Soft Computing: Philosophical, Mathematical, and ...

Keynote: The 6th World Conference on Soft Computing, UC Berkeley

Soft Computing: Philosophical, Mathematical, and Theoretical Foundations for Cognitive Robotics and eBrain Yingxu Wang, Prof., PhD, PEng, FWIF, FICIC, SMIEEE, SMACM Visiting professor, Information Systems Lab, Stanford University, USA President, International Institute of Cognitive Informatics and Cognitive Computing (ICIC) 1

2

Director, Laboratory for Computational Intelligence, Software Science, & Denotational Mathematics

Dept. of Electrical and Computer Engineering Schulich School of Engineering and Hotchkiss Brain Institute University of Calgary, Canada Visiting professor: UC Berkeley (08),Stanford (08|16), MIT (12) & Oxford (95) http://www.ucalgary.ca/icic/ [email protected] [email protected] WConSC’16, UC Berkeley, USA, May 22-25, 2016  Prof. Y. Wang 1

This talk is dedicated to Prof. Lotfi A. Zadeh

 The 95th Birthday of Prof. Lotfi Zadeh  The 50th anniversary of Fuzzy Logic & its Applications  The seminar work and great inspirations of Prof. Lotfi Zadeh

WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

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1. Introduction

► 1. Introduction 2. Fuzzy concept / semantic algebra 3. Fuzzy logical algebra 4. Cognitive robotics and soft computing 5. Conclusions

WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

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Fuzziness is Deeply Rooted in Human Sensory, Knowledge & Reasoning















Prof. Zadeh

The extended domains of numbers & human ability for abstraction ICIC, © Prof. Y. Wang

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a) Prof. Zedeh’s Philosophy: Fuzziness of Human Cognition & Reasoning • Human conceptual and reasoning fuzziness is rooted in: - The basic unit of expressions - Fuzzy concepts, semantics, and fuzzy relations - The fundamental mechanism of reasoning - Fuzzy vs. bivalent • Computational means have to be closer and suitable for the mechanisms of human reasoning - Mathematics: Fuzzy sets and fuzzy logic - Computing: Soft computing, CWW - Reasoning: Fuzzy expressions and laws ICIC,  Prof. Y. Wang

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b) Prof. Zadeh’s Mathematical Vision: Domains of Fuzzy Numbers  The Domain of Fuzzy Numbers

- The domain of fuzzy numbers  is a fuzzy set with the range of values of all fuzzy variables and associate degrees of memberships in the hyperstructure of the fuzzy mathematical discourse , i.e.:     U   {(, ) |  [, ]    [0,1]  } n |  (n .v|, n . |) n

E.g. x |  (619, 0.96)  619 | 0.96 ICIC,  Prof. Y. Wang

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c) Prof. Zadeh’s Fuzzy Logic: How True is True?  Extended the basic foundation of logics - that was believed no proposition is either true or false.     U  ( F , T )  {0,1}

T F

p |  ( p . v | , p .  p ( v | )|  1) e.g. q |  ( T |L , 1), s im ply q  T         U 

 { (  ,  ) |   ( F , T )  {0,1}    [0,1]   }  p |   (  p . v | ,  p .  p ( v | ) | )

T

F

 ( T |L , 0.8)  T | 0.8  e.g. q |     ( F |L , 0.2 )  F | 0.2 ICIC,  Prof. Y. Wang

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d) Prof. Zadeh’s Soft Computing: Towards the Next Generation of Computing Theories for Fuzzy Systems • Soft Computing “Soft Computing is a consortium of methodologies which play an important role in the conception, design and utilization of Intelligent Systems. The principal methodologies in Soft Computing are: fuzzy logic, neurocomputing, evolutionary computing, cyber computing, probability computing and machine learning.” [Zadeh, 2016] ICIC,  Prof. Y. Wang

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Recent Discoveries (I) Support Prof. Zadeh’s Assertion:

The General Fuzziness of Human Senses  All five senses of human are fuzzy according to their cognitive and mathematical models [Wang, 2008-2016]  Vision is fuzzy - The brain only memorizes analytic attributes of images rather than bit maps - e.g.: Can you remember exactly what you look like?  Hearing is fuzzy - The brain does not elicit quantitative details of sounds - e.g.: Can you recognize the same piano in symphonies? ICIC,  Prof. Y. Wang

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Recent Discoveries (IIa): The General Fuzziness of Human Knowledge (Wang et al., 2016)

– Fuzzy Intensions of Human Concepts

Performance Attributes Animals

Performance Attributes Realistic

Performance by Attributes of each Dictionary vs each Formal Concept (three categories) 1 0.8 0.6 0.4 0.2 0 1

2

3

4

5

6

7

8

9

10

1 0.8 0.6 0.4 0.2 0 1

2

3

4

Performance Attributes Abstract

Dictionary (index)

5

6

7

8

9

Dictionary (index) 1):Pen

1 0.8

2):Pencil

9):Animal

3):Stationery

10):Mammal

4):Computer

0.6

5):Printer

0.4

0 1

2

3

4

5

6

7

8

9

11):Feline

16):Knowledge

12):Cat

17):Information

13):Tiger

18):Cognition

7):Bank r iver

14):Lion

19):Intelligence

8):Bank s torage

15):Leopard

20):Science

6):Bank

0.2

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Sources 1.- Oxford American English Dictionary 2.- Collins Online Dictionary 3.- Wordnet 4.- Oxford British English Dictionary 5.- Dictionary.com and Thesaurus.com 6.- Merriam-Webster Online 7.- Macmillan British Online Dictionary and Thesaurus 8.- WordReference-Random House 9.- Cambridge English Online Dictionary / Encyclopedia of life 10.- Longman English Online Dictionary/ Encyclopedia.com

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Dictionary (index)

ICIC,  Prof. Y. Wang

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Recent Discoveries (IIb): The General Fuzziness of Human Knowledge (Wang et al., 2016)

– Fuzzy Extensions of Human Concepts

Performance Objects Animals

Performance Objects Realistic

Performance by Objects of each Dictionary vs each Formal Concept (three categories)

1 0.8 0.6 0.4 0.2 0 1

2

3

4

5

6

7

8

9

10

1 0.8 0.6 0.4 0.2 0 1

2

3

Dictionary (index) Performance Objects Abstract

4

5

Dictionary

6

7

8

9

(index)

1):Pen

1

2):Pencil

0.8

3):Stationery 4):Computer

0.6

5):Printer 0.4

6):Bank 7):Bank river

0.2

8):Bank s torage

0 1

2

3

4

5

Dictionary

6

7

8

9

9):Animal

10

Sources 1.- Oxford American English Dictionary 2.- Collins Online Dictionary 3.- Wordnet 4.- Oxford British English Dictionary 5.- Dictionary.com and Thesaurus.com 6.- Merriam-Webster Online 7.- Macmillan British Online Dictionary and Thesaurus 8.- WordReference-Random House 9.- Cambridge English Online Dictionary / Encyclopedia of life 10.- Longman English Online Dictionary/ Encyclopedia.com

10):Mammal 11):Feline

16):Knowledge

12):Cat

17):Information

13):Tiger

18):Cognition

14):Lion

19):Intelligence

15):Leopard

20):Science

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(index)

ICIC,  Prof. Y. Wang

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Related Experience

Visiting Professor (on sabbatical leaves)

Invited Talks

2. Fuzzy Concept Algebra / Semantic Algebra

1. Introduction ► 2. Fuzzy concept / semantic algebra 3. Fuzzy logical algebra 4. Cognitive robotics and soft computing 5. Conclusions

WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

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Prof. Zadeh’s Inspiration: Fuzzy Data vs. Fuzzy Knowledge • Fuzzy Arithmetic - fuzzy numbers - fuzzy data - fuzzy sets • Fuzzy Logic - fuzzy logical variables - fuzzy propositions - fuzzy reasoning • Soft Computing - fuzzy knowledge - fuzzy words / semantics - Computing with words (CW) ICIC,  Prof. Y. Wang

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2.1 Fuzzy Concepts • A fuzzy concept is a hyperstructure of language entities denoted by a 5-tuple encompassing the fuzzy sets of attributes, objects, internal relations, external relations, and qualifications i.e.: i , R o , Q , R )   ( C A,O  A  { ( a 1 ,  ( a 1 ) ) , ( a 2 ,  ( a 2 ) ) , ..., ( a n ,  ( a n ) ) }  Þ A   { ( o ,  ( o ) ) , ( o ,  ( o ) ) , ..., ( o ,  ( o ) ) }  Þ O O 1 1 2 2 m m i  O   R A  ÞR =

| |O

| A|

R R ((o j 1

i 1

j

, a i ),  (oi )   ( a j ))

o  C  C '  Þ R , C '  C  C '  U R 

|O |

R

k 1

' ) ,  , C { (C k

s

|   ( R k )   )} ,   Ro |

 A   A'|  A   A'|

  { ( q ,  ( q ) ) , ( q ,  ( q ) ) , ..., ( q ,  ( q ) ) ) }  Þ R Q 1 1 2 2 p p

E.g. 1. A Fuzzy Concept -

C (pen )

i  , O , R ) , R o ,Q C ( p en )  C (A       ) i'  = p e n ((A, m (A )), (O , m (O ), R , R o ' , Q C

C

ì ï  = {(a , m (a )), (a , m (a ), (a , m (a ), (a , m (a )} ï A ï 1 1 2 2 3 3 4 4 ï ï ï = {(w ritin g _ to o l , 1.0), (in k , 0.9), (n ib , 0.9), ï ï ï (in k _ co n ta in er , 0.8) } ï ï ï  = {(o , m (o )), (o , m (o ), (o , m (o ), (o , m (o )} ï O ï 1 1 2 2 3 3 4 4 ï ï = {(ba llp o in t, 1.0), ( fo un ta in , 1.0), = ï í ï ï ï ( pen cil , 0.9), (brush , 0.7 ) } ï ï i ï ´A  ï = R O ï ï ï  o ï = C ´ C' R ï ï ï  = Æ ï Q ï ï î ICIC,  Prof. Y. Wang

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E.g. 2. A Fuzzy Concept

 (man ) C -

i o      C(man)  C(A,O, R , R ,Q)    ((A , m  , m  = man (A)),(O (O), Ri ', Ro ',Q ) C

C

 = {(human _being, 1.0), (male, 1.0), (adult, 0.9)} ìA ï ï ï ï  ï O = {(American, 1.0), (Australasian, 1.0), ï ï ï (business_man, 1.0), ...} ï ï = ïí i '   ï R = O ´A ï ï ï   ´C ' o' ï = R C ï ï ï  =Æ ï Q ï ï î ICIC,  Prof. Y. Wang

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2.2 Fuzzy Semantics • The fuzzy semantics of an entity (noun) e,  , is an Q(e) equivalent fuzzy concept

 Ce

in U, i.e.:

   Q(e )  Q(e = C e )  i o    = C (A ,O , R , R ,Q ) e

e

e

e

e

e

• The rule of semantic deduction states that the semantics of a given fuzzy entity is comprehended in semantic analysis, if and only if its fuzzy semantics can be reduced onto a known fuzzy concept with determined membership and weight values. ICIC,  Prof. Y. Wang

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Fuzzy Semantics of Fuzzy Modifiers z

 t = {R (tk , w(tk ))} k =1

= {(t1, w(t1 )),(t2 , w(t2 )),..., (tz , w(tz )} 4

t(good ) = {R (tk (quality ), w(tk ))} k =1

= {(neutral , 0.1), (good, 0.6), (excellent, 0.8), (perfect, 1.0)} 6

t(old ) = {R (tk (age ), w(tk ))} k =1

= {([1-20], 0), ([21-30], 0.1), ([31-50], 0.4), ([51-65], 0.7), ([66-80], 0.9), ([>80], 1.0)} ICIC,  Prof. Y. Wang

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Fuzzy Semantics of Fuzzy Quantifiers

p

 d = {R(di , w(di ))} i =1

= {(d1, w(t1 )),(d2, w(d2 )),...,(dp , w(dp )} d = {(definitely _ not, -3.0), (imperfectly, -1.5), (somewhat, 0.5), (fairly, 1.2), (quite, 1.5), (excellently, 2.0), (extremely, 3.0)}, 1 £ |d| £ 3

ICIC,  Prof. Y. Wang

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Fuzzy Semantics with Composite Fuzzy Modifiers

(t' · C )  Q (C  |Q  = dt ) Q i o      ) = C (A,O, R , R ,Q | Q = dt i o       = C ((A, mA (A),(O, mO (O ), R , R ,(Q = dt )) i o     = C (A,O, R , R ,Q )

ICIC,  Prof. Y. Wang

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E.g. 3. Semantics of Composite Fuzzy Modifiers (dt  · C ) = Q (d (extrem ely ) · t (good ) · pen ) Q   , m , m  = extrem ely _ excellent _ pen ((A (A ), (O (O ), R i , R o ,  A

 O

 = t ' | t ' = d (extrem ely ) · t (excellent ) = 3.0 · 0.8)) (Q 0 0 ì ï ï  = { (writing _ tool , 1.0), (ink , 0.9), (nib , 0.9), ï A ï ï ï (ink _ container , 0.8) } ï ï ï  = {(ballpo int, 1.0), ( fountain , 1.0), ( pencil , 0.9), ï O ï ï ï =ï (brush , 0.7) } í ï ï i ´A  ï = R O ï ï ï o ï  ´ C' ï = R C ï ï ï  (extrem ely _ excellent ) = 2.4 ï Q ï ï î ICIC,  Prof. Y. Wang

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E.g. 4. Semantics of Composite Fuzzy Modifiers (dt  ·C ) = Q (d(quite) · t(old ) · man ) Q i o      = a_quite_old_man((A, m (A),(O, m (O), R , R ,  A

 O

 = t' | t' =  (Q d(quite) · t(60) = 1.5 · 0.7)) 0 0

 = {(human_being, 1.0), (male, 1.0), (adult, 0.9)} ì ï A ï ï ï  = {(American, 1.0), (Australia, 1.0), ï O ï ï ï ï (business _ man, 1.0), ...} ï =ï í i  ´A  ï R O = ï ï ï o   ï R = old _ man ´C ' ï ï ï (quite_old ) = 1.05 ï Q ï ï î ICIC,  Prof. Y. Wang

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2.3 The Fuzzy Semantic Model for Machine-enabled Humor Cognition  The mathematical model of humor (GPH)

[Wang, 2015]

 e '  F   J  $c, c Å ) T e  ICIC,  Prof. Y. Wang

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E.g. 5. J1 according to GPH • Example 1 (J1): A linguistic philosopher in a lecture at Oxford made the claim that although a double negative in English results in a positive meaning, there is no language in which a double positive implies a negation. Responding to this statement, “Yeah, yeah,” commented by anther philosopher in a dismissive tone.

J1  " modifiers M , (M ) º M : F M ´ M º M   T M ´ M = M ,  Å (i.e., “yeah, yeah?!”) ICIC,  Prof. Y. Wang

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)

E.g. 6. J2 according to GPH • Example 2 (J2): Jeremy made a speech at a wedding reception with Jenny, his six-year-old daughter. It was a great success for his funny stories in the speech. However, as soon as he had finished, Jenny told him she wanted to go home. On the way home, Jeremy asked Jenny why she was unhappy. She told him that she did not like to see so many people laughing at him!

J2  $ Jeremy (dad)and Jenny (young daughter) @ a wedding when Jeremy's jokes were successful but Jenny was unhappy: F Jenny didn't like Jeremy's jokes   Å  ) T Jenny didn't like people laughing at him  ICIC,  Prof. Y. Wang

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Once a Machine Tells: It‘s a Joke … !  Please don’t be surprised if you’d here the following from a cognitive robot or a cognitive system in the near future!

" :) It was a joke ! " ICIC,  Prof. Y. Wang

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3. Fuzzy Logical Algebra

1. Introduction 2. Fuzzy concept / semantic algebra ► 3. Fuzzy logical algebra 4. Cognitive robotics and soft computing 5. Conclusions

WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

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Prof. Zadeh’s Inspiration: Fuzzy Logic for Precisiation Fuzzy Reasoning

Logical Value

Logical Relation

Bivalent logic

Fuzzy logic [Prof. Zadeh]

FLA

 The extended philosophy of logic - from one to two dimensions  “To be, or not to be, that is the question.” ICIC,  Prof. Y. Wang

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3.1 Fuzzy Logical Propositions A fuzzy logical variable, p| , in  is a special logical variable, which is a pair of a truth value in bivalent type  and a confidence weight in the unit scale   = [0, 1]  , that asserts the degree of belief or  accuracy of the truth value, i.e.: 

       U   {(  ,  ) |   [ F , T ]    [0,1]  } p |  ( p.v | , p.  ( v | )| ) p  (T |L, 0.8)  T | 0.8  e.g. q|    ( F |L, 0.2 )  F | 0.2 ICIC,  Prof. Y. Wang

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The Universe of Discourse of Fuzzy Propositions  Let V, E, and P be finite sets of fuzzy logical variables, expressions, and propositions, respectively, and F a finite set of fuzzy logical operators on V, E, and/or P. Then the universe of discourse of fuzzy logical algebra, U, is a 4-tuple:

U (V, E, P, F ) V  V // FL variable (term)  P  E  f (V )  E // FL expression (formula)  P  g ( E )  g ( f (V ))  P // FL proposition  ICIC,  Prof. Y. Wang

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Framework of Fuzzy Logical Algebra (FLA)  The fuzzy logical algebra, FLA, in the universe of discourse of fuzzy propositions U is a triple:

 ,  , U ) FLA  (   .v ,   . ), ( r ,  l ), U ) = ((        P is a finite set of fuzzy propositions, and where 

  ( r ,  c ) a finite set of fuzzy algebraic operators in the categories of the relational and logical operations on the formal fuzzy propositions. ICIC,  Prof. Y. Wang

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Fuzzy Logical Variables

ICIC,  Prof. Y. Wang

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Typical Fuzzy Logical Variables

Compatibility between fuzzy & bivalent logical variables:

The Duality of Fuzzy Logical Variables

ICIC,  Prof. Y. Wang

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Fuzzy Logical Expressions

Fuzzy Logical Propositions

3.2 Fuzzy Logical Operators of FLA

    ·l = {, , }

ICIC,  Prof. Y. Wang

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Fuzzy Logical Conjunction   B   A

 = 

 (v ,  )  ( v A ,  A )    B B  ( v   v A  v B ,     A   B )

2

2

= (T, (0.9  0.5)) = (T, 0.45)

= ( v  , i =1

n

n

i 1

i 1

Xi

 (v  vA  vC' ,   A  (1 C' )) 3

= (T, (0.9  (1 0.1))) = (T, 0.81)

n

 m X ) i

i =1

n

 X   Xi  (v X ,   X |  X  1) i

i

i 1

n

C   A  ' P 3 3

  X i  i =1 n

  A  B  P 2  (v  vA  vB ,   A  B )

n

 (  v X i , 1) i 1 n

= (  X i , 1)

ICIC,  Prof. Y. Wang

i 1

39

i

Decay of Degree of Confidence in FL Conjunctions

ICIC,  Prof. Y. Wang

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Fuzzy Logical Disjunction  B   A

  

 (v ,  )  (v A ,  A )    B B  ( v  vA  vB ,   

1 (  A   B )) 2

1 (     B )) 4 4 2 A = (T, (0.9  0.5)/2)  (T, 0.7)

  X i  i =1 n

1 = ( v  , Xi n i =1

  B  P A 4  ( v  v A  v B ,    P P

n

n

n

å mX ) i =1

i

n 1  X   Xi  (vXi , Xi | Xi  1) n i 1 i 1 i 1 n

n

 ( vXi , 1)

C   ' P A 5 1 (    (1   C ' ))) 5 5 2 A = (T, (0.9  (1  0.1))/2)  (T, 0.9)  ( v  v A  v C ' ,    P P

ICIC,  Prof. Y. Wang

i 1 n

= ( X i , 1) i 1

41

Propagation of Degree of Confidence in FL Disjunctions

ICIC,  Prof. Y. Wang

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Fuzzy Logical Negation  (v , m )   (v , m )          ì (v , 1- m ) ï ï    ï   =í  ï  '(v ¢ , 1 - m ¢ ) ï ï   î

(   ) =  =  =(v , m )      Snow   _ is _ usually _ white   SINUW

A  =A (T, 0.9)  A '(F, 0.1)  (F, 0.1)  A '(T, 0.9) =A  C' = C'(F, 0.1)  C(T, 0.9)  = C'(T, 0.9)  C(F, 0.1) ICIC,  Prof. Y. Wang

(SIUW  T, SINUW   0.1)   0.9) (SIUW  F, SINUW  (SINUW  T, SINUW   0.1)   0.9)  (SINUW  F, SINUW 43

Reasoning with Fuzzy Variables and Prepositions

    C ausation : C  (E = C )  f x (C ) Analogy : "x Î S(x , mS (x )), p(x )  $a Î S(a , mS (a )), p(a )  $b Î S(b, m (b ))  a ¹ b, p(b ) S

  Deduction: "x Î S(x, mS (x )), p(x ) ß $a Î S(a, mS (a)),q(a) Inductio n : $ a Î S (a , m S (a )), p (a )  $ k , succ(k ) Î S ((k , m S (k )), (succ(k ), m S (succ(k ))), p (k )  p (succ(k ))  " x Î S(x , m (x )), p (x ) S

 A b d u c t io n : " x Î S ( x , m S ( x ) ) , x  p ( x )  $ a Î S (a , m S (a ) ) , a  p ( x )

4. Cognitive Robotics and Soft Computing

1. Introduction 2. Fuzzy concept / semantic algebra 3. Fuzzy logical algebra ► 4. Cognitive robotics and soft computing 5. Conclusions

WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

45

Fuzzy Systems Inspired by Prof. Zadeh’s Seminal Work • Fuzzy systems are human-like intelligent systems that are capable of dealing with abstract, inaccurate, incomplete, uncertain, and irregular inferences for complex information elicitation, cognition, processing, and generation. • The fuzzy properties of human centric systems stem from the fuzzy reasoning mechanisms of human brains.

• Paradigms of fuzzy systems - Fuzzy computing systems: Cognitive computers, fuzzy automata, fuzzy neural networks - Fuzzy control systems: Fuzzy control logic, fuzzy fault-tolerant systems, fuzzy diagnostic systems, … - Fuzzy inference systems: Fuzzy causations, fuzzy deduction/induction/ abduction/analogy - Fuzzy learning systems: Cognitive learn engine, cognitive knowledge bases - Fuzzy linguistics: Fuzzy concepts, fuzzy syntax, fuzzy semantics - Fuzzy mathematics: Fuzzy concept algebra, fuzzy semantic algebra, fuzzy truth algebra, fuzzy probability, … ICIC,  Prof. Y. Wang

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Cognitive Robots (CR) for Soft Computing Wang, 2011, IEEE MRA

• A cognitive robot (CR) is an autonomous robot that is capable of perception, inference, and learning mimicking the cognitive mechanisms of the brain.

ICIC, Prof. Y. Wang

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The Layered Reference Model of the Brain (LRMB)

Wang et al., IEEE TSMC-Sys, 2006

Cognitive Architecture of CR B n

[Ce re b ru m]

STM ( Wor king) [Fr on tal lo b e]

LTM ( Visua l) [O ccip ital lo b e]

LTM (K no wle dge) [T emp or al l ob e]

LTM (Expe rienc e/ep isode ) [ Pa rie tal l ob e]

Se nso ri es V isio n

Auditio n

Occ ipital lobe [ Vi sua l a re a] Temp oral

lobe

[ Auditory ar ea ] S me ll Ta ste To uch S timuli

Pa rieta l lobe [Soma t. a re a] Body stimu li [ Me du lla ] S BM

Be h avi ors

B- CPU M UX (a ttentio n sw itc h) [Hi pp oca mp us]

Pe rc ep tion Engin e [ Th ala mu s] Con sc ious En gine [ Hyp oth ala mu s]

ABM

Act ion d rive

[Pri mar y mo to r co rte x]

[P on s/ medulla ]

[Po n s]

Refle ctive ac tion s

CSM [Ce re b ell um]

S u rvival b eh aviors [sp in al co rd ]

M u sc le se rvos [ moto r neurons ]

E ye s F ac e Ar ms Legs … Othe rs

DM for CRs Expression

Category

Mathematical Means

(Linguistics)

Conventional Maths

Numerical Maths

Denotational Maths (Wang, 2000-2015)

Logic , 

, , , 

 Concept algebra  Semantic algebra  Fuzzy logical algebra  Big data algebra  Visual semantic algebra

To have Specify structures & (|) relations

Set theory

Vectors Matrixes

 System algebra  Relational algebra  Structure algebra

Describe behaviors & functions

Functions Calculus

Discrete functions Iterative functions Linear algebra

 Behavioral process algebra  Inference algebra  Induction algebra  Fuzzy probability algebra

Identify objects & status

To be (|=)

To do (|>)

ICIC, Prof. Y. Wang

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Cognitive Knowledge Base of CRs OAR = (O, A, R) O – object A – attribute R – relation

ICIC, Prof. Y. Wang

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Formal Models of Knowledge in CRs • Because the basic structure of knowledge is an abstract concept Ci in , the mathematical model of knowledge is a Cartesian product of power sets of formal concepts. n

K =  : XCi  i=1

n

XC

j

j=1

• Latest Discoveries in Knowledge Science [Wang, 2008-2016] - The basic structure of knowledge is a formal concepts - The basic unit of knowledge is a BIR (binary relation) - As a counterpart, the basic unit of information and data is a BIT (binary digits) ICIC, Prof. Y. Wang

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The Framework of Sentence Learning P ro c ess T ex t in p u t O b je c t i d e n ti f ic a tio n

C o n c ep t f o r m u la t io n

E x a m ple “ A c o g n it iv e c o m p u t e r

C

C

g

C

g

is a n in t e llig e n t co m p u t e r f o r k n o w le d g e p r o c e s s in g . ”

C

I



 C



p

K

P

K  P

I  C

C gC



IC



KP

C o n c ep t r e la tio n a n a ly s is

C gC



IC



KP

K n o w le d g e r e p r e s e n t a ti o n

C gC

KP IC C



sO AR M e m o r i z a t io n

C

g

C gC

K n o w led g e r e tr i e v a l

C

I IC

O AR’ K KP

P

 sO A R

O AR

Su b ject = { C gC } C o n c e p t = { C g , C , I , K , P , C g C , IC , K P } K n o w le d g e _ d if f e r e n t ia l = d ( O A R ) /d t = { C g , C , I, K , P , C g C , I C , K P } \ { C , K , P } = { C g , I , C g C , I C , K P }

Concept Elicitation by CR C 1+ ( pen )  C 1 (AS 1 , O S 1 , R1c , R1i , R1o ) ì ï AS 1 = {instrum ent (1.0), w riting (1.0) ï ï ï ï , ink (1.0), nib (0.5)} ï ï ï ï O S 1 = {ballpoint (1.0), fountain (0.9), ï ï = C 1+ ï quill (0.7), m arker (0.5)} í ï ï ï R1c = O S 1 ´ AS 1 ï ï ï ï R1i = K ´ AS 1 ï ï ï ï R o = O S1 ´ K ï î 1

Latest breakthrough in my Lab: - Machine (CR) may beat human in knowledge learning and concept generation !

ICIC, ©Prof. Y. Wang

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Sentence Synthesis C g C|S M  C g |S M  C|S M = (A| = {C g |S M .A| È C|S M. A|}, O | = C|S M.O | \ C g |S M. O| , c i’ o’ R | = O| ´ A |, R | , R | )

KP |S M  K |S M  P |S M = (A| = K |S M .A | È P |SM .A | , O| = P |S M .O | \ K |S M.O | , R c| = O | ´ A| , R i ’| , R o’ | )

C gC’|SM  (C g|SM  C|SM)  (I|SM  C|SM))  (K|SM  P|SM)

ICIC,  Prof. Y. Wang

55

Complex Learning by Recursive Algorithms C L E L ea rn in g M ech a n is m s  T h em eL e arn in g | PM  S en te n ceL earn i n g |P M  C o n cep t L ea rn in g |P M O b jec t_ I de n tif ica tio n | P M

C o n cep t _ F or m ula tio n | PM

K n o w l ed g e_ F u si o n |P M

DM: Contemporary Denotational Mathematics for Cognitive Robots & Cognitive Computers T he T he o reti ca l F ra m ew o r k o f D e no ta tio n a l M a the ma ti cs

N e w m a th em a t ic a l o b je cts

C o n ce p ts S em an tic s B eh av io rs C au s atio n s A lg o rith m s D e cis io n s G r an u le s

N e w e x pre s sio n a l ne ed s

F u zzy sy s tem s tru c tu re s A b s tra ct sy s tem b eh a vio rs K n o w led g e m an ip u latio n In te llig en ce g en e ratio n E fficie n t co mp le x ity h an d lin g

S ys tem s

R ig o r o u s s y s. mo d ellin g

H y p er s tru c tu res

E x p re s siv e sy s . d en o ta tio n

N ew m a th em a tica l pa ra d ig m s

C o n ce p t a lg eb r a S ys te m alg e b ra P ro c es s a lg e br a S em an tic a lg eb r a In fer en ce a lg e br a G r an u la r alg eb ra F u z z y lo gi ca l a lg e bra

T r u th a lg eb r a B ig d ata a lg eb r a V isu a l se m a n tic a lg e b ra

N ew a p plica t io n fie ld s

I nte llig en c e s cien ce K n o w led g e s cien ce B ra in s cien ce S ys te m s cien ce C o g n itive info rma tic s C o g n itive c o mp u te rs S o ftw a re s cien ce

Cognitive Robots and Cognitive Machine Learning • CR learns more rigorous concepts in its knowledge, which leads to better reasoning & discovery • CR learns common or professional knowledge faster than human does • CR learns and processes knowledge continually beyond the natural memory creation constraints of humans • CR may never forget a piece of learned knowledge once that has been cognized and memorized • Most excitingly, CA can directly transfer learned knowledge to peers without requiring re-learning because they use the same knowledge representation and manipulation mechanisms ICIC,  Prof. Y. Wang

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5. Conclusions

1. Introduction 2. Fuzzy concept / semantic algebra 3. Fuzzy logical algebra 4. Cognitive robotics and soft computing ► 5. Conclusions

WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

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Conclusions (1/2) • Prof. Zedah’s seminal work - Philosophy: human expressions, knowledge, and reasoning are fuzzy - Fuzzy sets: an extended number theory from  to  - Fuzzy logic: a higher order logic for nonbivalent reasoning - Soft computing: human-like computing, CW, and fuzzy systems

WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

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Conclusions (2/2) • Latest Advances in Basic Studies on DMs for SoftComp - Basic studies on Denotational Mathematics (DMs) • • • • •

Fuzzy arithmetic Fuzzy truth algebra Fuzzy probability algebra Fuzzy statistics Fuzzy concept / semantic algebra …

- Recent Discoveries • The law of fuzzy information fusion (FIF) • The basic unit of knowledge: BIR vs. BIT for information • The general fuzziness of human concept, semantics, knowledge, & inference - It’s inherited deeply in our concepts as the basic mechanism of reasoning • The theory of machine knowledge learning and systems WConSC’16, UC Berkeley, USA, May 23, 2016  Prof. Y. Wang

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Welcome to IEEE ICCI*CC 2016

ICCI*CCʼ16

15th

IEEE

2016

The 15th IEEE International Conference on

Cognitive Informatics & Cognitive Computing Aug. 22-23, 2016, Stanford Univ., USA http://www.ucalgary.ca/icci_cc/ICCICC2016