Looking for a Mathematical Theory of Knowledge

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Looking for a Mathematical Theory of Knowledge
Ruqian Lu Institute of Mathematics AMSS,
Academia Sinica
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Photographed by ESA on 25.03.2004
Or a small road on the top of the mountain?
(27.05.04)
Or a canal Into the Mi Yun Water
reservoir? (19.05.04)
This is the Chinese Great Wall? (11.05.04)
A Valley in the mountains with small streams ?
(01.06.04)
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Why all these misreadings?

• Isnt ESA having
• very advanced devices?

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Yes, They do
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Lots of advanced devices and instruments
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Lots of powerful computers
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Advanced devices bring massive information

• But not Knowledge!

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Knowledge is not the same as information.
Knowledge is information that has been pared,
shaped, interpreted, selected, and transformed
--E.Feigenbaum
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What is the Essence of Knowledge?

• Knowledge is
• Structured Information

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Power of Structureness

• We have only about 100 chemical elements
• But we have several millions of chemical
  compounds
• We have only 10 digits
• But we have a huge rich number theory
• That means connection is more important than
  connected elements!
• The essence of knowledge is its connecting
  mechanisms

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We are seeking a mathematical theory for
structureness

• Thats the
• Category Theory

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Knowledge Science meets Category Theory

• Category Theory has been successful in Uniting
  different branches of Mathematics
• In category theory, morphism is the major tool
  for representing structureness
• In order to describe different kinds of
  knowledge, we introduce types for morphisms
• We propose a typed category theory

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Heterogeneous Monoid

• A heterogeneous monoid is a heterogeneous algebra
  with unit element and binary associative
  operations only
• The operations have the form
• u x any ? any
• any x u ? any
• ai x bi ? ci

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Typed Category

• K (O, M, G) is called a typed category, where
• O is a class of objects
• M is a class of morphisms
• (O, M) is a category in usual sense
• G is a heterogeneous monoid of types
• Each morphism is attached with a type
• If ma,b and mb,c are morphisms between a,b and
  b,c, then there is an operation
• type (ma,b) x type (mb,c) ? type (ma,c), where
• ma,c is a morphism between a and c, and
• mb,c mb,c ma,c

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Typed Category with Structures

• A typed category C of sets, where to each set X a
  class of structures C(X) is assigned, is called a
  typed category with structures.
• A morphism f in C is called admissible, if it
  preserves the structure, i.e. if it can be
  considered as a map f (X, s) ? (Y, t), where s ?
  C(X) and t ? C(Y).
• It is called forgetful if t is strictly less
  structured than s, informative if s is strictly
  less structured than t.

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Knowledge Considered as Category

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Category and

• Knowledge Representation

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Knowledge as Category in General

• Every piece of knowledge is a typed category
  (with structures)
• Any knowledge processing
• Kp knowledge ? knowledge
• is a functor between two typed categories
• Knowledge in general is the category of all typed
  categories with these functors as morphisms

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Category of Knowledge bases

• Objects All knowledge bases
• Structures different representations
• Morphisms If knowledge base A is transformable
  to knowledge base B while keeping its knowledge
  content unchanged, then there is a morphism from
  A to B
• Identity morphism identity transformation
• Morphism composition transformation composition
• Associative law h (g f) (h g) f

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Structure
KB
KB
KB
KB
KB
Morphism
KB
KB
KB
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Category of Kripke Semantics

• Objects sets of possible worlds in sense of
  Kripke semantics
• Structures reachability relations
• Morphisms homomorphic maps from object A to
  object B.
• It is assumed that the reachability relation is
  reflexive and transitive.

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Category of Database States
Delete
Update
Insert
--- Johnson, Rosebrugh Wood
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Opposite Typed Category

• Let K (O, M, G) be a typed category.
• Its opposite category KOP (O, MOP, GOP) is
  defined as follows
• MOP fOP f ? M is the set of all reversed
  morphisms of K, where fOP has the additional
  constraints
• type (f) type (g) ?? type (f OP) type (gOP),
  uOP u

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Proposition

• For each a ? K,
• 1. Ma,a ? MOPa,a forms a group
• 2. Ga,a ? GOPa,a forms a group.

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Conceptual Graphs

• Objects symptoms, deceases, therapies,
  medicines, doctors, patients, ..
• Morphisms
• (deceases) showing (symptoms)
• (symptoms) due to (diseases)
• (deceases) need (therapies)
• (therapies) make use of (medicine)
• (therapies) prescribed by (doctors).

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Coequalizer
C
f
g
B
A
f
h
f
C
f h f g
unique morphism ? with ? f f.
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Coequalizer
Therapy1
C
need1
dueto1
Can replace
B
A
dueto2
need2
C
Symptoms
Therapy2
Deceases
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Equalizer
C
f
g
B
A
f
h
f
C
h f g f
unique morphism ? with f ? f.
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Equalizer
Sympton1
C
dueto1
need1
Caused by
B
A
need2
dueto2
C
Therapy
Sympton2
Diseases
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Category and

• Problem Complexity

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A Diagram
A1
A2
A3
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X
A Cone
A1
A2
A3
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Problem Reduction Cone

• Object problem class.
• Morphism if problem class A is reducible to
  problem class B, then there is a morphism from A
  to B.
• Type of morphism
• p polynomial time reducible
• np (only) exponential time reducible
• u reducible with a constant factor gt 0
• Morphism type composition
• p p p, p np np np np.

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Proposition

• The type set G
• p, np, u is an Abelian monoid

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X
Problem Reduction Cone
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Proposition

• If X ? A is polynomial for some A ? PD, then for
  all B ? C ? (X, PD), B ? C is polynomial.

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Polynomial Cone
p
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Polynomial Cone
p
p
p
p
p
p
p
p
p
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Proposition

• If A ? B is exponential for some A,B ? PD, then
  for all X ? C ? (X, PD), X ? C is exponential.

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X
Non Polynomial Cone
np
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X
Non Polynomial Cone
np
np
np
np
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X
Non Polynomial Cone
np
np
p
np
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A Limit
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Proposition

• Given a diagram PD, if there is at least one
  polynomial cone (co-cone) on it, then its limit
  (co-limit) is polynomial

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Category and

• Knowledge Acquisition

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Pseudo-Category

• A typed pseudo-category is similar to a typed
  category
• without guaranteeing the existence of morphism
  composition

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Operation on Pseudo-Categories

• The union U P ? Q of two pseudo-categories is
  defined as follows
• Any object A (any morphism f) belongs to U iff it
  belongs to at least one of P and Q.
• The intersection I P ? Q of two
  pseudo-categories is defined as follows
• Any object A belongs to I iff it belongs to both
  P and Q
• Any morphism f belongs to I iff it belongs to
  both P and Q and its domain and codomain also
  belong to I.

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Proposition

• The union and intersection of pseudo-categories
  are also pseudo-categories.

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Pseudo-Functor

• A typed pseudo-functor F P ? Q between two
  pseudo-categories is similar to a typed functor
  with the limitation that
• Fh Fg Ff iff g f h in P
• F is called faithful if F is injective and
  preserves the structure of objects.

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Repair of Pseudo-Category

• Given a category K. A pseudo-category P is said
  to be repairable with respect to K if there is a
  faithful pseudo-functor F P ? K.
• We say FP is embedded in K and K is a repair of
  P.
• If K is a subcategory contained in K and FP is
  also embedded in K than K is a coarser repair
  of P and K is a finer repair of P.
• Each coarsest repair is called a primitive
  repair.

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Repairability of Pseudo-Category

• With respect to category K, pseudo-category P is
• Inconsistent, if P is not repairable by K,
• Incomplete, if F(P) is not equal to any repair of
  P in K,
• Ambiguous, if there are at least two finest
  repairs F1(P) and F2(P) of P in K, such that
  there is no natural transformation between them
• Similar only, if there is at least one morphism
  f of P, such that type (f) ? type (F(f))

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P-functor F2
P-C
C
P-functor F1
S-C
S-C
S-C
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Joint Repairability

• With respect to category K, two pseudo-categories
  P and Q are
• Incompatible, if there is no typed pseudo-functor
  F, such that F(P?Q) has a repair in K,
• Redundant, if each repair F(P) has a common part
  with each repair G(Q), where F and G are typed
  pseudo-functors,
• Self-complementing, if F(P?Q), but not G(P) nor
  H(Q) alone, is a subcategory of K for some F and
  any G and H.

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Proposition

• All repairs of P with respect to K and the
  contain relation,
• 1. form a partial order of categories.
• 2. form themselves a category

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Proposition

• If P can be repaired by K with two
  pseudo-functors F1 and F2, then each natural
  transformation between them can be extended
  uniquely to a functor between any two primitive
  repairs R(F1P) and R(F2P).
• We call it the natural functor between repairs.

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Intuitionally

• if two pseudo-functors are natural to each
  other,
• then their primitive repairs
• behave in the same way.

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Conclusion

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Casual and individual application of
category theory
CS problem3
CS problem2
CS problem1

algebra
logic
topology
..
Category Theory
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Dream of Mathematicians
Mathematics
Set Theory
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Proposal of Category Theorists
Mathematics
Category Theory
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Our Proposal
Mathematics
Knowledge Science
Category Theory
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Acknowledgement

• We thank Prof. Mingsheng Ying for valuable
  discussions, critical comments and very helpful
  suggestions

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Questions and Comments are welcome
Give up wrong points of view
Accept good suggestions
Get a better theory of knowledge science
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Questions and Comments

• Are welcome