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Belief Augmented Frames 14 June 2004

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- 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

- 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.

- 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

- 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.

- [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.

- 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.

- 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.

- Intuitively represents relationships between objects using slot-filler pairs.

- Frames are powerful form of representation:
- Daemons
- Small programs that are invoked when a frame is instantiated or when a slot is filled.

- Daemons

- 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.

- 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 φTand φFfrom different sources.
- Have different chains of reasoning for φT and φF.

- 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).

- 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

- The Degree of Inclination is defined as:
- DI = T - F

- DI is in the range of [-1, 1].
- One possible interpretation of DI:

- The Degree of Inclination DI can be re-mapped 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 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]

- 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.

- 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

- 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.

- 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 TBand FB

- Detailed study of this topic still to be made.

- E.g. categorical defaults:

- BAF-Logic properties that are identical to Propositional Logic:
- Associativity, Commutativity, Distributivity, Idempotency, Absorption, De-Morgan’s Theorem, - elimination.

- 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

- In particular, some of the properties hold true only if the constituent propositions are at least “probably true” or “probably false”

- For example, P and P Q must both be at least probably true for Q to not be false.
- If DIPand DIP Qare less than 0.5, DIQmight 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.

- 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 DIPand DIPQare less than 0.5
- In actual fact, the magnitude of DIPand DIP Qdon’t both have to be 0.5. Only their average magnitudes must be 0.5.

- 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 TPat time 0, TP,1at 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, *

- 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.

- 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 BR, TONE , FONE)}
- We want to derive TR, 1, FR, 1.

- Combining our clauses regarding R, we obtain:
- R = (A B) (A B)
- = A B ( A B)

- R = (A B) (A B)
- With De-Morgan’s Theorem we can derive R:
- R= A B (A B)

- 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

- We begin with default values for R:
- TR,0= TDEF
= 0.0

- FR,0= FDEF
= 0.0

- TR,0= TDEF
- This gives us the following attributes:

- Deriving the new belief values with = 0.4
- TR,1 = 0.4 * 0.0 + (1.0 – 0.4) * 0.7
= 0.42

- FR,1 = 0.4 * 0.0 + (1.0 – 0.4) * 0.2
= 0.12

- TR,1 = 0.4 * 0.0 + (1.0 – 0.4) * 0.7
- This gives us:

- 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

- Combining our clauses regarding R, we obtain:
- R = (A B) (B C) (A B)
= A B C ( A B)

- R = (A B) (B C) (A B)
- With De-Morgan’s Theorem we can derive R:
- R= A B C (A B)

- 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

- Updating the beliefs:
- TR,2 = 0.4 * 0.42 + (1.0 – 0.4) * 0.2
= 0.288

- FR,2 = 0.4 * 0.12 + (1.0 – 0.4) * 0.7
= 0.468

- TR,2 = 0.4 * 0.42 + (1.0 – 0.4) * 0.2
- This gives us:

- 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,2enters into the negative range.

- Belief measures to quantify:
- The existence of the object/concept represented by the frame.
- The existence of relations between 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)}

- Assuming that = 0, we can derive:
Tsam,son,bob = min(0.6, 0.8, 0.7)

= 0.6

Fsam,son,bob = max(0.2, 0.1, 0.1)

= 0.2

DIsam,son, bob = 0.4

Plsam, son, bob= 0.8

Igsam, son, bob= 0.2

- 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.

- 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.

- 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].

- 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

- E.g. to model a person being helpful

- Initial belief masses may be set using fuzzy-sets.

- 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.

- Can model text classification as a BAF problem:
- In BAF-Logic the jth document Dijin 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}

- In BAF-Logic the jth document Dijin the document class ci is taken to be a conjunction of terms tk:

- 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) | tkoccurs in at least % of documents Djin class ci}
- Tik = minj Tijk
- Fik = maxlmaxj Tljk, l i

- 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.

- Ti, unk = min( Ti0, Ti1, …max( Fi0, Fi1, …))
- Fi, unk = max( Fi0, Fi1, …min( Ti0, Ti1, …))

- DIi, unk = Ti, unk - Fi, unk

- win = argmax DIi,unk

- 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.

- 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.

- 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.

- Similar to Experiment I

- 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.

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.

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.

- Currently:
- Developing a BAF Reasoning Engine

- Future:
- Dialog Management using BAFs
- Automatic Text Classification
- AI Engine for Game Playing

- 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.

- [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

- [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.