A Structural Characterization of Temporal Dynamic Controllability Paul Morris NASA Ames Research Center Moffett Field, CA 94035, U.S.A.

Simple Temporal Networks (Dechter, Meiri, Pearl 1991) Special type of CSP Variables: Event times Binary Constraints: Event separations Issue: Determine consistency [10,20] AB The time of B is somewhere between 10 and 20 after A.

Temporal Constraints in Planning: Requirements versus Contingencies [10,20] AB Want B to occur at some time between 10 and 20 after A. [10,20] AB Know B will occur at a time between 10 and 20 after A. (precise time is uncertain) Requirement Agent must somehow ensure bounds are satisfied Contingency Nature ensures bounds are satisfied Nature chooses B

Contingencies are Observable Before a contingent link finishes, we can observe that it has not finished yet. After a contingent link finishes, we can observe how long it took.

STN with Uncertainty (STNU) (Vidal & Ghallab 1996, etc.) C D B A [1,2] [4,5] [2,3] C D B A [1,2] [4,4] [2,3] STN with contingent link Projection1 Projection2 [5,5] C D B A [1,2][2,3] Courtesy Morris, Muscettola, & Vidal IJCAI-2001 min max

STN/STNU Solutions STN Problem Solution = Schedule STNU Dynamic Controllability Problem Solution = Dynamic Strategy — mapping from projections to schedules — depends only on past — must not depend on future Courtesy Morris, Muscettola, & Vidal IJCAI-2001

Example ABCD [2,4] [-2,+2] [0,1] Strategy: execute A and C simultaneously Courtesy Morris, Muscettola, & Vidal IJCAI-2001

Example ABCD [2,4] [-2,+2] [0,1] Strategy: execute A and C simultaneously AB CD [-2,+2] [0,0] [2,4] tighten Courtesy Morris, Muscettola, & Vidal IJCAI-2001

Example Wait Tightening AB [0,4] C [-2,+2] C must wait for B or until 2 units after A Wait-for-event with timeout AB [0,4] C [-2,+2] tighten Wait(B;2)

STN/STNU Solutions STN Problem Solution = Schedule STNU Dynamic Controllability Problem Solution = Tightened Network Courtesy Morris, Muscettola, & Vidal IJCAI-2001

Previous & New Results STNU Dynamic Controllability Problem Morris, Muscettola & Vidal (IJCAI01) Tractable; pseudo-polynomial (bounded values) Morris & Muscettola (AAAI05) Strong-polynomial O(N^5) This paper: Strong-polynomial O(N^4) Courtesy Morris, Muscettola, & Vidal IJCAI-2001

Approach Element1: simplify WLOG AB [x,y] UB [0,y-x] A [x,x]

Approach Element2: mimic STN ideas Distance graph Paths of edges Consistent  no neg cycle Neg cycle  O(N) neg cycle Bellman-Ford O(N) cutoff Labeled distance graph paths of label-edges?? DC  no ?? cycle ?? cycle  simpler ?? ??? cutoff STN Methods STNU Methods

Labeled Distance Graph (aaai05) AB [0,y] A [x,y] B A B A B y -x b:0 B:-y A Wait(B;z) C A C B:-z

Labeled-Path Distance Add up edge weights, ignoring labels B c:y A x D E:z Path Distance = x+y+z Theorem???: DC iff no labeled negative cycle?? NOT TRUE!! Theorem: DC iff no allowed labeled negative cycle  C

Allowed Paths: Moat Edges A b:0 CBD first D with BD negative moat edge for AB A b:0 CBA B:-u Unusable moat edge

Allowed Paths: Moat Edges A b:0 CBD Usable moat edge Derived edge same path distance lower-case free

Dynamic Controllability Characterization Theorem: DC if and only if no allowed negative cycle A path is allowed if every lower-case edge has a usable moat edge in the path (These paths are semi-reducible: they can be transformed into paths without lower-case edges)

Nesting Lemma subpaths cannot partially overlap. The subpaths are either nested or disjoint. Thus they lead to a parenthesization of a path. b:0 1 D:-3 d:0 3 B:-2 b:0 -2 A  B  D  C  D  B  A  B  E Depth of nesting = 2

Negative Cycle Simplification x y If x+y < 0 then x < 0 or y < 0 Can eliminate repetitions in STN negative cycles BUT x y lc edge moat edge Cannot eliminate all repetitions in STNU allowed negative cycles

Negative Cycle Simplification HOWEVER: AABBCEDF Nested repetitions CAN be eliminated!  neg cycle with depth of nesting at most O(K). nested repetition (K = no. of contingent links)

Outline of DC Test Algorithm Match up usable pairs from the inside out; add derived edge. Each iteration eliminates one level of nesting. Detects allowed neg. cyc. in O(K) iterations. Each iteration propagates forward from each contingent link. Overall O(N^4) complexity.

DC Execution Needs Test if network is DC (previous slides) Regress waits as far as they will go Bypass certain lower-case edges –lower-case edge followed by negative ordinary path Ensures contingent edges will not be squeezed during execution.

Regression Paths: Tower Edges A B:-x CBD nearest D with DA non-negative tower edge for BA D:-3 2 b:0 B:-1 2 C  D  B  A  B  E Precursor path of derived wait can be parenthesized

No Nested Repetitions in Wait Derivations AABBCEDF Upper-case nested repetition would have to be negative Would be allowed negative cycle Cannot occur if verified DC upper-case nested repetition negative Regression of waits can be done in O(N^4) time

DC Execution Needs Test if network is DC (previous slides) –O(N^4) complexity  Regress waits as far as they will go –O(N^4) complexity  Bypass certain lower-case edges –lower-case edge followed by negative ordinary path –also O(N^4) complexity

Conclusions Simplification: contingent links with zero lower bound Dynamic Controllability Structural Characterization: DC  no negative cycle of specific type Parenthesization based on moat edges Can eliminate nested repetitions, limit depth Strong polynomial O(N^4) algorithm for DC –Compare: STN consistency testing is O(N^3)