A New Frontier in ComputationComputation with Information Described in Natural Language

1
A New Frontier in ComputationComputation with
Information Described in Natural Language Lotfi
A. Zadeh Computer Science Division Department
of EECSUC Berkeley NIH, Bethesda, MD February
8, 2007 URL http//www-bisc.cs.berkeley.edu URL
http//www.cs.berkeley.edu/zadeh/ Email
Zadeh_at_eecs.berkeley.edu
2
PREAMBLE

• In conventional modes of computation, the objects
  of computation are values of variables. In
  computation with information described in natural
  language, or NL-Computation for short, the
  objects of computation are not values of
  variables but states of information about the
  values of variables, with the added assumption
  that information is described in natural
  language.
• The importance of NL-Computation derives from the
  fact that much of human knowledge, and
  particularly world knowledge, is expressed in
  natural language.

3
BASIC IDEA
?Z f(X, Y)

• Conventional computation
• given value of X
• given value of Y
• given f
• compute value of Z

4
CONTINUED
Z f(X, Y)

• NL-Computation
• given NL(X) (information about the value of X
  described in natural language) X
• given NL(Y) (information about the values of Y
  described in natural language) Y
• given NL(X, Y) (information about the values of
  X and Y described in natural language) (X, Y)
• given NL (f) (information about f described in
  natural language) f
• computation NL(Z) (information about the value
  of Z described in natural language) Z

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EXAMPLE (AGE DIFFERENCE)

• Z Age(Vera) - Age(Pat)
• Age(Vera) Vera has a son in late twenties and a
  daughter in late thirties
• Age(Pat) Pat has a daughter who is close to
  thirty. Pat is a dermatologist. In practice for
  close to 20 years
• NL(W1) (relevant information drawn from world
  knowledge) child bearing age ranges from about 16
  to about 42
• NL(W2) age at start of practice ranges from
  about 20 to about 40
• Closed (circumscribed) vs. open (uncircumscribed)
• Open augmentation of information by drawing on
  world knowledge is allowed

6
EXAMPLE (NL(f))

• Yf(X)
• NL(f) if X is small then Y is small
• if X is medium then Y is large
• if X is large then Y is small
• NL(X) usually X is medium
• ?NL(Y)

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EXAMPLE (balls-in-box)

• a box contains about 20 black and white balls.
  Most are black. There are several times as many
  black balls as white balls. What is the number of
  white balls?
• EXAMPLE (chaining)
• Overeating causes obesity
• Obesity causes high blood pressure
• I overeat. What is the probability that I will
  develop high blood pressure?

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KEY OBSERVATIONSCOMPUTATION, PRECISIATION AND
UNDERSTANDING

• Precisiation is a prerequisite to computation
• Understanding is a prerequisite to precisiation
• NL-Computation presupposes understanding of given
  information
• Example
• Use with adequate ventilation. I understand what
  you mean, but can you be more precise?

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KEY OBSERVATIONS--PERCEPTIONS

• A natural language is basically a system for
  describing perceptions
• Perceptions are intrinsically imprecise,
  reflecting the bounded ability of human sensory
  organs, and ultimately the brain, to resolve
  detail and store information
• Imprecision of perceptions is passed on to the
  natural languages which is used to describe them
• Natural languages are intrinsically imprecise

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INFORMATION
measurement-based numerical
perception-based linguistic

• it is 35 C
• Over 70 of Swedes are taller than 175 cm
• probability is 0.8

• It is very warm
• most Swedes are tall
• probability is high
• it is cloudy
• traffic is heavy
• it is hard to find parking near the campus

• measurement-based information may be viewed as a
  special case of perception-based information
• perception-based information is intrinsically
  imprecise

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NL-capability

• In the computational theory of perceptions (Zadeh
  1999) the objects of computation are not
  perceptions per se but their descriptions in a
  natural language
• Computational theory of perceptions (CTP) is
  based on NL-Computation
• Capability to compute with perception-based
  information capability to compute with
  information described in a natural language
  NL-capability.

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KEY OBSERVATIONNL-incapability

• Existing scientific theories are based for the
  most part on bivalent logic and
  bivalent-logic-based probability theory
• Bivalent logic and bivalent-logic-based
  probability theory do not have NL-capability
• For the most part, existing scientific theories
  do not have NL-capability

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BASIC STRUCTURE OF NL-COMPUTATION
Basically, NL-Computation is a system of
computation in which the objects of computation
are predicates and propositions drawn from a
natural language
COMPUTATION
PRECISIATION
NL
Pre1(p)
Pren(p)
p q
information
solution
reduction
ans(q/p)
question
Pre1(q)
Pren(q)
final solution
reduction to a standard problem

• NL-Computation is closely related to Computing
  with Words

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KEY IDEAS IN NL-COMPUTATION

• FUNDAMENTAL THESIS
• Information closed generalized constraint
• proposition is a carrier of information
• MEANING POSTULATE
• proposition closed generalized constraint
• predicate open generalized constraint
• NL-Computation is reduced to computation with
  generalized constraints. NL-Computation is based
  on fuzzy logic.

15
FUZZY LOGICKEY POINTS

• Humans have a remarkable capability to reason and
  make decisions in an environment of imprecision,
  uncertainty and partiality of knowledge, truth
  and class membership. The principal objective of
  fuzzy logic is formalization/mechanization of
  this capability.

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WHAT IS FUZZY LOGIC?

• Fuzzy logic is not fuzzy logic
• Fuzzy logic is a precise logic of reasoning and
  decision making based on information which is
  imprecise, uncertain, incomplete and partially
  true.
• The principal distinguishing features of fuzzy
  logic are
• In fuzzy logic everything is, or is allowed to be
  graduated, that is, be a matter of degree or,
  equivalently fuzzy
• In fuzzy logic everything is, or is allowed to be
  granulated, with a granule being a clump of
  points drawn together by indistinguishability,
  similarity or proximity
• Graduation and granulation have a position of
  centrality in human cognition

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ANALOGY

• In bivalent logic, one draws with a ballpoint pen
• In fuzzy logic, one draws with a spray pen which
  has an adjustable and precisely defined spray
  pattern
• This simple analogy suggests many mathematical
  problems
• What is the maximum value of f?

Y
X
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THE CONCEPTS OF PRECISIATION AND
COINTENSIVE PRECISIATION
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PRECISIATION AND COMPUTATION--LOOKAHEAD

• Example
• Robert left his office at time a. Usually it
  takes him about b to get home. At what time
  would he be expected to arrive?
• Precisiation of problem
• Za (usually) b

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PRECISIATION OF approximately a, a
?
1
singleton
s-precisiation
0
x
a
?
1
cg-precisiation
interval
0
a
x
fuzzy interval
g-precisiation
?
type 2 fuzzy interval
0
a
x
fuzzy graph
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PRECISIATION OF usually (b)

• Time of arrival, v, is a random variable with
  probability density p.

p(v)
p(v-u)
v
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WHAT IS PRECISE?
PRECISE
v-precise
m-precise

• precise value
• p X is a Gaussian random variable with mean m
  and variance ?2. m and ?2 are precisely defined
  real numbers
• p is v-imprecise and m-precise
• p X is in the interval a, b. a and b are
  precisely defined real numbers
• p is v-imprecise and m-precise

precise meaning
m-precise mathematically well-defined
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PRECISIATION AND IMPRECISIATION
v-imprecisiation
1
1
0
0
a
v-precisiation
a
x
x
m-precise
m-precise
v-precise
v-imprecise
1
If X is small then Y is small If X is medium then
Y is large If X is large then Y is small
v-imprecisiation
0
x
v-imprecise
v-precise
m-imprecise
m-precise
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v-IMPRECISIATION
v-imprecisiation
forced
deliberate

• forced V is not known precisely
• deliberate V need not be known precisely
• v-imprecisiation principle Precision carries a
  cost. If there is a tolerance for imprecision,
  exploit it by employing v-imprecisiation to
  achieve lower cost, robustness, tractability,
  decision-relevance and higher level of confidence

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IMPRECISIATION/ SUMMARIZATION OF FUNCTIONS
L
If X is small then Y is small If X is medium then
Y is large If X is large then Y is small
v-imprecisiation
M
summarization
S
0
S
M
L
If X is small then Y is small If X is medium then
Y is large If X is large then Y is small
(X, Y) is small ? small medium ? large large
? small
mm-precisiation
fuzzy graph
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PRECISIATION/IMPRECISIATION PRINCIPLE
(Zadeh 2005)

• a approximately a
• simple version

f(a) f(a)
Y
Y
X
X
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PRECISE SOLUTION
level set
undominated
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MODALITIES OF m-PRECISIATION
m-precisiation
mh-precisiation
mm-precisiation
machine-oriented
human-oriented
m-precisiation
l precisiand of l (Pre(l))
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BASIC STRUCTURE OF DEFINITIONS
definiens
definiendum (idea/perception)
concept
mh-precisiand
mm-precisiand
mh-precisiation
mm-precisiation
cointension
cointension
cointension wellness of fit of meaning
Declining market with expectation of further
decline We classify a bear market as a 30
percent decline after 50 days, or a 13 percent
decline after 145 days. (Robert Shuster)
mh-precisiation
bear market
mm-precisiation
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EXAMPLES MOUNTAIN, CLUSTER, STABILITY
mh-precisiation
A natural raised part of the earths surface,
usually rising more or less abruptly, and larger
than a hill
mountain
mm-precisiation
?
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CONTINUED
mh-precisiation
A number of things of the same sort gathered
together or growing together bunch
cluster
mm-precisiation
?

• the concepts of mountain and cluster are
  PF-equivalent, that is, have the same deep
  structure

mh-precisiation
The capacity of an object to return to
equilibrium after having been displaced
stability
mm-precisiation
Lyapuonov definition
mm-precisiation
fuzzy stability definition
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GRANULATION REVISITED

• Granulation plays a key role in human cognition
• In human cognition, v-imprecisiation is followed
  by mh-precisiation. Granulation is
  mh-precisiation-based
• In fuzzy logic, v-imprecisiation is followed by
  mm-precisiation. Granulation is
  mm-precisiation-based

• mm-precisiation-based granulation is a major
• contribution of fuzzy logic. No other logical
  
• system offers this capability

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DIGRESSIONEXTENSION VS. INTENSION

• extension and intension are concepts drawn from
  logic and linguistics
• basic idea
• object (name (attribute1, value1), ,
  (attribute n, value n))
• more compactly
• object (name, (attribute, value))
• n-ary n-ary

name
attribute name
attribute value
object
attribute name
attribute value
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OPERATIONS ON OBJECTS
function
name-based
extensional
definition
object
intensional
definition
attribute-based
(algorithmic)
object (Michael, (gender, male), , (age,
25)) son (Michael) Ron
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PRINCIPAL MODES OF DEFINITION

• Extension name-based meaning
• Intension attribute-based meaning
• Extensional Pu1, , un e-meaning of P
• Ostensive Pu, uk, ul o-meaning of P
• Intensional PuP(u), i-meaning of P

exemplars
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THE CONCEPT OF COINTENSION

• p, q are predicates or propositions
• CI(p,q) cointension of p and q degree of match
  between the i-meanings of p and q
• q is cointensive w/n to p if GI(p, q) is high
• A definition is cointensive if CI(definiendum,
  definiens) is high
• In practice, CI(p,q) is frequently associated
  with o-meaning of p and i-meaning of q
• The o-meaning of the definiendum is
  perception-based

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THE CONCEPT OF COINTENSIVE PRECISIATION

• Precisiation of a concept or proposition, p, is
  cointensive if Pre(p) is cointensive with p.
• Example bear market
• We classify a bear market as a 30 percent
  decline after 50 days, or a 13 percent decline
  after 145 days. (Robert Shuster)
• This definition is clearly not cointensive

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KEY POINTS

• Precisiandmodel of meaning
• In general, p, may be precisiated in many
  different ways, resulting in precisiands Pre1(p),
  , Pren(p), each of which is associated with the
  degree, CIi, of cointension of Prei(p), i 1, ,
  n. In general, CIi is context-dependent.
• Precisiation is necessary but not sufficient
• To serve its pupose, precisiation must be
  cointensive
• Cointensive precisiation is a key to
  mechanization of natural language understanding

precisiation1
Pre1(p) C1
precisiation2
p
Pre2(p) C2
precisiationn
Pren(p) Cn
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AN IMPORTANT IMPLICATION FOR SCIENCE

• It is a deep-seated tradition in science to
  employ the conceptual structure of bivalent logic
  and probability theory as a basis for formulation
  of definitions of concepts. What is widely
  unrecognized is that, in reality, most concepts
  are fuzzy rather than bivalent, and that, in
  general, it is not possible to formulate a
  cointensive definition of a fuzzy concept within
  the conceptual structure of bivalent logic and
  probability theory.

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EXAMPLES OF FUZZY CONCEPTS WHOSE STANDARD,
BIVALENT-LOGIC-BASED DEFINITIONS ARE NOT
COINTENSIVE

• stability
• causality
• relevance
• bear market
• recession
• mountain

• independence
• stationarity
• cluster
• grammar
• risk
• linearity

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A GLIMPSE INTO THE FUTURE

• To formulate a cointesive definition of a fuzzy
  concept it is necessary to employ fuzzy logic
• Replacement of bivalent-logic-based definitions
  with fuzzy-logic-based definitions is certain to
  take place but it will be a slow process
• Fuzzy-logic-based definitions will be targeted
  (customized)

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CHOICE OF PRECISIAND

• Cointension and tractability are contravariant
• To be tractable, precisiation should not be
  complex
• An optimal choice is one which achieves a
  compromise between tractability and cointension

cointension
tractability
complexity
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THE KEY IDEA MEANING POSTULATE

• In NL-computation, a proposition, p, is
  precisiated by expressing its meaning as a
  generalized constraint. In this sense, the
  concept of a generalized constraint serves as a
  bridge from natural languages to mathematics.

NL
Mathematics
p
p (GC(p))
precisiation
generalized constraint

• The concept of a generalized constraint is the
  
• centerpiece of NL-computation

44
THE CONCEPT OF A GENERALIZED CONSTRAINT
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PREAMBLE

• The centerpiece of fuzzy logic is the concept of
  a generalized constraint. Constraints are
  ubiquitous. In scientific theories,
  representation of constraints is generally over
  simplified. Over simplification of constraints is
  a necessity because existing constrained
  definition languages have a very limited
  expressive power. The concept of a generalized
  constraint is intended to provide a basis for
  construction of a maximally expressive constraint
  definition language which can also serve as a
  meaning representation/precisiation language for
  natural languages.

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GENERALIZED CONSTRAINT (Zadeh 1986)

• Bivalent constraint (hard, inelastic,
  categorical)

X ? C
constraining bivalent relation

• Generalized constraint on X GC(X)

GC(X) X isr R
constraining non-bivalent (fuzzy) relation
index of modality (defines semantics)
constrained variable
r ? ? ? ? blank p v u rs
fg ps
bivalent
primary

• open GC(X) X is free (GC(X) is a predicate)
• closed GC(X) X is instantiated (GC(X) is a
  proposition)

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CONTINUED

• constrained variable
• X is an n-ary variable, X (X1, , Xn)
• X is a proposition, e.g., Leslie is tall
• X is a function of another variable Xf(Y)
• X is conditioned on another variable, X/Y
• X has a structure, e.g., X Location
  (Residence(Carol))
• X is a generalized constraint, X Y isr R
• X is a group variable. In this case, there is a
  group, G (Name1, , Namen), with each member of
  the group, Namei, i 1, , n, associated with an
  attribute-value, hi, of attribute H. hi may be
  vector-valued. Symbolically

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CONTINUED

• G (Name1, , Namen)
• GH (Name1/h1, , Namen/hn)
• GH is A (µA(hi)/Name1, , µA(hn)/Namen)
• Basically, GH is a relation and GH is A is a
  fuzzy restriction of GH
• Example
• tall Swedes SwedesHeight is tall

49
SIMPLE EXAMPLES

• Check-out time is 1 pm, is an instance of a
  generalized constraint on check-out time
• Speed limit is 100km/h is an instance of a
  generalized constraint on speed
• Vera is a divorcee with two young children, is
  an instance of a generalized constraint on Veras
  age

50
GENERALIZED CONSTRAINTMODALITY r
X isr R
r equality constraint XR is abbreviation of
X isR r inequality constraint X
R r? subsethood constraint X ? R r
blank possibilistic constraint X is R R is the
possibility distribution of X r v veristic
constraint X isv R R is the verity distributio
n of X r p probabilistic constraint X isp R R
is the probability distribution of X
Standard constraints bivalent possibilistic,
bivalent veristic and probabilistic
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CONTINUED
r bm bimodal constraint X is a random
variable R is a bimodal distribution r rs
random set constraint X isrs R R is the set-
valued probability distribution of X r fg fuzzy
graph constraint X isfg R X is a function
and R is its fuzzy graph r u usuality
constraint X isu R means usually (X is R) r g
group constraint X isg R means that R constrains
the attribute-values of the group
52
PRIMARY GENERALIZED CONSTRAINTS

• Possibilistic X is R
• Probabilistic X isp R
• Veristic X isv R
• Primary constraints are formalizations of three
  basic perceptions (a) perception of possibility
  (b) perception of likelihood and (c) perception
  of truth
• In this perspective, probability may be viewed as
  an attribute of perception of likelihood

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EXAMPLES POSSIBILISTIC

• Monika is young Age (Monika) is young
• Monika is much younger than Maria
• (Age (Monika), Age (Maria)) is much younger
• most Swedes are tall
• ?Count (tall.Swedes/Swedes) is most

X
R
X
R
R
X
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EXAMPLES PROBABILISITIC

• X is a normally distributed random variable with
  mean m and variance ?2
• X isp N(m, ?2)
• X is a random variable taking the values u1, u2,
  u3 with probabilities p1, p2 and p3, respectively
• X isp (p1\u1p2\u2p3\u3)

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EXAMPLES VERISTIC

• Robert is half German, quarter French and quarter
  Italian
• Ethnicity (Robert) isv (0.5German
  0.25French 0.25Italian)
• Robert resided in London from 1985 to 1990
• Reside (Robert, London) isv 1985, 1990

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STANDARD CONSTRAINTS

• Bivalent possibilistic X ? C (crisp set)
• Bivalent veristic Ver(p) is true or false
• Probabilistic X isp R
• Standard constraints are instances of generalized
  constraints which underlie methods based on
  bivalent logic and probability theory

57
GENERALIZED CONSTRAINTSEMANTICS
A generalized constraint, GC, is associated with
a test-score function, ts(u), which associates
with each object, u, to which the constraint is
applicable, the degree to which u satisfies the
constraint. Usually, ts(u) is a point in the unit
interval. However, if necessary, it may be an
element of a semi-ring, a lattice, or more
generally, a partially ordered set, or a bimodal
distribution. example possibilistic constraint,
X is R X is R Poss(Xu) µR(u) ts(u) µR(u)
58
TEST-SCORE FUNCTION

• GC(X) generalized constraint on X
• X takes values in U u
• test-score function ts(u) degree to which u
  satisfies GC
• ts(u) may be defined (a) directly (extensionally)
  as a function of u or indirectly (intensionally)
  as a function of attributes of u
• intensional definitionattribute-based
  definition
• example (a) Andrea is tall 0.9
• (b) Andreas height is 175cm µtall(175)0.9
  Andrea is 0.9 tall

59
CONSTRAINT QUALIFICATION

• p isr R means r-value of p is R
• in particular
• p isp R Prob(p) is R (probability
  qualification)
• p isv R Tr(p) is R (truth (verity)
  qualification)
• p is R Poss(p) is R (possibility
  qualification)
• examples
• (X is small) isp likely ProbX is small
  is likely
• (X is small) isv very true VerX is small
  is very true
• (X isu R) ProbX is R is usually

60
GENERALIZED CONSTRAINT LANGUAGE (GCL)

• GCL is an abstract language
• GCL is generated by combination, qualification,
  propagation and counterpropagation of generalized
  constraints
• examples of elements of GCL
• X/Age(Monika) is R/young (annotated element)
• (X isp R) and (X,Y) is S)
• (X isr R) is unlikely) and (X iss S) is likely
• If X is A then Y is B
• the language of fuzzy if-then rules is a
  sublanguage of GCL
• deduction generalized constraint propagation and
  counterpropagation

61
CONSTRAINTS
generalized constraints
primary constraints
standard constraints

• generalized X isr R , r possibilistic,
  probabilistic, veristic, random
  set, usuality, group,
• primary possibilistic, probabilistic, veristic
• standard bivalent possibilistic, probabilistic,
  bivalent veristic
• existing scientific theories are based on primary
  constraints

62
PRECISIATION TRANSLATION INTO GCLBASIC
STRUCTURE
NL
GCL
p
p
precisiation
precisiand of p GC(p)
translation
generalized constraint

• annotation
• p X/A isr R/B GC-form of p
• example
• p Carol lives in a small city near San
  Francisco
• X/Location(Residence(Carol)) is R/NEARCity ?
  SMALLCity

63
EXAMPLE

• p Speed limit is 100 km/h

poss
cg-precisiation r blank (possibilistic)
p
speed
100
110
poss
g-precisiation r blank (possibilistic)
p
100
110
prob
g-precisiation r p (probabilistic)
p
speed
100
110
64
CONTINUED
prob
g-precisiation r bm (bimodal)
p
100
110
120
speed
If Speed is less than 110, Prob(Ticket) is
low If Speed is between 110 and 120,
Prob(Ticket) is medium If Speed is greater than
120, Prob(Ticket) is high
65
A BASIC CONCEPT IN NL-COMPUTATION GRANULAR VALUE
A
granular value of X
a
singular value of X
universe of discourse

• singular X is a singleton
• granular X isr A granule
• a granule is defined by a generalized constraint
• example
• X unemployment
• a 7.3
• A high

66
GC-BASED DEFINITION OF GRANULAR VALUE

• X is a singular value
• X is A granular value
• A is defined as a generalized constraint
• example
• X is small granular value

singleton
granule
fuzzy set
67
GRANULAR COMPUTING (GrC) REVISITED

• The objects of computation in granular computing
  are granular values of variables and parameters
• Granular computing has a position of centrality
  in fuzzy logic
• Granular computing plays a key role in
  precisiation and deduction
• Informally
• granular computingballpark computing

68
GRANULAR COMPUTING AND DEDUCTION

• The principal rule of deduction in fuzzy logic is
  the Extension Principle (Zadeh 1965, 1975).

f(X) is A g(X) is B
subject to
69
CONTINUED

• Generalized extension principle
• Zf(X,Y) singular values
• Zf(X,Y) granular values

extension
70
EXAMPLE

• p most Swedes are tall
• p ?Count(tall.Swedes/Swedes) is most
• further precisiation
• X(h) height density function (not known)
• X(h)du fraction of Swedes whose height is in h,
  hdu, a ? h ? b

71
PRECISIATION AND CALIBRATION

• µtall(h) membership function of tall (known)
• µmost(u) membership function of most (known)

?height
?most
1
1
0
0
height
fraction
0.5
1
1
X(h)
height density function
0
h (height)
b
a
72
CONTINUED

• fraction of tall Swedes
• constraint on X(h)

is most
granular value
73
DEDUCTION
q How many Swedes are short q is ?
Q deduction is
most given
is ? Q needed

• Frege principle of compositionalityprecisiated
  version
• precisiation of a proposition requires
  precisiations
• (calibrations) of its constituents

74
CONTINUED
deduction
given
is ? Q needed
solution
subject to
75
CONTINUED
q What is the average height of Swedes? q
is ? Q deduction is most
is ? Q
76
LOOKAHEAD--PROTOFORMAL DEDUCTION

• Example
• most Swedes are tall 1/n?Count(GH is R)
  is Q

Height
77
PROTOFORMAL DEDUCTION RULE
1/n?Count(GH is R) is Q
1/n?Count(GH is S) is T
?i µR(hi) is Q
?i µS(hi) is T
µT(v) suph1, , hn(µQ(?i µR(hi))
subject to
v ?i µS(hi)
values of H h1, , hn
78
PROTOFORM LANGUAGE AND PROTOFORMAL DEDUCTION
PFL
79
THE CONCEPT OF A PROTOFORM
PREAMBLE

• As we move further into the age of machine
  intelligence and automated reasoning, a daunting
  problem becomes harder and harder to master. How
  can we cope with the explosive growth in
  knowledge, information and data. How can we
  locateand infer fromdecision-relevant
  information which is embedded in a large
  database.
• Among the many concepts that relate to this
  issue there are four that stand out in
  importance search, precisiation and deduction.
  In relation to these concepts, a basic underlying
  concept is that of a protoforma concept which is
  centered on the confluence of abstraction and
  summarization

80
WHAT IS A PROTOFORM?

• protoform abbreviation of prototypical form
• informally, a protoform, A, of an object, B,
  written as APF(B), is an abstracted summary of B
• usually, B is lexical entity such as proposition,
  question, command, scenario, decision problem,
  etc
• more generally, B may be a relation, system,
  geometrical form or an object of arbitrary
  complexity
• usually, A is a symbolic expression, but, like B,
  it may be a complex object
• the primary function of PF(B) is to place in
  evidence the deep semantic structure of B

81
CONTINUED
object space
object p
protoform space
summary of p
protoform
summarization
abstraction
S(p)
A(S(p))
PF(p)

• PF(p) abstracted summary of p
• deep structure of p
• protoform equivalence
• protoform similarity

82
PROTOFORMS
object space
protoform space
PF-equivalence class

• at a given level of abstraction and
  summarization, objects p and q are PF-equivalent
  if PF(p)PF(q)
• example
• p Most Swedes are tall Count (A/B) is Q
• q Few professors are rich Count (A/B) is Q

83
EXAMPLES
instantiation

• Monika is young Age(Monika) is young A(B) is C
• Monika is much younger than Robert
• (Age(Monika), Age(Robert) is much.younger
• D(A(B), A(C)) is E
• Usually Robert returns from work at about 615pm
• ProbTime(Return(Robert) is 615 is usually
• ProbA(B) is C is D

abstraction
usually
615
Return(Robert)
Time
84
CONTINUEDEXTENSION VS INTENSION
Q As are Bs
(attribute-free extension)

• most Swedes are tall

Count(GH is A) is Q
(attribute-based intension)
85
EXAMPLES
gain
Alan has severe back pain. He goes to see a
doctor. The doctor tells him that there are two
options (1) do nothing and (2) do surgery. In
the case of surgery, there are two possibilities
(a) surgery is successful, in which case Alan
will be pain free and (b) surgery is not
successful, in which case Alan will be paralyzed
from the neck down. Question Should Alan elect
surgery?
2
1
0
option 2
option 1
Y
Y
object
i-protoform
X
0
X
0
86
PROTOFORMAL DEDUCTION
NL
GCL
PFL
p q
p q
p q
precisiation
summarization
precisiation
abstraction
WKM
DM
r
World Knowledge Module
a
answer
deduction module
87
PROTOFORMAL DEDUCTION

• Rules of deduction in the Deduction Database
  (DDB) are protoformal
• examples (a) compositional rule of inference

X is A (X, Y) is B Y is AB
symbolic
computational
(b) Extension Principle
X is A Y f(X) Y f(A)
Subject to
symbolic
computational
88
RULES OF DEDUCTION

• Rules of deduction are basically rules governing
  generalized constraint propagation
• The principal rule of deduction is the extension
  principle

X is A f(X,) is B
Subject to
computational
symbolic
89
GENERALIZATIONS OF THE EXTENSION PRINCIPLE
information constraint on a variable
f(X) is A g(X) is B
given information about X
inferred information about X
subject to
90
CONTINUED
f(X1, , Xn) is A g(X1, , Xn) is B
Subject to
(X1, , Xn) is A gj(X1, , Xn) is Yj , j1,
, n (Y1, , Yn) is B
Subject to
91
EXAMPLE OF DEDUCTION

• p Most Swedes are much taller than most Italians
• q What is the difference in the average height
  of Swedes and Italians?
• Solution
• Step 1. precisiation translation of p into GCL
• S S1, , Sn population of Swedes
• I I1, , In population of Italians
• gi height of Si , g (g1, , gn)
• hj height of Ij , h (h1, , hn)
• µij µmuch.taller(gi, hj) degree to which Si is
  much taller than Ij

92
CONTINUED

• Relative ?Count of Italians in relation to
  whom Si is much taller
• ti µmost (ri) degree to which Si is much
  taller than most Italians
• v Relative ?Count of Swedes who are
  much taller than most Italians
• ts(g, h) µmost(v)
• p generalized constraint on S and I
• q d

93
CONTINUED

• Step 2. Deduction via Extension Principle

subject to
94
DEDUCTION PRINCIPLE

• Precisiate query
• Precisiate query-relevant information
• Employ constraint propagation (Extension
  Principle) to deduce the answer to query
• example
• q What is the average height of Swedes?
• Assume that P is a population of Swedes,
  P(Name1, , Namen), with hiHeight(Namei), i1,
  , n.

95
CONTINUED

• q (h1hn)
• (qri) I Most Swedes are tall
• I (µtall(h1)µtall(hn) is most
• GC(h) (µmost( (?iµtall(hi)) , h (hi, ,
  hn)

96
CONTINUED

• constraint propagation
• (µmost( (?iµtall(hi))
• Ave(h) ?ihi
• Extension Principle
• (µAve(h)(v) suph(µmost( ?iµtall(hi)) ,
  (h1hn)
• subject to
• v ?ihi

97
DEDUCTION PRINCIPLEGENERAL FORMULATION

• Point of departure question, q
• Data D (X1/u1, , Xn/un)
• ui is a generic value of Xi
• Ans(q) answer to q
• If we knew the values of the Xi, u1, , un, we
  could express Ans(q) as a function of the ui
• Ans(q)g(u1, ,un) u(u1, , un)
• Our information about the ui, I(u1, , un) is a
  generalized constraint on the ui. The constraint
  is defined by its test-score function
• f(u)f(u1, , un)

98
CONTINUED

• Use the extension principle

subject to
99
MODULAR DEDUCTION DATABASE
POSSIBILITY MODULE
PROBABILITY MODULE
FUZZY ARITHMETIC MODULE
agent
SEARCH MODULE
FUZZY LOGIC MODULE
EXTENSION PRINCIPLE MODULE
100
SUMMATION

• The concept of GC-computation is the centerpiece
  of NL-computation. The point of departure in
  NL-computation is the key idea of representing
  the meaning of a proposition drawn from a natural
  language, p, as a generalized constraint. This
  mode of representation may be viewed as
  precisiation of p, with the result of
  precisiation being a precisiand, p, of p. Each
  precisiand is associated with a measure, termed
  cointension, of the degree to which the intension
  of p is a good fit to the intension of p. A
  principal desideratum of precisiation is that the
  resulting precisiand be cointensive. The concept
  of cointensive precisiation is a key to
  mechanization of natural language understanding.
• The concept of NL-computation has wide-ranging
  ramifications, especially within human-centric
  fields such as economics, law, linguistics and
  psychology

101
RELATED PAPERS

• K. Schmucker, Fuzzy Sets, Natural Language
  Computations and Risk Analysis, Computer Science
  Press, Rockville, MD, 1984.
• RELATED PAPERS BY L.A.Z IN REVERSE CHRONOLOGICAL
  ORDER
• Generalized theory of uncertainty (GTU)principal
  concepts and ideas, Computational Statistics and
  Data Analysis 51, 15-46, 2006.
• Precisiated natural language (PNL), AI Magazine,
  Vol. 25, No. 3, 74-91, 2004.
• Toward a perception-based theory of probabilistic
  reasoning with imprecise probabilities, Journal
  of Statistical Planning and Inference, Elsevier
  Science, Vol. 105, 233-264, 2002.
• A new direction in AItoward a computational
  theory of perceptions, AI Magazine, Vol. 22, No.
  1, 73-84, 2001.

102
CONTINUED

• From computing with numbers to computing with
  words --from manipulation of measurements to
  manipulation of perceptions, IEEE Transactions on
  Circuits and Systems 45, 105-119, 1999.
• Some reflections on soft computing, granular
  computing and their roles in the conception,
  design and utilization of information/intelligent
  systems, Soft Computing 2, 23-25, 1998.
• Toward a theory of fuzzy information granulation
  and its centrality in human reasoning and fuzzy
  logic, Fuzzy Sets and Systems 90, 111-127, 1997.

103
CONTINUED

• Outline of a computational approach to meaning
  and knowledge representation based on the concept
  of a generalized assignment statement,
  Proceedings of the International Seminar on
  Artificial Intelligence and Man-Machine Systems,
  M. Thoma and A. Wyner (eds.), 198-211.
  Heidelberg Springer-Verlag, 1986.
• Precisiation of meaning via translation into
  PRUF, Cognitive Constraints on Communication, L.
  Vaina and J. Hintikka, (eds.), 373-402.
  Dordrecht Reidel, 1984.
• Fuzzy sets and information granularity, Advances
  in Fuzzy Set Theory and Applications, M. Gupta,
  R. Ragade and R. Yager (eds.), 3-18. Amsterdam
  North-Holland Publishing Co., 1979.