Belief Augmented Frames for
Knowledge Representation
Colin Tan,
Department of Computer Science,
School of Computing,
National University of Singapore.
Belief Augmented Frames
Motivation
• Frames
– Flexible, intuitive way of representing
knowledge.
– Frames represent an entity or a concept
– Frames consist of slots with values
• Represents relations between current frame and
other frames.
– Slots have events attached to them
• Can invoke procedures (“daemons”) whenever a
slot’s value is changed, removed etc.
Belief Augmented Frames
Motivation
• In the original definition of frames, slotvalue pairs are “definite”.
• One improvement is to introduce
uncertainties into these relations.
Belief Augmented Frames
Motivation
• Statistical representations are not always ideal:
– If we are p% certain that a fact is true, this doesn’t
mean that we are 1-p% certain that it is false.
• Various uncertainty reasoning methods introduced
to address this:
– Dempster-Schafer Theory
– Transferrable Belief Model
– Probabilistic Argumentation Systems
Belief Augmented Frames
Motivation
• By combining uncertainty measures with
frames:
– Uncertainties in slot-value pair assignments
provide frames with greater expressiveness and
reasoning ability.
– Frames offer a neat intuitive structure for
reasoning uncertain relations.
Modeling Uncertainty in
Belief Augmented Frames
• Uncertainty not only on slot-value pair
assignments, but also on the existence of the
concept/object represented by the frame.
• The belief mass Tf is called the
“Supporting Mass”, and it is the degree of
support that the fact f is true.
• Likewise Ff is the Refuting Mass, and it is
the degree of support that the fact f is false.
Modeling Uncertainty in
Belief Augmented Frames
• In general:
• 0  Tf , Ff  1
• Tf is fully independent of Ff
• Tf + Ff  1
• The Degree of Inclination Dif is defined as:
– Dif = Tf - Ff
• The Plausibility plf is defined as:
– Plf = 1 - Ff
Combining Belief Masses
• Fuzzy-logic style min-max functions are
used to combine belief masses from
different facts.
• Given two facts P and Q:
– Conjunctions
• TPQ = min(TP, TQ)
• FPQ = max(FP, FQ)
Combining Belief Masses
• Given two facts P and Q:
– Disjunctions
• TPQ = max(TP, TQ)
• FP  Q = min(FP, FQ)
– Negation
• TP = FP
• FP = TP
Reasoning Example
• Suppose we have the following facts:
– P:-A   B, P:-A  C  B, P:-A  B
– Supose also that A, B and C are three facts
defined as follows (Fact, Tfact, Ffact)
• (A, 0.3, 0.4), (B, 0.9, 0.2), (C, 0.8, 0.3)
– Then to evalute P:
• P =((A   B)  (A  C  B))  (A  B)
• P = ((A   B)  (A  C  B))  (A  B)
= ((A  B)  (A  C  B))  (A  B)
Reasoning Example
– P =((A   B)  (A  C  B))  (A  B)
• TP = min(max(min(TA, FB), min(FA, TC, FB)),
max(FA, TB))
= min(max(min(0.3, 0.2), min(0.4, 0.8, 0.2)),
max(0.4, 0.9))
= min(max(0.2, 0.2), max(0.4, 0.9))
= min(0.2, 0.9)
= 0.2
Reasoning Example
– P = ((A  B)  (A  C  B))  (A  B)
FP = max(min(max(FA, TB), max(TA, FC, TB)),
min(TA, TB))
= max(min(max(0.4, 0.9), max(0.3, 0.3, 0.9)),
min(0.3, 0.9))
= max(min(0.9, 0.9), min(0.3, 0.9))
= 0.9
DIP = 0.2 – 0.9 = -0.7
PlsP = 1 – 0.9 = 0.1
Major BAF Operations
• add_concept(con, ex_t, ex_f)
– Creates a new frame named “con” representing a
concept or an object. ex_t and ex_f are respectively the
supporting and refuting masses for the existence of
“con”.
• add_concept_inh(con, par, ex_t, ex_f, in_t, in_f)
– Creates a new frame “con” inheriting all the slot-value
assignments of a parent frame “par”.
– Sets up “isChild” and “isParent” relations between the
new and parent frames. in_t and in_f are respectively
the supporting and refuting masses for these relations.
Major BAF Operations
• add_rel(frame, slot, value, r_t, r_f)
– Adds a new relationship between current frame “frame”
and another frame “value”, with the relation given in
“slot” r_t and r_f are respectively the supporting and
refuting masses for this relation.
• abstract(frame, frame_set)
– Creates a new frame “frame” consisting of slot-value
assignments found in every frame in the given set
“frame_set”.
– The new frame will thus contain all the slot-value
assignments common to every frame in the set.
Daemons
• Daemons, or event handlers, may be
attached to slots to respond to any of the
following events.
– NewValue:A new value is being added to a slot.
– DelValue: An existing value is being deleted.
– UpdValue: The belief masses for an existing
value is being updated.
– NeedValue: An empty slot needs a value.
Daemons
• Demons respond with:
–
–
–
–
dm_Ignore: No action to be taken
dm_Rule: Execute a returned rule set
dm_Assign: Assign a returned value to the slot
dm_Delete: Remove existing value from slot
• dm_Rule can be returned together with
dm_Assign, dm_Delete and dm_Ignore.
• dm_Assign has two modes:
– Exclusive: All previous slot assignments are removed
– Inclusive: Returned value is attached to previous
values.
Rules
• Rules may be attached to modify the
existence or relation belief values. E.g.
– rel(a, rel1, b) :- rel(c,rel2,d), rel(c,rel3,e)
– The Supporting and Refuting belief masses for
a rel1 b will depend on the Supporting and
Refuting belief masses for the right hand side.
– Rules are “Prolog-like”, everything on RHS is
taken to be a conjunction.
– Identical named rules are taken to be
conjunctions, and the final belief values are
evaluated as in the earlier example.
Application
• Text Classification
– Compute term frequency ft from all documents
in a class.
– Let fti be the term frequency of term t in
document i in class c. Let ftk be the term
frequency of term t in document k of class c’,
where c’  c
• cT = mini(fti)
• cF = maxk(ftk)
Application
• Text Classification
– For an unseen document dm, extract all terms tm.
– Compute:
• Tc = min(TCt)
• Fc = max(FCt)
– Here TCt, FCt = Support and Refuting mass for
term t in class C, as computed earlier.
– Compute DIC = TCt  FCt
– The document is classified as class cmax =
argmaxC(DIC)
Application
• Text Classification
– Trained on 400 news articles broken into 5
categories.
– Tested on training set (inside test) and unseen
set of 100 articles.
– Evaluation of BAF vs. Naïve Bayes
Results – Inside Test
Naive Bayes vs. BAF - Inside Test
100
Accuracy (%)
90
80
70
60
50
40
30
20
10
0
Naïve Bayes
BAF
Naïve Bayes
BAF
None
AddOne
Jperks
56.31
85.1
89.89
51
88.4
88.4
Smoothing
Results – Outside Test
Naive Bayes vs. BAF - Outside Test
90
80
70
Accuracy (%)
60
50
Naïve Bayes
40
BAF
30
20
10
0
Naïve Bayes
BAF
None
AddOne
Jperks
36.54
64.42
67.3
64
83.7
83.7
Smoothing
Results - Analysis
• BAF outperforms Naïve Bayes for all
outside tests.
• BAF performs slightly less well than Naïve
Bayes without smoothing or with JeffreysPerks smoothing for inside tests
• Results are promising – Use thesaurus to
improve results?
Difficulties
• Semantics of slots is ill-defined
– There is no fixed way to use the slots to
represent relations between frames.
– This complicates the modeling of real-world
English sentences. E.g.
• The black cat stole the purse.
– Should this be modeled as stole(subject: cat,
object: purse), cat(action:stole, object:purse),
purse(action:stolenby, subject:cat)?
Difficulties
• Many ways to derive a-priori Supporting
and Refuting masses.
– Some ways might be better than others.
• Separation of Supporting and Refuting
masses introduces additional problems that
can make modeling awkward and counterintuitive:
– E.g. plsf < Tf , when Tf , Ff > 0.5
Difficulties
• The range for the sum of Tf ,andFf falls in
[0, 2] instead of [0, 1]. This is again
counter-intuitive.
Future Work
• A model for measuring the quality of the
knowledge in the BAF knowledge base
should be developed.
• To apply BAF to more problems to study its
behavior, especially against established
methods.