Belief Augmented Frames Belief Augmented Frames 14 June 2004 Colin Tan

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 E 1 and E 2 are 0.1 and 0.3 respectively, then the ignorance is (1 – ( )) = 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

Belief Representation in Belief Augmented Frames Beliefs are represented by two masses: – φ T : Belief mass supporting a proposition. – φ F : Belief mass refuting a proposition. – In general φ T + φ F  1 Room to model ignorance of the facts. Separate belief masses allow us to: – Draw φ T and φ F from different sources. – Have different chains of reasoning for φ T and φ F.

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:

Utility Value 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, 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 IntervalInterpretation [0, 1] Complete ignorance. [0, 0] The evidence provided completely refutes the fact. [1, 1] The evidence provided completely supports the fact. [  T, Pl] 0 <  T, Pl < 1 Pl   T The evidence both supports and refutes the fact. [  T, Pl] 0 <  T, Pl < 1 Pl <  T 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  T A,  F A,  T B, and  F B.

Reasoning with BAFs AND, OR, NOT A  B: –  T A  B = min(  T A,  T B ) –  F A  B = max(  F A,  F B ) A  B: –  T A  B = max(  T A,  T B ) –  F A  B = min(  F A,  F B )  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  T DEF and  F DEF respectively. – Conventionally,  T DEF =  F DEF = 0 Two special default values: –  T ONE = 1,  F ONE = 0 –  T ZERO = 0,  F ZERO = 1 Used for defining contradiction and tautology.

Default Reasoning in BAF Other default reasoning models are possible too. – E.g. categorical defaults: : (A,  T A,  F A )  (B,  T B,  F B ) / (B,  T B,  F B ) Semantics: –Given a knowledge base KB. –If KB :- A and KB :-/-  B, infer B with supporting and refuting masses  T B and  F B – 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. |DI P |  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 DI P and DI P  Q are less than 0.5, DI Q 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 DI Q < 0. A possible solution is to set {  T Q =  T DEF,  F Q =  F DEF } when DI P and DI P  Q are less than 0.5 In actual fact, the magnitude of DI P and DI P  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.  T P,0 is the value of  T P at time 0,  T P,1 at time 1 etc. At time t, given a proposition P with masses  T P, t and  F P,t, suppose we derive masses  T P, * and  F P, *, then the new belief masses at time t+1 are: –  T P, t+1 =   T P, t + (1-  )  T P, * –  F P, t+1 =   F P, t + (1-  )  F P, *

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,  T ONE,  F ONE, ), (A   B   R,  T ONE,  F ONE )} – We want to derive  T R, 1,  F R, 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  T R,* = min(  T A,  T B, max(  F A,  T B )) = min(0.7, 0.9, max(0.2, 0.9)) = min(0.7, 0.9, 0.9) = 0.7  F R,* = max(  F A,  F B, min(  T A,  F B )) = 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: –  T R,0 =  T DEF = 0.0 –  F R,0 =  F DEF = 0.0 This gives us the following attributes:

An Example MeasureValue DI R, Pl R,0 1.0 Ig R,0 1.0 EI R,0 [0.0, 1.0]

An Example Deriving the new belief values with  = 0.4 –  T R,1 = 0.4 * (1.0 – 0.4) * 0.7 = 0.42 –  F R,1 = 0.4 * (1.0 – 0.4) * 0.2 = 0.12 This gives us:

An Example MeasureValue DI R, – 0.12 = 0.30 Pl R,1 1.0 – 0.12 = 0.88 Ig R, – 0.42 = 0.46 EI R,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 With more knowledge available about whether R is true, we also see the plausibility falling from 1.0 to 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  T R,* = min(  T A,  T B,  T C, max(  F A,  T B )) = min(0.7, 0.9, 0.2, max(0.2, 0.9)) = min(0.7, 0.9, 0.2, 0.9) = 0.2  F R,* = max(  F A,  F B,  F C, min(  T A,  F B )) = 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: –  T R,2 = 0.4 * (1.0 – 0.4) * 0.2 = –  F R,2 = 0.4 * (1.0 – 0.4) * 0.7 = This gives us:

An Example MeasureValue DI R, – = Pl R,2 1.0 – = Ig R, – = EI R,2 [0.288, 0.532]

An Example Here the new evidence that B  C  R fails to support R, because C is not true (DI C = -0.5) Hence the plausibility of R falls from 0.88 to 0.532, while the truth value DI R,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

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:  T sam,son,bob = min(0.6, 0.8, 0.7) = 0.6  F sam,son,bob = max(0.2, 0.1, 0.1) = 0.2 DI sam,son, bob = 0.4 Pl sam, son, bob = 0.8 Ig sam, son, bob = 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 –S helpful = {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: –  T kenny_helpful = 0.75 –  F kenny_helpful = = 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 D ij in the document class c i is taken to be a conjunction of terms t k : D ij = t ij0  t ij1  …  t ij(n-1) – Each term and document is related by a set of relations: R ijk = {(D ij, term, t k,  T ijk,  F ijk ) | t k is a term in D ij }

Application Examples Text Classification Given a set of documents D in class c i, we apply the abstract operator to produce the set of relations characterizing c i. – v = (S i0, S i1, S i2, … S i(m-1) ) Each S ik is the relation: – S ik = {(c i, term, t k,  T ik,  F ik ) | t k occurs in at least  % of documents D j in class c i } –  T ik = min j  T ijk –  F ik = max l max j  T ljk, l  i

Application Examples Text Classification –  T ik is our belief that the term t k implies that the document belongs to class c i. –  F ik is our belief that the term t k implies that the document belongs to some other class c l. Given an unseen document D u, we derive the keyword terms t unk, k. We can derive the following masses that support and refute the proposition that D u belongs to class c i.

Application Examples Text Classification –  T i, unk = min(  T i0,  T i1, … max(  F i0,  F i1, … )) –  F i, unk = max(  F i0,  F i1, … min(  T i0,  T i1, … )) From this we derive the degree of inclination using the standard definition: – DI i, unk =  T i, unk -  F i, unk We choose the class with the largest DI as the winner. – win = argmax DI i,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

Classification Results Outside Testing

Classification Results Overall

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”, 2 nd International Conference on Computational Intelligence, Robotics and Autonomous Systems, Singapore, C. K. Y. Tan, K. T. Lua, “Belief Augmented Frames for Knowledge Representation in Spoken Dialogue Systems”, 1 st International Indian Conference on Artificial Intelligence, Hyderabad, India, 2003.

Publications C. K. Y. Tan, “Text Classification using Belief Augmented Frames”, 8 th Pacific Rim International Conference on Artificial Intelligence, Auckland, 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 , [Dempster67] A. P. Dempster, “Upper and Lower Probabilities Induced by a Multivalued Mapping”, The Annals of Mathematical Statistics Vol 38 No 2, pp , 1967

References [Pollock00] J. L. Pollock, A. S. Gilles, “Belief Revision and Epistemology”, Synthese 122, pp 69-92, [Tan03] C. K. Y. Tan, K. T. Lua, “Discourse Understanding with Discourse Representation Theory and Belief Augmented Frames”, 2 nd International Conference on Computational Intelligence, Robotics and Autonomous Systems, Singapore, 2003.