Consistency Methods for Temporal Reasoning

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Consistency Methods for Temporal Reasoning

• Lin XU
• Constraint Systems Laboratory
• Advisor Dr. B.Y. Choueiry
• April, 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 task
• takes 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 800 a.m. Today, he gets up
between 730 and 740 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 ? 5?Pb-Pa?10 )
• Solution A value for each variable such that all
  temporal constraints are satisfied

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More complex example
Tom has class at 800 a.m. Today, he gets up
between 730 and 740 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
ltnew treegt 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
?Our 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
?STP
PPC
Temporal graph
F-W
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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 ?AC
Remove inconsistent intervals from the label of
edge before search.
Polynomial number of polynomial-size ternary
constraints
One global, exponential size constraint
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?AC reduces size of TCSP
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Advantages of ?AC

• 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

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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

• Restricts effort to new bi-connected component

• Reduces effort of consistency checking
• Does not affect of nodes visited in BT-TCSP

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Edge Ordering in BT-TCSP

• Repeat your graph

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EdgeOrd Heuristic
Order the edges using triangle
adjacency Priority list is a by-product of
triangulation
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Advantages of EdgeOrd

• Localized backtracking
• Automatic decomposition of the constraint graph
• ? no need for AP

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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