Constraint Propagation Algorithms for Temporal Reasoning

Marc Vilain, Henry Kautz (AAAI 1986)Presentation by Lin XU

March 4, 2002

Introduction

Many representation schemes have been proposed for temporal reasoning. Most attractive is Allen’s algebra of temporal intervals.

There are several ways to use this algebra. Allen and Koomen rely heavily on time interval

algebra for reasoning about the ordering of actions. This is an elegant approaches, but compromised by

computational complexity of interval algebra. This paper addresses computational aspects and

compares Allen’s algebra and time point algebra.

Results

Determining consistency of statements in interval algebra is NP-hard. Allen’s polynomial-time algorithm is sound but not complete.

In time point algebra:constraint propagation is sound and complete. Time: O(n3) and Space: O(n2)

Time point algebra is proposed as a restricted form of interval algebra

Interval Algebra

Allen’s interval algebra admits 213

possible relations.

All of those relations related to 13 simple relations.

Simple relation and relation vector

Vectors are disjunction of simple relations

Two operations on vectors: + and x

Addition

Given two vectors that apply to the same pair of intervals,

the addition operation “intersects” these vectors (i.e., logical and) V1=(BEFORE MEETS OVERLAPS)

V2=(OVERLAPS STARTS DRING) V1+V2=(OVERLAPS)

Multiplication

Applies to two vectors defined over three A, B, and C

Vector V1 relates the pair of intervals (A, B) and Vector V2 relates the pair (B, C)

The product V1xV2 is the least restrictive relation between A and C.

V1=(BEFORE MEETS OVERLAPS)v2=(BEFORE MEETS)

V1V2=(BEFORE)

Multiplication: example

Closure in Interval Algebra Theorem 1: Time

complexity of the close procedure: O(n3).

This propagation is sound but incomplete.

Intractability of the Interval Algebra

Computing closure of assertions is NP-hard. Theorem 2: Determining the satisfiability of

a set of assertions in the interval algebra is NP-hard.

Theorem 3: Determining the satisfiability of assertions in the interval algebra and determining their closure are equivalent.

Consequences of Intractability

Several author give us exponential-time algorithms that computer the closure a assertions in the interval algebra.

Interval algebra is not useless Limit to small database. Accept its incompleteness. Choose another temporal representation.

A Point Temporal Algebra

Three possible relations: precedes, same, follows As with intervals, points are related to each

other through relation vectors.

Computing Closure in the Point Algebra

We can directly adapt the constraint propagation algorithm.

The algorithm runs with time complexity O(n3).

It is sound and complete.

Relating Interval and Point algebras

Single time points by themselves aren’t sufficient to express natural language.

Many interval relations can be encoded in the point algebra.

Interval algebra: A (DURING) B

Point algebra: A- (FELLOWS) B-A+ (PRECEDES) B+A- (PRECEDES) a+B- (PRECEDES) B+

Relating the Interval and Point algebras

Time Point scheme can represent all unambiguous relations and some ambiguous relations but not all.

Consequences of These Result

What lesson may we learn from such an analysis?

Understanding of computational advantages and disadvantages of different representation languages.Compromise between expressiveness and

tractability

Questions?

Thank You!