 1,?,?, Y =?,?,?, X = 1,?,?, Y = 3,?,?, X = 1, 0,?, Y = 2,?,?, X = 1, 1,?, Y = 2,?,?, 23

24 Searching A branch is failed when there is no solution in the subtree No solution! X 24

25 Searching Dominance: subtrees having equivalent search space up to renaming of time points No solution! X equivalent search space X 25

26 Searching Theorem Each branch in a search tree is finite and must either (a) end in failure or (b) contain search states dominated by an earlier state. The search must terminate. No solution! X 26

27 Solutions to Stream CSP A solution set of Stream CSP is an omegaregular language which can be represented by a Büchi automaton The automaton representing the solution set is isomorphic to the search Search Automaton X

28 Solutions to StCSP Theorem (Sound and Complete) The automaton constructed from a complete tree search contains only all the solutions to the problem. Search Tree Automaton X X X 28

29 Consistency Notions Enforcing consistency to reduce search space Similar to searching, consistency enforcement can take infinite time Consistency on a finite window of time: Prefix-k Consistency t=0 X?????????? Y?????????? e.g. when k=3 t=1 X 1????????? Y 3????????? t=2 X 1 5???????? Y 3 2???????? Time

30 Experimental Result : Juggling Run time, number of fails for simulating juggling for n balls and maximum m unit of force No Consistency Prefix-1 Prefix-2 Prefix-3 (n,m) Time Fails Time Fails Time Fails Time Fails (3,3) (3,4) , , ,277 (4,4) , (5,5) 2, , (6,6) - - 5, , , The `- marks 100 min timeout. 30

31 Conclusion Proposed a framework of constraint satisfaction on infinite streams Language for model specification Search algorithm Consistency notion Used to model problems which involve nonterminating sequences 31

32 Future Work Introduce other stream constraints other consistency notions Improvement on implementation of solver Stream Constraint Optimization Problems Stream CSP involving external input streams 32