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Managing Uncertainty in Semistructured Databases

and Spatiotemporal Databases

- Edward Hung
- University of Maryland, College Park
- PhD Proposal Oral Defense, Apr 22, 2003

Outline

- Motivating examples
- PXML data model
- Semantics
- Algebra
- Aggregation
- PIXML data model
- Uncertain spatiotemporal databases
- Related work

Motivating Example 1

- Bibliographic applications, citation index, e.g.

Citeseer, DBLP - automatic information extraction techniques ?

uncertainty (e.g., Fuhr, Buckley, Salton) - is it a reference?
- a conference paper, a journal article, etc?
- author? title? year?
- different names of the same author?

Motivating Example 2

- Surveillance applications monitoring a region of

battlefield - Image processing system identifies vehicles in

convoys appearing in the region at different

times - Convoys
- timestamp
- tanks, trucks, etc
- Uncertainty
- number of vehicles
- Category and identity of a vehicle, e.g., a tank?

T-72?

Example Queries

- we are only interested in titles of books but not

the publishers or locations - we are not sure there exists a book called XML

handbook or not, but we are interested to

consider the cases that it exists - we have two instances with data obtained from two

sources and we want to combine them - what is the probability that the book XML

handbook exists in the database?

Motivating Examples

- Semistructured data model
- General hierarchical structure is known.
- The schema is not fixed
- Number of authors/vehicles
- Properties of authors/vehicles
- My work store uncertain information in

probabilistic environments.

Semistructured Data Model

PXML Data Model

- Uncertainty
- Existence of sub-objects
- Number of sub-objects
- Identity of the sub-objects

PXML Data Model (Cardinality)

- Example of cardinality

card(B1, author)1,2

Weak Instance W Semistructured Instance card

PXML Data Model (Weak Instance)

- Example of a weak instance W

card(R,book)2,3

card(B1, author)1,2

card(B2, author)2,2

card(B3, author)1,1

card(B3, title)1,1

PXML Data Model

- Example of an instance compatible with W

card(R,book)2,3

card(B1, author)1,2

card(B2, author)2,2

card(B3, author)1,1

card(B3, title)1,1

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

card(B1, author)1,2

The set of all potential child set of B1, PC(B1)

A1, A2,

A1,

A2

Probabilistic Instance I Weak Instance W

local interpretation (p)

For non-leaf objects (e.g., B1), local

interpretation (p(B1)) returns an object

probability function (OPF), which is a mapping w

PC(B1) ? 0,1 s.t. w is a valid probability

distribution.

card(B1, author)1,2

conditional prob. distribution over its potential

child sets given that it exists

p(B1)(A1, A2) 0.5

p(B1)(A1) 0.3

p(B1)(A2) 0.2

Probabilistic Instance I Weak Instance W

local interpretation (p)

For leaf objects (e.g., T2), local interpretation

(p(T2)) returns an value probability function

(VPF), which is a mapping w from the domain of

type of T2 to 0,1 s.t. w is a legal probability

distribution.

p(T2)(XML Black Book) 0.2

p(T2)(XML Book) 0.3

p(T2)(XML) 0.5

Semantics (Local Interpretation)

- Here the local interpretation assigns the

probability to each possible set of children of

each non-leaf object in a local manner. - More independence assumptions are possible to

make the representation more compact - e.g. independence between authors and titles.
- e.g. all authors are all indistinguishable (e.g.,

no information about names of authors of a book).

Semantics (Global Interpretation)

- Local interpretation for efficient computation
- Now we are going to assign probabilities of each

compatible instance globally, which is more

intuitive.

Semantics (Global Interpretation)

- Interpretation
- Global interpretation, P
- a mapping from D(W) (the set of semistructured

instances compatible with W) to 0,1 s.t.

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2

0.15

0.3

0.05

0.09

0.03

0.18

Semantics (Local ? Global)

- Given a semistructured instance S compatible with

a weak instance W and a local interpretation p

for W - Pp(S)Õo S p(o)(CS(o))
- CS(o) is the actual set of children of o
- Theorem
- Pp is a global interpretation for W

Semantics

S1a

p(B1)(A1)0.6

- Example
- Pp (S1a)
- p(R)(B1, B2) x p(B1)(A1) x p(B2)(A2,

A3)0.5 x 0.6 x 10.3

p(R)(B1, B2)0.5

p(B2)(A2, A3)1

Semantics (Global ? Local)

- Theorem
- Given a global interpretation P, if the

probability of any potential child of an object o

is independent of non-descendants of o, then

there exists a local interpretation p such that

Pp P

Semantics (Local ?? Global)

- I have defined operators to convert between local

and global interpretations.

Semantics (Local ?? Global)

- Theorems (Reversibility)
- The conversions from local to global

interpretation is correct. - Under the conditional independence (of

non-descendants ) assumption, the conversions

from global to local interpretation is correct. - The conversion between local and global

interpretations is reversible.

Algebra

- Operators
- Projection
- Selection
- Cross-product
- Path expression
- o.l1.l2ln

R.book.author

Algebra

- Operators
- Projection
- Selection
- Cross-product
- Path expression
- o.l1.l2ln

R.book.author

Algebra

- Example of a probabilistic instance I

card(R,book)2,3 p(R)(B1,B2)0.5 p(R)(B1,B3)

0.3 p(R)(B2,B3)0.2

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

card(B3, title)1,1 p(B3)(A3,T2)1

Algebra (Projection)

Semistructured Instance

- Ancestor projection ( )
- e.g., we are only interested in authors but not

other details

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2

0.15

0.3

0.05

0.09

0.03

0.18

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2

0.15

0.3

0.05

0.09

0.03

0.18

- More efficient to compute locally
- input probabilistic instance

card(R,book)2,3 p(R)(B1,B2)0.5 p(R)(B1,B3)

0.3 p(R)(B2,B3)0.2

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

card(B3, title)1,1 p(B3)(A3,T2)1

- output probabilistic instance

card(R,book)2,3 p(R)(B1,B2)0.5 p(R)(B1,B3)

0.3 p(R)(B2,B3)0.2

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

p(B3)(A3)1

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2

0.15

0.3

0.05

0.09

0.03

0.18

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2

0.15

0.3

0.05

0.09

0.03

0.18

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.30.150.050.5

0.180.090.030.20.5

- input probabilistic instance

card(R,book)2,3 p(R)(B1,B2)0.5 p(R)(B1,B3)

0.3 p(R)(B2,B3)0.2

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

card(B3, title)1,1 p(B3)(A3,T2)1

- output probabilistic instance

card(R,book)0,1 p(R)()0.5(0.30.2) x prob.

of B3 has no child 0.5 0.5x0

0.5 p(R)(B3)(0.30.2) x prob. of B3 has a

child 0.5 x 1 0.5

card(B3, title)1,1 p(B3)(A3)1

- Experiments
- a few seconds for 300K objects and 10M OPF

entries - By measuring the slopes,
- running time is approximately linear to the

number of objects (selected objects and their

ancestors) - time to update the OPF entries of an object o is

sub-quadratic to the number of OPF entries

Algebra (Selection)

- Selection ( )
- e.g., we are not sure whether there exists T2 as

a title of some book, but we are interested to

keep the possible cases where the title T2 really

exists - R.book.title T2

Algebra (Selection)

- Selection ( )
- object selection condition
- e.g., we know that a particular author A1exists
- R.book.author A1
- value selection condition
- e.g., R.book.title XML

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2

0.15

0.3

0.05

0.09

0.03

0.18

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2

0.15

0.3

0.05

0.09

0.03

0.18

- D(W)
- the set of all semistructured instances

compatible with the weak instance W

0.2/0.50.4

0.09/0/50.18

0.03/0/50.06

0.18/0.50.36

- input probabilistic instance

card(R,book)2,3 p(R)(B1,B2)0.5 p(R)(B1,B3)

0.3 p(R)(B2,B3)0.2

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

card(B3, title)1,1 p(B3)(A3,T2)1

- output probabilistic instance

card(R,book)2,3 p(R)(B1,B3)0.3/0.50.6 p(R)

(B2,B3)0.2/0.50.4

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

card(B3, title)1,1 p(B3)(A3,T2)1

Algebra (Cross product (x))

e.g., we want to combine two instances (of

information obtained from two sources) into one

card(R, book)1,1 p(R)(B1)0.2 p(R)(B2)0.8

I1 I2

card(R, book)1,1 p(R)(B3)0.3 p(R)(B4)0.7

card(R, book)2,2

I1 x I2

p(R)(B1,B3)0.2 x 0.3 0.06 p(R)(B1,B4)0.2

x 0.7 0.14 p(R)(B2,B3)0.8 x 0.3

0.24 p(R)(B2,B4)0.8 x 0.7 0.56

Probabilistic point query

- returns the probability that a given object

satisfies a given path expression

- Example of a probabilistic instance I

card(R,book)2,3 p(R)(B1,B2)0.5 p(R)(B1,B3)

0.3 p(R)(B2,B3)0.2

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

card(B3, title)1,1 p(B3)(A3,T2)1

P(R.book.authorA1)

probability that A1 is an author of some book?

(0.60.1)

x (0.50.3)

0.7 x 0.8 0.56

card(R,book)2,3 p(R)(B1,B2)0.5 p(R)(B1,B3)

0.3 p(R)(B2,B3)0.2

card(B1, author)1,2 p(B1)(A1)0.6 p(B1)(A2)

0.3 p(B1)(A1,A2)0.1

card(B2, author)2,2 p(B2)(A2,A3)1

card(B3, author)1,1

card(B3, title)1,1 p(B3)(A3,T2)1

Other Work Done

- Implementation of a prototype
- Experiment
- Execution time is linear to the total number of

ipf entries, i.e., the instance size - A paper accepted by ICDE

Aggregation

- Example aggregate query count(S1.convoy.truck)
- Example of a probabilistic instance

S1 convoy1 0.5 convoy2 0.3 convoy1,convoy2

0.2

convoy1 ts1,truck1,tank1 0.2 ts1,tank1,tank2

0.8

convoy2 ts2,truck3,truck4 0.3 ts2,truck4

0.7

Aggregation

- Example aggregate query count(S1.convoy.truck)
- Example of a probabilistic instance

S1 convoy1 0.5 convoy2 0.3 convoy1,convoy2

0.2

convoy1 ts1,truck1,tank1 0.2 ts1,tank1,tank2

0.8

convoy1 P(count0)0.8 P(count1)0.2

convoy2 ts2,truck3,truck4 0.3 ts2,truck4

0.7

convoy2 P(count1)0.7 P(count2)0.3

Aggregation

- Query count(S1.convoy.truck)

S1 convoy1 0.5 convoy2 0.3 convoy1,convoy2

0.2

P(count0)0.50.8 P(count1)0.50.2

convoy1 P(count0)0.8 P(count1)0.2

convoy2 P(count1)0.7 P(count2)0.3

Aggregation

- Query count(S1.convoy.truck)

S1 convoy1 0.5 convoy2 0.3 convoy1,convoy2

0.2

P(count0)0.50.8 P(count1)0.50.2

0.30.7 P(count2)0.30.3

convoy1 P(count0)0.8 P(count1)0.2

convoy2 P(count1)0.7 P(count2)0.3

Aggregation

- Query count(S1.convoy.truck)

S1 convoy1 0.5 convoy2 0.3 convoy1,convoy2

0.2

P(count0)0.50.8 P(count1)0.50.2 0.30.7

0.20.80.7 P(count2)0.30.3 0.20.80.3

0.20.20.7 P(count3)0.20.20.3

convoy1 P(count0)0.8 P(count1)0.2

convoy2 P(count1)0.7 P(count2)0.3

Aggregation

- Query count(S1.convoy.truck)

- Worst-case number of aggregate values is

exponential in the number of selected objects! - Thus, pruning is used to prune aggregate values

with very low probability.

P(count0)0.4 P(count1)0.422

P(count2)0.166 P(count3)0.012

PIXML

- Interval probability (ipf) instead of point

probability (OPF) to represent the local

probability of sets of children given the parent

exists. - A sound and complete operational semantics for

processing a query to obtain objects satisfying

the a query with occurrence probabilities

exceeding a threshold for all possible satisfying

interpretations.

Probabilistic Instance I Weak Instance W ipf

ipf(convoy2, ts2, truck3 , truck4)0.2, 0.3

ipf(convoy2, ts2, truck3)0.3, 0.5

ipf(convoy2, ts2, truck4)0.2, 0.4

PIXML Semantics (Local Interpretation)

- Interpretation
- Local interpretation, p
- a mapping from the set of non-leaf objects to

OPFs - Example
- p(convoy2) wconvoy2
- A local interpretation p satisfies a

probabilistic instance I iff for every non-leaf

object, p returns an OPF that is a probability

distribution w.r.t. PC(o) over ipf.

Probabilistic Instance I Weak Instance W ipf

p(convoy2)(ts2, truck3, truck4) 0.2

ipf(convoy2, ts2, truck3 , truck4)0.2, 0.3

p(convoy2)(ts2, truck3) 0.4

ipf(convoy2, ts2, truck3)0.3, 0.5

p(convoy2)(ts2, truck4) 0.4

ipf(convoy2, ts2, truck4)0.2, 0.4

PIXML Query Language

- Example of a query
- val(S1.convoy.tank) T80
- r-answer to the query Q on a probabilistic

instance I is the set of objects o that - satisfy Q
- and the sum of probabilities of compatible

instances containing o is greater than or equal

to r for all possible interpretations (that

satisfy I).

Operational Semantics

- Identify all objects that satisfy query Q.
- Check which object has an occurrence probability

exceeding the threshold r w.r.t. all global

interpretations (that satisfy the prob.

instance). - Compute the minimal occurrence probability of

every object identified in step 1 in polynomial

time. - Theorem
- Our operational semantics is sound and complete.

Example of Operational Semantics

- Query val(S1.convoy.tank) T80
- Example of a probabilistic instance
- tank1 is the only candidate

S1 convoy1,convoy2 1,1

convoy1 ts1,truck1,tank1 0.2,0.7 ts1,truck1,

tank2 0.3,0.8

convoy2 ts2,truck3 0.3,0.6 ts2,truck4

0.4,0.7

Local Interpretation convoy1 ts1,truck1,tank1

ts1,truck1,tank2 convoy2 ts2,truck3

ts2,truck4 P(S1a) P(S1b) P(S1c) P(S1d)

S1a

0.2 0.8 0.3 0.7 0.06 0.14 0.24 0.56

0.7 0.3 0.6 0.4 0.42 0.28 0.18 0.12

S1b

S1c

0.2

0.7

S1d

Possible that Infinitely Many Interpretations

Satisfy the Probabilistic Instance!

Example of Operational Semantics

S1 cex(convoy1) min. conditional probability

of occurrence of convoy1 minimize

p(convoy1,convoy2) subject to 1 lt

p(convoy1,convoy2) lt 1

convoy1 cex(tank1) min. conditional probability

of occurrence of tank1 minimize

p(ts1,truck1,tank1) subject to 0.2 lt

p(ts1,truck1,tank1) lt 0.7 0.3 lt

p(ts1,truck1,tank2) lt 0.8

cex(tank1) min p(ts1,truck1,tank1) 0.2

min. computed occurrence probability of tank1

cop(tank1) cex(convoy1) X cex(tank1) 1 X 0.2

0.2

cex(convoy1) min p(convoy1,convoy2) 1

Example of Operational Semantics

- if r lt 0.2, then r-answer of the query

val(S1.convoy.tank) T80 is tank1

otherwise, r-answer is empty.

Uncertain Spatiotemporal Databases

- Applications
- personal mobile locating (Global Positioning

System in cars, personal locators, etc)

measurement error - traffic monitoring delay in updates or periodic

updates - weather forecast (predict the path of a typhoon)

uncertain in prediction - prediction programs in surveillance applications

uncertainty in prediction of the paths of convoys - Different approaches are suitable for different

applications (e.g., what kind of information can

be obtained, or preferred to store? position,

speed, or path?)

Uncertain Spatiotemporal Databases

- Approach 1
- time as another spatial dimension ? uncertainty

problem in high dimensional spatial databases - e.g., time and place (hospitals) of birth of

every person ? a point in space-time - a probability distribution over a space-time

region where an object/event may be found (i.e.

sum over the region 1) - modify existing spatial structures to support

uncertainty, e.g. R-tree

- e.g. integers x, y, t
- uniform distribution P(x,y,t 0ltx,y,tlt4) 1/27
- P(x,y,t x,y,t 1 or 3) 1/52
- P(x,y,t x,y,t 2) 1/2

t

x

x

y

Uncertain Spatiotemporal Databases

- Approach 2
- an object/event may be found in several (possibly

overlapping) space time regions with interval

probabilities

t

L1,U1

L4,U4

x

L2,U2

y

L3,U3

Uncertain Spatiotemporal Databases

- Approach 2

- for a particular interpretation (actual prob

dist. of objects over the whole space-time), - prob in some point in Ri (at some time t) is in

Li, Ui - prob in Ri at time t sum of prob in all points

in Ri at time t - disjunction of prob in all regions at any

particular time t is not greater than 1 - disjunction of prob in region Ri over all time is

within Li, Ui

t

L1,U1

L4,U4

x

L2,U2

y

L3,U3

Uncertain Spatiotemporal Databases

- Approach 3
- a probability distribution over possible paths or

possible velocities of a moving object

t

pdf over possible paths P10.5 P20.3 P30.2

pdf over possible velocities

P3

x

P2

P1

y

e.g. a surveillance application estimates the

current velocity of a tank or even predicts its

possible paths

Uncertain Spatiotemporal Databases

- Approach 4
- a probability distribution over a region of a

moving object at a particular (discrete or

continuous) time

t

pdft

f(t)pdft

pdft-1

When the system gets update of position as time

goes, f(t) may be updated (for future time t)

x

y

e.g., p(x,y,t) 1/ (pitt) for (xxyy) lt tt

p(x,y,t) 0 otherwise

Related Work

- Semistructured Probabilistic Objects (SPOs)

(Dekhtyar, Goldsmith, Hawkes, in SSDBM, 2001) - SPO express contexts (not random variables) in a

semistructured manner - PXML data model stores XML data AND probabilistic

information.

Related Work

- ProTDB (Nierman, Jagadish, in VLDB, 2002)
- Independent probabilities assigned to each child

vs arbitrary distributions over sets of children - Tree-structured
- My model theory provides two formal semantics
- I propose a set of algebraic operators,

aggregations - I extend my model to deal with interval

probabilities with a query language

Related Work

- MOST model (Sistla, Wolfson, Chamberlain, et al.)
- a moving object data model
- use lower and upper bounds to represent uncertain

data (e.g. position, speed) without using any

probability distribution - propose a query language FTL
- an algorithm to process a limited class of FTL

queries (either without uncertainty or objects

with uncertain speed moving on fixed routes) - propose indexing and update policy

Summary

- PXML data model
- Semistructured instance
- Weak instance (add cardinality)
- Probabilistic instance (add opf)
- Semantics
- Local and Global Interpretation
- Algebra
- Projection, selection, cross product
- Aggregation

Summary

- PIXML
- Interval probability
- Query
- Uncertain spatiotemporal databases
- probability distribution over a region in

space-time - possible regions with interval probabilities
- probability distribution over possible paths or

velocities - probability distribution over a region of a

moving object at a particular time

Related Work

- Algebras TAX, SAL
- TAX (Jagadish, Lakshmanan, Srivastava, 2001)
- use pattern tree to extract subsets of nodes, one

for each embedding of pattern tree. - fixed number of children
- SAL (Beeri, Tzaban, 1999)
- bind objects to variables
- original structure is totally lost

Related Work

- Bayesian net (Pearl, 1988)
- random variables (probability of events)
- ours existence of children requires existence of

parents

PIXML

- Interval probability (ipf) instead of point

probability (OPF) to represent the local

probability of sets of children given the parent

exists.

Probabilistic Instance I Weak Instance W ipf

ipf(convoy2, ts2, truck3 , truck4)0.2, 0.3

ipf(convoy2, ts2, truck3)0.3, 0.5

ipf(convoy2, ts2, truck4)0.2, 0.4

PIXML Semantics (Local Interpretation)

- Interpretation
- Local interpretation, p
- a mapping from the set of non-leaf objects to

OPFs - Example
- p(convoy2) wconvoy2
- A local interpretation p satisfies a

probabilistic instance I iff for every non-leaf

object, p returns an OPF that is a probability

distribution w.r.t. PC(o) over ipf.

Probabilistic Instance I Weak Instance W ipf

p(convoy2)(ts2, truck3, truck4) 0.2

ipf(convoy2, ts2, truck3 , truck4)0.2, 0.3

p(convoy2)(ts2, truck3) 0.4

ipf(convoy2, ts2, truck3)0.3, 0.5

p(convoy2)(ts2, truck4) 0.4

ipf(convoy2, ts2, truck4)0.2, 0.4

PIXML Query Language

- Example of a query
- val(S1.convoy.tank) T80
- r-answer to the query Q on a probabilistic

instance I is the set of objects o that - satisfy Q
- and the sum of probabilities of compatible

instances containing o is greater than or equal

to r for all possible interpretations (that

satisfy I).

Operational Semantics

- Identify all objects that satisfy query Q.
- Check which object has an occurrence probability

exceeding the threshold r w.r.t. all global

interpretations (that satisfy the prob.

instance). - Compute the minimal occurrence probability of

every object identified in step 1 in polynomial

time. - Theorem
- Our operational semantics is sound and complete.

Example of Operational Semantics

- Query val(S1.convoy.tank) T80
- Example of a probabilistic instance
- tank1 is the only candidate

S1 convoy1,convoy2 1,1

convoy1 ts1,truck1,tank1 0.2,0.7 ts1,truck1,

tank2 0.3,0.8

convoy2 ts2,truck3 0.3,0.6 ts2,truck4

0.4,0.7

Local Interpretation convoy1 ts1,truck1,tank1

ts1,truck1,tank2 convoy2 ts2,truck3

ts2,truck4 P(S1a) P(S1b) P(S1c) P(S1d)

S1a

0.2 0.8 0.3 0.7 0.06 0.14 0.24 0.56

0.7 0.3 0.6 0.4 0.42 0.28 0.18 0.12

S1b

S1c

0.2

0.7

S1d

Possible that Infinitely Many Interpretations

Satisfy the Probabilistic Instance!

Example of Operational Semantics

S1 cex(convoy1) min. conditional probability

of occurrence of convoy1 minimize

p(convoy1,convoy2) subject to 1 lt

p(convoy1,convoy2) lt 1

convoy1 cex(tank1) min. conditional probability

of occurrence of tank1 minimize

p(ts1,truck1,tank1) subject to 0.2 lt

p(ts1,truck1,tank1) lt 0.7 0.3 lt

p(ts1,truck1,tank2) lt 0.8

cex(tank1) min p(ts1,truck1,tank1) 0.2

min. computed occurrence probability of tank1

cop(tank1) cex(convoy1) X cex(tank1) 1 X 0.2

0.2

cex(convoy1) min p(convoy1,convoy2) 1

Example of Operational Semantics

- if r lt 0.2, then r-answer of the query

val(S1.convoy.tank) T80 is tank1

otherwise, r-answer is empty.