Computing Applicability Conditions for Plans with Loops

Transcription

1 Abacus Programs Translation Examples Computing Applicability Conditions for Plans with Loops Siddharth Srivastava Neil Immerman Shlomo Zilberstein Department of Computer Science, University of Massachusetts Amherst Twentieth International Conference on Automated Planning and Scheduling 15 May, 2010 Siddharth Srivastava Applicability Conditions for Plans with Loops 1

2 Abacus Programs Translation Examples Plans with Loops Siddharth Srivastava Applicability Conditions for Plans with Loops 2

3 Abacus Programs Translation Examples Plans with Loops Loops = smaller, more general plans; allow unknown quantities Siddharth Srivastava Applicability Conditions for Plans with Loops 2

4 Abacus Programs Translation Examples Plans with Loops: Power D C A B [Bonet et al., 2009] Siddharth Srivastava Applicability Conditions for Plans with Loops 3

5 Abacus Programs Translation Examples Plans with Loops: Power D C A B Correct numbers of iterations can reach any point in the interior Siddharth Srivastava Applicability Conditions for Plans with Loops 3

6 Abacus Programs Translation Examples Plans with Loops: Risk G S Siddharth Srivastava Applicability Conditions for Plans with Loops 4

7 Abacus Programs Translation Examples Plans with Loops: Risk G S Never reaches G! A plan with loops may traverse almost the entire state space without solving the problem Siddharth Srivastava Applicability Conditions for Plans with Loops 4

8 Abacus Programs Translation Examples Loop Effects and Preconditions Why analyze loops of actions? During construction: utility, safety of potential loops After construction: will plan Π solve problem instance x? Longstanding issues of plan reuse Triangle tables, macro operators: linear sequences [Fikes et al., 1972] Siddharth Srivastava Applicability Conditions for Plans with Loops 5

9 Abacus Programs Translation Examples Existing Approaches [Levesque, 2005, Winner and Veloso, 2007, Bonet et al., 2009] Plan Generation Observe working plans Extract repetitive patterns and make loops Loop utility justified largely by prior experience Plan Execution Apply on any problem instance No preconditions; instantiation may traverse the entire state space! (EX: [Levesque, 2005]) Fundamental problems behind these limitations: termination, reachability in plans with loops. Siddharth Srivastava Applicability Conditions for Plans with Loops 6

10 Abacus Programs Translation Examples Overview Abacus Programs Linear Abacus Programs Abacus Programs with Simple Loops Abacus Programs with Complex Loops Theoretical Results Translating Plans to Abacus Programs Empirical Results Siddharth Srivastava Applicability Conditions for Plans with Loops 7

11 Abacus Programs Definition R finite set of registers S finite set of states s 0 initial state s h halt state l S A Actions (A): Inc(r, s): r + +; goto s Dec(r, s 1, s 2 ): if r = 0 goto s 1 else r ; goto s 2 Turing-complete = reachability is undecidable Siddharth Srivastava Applicability Conditions for Plans with Loops 8

12 Linear Abacus Programs Can Efficiently Find Cumulative action effects Branch Conditions Siddharth Srivastava Applicability Conditions for Plans with Loops 9

13 Linear Abacus Programs Notation Can Efficiently Find Cumulative action effects Branch Conditions R R 1, R 2,... R k Ai A(i) R i Ai s change vector Index of Ai s register R i 1...m A1 + + Am Siddharth Srivastava Applicability Conditions for Plans with Loops 9

14 Linear Abacus Programs Notation Can Efficiently Find Cumulative action effects Branch Conditions R R 1, R 2,... R k Ai A(i) R i Ai s change vector Index of Ai s register R i 1...m A1 + + Am Siddharth Srivastava Applicability Conditions for Plans with Loops 9

15 Linear Abacus Programs: Preconditions Necessary and Sufficient Conditions Sequence of n actions; F = final register values ( R i 1 ) A(i) 0, i = 1... n F = R + 1..n Siddharth Srivastava Applicability Conditions for Plans with Loops 10

16 Abacus Programs with Simple Loops Siddharth Srivastava Applicability Conditions for Plans with Loops 11

17 Abacus Programs with Simple Loops LoopIneq( R 0 ) ( R i 1 ) A(i) 0 i = 1... n Siddharth Srivastava Applicability Conditions for Plans with Loops 11

18 Abacus Programs with Simple Loops LoopIneq( R 0 ) ( R i 1 ) A(i) 0 i = 1... n R x = R 0 + x 1..n Siddharth Srivastava Applicability Conditions for Plans with Loops 11

19 Abacus Programs with Simple Loops LoopIneq( R 0 ) ( R i 1 ) A(i) 0 R x = R 0 + x 1..n i = 1... n For l complete iterations: LoopIneq( R 0 ) LoopIneq( R 1 ) LoopIneq( R l 1 ) LoopIneq( R 0 ) LoopIneq( R l 1 ) Siddharth Srivastava Applicability Conditions for Plans with Loops 11

20 Abacus Programs with Simple Loops LoopIneq( R 0 ) ( R i 1 ) A(i) 0 R x = R 0 + x 1..n i = 1... n For l complete iterations: LoopIneq( R 0 ) LoopIneq( R 1 ) LoopIneq( R l 1 ) LoopIneq( R 0 ) LoopIneq( R l 1 ) F = R l = R 0 + l 1...n Siddharth Srivastava Applicability Conditions for Plans with Loops 11

21 Abacus Programs with Simple Loops: Preconditions Necessary and Sufficient Conditions For l complete iterations of a loop with n actions LoopIneq( R 0 ) LoopIneq( R l 1 ) F = R l = R 0 + l 1...n Siddharth Srivastava Applicability Conditions for Plans with Loops 12

22 Recall: Approach Translate plans with loops into abacus programs (preserve structure) Design algorithms for finding preconditions of classes of abacus programs Need to represent sensing actions. Siddharth Srivastava Applicability Conditions for Plans with Loops 13

23 Non-deterministic Actions NSet Action NSet(r, s 1, s 2 ) or set r to 0; goto s 1 S set r to 1; goto s 2 S Siddharth Srivastava Applicability Conditions for Plans with Loops 14

24 Non-deterministic Actions NSet Action NSet(r, s 1, s 2 ) or set r to 0; goto s 1 S set r to 1; goto s 2 S Abacus programs don t need NSet for computational power (Turing-complete). NSet allows in-place translation of plans with sensing actions. Find preconditions in terms of effect counts Siddharth Srivastava Applicability Conditions for Plans with Loops 14

25 Simple Loops with Shortcuts Siddharth Srivastava Applicability Conditions for Plans with Loops 15

26 Simple Loops with Shortcuts More commonly considered nested loops Siddharth Srivastava Applicability Conditions for Plans with Loops 15

27 Simple Loops with Shortcuts More commonly considered nested loops...can be translated by changing the start node. Siddharth Srivastava Applicability Conditions for Plans with Loops 15

28 Simple Loops with Shortcuts Siddharth Srivastava Applicability Conditions for Plans with Loops 15

29 Monotonicity Monotone Shortcuts The net change on a register due to every simple loop created by shortcuts must be in the same direction (positive/negative). Siddharth Srivastava Applicability Conditions for Plans with Loops 16

30 Simple Loops with Shortcuts: Computing Preconditions Suppose: m simple loops; k 1,..., k m iterations F = R 0 + m i=0 k i loop i k i iterations of loop i: no register can fall below zero. Need: least intermediate value of every register 0. Siddharth Srivastava Applicability Conditions for Plans with Loops 17

31 Constraining Minimum Register Values Constraints on Min Values m simple loops; k 1,..., k m iterations R j R + : R 0 j + δ bj 0 R j R : R 0 j + m i=0 k i loop i +δ bj loop bj 0 Notation R + R registers with net positive change registers with net negative change ĵ index of loop with greatest partial negative change on R j. Siddharth Srivastava Applicability Conditions for Plans with Loops 18

32 Order Dependence Constraints on Minimum Register Values R j R + : R 0 j + δ bj 0 R j R : R 0 j + m i=0 k i loop i +δbj loop bj 0 Example loop 1 increase R 1 by 3. loop 2 decrease R 1 by 2, then increase it by 5. Efficiently expressing general order dependent constraints: open problem! Siddharth Srivastava Applicability Conditions for Plans with Loops 19

33 Order Dependence Constraints on Minimum Register Values R j R + : R 0 j + δ bj 0 R j R : R 0 j + m i=0 k i loop i +δbj loop bj 0 Example loop 1 increase R 1 by 3. loop 2 decrease R 1 by 2, then increase it by 5. Constraint R (sufficient condition) Efficiently expressing general order dependent constraints: open problem! Siddharth Srivastava Applicability Conditions for Plans with Loops 19

34 Order Dependence Constraints on Minimum Register Values R j R + : R 0 j + δ bj 0 R j R : Example R 0 j + m i=0 k i loop i +δbj loop bj 0 loop 1 increase R 1 by 3. loop 2 decrease R 1 by 2, then increase it by 5. Constraint R (sufficient condition) Accurate Conditions Order dependent! Efficiently expressing general order dependent constraints: open problem! Siddharth Srivastava Applicability Conditions for Plans with Loops 19

35 Equality Constraints, Order Dependence Non-spurious equality constraint: inherently order dependent Equality constraints, δ bj loop bj induce order dependence (details in paper). Siddharth Srivastava Applicability Conditions for Plans with Loops 20

36 Main Results Let S be a node in an abacus program Π, all of whose strongly connected components are simple loops or simple loops with monotone shortcuts; F: a vector of register values. Theorem We can compute a disjunction of linear constraints on the initial register values ( R 0 ) for reaching S with F. The computed conditions are necessary and sufficient if every simple loop with shortcuts is order independent, and sufficient otherwise. Siddharth Srivastava Applicability Conditions for Plans with Loops 21

37 Abacus Programs Translation Examples Translating Plans To Abacus Programs Siddharth Srivastava Applicability Conditions for Plans with Loops 22

38 Abacus Programs Translation Examples Translating Plans To Abacus Programs #{obj, atl1}=1 movetl1() Changes in role counts: #{obj, atl1, int}++ #{obj, atl1} #{obj, atl2, int}++ #{obj, atl1, int} #{obj, atl2}++ #{obj, atl2, int} c {obj, atl1} {obj, atl1, int} {obj, atl2, int} {truck, atl1} {obj, atl1} choose c: obj(c) atl1(c) #{obj, atl1}>1 {truck, atl1} {obj, atl1} {truck, atl1} {truck, atl2} loadt(c) {obj, atl1} movetl2() {obj, atl1} unloadt() {truck, atl2} {obj, atl1} {obj, atl2} {obj, atl2} {obj, atl2} {obj, atl2} {obj, atl2} S 1 S 2 S 3 S 4 S 5 {truck, atl1} c {obj, atl1} {obj, atl2} S 6 Works for all unary, and a class of binary domains Siddharth Srivastava Applicability Conditions for Plans with Loops 22

39 Abacus Programs Translation Examples Empirical Results s f 3 = mf 3 = 7 i=0 k i; s f 1 = s0 1 7 i=0 k i k 0 k 5 k 6 k 7 = 0 Siddharth Srivastava Applicability Conditions for Plans with Loops 23

40 Abacus Programs Translation Examples Empirical Results Time Taken to Compute Preconditions Problem Time (s) Problem Time(s) Accumulator 0.01 Prize-A(7) 0.02 Corner-A 0.00 Recycling 0.02 Diagonal 0.01 Striped Tower 0.02 Hall-A 0.01 Transport 0.01 Prize-A(5) 0.01 Transport (conditional) 0.06 Siddharth Srivastava Applicability Conditions for Plans with Loops 24

41 Abacus Programs Translation Examples Conclusion An approach for computing summarized effects of loops of actions Use during construction and for precondition evaluation Future work: Greater translation to abacus programs (counts of properties) Further categorization of tractable classes Expression of order dependent constraints Use of symbolic precondition evaluation for non-numeric branches. Siddharth Srivastava Applicability Conditions for Plans with Loops 25

42 Abacus Programs Translation Examples Complete Preconditions Preconditions for a Simple Loop with Shortcuts l = P m i=0 k i; F = R 0 + P m i=0 k i loop i R j R : R j R + : W x=0 j,...,m j ki<x = 0; k x 0; R 0 j + P i x k i loop i + δ x loop x 0 Wx=0 j,...,m j ki<x = 0; k x 0; R 0 j + δ x 0 Siddharth Srivastava Applicability Conditions for Plans with Loops 26

43 Abacus Programs Translation Examples References I Bonet, B., Palacios, H., and Geffner, H. (2009). Automatic derivation of memoryless policies and finite-state controllers using classical planners. In Proc. of the 19th International Conf. on Artificial Intelligence Planning and Scheduling. Fikes, R., Hart, P., and Nilsson, N. (1972). Learning and Executing Generalized Robot Plans. Technical report, AI Center, SRI International. Levesque, H. J. (2005). Planning with loops. In Proc. of IJCAI, pages Siddharth Srivastava Applicability Conditions for Plans with Loops 27

44 Abacus Programs Translation Examples References II Winner, E. and Veloso, M. M. (2007). LoopDISTILL: Learning domain-specific planners from example plans. In Workshop on AI Planning and Learning, ICAPS. Siddharth Srivastava Applicability Conditions for Plans with Loops 28