Belief Augmented Frames
14 June 2004
Colin Tan
ctank@comp.nus.edu.sg
http://www.comp.nus.edu.sg/~ctank
Motivation
• Primary Objective:
–
To study how uncertain and defeasible
knowledge may be integrated into a
knowledge base.
• Main Deliverable:
–
A system of theories and techniques that
allow us to integrate new knowledge we
have gained, and to use this knowledge to
make better inferences
Proposed Solution
• A frame-based reasoning system
augmented with belief measures.
–
Frame-based system to structure
knowledge and relations between entities.
– Belief measures provide uncertain
reasoning on existence of entities and the
relationships between them.
Why Belief Measures?
• Statistical Measures
–
Standard tool for modeling uncertainty.
– Essentially, if the probability that a
proposition E is true is p, then the
probability of that E is false is 1-p.
• P(E) = p
• P(not E) = 1-p
Why Belief Measures?
• This relationship between P(E) and
P(not E) introduces a problem:
–
This relationship essentially leaves no
room for ignorance. Either the proposition
is true with a probability of p, or it is false
with a probability of 1-p.
– This can be counter-intuitive at times.
Why Belief Measures?
• [Shortliffe75] cites a study in which,
given a set of symptoms, doctors were
willing to declare with certainty x that a
patient was suffering from a disease D,
yet were unwilling to declare with
certainty 1-x that the patient was not
suffering from D.
Why Belief Measures?
• To allow for ignorance our research
focuses on belief measures.
• The ability to model ignorance is
inherent in belief systems.
–
E.g. in Dempster-Shafer Theory
[Dempster67], if our belief in E1 and E2
are 0.1 and 0.3 respectively, then the
ignorance is (1 – (0.1 + 0.3)) = 0.6.
Why Frames?
• Frames are a powerful form of
representation.
–
Intuitively represents relationships
between objects using slot-filler pairs.
• Simple to perform reasoning based on
relationships.
–
Hierarchical
• Can perform generalizations to create general
models derived from a set of frames.
Why Frames?
• Frames are powerful form of
representation:
–
Daemons
• Small programs that are invoked when a
frame is instantiated or when a slot is filled.
Combining Frames with
Uncertainty Measures
• Augmenting slot-value pairs with
uncertainty values.
–
Enhance expressiveness of relationships.
– Can now do reasoning using the
uncertainty values.
• A Belief Augmented Frame (BAF) is a
frame structure augmented with belief
measures.
Example BAF
Donkey
0.6, 0.3
color
1.0, 0.0
Grey,
1.0, 0.0
owns
0.7, 0.2
Alice,
1.0, 0.0
walks
0.9, 0.1
Blue,
1.0, 0.0
color
1.0, 0.0
Dog
0.9, 0.0
location
1.0, 0.0
Bay,
1.0, 0.0
Belief Representation in
Belief Augmented Frames
• Beliefs are represented by two masses:
–
–
–
φT: Belief mass supporting a proposition.
φF: Belief mass refuting a proposition.
In general φT + φF  1
• Room to model ignorance of the facts.
• Separate belief masses allow us to:
Draw φT and φF from different sources.
– Have different chains of reasoning for φT and φF.
–
Belief Representation in
Belief Augmented Frames
• This ability to derive the refuting masses
from different sources and chains of
reasoning is unique to BAF.
–
–
In Probabilistic Argumentation Systems (the
closest competitor to BAF) for example, p(not E)
= 1 – p(E).
Possible though to achieve this in Dempster
Shafer Theory through the underlying
mechanisms generating m(E) and m(not E).
Belief Representation in
Belief Augmented Frames
• BAFs however give a formal framework for
deriving T and F
–
BAF-Logic, a complete reasoning system for
BAFs.
• BAFs provide a formal framework for Frame
operations.
–
E.g. how to generalize from a given set of
frames.
• BAF and DST can in fact be complementary:
–
BAF as a basis of generating masses in DST
Degree of Inclination
• The Degree of Inclination is defined as:
–
DI = T - F
• DI is in the range of [-1, 1].
• One possible interpretation of DI:
-1
False
-0.75
Most
Probably
False
-0.5
Probably
False
-0.25
Likely
False
0
Ignorant
0.25
Likely
True
0.5
Probably
True
0.75
Most
Probably
True
1
True
Utility Value
• The Degree of Inclination DI can be remapped to the range [0, 1] through the
Utility function:
–
U = (DI + 1) / 2
– By normalizing U across all relevant
propositions it becomes possible to use U
as a statistical measure.
Plausibility, Ignorance,
Evidential Interval
• Plausibility pl is defined as:
pl = 1 - F
• Ignorance ig is defined as:
ig = pl – T
= 1 – (T + F)
• The Evidential Interval EI is defined to
be the range
EI =[T, pl]
Interpreting the Evidential
Interval
Evidential Interval
Interpretation
[0, 1]
[0, 0]
Complete ignorance.
[1, 1]
The evidence provided completely
supports the fact.
[T, Pl] 0 < T, Pl < 1
Pl  T
[T, Pl] 0 < T, Pl < 1
Pl < T
The evidence both supports and
refutes the fact.
The evidence provided completely
refutes the fact.
The evidence supporting the fact
exceeds the plausibility of the fact.
I.e. the evidence is contradictory.
Reasoning with BAFs
• Belief Augmented Frame Logic, or
BAF-Logic, is used for reasoning with
BAFs.
• Throughout the remainder of this
presentation, we will consider two
propositions A and B, with supporting
and refuting masses TA, FA, TB, and
FB.
Reasoning with BAFs
AND, OR, NOT
• A  B:
–
–
TA B = min(TA, TB)
FA B = max(FA, FB)
• A  B:
–
–
TA  B = max(TA, TB)
FA  B = min(FA, FB)
•  A:
–
–
T A = F A
F A = T A
Default Reasoning in BAF
• When the truth of a proposition is unknown,
then we set the supporting and refuting
masses to TDEF and FDEF respectively.
–
Conventionally, TDEF = FDEF = 0
• Two special default values:
–
–
TONE = 1 , FONE = 0
TZERO = 0 , FZERO = 1
• Used for defining contradiction and
tautology.
Default Reasoning in BAF
• Other default reasoning models are
possible too.
–
E.g. categorical defaults:
• : (A, TA , FA)  (B, TB , FB) / (B, TB , FB)
• Semantics:
– Given a knowledge base KB.
– If KB :- A and KB :-/-  B, infer B with supporting
and refuting masses TB and FB
–
Detailed study of this topic still to be
made.
BAF and Propositional Logic
• BAF-Logic properties that are identical
to Propositional Logic:
–
Associativity, Commutativity, Distributivity,
Idempotency, Absorption, De-Morgan’s
Theorem, - elimination.
BAF and Propositional Logic
• Other properties of Propositional Logic
work slightly differently in BAF-Logic.
–
In particular, some of the properties hold
true only if the constituent propositions are
at least “probably true” or “probably false”
• I.e. |DIP |  0.5
BAF and Propositional Logic
• For example, P and P Q must both
be at least probably true for Q to not be
false.
–
If DIP and DIP Q are less than 0.5, DIQ
might end up < 0.
• For  - elimination, P  Q must be
probably true, and P must be probably
false, before we can infer that Q is not
false.
BAF and Propositional Logic
• This can lead to unexpected reasoning
results.
–
E.g. P, P Q are not false, yet DIQ < 0.
• A possible solution is to set {TQ =  TDEF ,
FQ =  FDEF} when DIP and DIPQ are less
than 0.5
• In actual fact, the magnitude of DIP and DIP
Q don’t both have to be  0.5. Only their
average magnitudes must be  0.5.
Belief Revision
• Beliefs are not static. We need a mechanism
to update beliefs [Pollock00].
• To track the revision of belief masses, we
add a subscript t to time-stamp the masses.
–
E.g. TP,0 is the value of TP at time 0, TP,1 at
time 1 etc.
• At time t, given a proposition P with masses
TP, t and FP,t, suppose we derive masses
TP, * and FP, *, then the new belief masses at
time t+1 are:
–
–
TP, t+1 =  TP, t + (1-  ) TP, *
FP, t+1 =  FP, t + (1-  ) FP, *
Belief Revision
• Intuitively, this means that we give a
credibility factor  to the existing
masses, and (1-  ) to the derived
masses.
•  therefore controls the rate at which
beliefs are revised, given new
evidence.
An Example
• Given the following propositions in your
knowledge base:
–
KB = {(A, 0.7, 0.2), (B, 0.9, 0.1), (C, 0.2,
0.7), (A B R, TONE , FONE,), (A  B
R, TONE , FONE)}
– We want to derive TR, 1, FR, 1.
An Example
• Combining our clauses regarding R,
we obtain:
–
R = (A  B)   (A   B)
• = A  B  ( A  B)
• With De-Morgan’s Theorem we can
derive  R:
–
 R= A   B  (A   B)
An Example
• TR,* = min(TA , TB , max(FA , TB ))
= min(0.7, 0.9, max(0.2, 0.9))
= min(0.7, 0.9, 0.9)
= 0.7
• FR,* = max(FA , FB , min(TA , FB ))
= max(0.2, 0.1, min(0.7, 0.1))
= max(0.2, 0.1, 0.1)
= 0.2
An Example
•
•
We begin with default values for R:
–
TR,0 = TDEF
–
= 0.0
= FDEF
= 0.0
FR,0
This gives us the following attributes:
An Example
Measure
Value
DIR, 0
0.0
PlR,0
1.0
IgR,0
1.0
EIR,0
[0.0, 1.0]
An Example
• Deriving the new belief values with  =
0.4
–
TR,1 = 0.4 * 0.0 + (1.0 – 0.4) * 0.7
–
= 0.42
= 0.4 * 0.0 + (1.0 – 0.4) * 0.2
= 0.12
FR,1
• This gives us:
An Example
Measure
Value
DIR, 1
0.42 – 0.12 = 0.30
PlR,1
1.0 – 0.12 = 0.88
IgR,1
0.88 – 0.42 = 0.46
EIR,1
[0.42, 0.88]
An Example
• We see that with our new information
about R, our ignorance falls from 1.0
(total ignorance) to 0.46. With more
knowledge available about whether R
is true, we also see the plausibility
falling from 1.0 to 0.88.
• Further, suppose it is now known that:
–
B C R
An Example
• Combining our clauses regarding R,
we obtain:
–
R = (A  B)  (B  C)  (A   B)
= A  B  C  ( A  B)
• With De-Morgan’s Theorem we can
derive  R:
–
 R= A   B   C  (A   B)
An Example
•  TR,* = min(TA , TB , TC , max(FA , TB ))
= min(0.7, 0.9, 0.2, max(0.2, 0.9))
= min(0.7, 0.9, 0.2, 0.9)
= 0.2
• FR,* = max(FA , FB , FC , min(TA , FB ))
= max(0.2, 0.1, 0.7, min(0.7, 0.1))
= max(0.2, 0.1, 0.7, 0.1)
= 0.7
An Example
• Updating the beliefs:
–
TR,2 = 0.4 * 0.42 + (1.0 – 0.4) * 0.2
–
= 0.288
= 0.4 * 0.12 + (1.0 – 0.4) * 0.7
= 0.468
FR,2
• This gives us:
An Example
Measure
Value
DIR, 2
0.288 – 0.468 = -0.18
PlR,2
1.0 – 0.468 = 0.532
IgR,2
0.532 – 0.288 = 0.244
EIR,2
[0.288, 0.532]
An Example
• Here the new evidence that B C R
fails to support R, because C is not
true (DIC = -0.5)
• Hence the plausibility of R falls from
0.88 to 0.532, while the truth value
DIR,2 enters into the negative range.
Integrating Belief Measures
with Frames
• Belief measures to quantify:
–
The existence of the object/concept
represented by the frame.
– The existence of relations between frames
Frames with Belief Measures
Camp
(1.0, 0.0)
Baby
(0.75, 0.25)
IsA
At
(1.0, 0.0)
Ike
(1.0, 0.0)
Brother
Kyle
Friend
(0.8, 0.2)
Likes
Kicked
(0.9, 0.1)
(0.9, 0.1)
Kenny
Helped
(0.75, 0.25)
Insults
At
Stan
(0.9, 0.1)
(0.75, 0.25)
StudiesAt
Friend
(1.0, 0.0)
South Park
Integrating Belief Measures
with Frames
• Deriving Belief Values
–
BAF-Logic statements can be used to derive
belief measures.
• For example, suppose we propose that:
–
–
Sam is Bob’s son if Sam is male and Bob has a
child.
Within our knowledge base, we have {(Sam is
male, 0.6, 0.2), (Bob has child, 0.8, 0.1), (Sam is
male  Bob has child  Sam is Bob’s Son, 0.7,
0.1)}
Integrating Belief Measures
with Frames
• Assuming that  = 0, we can derive:
 Tsam,son,bob
 Fsam,son,bob
DIsam,son, bob
Plsam, son, bob
Igsam, son, bob
= min(0.6, 0.8, 0.7)
= 0.6
= max(0.2, 0.1, 0.1)
= 0.2
= 0.4
= 0.8
= 0.2
Integrating Belief Measures
with Frames
• Daemons
–
Can be activated based on belief masses,
DI, EI, Ig and Pl values.
– Can act on DI, EI, Ig, Pl values for further
processing.
• E.g. if it is likely that Sam is Bob’s son, and if
the ignorance is less than 0.2, create a new
frame School, and set Sam, Student, School
relationship.
Frame Operations
• add_frame, del_frame, add_rel, etc. etc.
• More interesting operations include abstract:
–
–
–
–
Given a set of frames
Create a super-frame that is the parent of the set
of frames.
Copy relations that occur in at least  % of the
set of frames to the superframe.
Set the belief masses to be a composition of all
the belief masses in the set for that relation.
Application Examples
Discourse Understanding
• Discourse can be translated to a
machine understandable form before
being cast as BAFs.
• Discourse Representation Structures
(DRS) are particularly useful.
–
Algorithm to convert from DRS to BAF is
trivial [Tan03].
Application Examples
Discourse Understanding
• Setting Belief Masses
–
Initial belief masses may be set using
fuzzy-sets.
• E.g. to model a person being helpful
– Shelpful = {1.0/”invaluable”, 0.75/”very helpful”,
0.5/”helpful”, 0.25/”unhelpful”, 0.0/”uncooperative”}
• If we say that Kenny is very helpful, we can
set:
– Tkenny_helpful = 0.75
– Fkenny_helpful = 1.0 - 0.75= 0.25
Application Examples
Discourse Understanding
• Further propositions and rules may be
inserted into the knowledge base to
perform reasoning on the initial belief
masses.
• Propositions and rules modeled as
prolog clauses.
Application Examples
Text Classification
• Can model text classification as a BAF
problem:
–
In BAF-Logic the jth document Dij in the
document class ci is taken to be a conjunction of
terms tk:
• Dij = tij0  tij1  …  tij(n-1)
–
Each term and document is related by a set of
relations:
• Rijk = {(Dij, term, tk,  Tijk,  Fijk) | tk is a term in Dij}
Application Examples
Text Classification
• Given a set of documents D in class ci, we
apply the abstract operator to produce the
set of relations characterizing ci.
–
v = (Si0, Si1, Si2, … Si(m-1))
• Each Sik is the relation:
Sik = {(ci, term, tk,  Tik,  Fik) | tk occurs in at least
 % of documents Dj in class ci}
–  Tik = minj  Tijk
–  Fik = maxlmaxj  Tljk, l  i
–
Application Examples
Text Classification
–
 Tik is our belief that the term tk implies that the
–
document belongs to class ci.
 Fik is our belief that the term tk implies that the
document belongs to some other class cl.
• Given an unseen document Du, we derive
the keyword terms tunk, k. We can derive the
following masses that support and refute the
proposition that Du belongs to class ci.
Application Examples
Text Classification
–
–
Ti, unk = min( Ti0,  Ti1, …max( Fi0,  Fi1, …))
Fi, unk = max( Fi0,  Fi1, …min( Ti0,  Ti1, …))
• From this we derive the degree of inclination
using the standard definition:
–
DIi, unk = Ti, unk - Fi, unk
• We choose the class with the largest DI as
the winner.
–
win = argmax DIi,unk
Text Classification
Experiment I
• Corpus used: 20 Newsgroups
–
–
–
20,000 USENET articles culled from 20
newsgroups.
19,600 articles to train classifiers, 400 to test.
Relatively poor performance from classifiers due
to nature of USENET postings.
• Jeffreys-Perks Law used to smoothen
statistics.
Classification Results
Inside Testing
Text Classification - Inside Test
Classification Accuracy (%)
100
90
80
70
60
NBAYES
50
BAF
40
PAS
30
20
10
0
0%
10%
Degree of Abstraction
20%
Classification Results
Outside Testing
Text Classification - Outside Test
Classification Accuracy (% )
70
60
50
NBAYES
BAF
PAS
40
30
20
10
0
0%
10%
Degree of Abstraction
20%
Classification Results
Overall
Classification Results - Overall
Classification Accuracy (% )
80
70
60
50
NBAYES
BAF
PAS
40
30
20
10
0
0%
10%
Degree of Abstraction
20%
Text Classification
Analysis
• Both BAF and Probabilistic
Argumentation Systems (PAS) perform
better than Naïve Bayes (NBAYES).
• BAF performs significantly better than
PAS for unseen documents.
• However performance for seen
documents is mixed. PAS and BAF
appear to have similar performance.
Text Classification
Experiment II
• Corpus Used: Reuters Newswire articles
–
–
2,000 articles in 25 categories for training.
500 articles for testing.
• Results:
–
Similar to Experiment I
• Compared with PAS, mixed performance for seen data.
• Superior performance for unseen data.
• PAS and BAF both have superior performance to Naïve
Bayes.
Text Classification
Conclusions
• Both BAF and PAS perform better than
Naïve Bayes.
• BAF and PAS have similar
performance for seen data.
• BAF has better performance over PAS
for unseen data.
Publications
C. K. Y. Tan, K. T. Lua, “Discourse Understanding
with Discourse Representation Theory and Belief
Augmented Frames”, 2nd International
Conference on Computational Intelligence,
Robotics and Autonomous Systems, Singapore,
2003.
C. K. Y. Tan, K. T. Lua, “Belief Augmented Frames
for Knowledge Representation in Spoken
Dialogue Systems”, 1st International Indian
Conference on Artificial Intelligence, Hyderabad,
India, 2003.
Publications
C. K. Y. Tan, “Text Classification using Belief
Augmented Frames”, 8th Pacific Rim International
Conference on Artificial Intelligence, Auckland,
2004.
C. K. Y. Tan, “Belief Augmented Frames”, Doctoral
Thesis, Department of Computer Science,
School of Computing, National University of
Singapore, 2003.
Current and Future Work
• Currently:
–
Developing a BAF Reasoning Engine
• Future:
–
Dialog Management using BAFs
– Automatic Text Classification
– AI Engine for Game Playing
Conclusion
• Use of belief measures to quantify
uncertainty.
–
Room for ignorance
• Use of Frames to organize knowledge.
–
Frames represent objects or ideas in the world.
– Slot-filler pairs represent relations between
frames.
– Relations are weighted by belief measures.
References
• [Shortliffe75] E. H. Shortliffe, B. G.
Buchanan, “A Model of Inexact Reasoning in
Medicine”, Mathematical Biosciences Vol 23,
pp 351-379, 1975.
• [Dempster67] A. P. Dempster, “Upper and
Lower Probabilities Induced by a Multivalued
Mapping”, The Annals of Mathematical
Statistics Vol 38 No 2, pp 325-339, 1967
References
• [Pollock00] J. L. Pollock, A. S. Gilles, “Belief
Revision and Epistemology”, Synthese 122,
pp 69-92, 2000.
• [Tan03] C. K. Y. Tan, K. T. Lua, “Discourse
Understanding with Discourse
Representation Theory and Belief
Augmented Frames”, 2nd International
Conference on Computational Intelligence,
Robotics and Autonomous Systems,
Singapore, 2003.