Download Qualitative Spatial- Temporal Reasoning

Qualitative SpatialTemporal Reasoning
Jason J. Li
Advanced Topics in A.I.
The Australian National University
Spatial-Temporal Reasoning
•
Space is ubiquitous in intelligent
systems
– We wish to reason, make predictions,
and plan for events in space
– Modelling space is similar to modelling
time.
Quantitative Approaches
•
Spatial-temporal configurations can be
described by specifying coordinates:
–
–
–
At 10am object A is at position (1,0,1), at 11am it
is at (1,2,2)
From 9am to 11am, object B is at (1,2,2)
At 11am object C is at (13,10,12), and at 1pm it
is at (12,11,12)
A Qualitative Perspective
•
Often, a qualitative description is more
adequate
– Object A collided with object B, then
object C appeared
– Object C was not near the collision
between A and B when it took place
Qualitative Representations
•
Uses a finite vocabulary
–
•
•
A finite set of relations
Efficient when precise information is not
available or not necessary
Handles well with uncertainty
–
Uncertainty represented by disjunction of
relations
Qualitative vs. Fuzzy
•
•
Fuzzy representations take
approximations of real values
Qualitative representations make only
as much distinctions as necessary
– This ensures the soundness of
composition
Qualitative Spatial-Temporal
Reasoning
•
•
Represent space and time in a
qualitative manner
Reasoning using a constraint calculus
with infinite domains
– Space and time is continuous
Trinity of a Qualitative Calculus
•
•
•
Algebra of relations
Domain
Weak-Representation
Algebra of Relations
•
Formally, it’s called Nonassociatve Algebra
–
–
Relation Algebra is a subset of such algebras
that its composition is associative
It prescribes the constraints between elements
in the domain by the relationship between them.
Algebra of Relations
•
It usually has these operations:
––
Composition:
Composition:
••
––
Converse:
Converse:
••
––
IfIf A
A is
is related
related to
to B,
B, what
what is
is B’s
B’s relation
relation to
to A
A
Intersection/union:
Intersection/union:
••
––
IfIf A
A is
is related
related to
to B,
B, B
B is
is related
related to
to C,
C, what
what is
is A
A to
to C
C
Defined
Defined set-theoretically
set-theoretically
Complement:
Complement:
••
A
A is
is not
not related
related to
to B
B by
by Rel_A,
Rel_A, then
then what
what is
is the
the relation?
relation?
Example – Point Algebra
•
•
Points along a line
Composition of
relations
–
–
–
–
{<} ; {=} = {<}
{<,=} ; {<} = {<}
{<,>} ; {<} = {<,=,>}
{<,=} ; {>,=} = {=}
Example – RCC8
Domain
•
•
The set of spatial-temporal objects we
wish to reason
Example:
– 2D Generic Regions
– Points in time
Weak-Representation
•
How the algebra is mapped to the
domain (JEPD)
– Jointly Exhaustive: everything is related
to everything else
– Pairwise Disjoint: any two entities in the
domain is related by an atomic relation
Mapping of Point Algebra
•
Domain: Real values
–
–
–
•
Between any two value there is a value
We say the weak representation is a
representation
Any consistent network can be consistently
extended
Domain: Discrete values (whole numbers)
–
Weak representation not representation
Network of Relations
•
•
•
•
•
Always complete graphs (JEPD)
Set of vertices (VNN) and label of edges (LNN)
Vertice VNN(i) denotes the ithth spatial-temporal
variable
Label LNN(i,j) denote the possible relations between
the two variables VNN(i), VNN(j)
A network M is a subnetwork of another network N
iff all nodes and labels of M are in N
Example of Networks
•
•
•
Greece is part of EU
and on its boarder
Czech Republic is
part of EU and not on
its boarder
Russia is externally
connected to EU and
disconnected to
Greece
Example of Networks
Czech
NTPP
U
EC
EU
Russia
U
TPP
Greece
DC
Path-Consistency
•
•
Any two variable assignment can be
extended to three variables
assignment
Forall 1 <= i, j, k <= n
– Rij = Rij ∩ Rik ; Rkj
Example of Path-Consistency
Czech
NTPP
U
EC
EU
Russia
U
TPP
Greece
DC
Example of Path-Consistency
Conv(NTPP) = NTPPi
EC ; NTPPi = DC
Czech
NTPP
DC
EC
EU
TPP
Russia
U
Greece
DC
Example of Path-Consistency
Conv(DC) = DC
DC ; DC = U
Czech
NTPP
DC
EC
EU
Russia
U
TPP
Greece
DC
Example of Path-Consistency
TPP ; NTPPi
= {DC,EC,PO,TPPi, NTPPi}
Conv(NTPP) = NTPPi
Czech
NTPP
DC
EC
EU
Russia
DC
TPP
Greece
DC,EC,PO,TPPi,NTPPi
Example of Path-Consistency
•
From the information given, we were
able to eliminate some possibilities of
the relation between Czech and
Greece
Consistency
•
A network is consistent iff
– There is an instantiation in the domain
such that all constraints are satisfied.
Consistency
•
A nice property of a calculus, would be that
path-consistency entails consistency for
CSPs with only atomic constraints.
–
If all the transitive constraints are satisfied, then
it can be realized.
••
••
RCC8,
RCC8, Point
Point Algebra
Algebra all
all have
have this
this property
property
But
But many
many do
do not…
not…
Path-Consistency and
Consistency
•
Path-consistency is different to
(general) consistency
– Consider 5 circular disks
– All externally connected to
each other
– This is PC, but not Consistent!
Important Problems in Qualitative
Spatial-Temporal Reasoning
•
A very nice property of a qualitative calculus
is that if path-consistency entails
consistency
–
–
–
If the network is path-consistent, then you can
get an instantiation in the domain
Usually, it requires a manual proof
Any way to do it automatically?
Important Problems in Qualitative
Spatial-Temporal Reasoning
•
Computational Complexity
– What is the complexity for deciding
consistency?
•
P? NP? NP-Hard? P-SPACE? EXP-SPACE?
Important Problems in Qualitative
Spatial-Temporal Reasoning
•
Unified theory of spatial-temporal reasoning
–
Many spatial-temporal calculi have been
proposed
••
–
Point
Point Algebra,
Algebra, Interval
Interval Algebra,
Algebra, RCC8,
RCC8, OPRA,
OPRA, STAR,
STAR,
etc.
etc.
How do we combine efficient reasoning calculi
for more expressive queries.
Important Problems in Qualitative
Spatial-Temporal Reasoning
•
Unified theory of spatial-temporal
reasoning
– Some approaches combines two calculi
to form a new calculi, with mixed results
•
•
•
IA (PA+PA), INDU (IA + Size), etc
BIG Calculus containing all information?
Meta-reasoning to switch calculi?
Important Problems in Qualitative
Spatial-Temporal Reasoning
•
Qualitative representations may have
different levels of granularity
–
How coarse/fine you want to define the relations
••
–
–
Do
Do you
you care
care PP
PP vs.
vs. TPP?
TPP?
What resolution do you want your representation?
What level of information do you want to use?
Important Problems in Qualitative
Spatial-Temporal Reasoning
•
Spatial Planning
–
–
–
Most automated planning problems ignore
spatial aspects of the problem
Most real-life applications uses an ad-hoc
representation for reasoning
How do we use make use of efficient reasoning
algorithms to better plan for spatial-change
Solving Complexity
•
•
If path-consistency decide consistency,
the problem is polynomial
If not, then some complexity proof is
required
– Transform the problem to one of the
known problems
Solving Complexity
•
Show NP-Hardness, you need to show 1-1
transformation for a subset of the problems
to a known NP-Complete Problem
–
–
Deciding consistency for some spatial-temporal
networks
Deciding the Boolean satisfiability problem (3SAT)
Transforming Problem
•
Boolean satisfiability
problem has
––
––
––
•
Variables
Variables
Literals
Literals
Constraints
Constraints
Transform each
component to spatial
networks
Transforming Problem
– Show deciding consistency is same as
deciding consistency for SAT problem,
and vice versa
– Program written to do this automatically
(Renz & Li, KR’2008)
Summary
•
•
•
•
Qualitative Spatial-Temporal Reasoning uses
constraint networks of infinite domains
It reasons with relations between entities, and
make only as few distinctions as necessary
It is useful for imprecise / uncertain information
Many open questions / problems in the field.
Further Reading
••
••
••
••
••
A.
A. G.
G. Cohn
Cohn and
and J.
J. Renz,
Renz, Qualitative
Qualitative Spatial
Spatial Representation
Representation and
and
Reasoning,
in:
F.
van
Hermelen,
V.
Lifschitz,
B.
Porter,
eds.,
Handbook
Reasoning, in: F. van Hermelen, V. Lifschitz, B. Porter, eds., Handbook of
of
Knowledge
Representation,
Elsevier,
551-596,
2008.
Knowledge Representation, Elsevier, 551-596, 2008.
J.
J. J.
J. Li,
Li, T.
T. Kowalski,
Kowalski, J.
J. Renz,
Renz, and
and S.
S. Li,
Li, Combining
Combining Binary
Binary Constraint
Constraint
Networks
Networks in
in Qualitative
Qualitative Reasoning,
Reasoning, Proceedings
Proceedings of
of the
the 18th
18th European
European
Conference
Patras, Greece,
Greece, July
July 2008,
2008,
Conference on
on Artificial
Artificial Intelligence
Intelligence (ECAI'08),
(ECAI'08), Patras,
515-519.
515-519.
G.
G. Ligozat,
Ligozat, J.
J. Renz,
Renz, What
What is
is aa Qualitative
Qualitative Calculus?
Calculus? A
A General
General
Framework,
8th
Pacific
Rim
International
Conference
on
Framework, 8th Pacific Rim International Conference on Artificial
Artificial Intelligence
Intelligence
(PRICAI'04),
Auckland, New
New Zealand,
Zealand, August
August 2004,
2004, 53-64
53-64
(PRICAI'04), Auckland,
J.
J. Renz,
Renz, Qualitative
Qualitative Spatial
Spatial Reasoning
Reasoning with
with Topological
Topological Information,
Information,
LNCS
LNCS 2293,
2293, Springer-Verlag,
Springer-Verlag, Berlin,
Berlin, 2002.
2002.
The
The above
above can
can all
all be
be accessed
accessed at
at http://www.jochenrenz.info
http://www.jochenrenz.info