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Managing Uncertainty in Semistructured Databases and Spatiotemporal Databases

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Title: Managing Uncertainty in Semistructured Databases and Spatiotemporal Databases


1
Managing Uncertainty in Semistructured Databases
and Spatiotemporal Databases
  • Edward Hung
  • University of Maryland, College Park
  • PhD Proposal Oral Defense, Apr 22, 2003

2
Outline
  • Motivating examples
  • PXML data model
  • Semantics
  • Algebra
  • Aggregation
  • PIXML data model
  • Uncertain spatiotemporal databases
  • Related work

3
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?

4
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?

5
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?

6
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.

7
Semistructured Data Model
8
PXML Data Model
  • Uncertainty
  • Existence of sub-objects
  • Number of sub-objects
  • Identity of the sub-objects

9
PXML Data Model (Cardinality)
  • Example of cardinality

card(B1, author)1,2
Weak Instance W Semistructured Instance card
10
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
11
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
12
  • D(W)
  • the set of all semistructured instances
    compatible with the weak instance W

13
card(B1, author)1,2
The set of all potential child set of B1, PC(B1)
A1, A2,
A1,
A2
14
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
15
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
16
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).

17
Semantics (Global Interpretation)
  • Local interpretation for efficient computation
  • Now we are going to assign probabilities of each
    compatible instance globally, which is more
    intuitive.

18
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.

19
  • 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
20
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

21
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
22
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

23
Semantics (Local ?? Global)
  • I have defined operators to convert between local
    and global interpretations.

24
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.

25
Algebra
  • Operators
  • Projection
  • Selection
  • Cross-product
  • Path expression
  • o.l1.l2ln

R.book.author
26
Algebra
  • Operators
  • Projection
  • Selection
  • Cross-product
  • Path expression
  • o.l1.l2ln

R.book.author
27
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
28
Algebra (Projection)
Semistructured Instance
  • Ancestor projection ( )
  • e.g., we are only interested in authors but not
    other details

29
  • 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
30
  • 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
31
  • 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
32
  • 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
33
  • 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
34
  • 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
35
  • 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
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
37
  • 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
38
  • 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

39
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

40
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

41
  • 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
42
  • 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
43
  • 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
44
  • 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
45
  • 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
46
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
47
Probabilistic point query
  • returns the probability that a given object
    satisfies a given path expression

48
  • 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
49
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
50
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

51
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
52
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
53
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
54
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
55
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
56
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
57
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.

58
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
59
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.

60
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
61
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).

62
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.

63
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
64
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!
65
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
66
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.

67
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?)

68
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
pdf
x
x
y
69
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
70
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
71
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
72
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
73
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.

74
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

75
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

76
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

77
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

78
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

79
Related Work
  • Bayesian net (Pearl, 1988)
  • random variables (probability of events)
  • ours existence of children requires existence of
    parents

80
PIXML
  • Interval probability (ipf) instead of point
    probability (OPF) to represent the local
    probability of sets of children given the parent
    exists.

81
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
82
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.

83
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
84
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).

85
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.

86
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
87
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!
88
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
89
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.
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