Consistency Methods for Temporal Reasoning Lin XU Constraint Systems Laboratory Advisor: Dr. B.Y. Choueiry April, 2003 Supported by a grant from NASA-Nebraska,

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.

Lin XuTuesday, April 15, 2003 2

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


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


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)


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?


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


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).


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?


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


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]


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


BT search for meta-CSP

<new tree> big


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


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?


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








Algorithms for solving the STP

Graph Cost Consistency

Minimality

F-W/PC Complete (n3) Yes Yes

DPC Not necessarily

O (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


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


STP [TIME 03]

STP considers the temporal graph as composed by triangles instead of edges

Temporal graph F-W STPPPC


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


Finding the minimal STP


Determining consistency of STP








Filtering algorithm: ACRemove inconsistent intervals from the label of edge before search.

One global, exponential size constraint

Polynomial number of polynomial-size ternary constraints


AC reduces size of TCSP


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








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


New cycle check (NewCyc)

Checks presence of new cycles O (|E |) Checks consistency only if a new cycle is added


Advantages of NewCyc

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

Restricts effort to new bi-connected component


Edge Ordering in BT-TCSP Repeat your graph


EdgeOrd Heuristic

Order the edges using ‘triangle adjacency’

Priority list is a by-product of triangulation


Localized backtracking

Automatic decomposition of the constraint graph

no need for AP

Advantages of EdgeOrd


Experimental evaluations With/without: AC, AP, NewCyc,

EdgeOrd


Number of solutions


Nodes visited (without AC)


Nodes visited (after AC)


CC for DPC-TCSP (without AC)


CC for DPC-TCSP (after AC)


CC for PPC-A-TCSP (without AC)


CC for PPC-A-TCSP (after AC)


CC for STP-TCSP BEST


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]


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


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


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


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


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


The End

Thank you for your attention

Questions & comments are welcome

Documents

Consistency Methods for Temporal Reasoning Lin XU Constraint Systems Laboratory Advisor: Dr. B.Y. Choueiry April, 2003 Supported by a grant from NASA-Nebraska,