Consistency Methods for Temporal Reasoning
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Consistency Methods for Temporal Reasoning
Lin XUConstraint Systems Laboratory
Advisor: Dr. B.Y. ChoueiryApril, 2003
Supported by a grant from NASA-Nebraska, CAREER Award #0133568, and a gift from Honeywell Laboratories.
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Outline Temporal Reasoning
motivation & background Simple Temporal Problem (STP) & Temporal
Constraint Satisfaction Problem (TCSP) what are they & how to solve them
Contribution: 3 research questions their solutions empirical evidence
Summary & future directions for research
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Time, always time! Tom wants to serve tea
Clear tea pot 3 min Clear tea cups 10 min Boil water 15 min
With little reasoning, the tasktakes 18 min instead of 28 min
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Temporal Reasoning in AI
Temporal Logic Temporal Networks
Qualitative: interval algebra, point algebra Before, after, during, etc.
Quantitative: temporal constraint networks Metric: 10 min before, during 15 min, etc. Simple TP (STP) & Temporal CSP (TCSP)
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Temporal Network: example
Tom has class at 8:00 a.m. Today, he gets up between 7:30 and 7:40 a.m. He prepares his breakfast (10-15 min). After breakfast (5-10 min), he goes to school by car (20-30 min). Will he be on time for class?
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Simple Temporal Network (STP)
Variable: Time point for an event Domain: A set of real numbers Constraint: distance between time points ( [5, 10] 5Pb-
Pa10 ) Solution: A value for each variable such that all
temporal constraints are satisfied
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More complex exampleTom has class at 8:00 a.m. Today, he gets up between 7:30 and 7:40 a.m. He either makes his breakfast himself (10-15 min), or gets something from a local store (less than 5 min). After breakfast (5-10 min), he goes to school either by car (20-30 min) or by bus (at least 45 min).
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Possible questions
Can Tom arrive school in time for class? Is it possible for Tom to take the bus? If Tom wanted to save money by
making breakfast for himself and taking the bus, when should he get up?
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Temporal CSP
Constraint: a disjunction of intervals [10, 15] [0, 5] Rest, same as STP
Variable: Time point for an event Domain: A set of real numbers Solution: Each variable has a value that satisfies all temporal
constraints
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Temporal Networks: STP & TCSP
Simple temporal problem (STP) One interval per constraint Can be solved in polynomial time Floyd-Warshall F-W algorithm (all-pairs shortest-paths)
Temporal Constraint Satisfaction Problem (TCSP) A disjunction of intervals per constraint is NP-hard Solved with Backtrack search (BT-TCSP) [Dechter]
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Solving the TCSP
Formulate TCSP as a meta-CSP: Given
Variables: Edges in constraint network Domains of variables: edge labels in constraint network A unique global constraint ( checking consistency of an
STP) Find all solutions to the meta-CSP
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BT search for meta-CSP
<new tree> big
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Solving the TCSP
Requires finding all solutions to the meta-CSP Every node in the search tree is an STP to be solved An exponential number of STPs to be solved
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Questions addressed Is there a better algorithm for STP than F-W?
exploiting topology of the constraint graph exploiting semantic properties of the temporal constraints
Is there a consistency filtering algorithm for reducing the size of TCSP?
Can we improve performance of BT-TCSP By using a better STP solver? By avoiding to check STP consistency at every node? By exploiting the topology of the constraint graph?
again! By finding a ‘good’ variable ordering heuristic?
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Contributions Two new algorithms for solving STP
Partial Path Consistency [adapted from Bliek & Sam-Haroud] STP [Xu & Choueiry, TIME 03]
A new algorithm for filtering TCSP AC [Xu & Choueiry, submitted]
Three heuristics to improve search Articulation points (AP) [classical, never tested] New cycle check (NewCyc) [Xu & Choueiry, submitted] Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]
Random generators: 2 for STP & 2 for TCSP
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Contributions Two new algorithms for solving STP
Partial Path Consistency [adapted from Bliek & Sam-Haroud] STP [Xu & Choueiry, TIME 03]
A new algorithm for filtering TCSP AC [Xu & Choueiry, submitted]
Three heuristics to improve search Articulation points (AP) [classical, never tested] New cycle check (NewCyc) [Xu & Choueiry, submitted] Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]
Random generators: 2 for STP & 2 for TCSP
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Algorithms for solving the STP
Graph Cost Consistency
Minimality
F-W/PC Complete (n3) Yes YesDPC Not
necessarilyO
(nW*(d)2)very cheap
Yes No
PPC Triangulated O (n3)usually
cheaper than F-W/PC
Yes Yes
STP Triangulated Always cheaper than
PPC
Yes YesOur approach requires triangulation of the constraint graph
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Partial Path Consistency (PPC)
Known features of PPC [Bliek & Sam-Haroud, 99] Applicable to general CSPs Triangulates the constraint graph In general, resulting network is not minimal For convex constraints, guarantees minimality
(same as F-W, but much cheaper in practice) Adaptation of PPC to STP [this thesis]
Constraints in STP are bounded difference, thus convex, PPC results in the minimal network
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STP [TIME 03]
STP considers the temporal graph as composed by triangles instead of edges
Temporal graph F-W STPPPC
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Advantages of STP A finer version of PPC Cheaper than PPC and F-W Guarantees the minimal network Automatically decomposes the graph
into its bi-connected components binds effort in size of largest component allows parallellization
Best known algorithm for solving STP use it in BT-TCSP where it is applied an exponential number of times
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Finding the minimal STP
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Determining consistency of STP
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Contributions Two new algorithms for solving STP
Partial Path Consistency [adapted from Bliek & Sam-Haroud] STP [Xu & Choueiry, TIME 03]
A new algorithm for filtering TCSP AC [Xu & Choueiry, submitted]
Three heuristics to improve search Articulation points (AP) [classical, never tested] New cycle check (NewCyc) [Xu & Choueiry, submitted] Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]
Random generators: 2 for STP & 2 for TCSP
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Filtering algorithm: ACRemove inconsistent intervals from the label of edge before search.
One global, exponential size constraint
Polynomial number of polynomial-size ternary constraints
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AC reduces size of TCSP
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It is powerful, especially under high density
It uses special, poly-size data structures It is sound, effective, and cheap O (n |E |
k3) We show how to make it optimal [to be
proved] It uncovers a phase transition in TCSP
Advantages of AC
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Contributions Two new algorithms for solving STP
Partial Path Consistency [adapted from Bliek & Sam-Haroud] STP [Xu & Choueiry, TIME 03]
A new algorithm for filtering TCSP AC [Xu & Choueiry, submitted]
Three heuristics to improve search Articulation points (AP) [classical, never tested] New cycle check (NewCyc) [Xu & Choueiry, submitted] Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]
Random generators: 2 for STP & 2 for TCSP
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Articulation points (AP) Decompose the graph into bi-connected components Solve each of them independently Binds the total cost by the size of largest component Classical solution, never implemented or tested
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New cycle check (NewCyc) Checks presence of new cycles O (|E |) Checks consistency only if a new cycle is added
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Advantages of NewCyc
Reduces effort of consistency checking Does not affect # of nodes visited in BT-TCSP
Restricts effort to new bi-connected component
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Edge Ordering in BT-TCSP Repeat your graph
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EdgeOrd HeuristicOrder the edges using ‘triangle adjacency’Priority list is a by-product of triangulation
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Localized backtracking
Automatic decomposition of the constraint graph
no need for AP
Advantages of EdgeOrd
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Experimental evaluations With/without: AC, AP, NewCyc,
EdgeOrd
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Number of solutions
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Nodes visited (without AC)
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Nodes visited (after AC)
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CC for DPC-TCSP (without AC)
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CC for DPC-TCSP (after AC)
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CC for PPC-A-TCSP (without AC)
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CC for PPC-A-TCSP (after AC)
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CC for STP-TCSP BEST
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Random generators STP generators
Implemented two new Tested three
GenSTP-1 [Xu & Choueiry, submitted] GenSTP-2 [Courtesy of Ioannis Tsamardinos] SPRAND (sub-class of SPLIB) [Public domain]
TCSP generator Implemented two new Tested 1: GenTCSP-1 [Xu & Choueiry, submitted]
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Output from thesis 1 paper accepted in TIME-ICTL
2003 2 papers submitted to CP 2003 2 papers submitted to IJCAI 2003
workshop on Spatial & Temporal Reasoning
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Answers to Question I Is there a better algorithm for STP than F-W?
Exploiting topology: AP improves any STP solver
Constraint semantic: convexity STP is more efficient than F-W and PPC
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Answer to Question II Is there a consistency filtering algorithm
for reducing the size of TCSP?
AC reduces the size of meta-CSP by eliminating intervals from the domain of edge
Effective, cheap, almost optimal
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Answers to Question III Can we improve the performance of BT-TCSP
by using a better STP solver? Yes, STP is better than DPC to reduce cost of BT
By avoiding to check STP consistency at every node? Yes, NewCyc avoids unnecessary checks & localizes
updates
By exploiting the topology of the constraint graph? Yes, using articulation points
By finding a good variable ordering heuristic We propose EdgeOrd, significantly reduces cost of search
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Future work Improve AC, establish optimality Integrate AC
with ULT (a closure algorithm) with search, as in forward-checking
Exploit interchangeability in BT-TCSP, best method for finding all solution
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The End Thank you for your attention Questions & comments are welcome