Expert Systems

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Expert Systems

• Linguistic variables a quintuple (x,T(x),U,G,
  )
• X is the name of the variable
• T denotes the term set of x, that is, the set of
  names of linguistic values of x, with each value
  being a fuzzy variable denoted by x and ranging
  over a universe of discourse U which is
  associated with the base variable u
• G is a syntactic rule (grammar) for generating
  the name, X, of values of x
• M is a semantic rule for associating with each X
  its meaning M(X) is a fuzzy subset of U.

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Approximate reasoning

• Generalized modus ponens
• Premise A is true
• Implication If A then B
• Conclusion B is ture
• Allow statements that are characterized by fuzzy
  sets.
• Relax the identity of the Bs in the implication
  and the conclusion.
• Premise x is A
• Implication If x is A then y is B
• Conclusion y is B

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• Zadehs compositional rule of inference Let
  R(x), R(x,y) and R(y) be fuzzy relations in X,
  XxY, and Y respectively, which act as fuzzy
  restrictions on x, (x,y), and y, respectively.
  Let A and B denote particular fuzzy sets in X and
  XxY. Then the compositional rule of inference
  asserts, that the solution of the relational
  assignment equations R(x) A, R(x,y) B is
  given by R(y) A o B.

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Fuzzy Implications

• Ordering of fuzzy implications Fig. 11.2.

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Selection of Fuzzy Implications

• It must satisfy the following formula

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Multiconditional Approximate Reasoning

• Disjunctive if-then rules
• Conjunctive if-then rules

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WHAT
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• Deduction. Logical reasoning in which conclusion
  must follow from their premises.
• Induction. Inference from the specific case to
  the general.
• Intuition. No proven theory. The answer just
  appears, possibly by unconsciously recognizing an
  underlying pattern. Expert systems do not
  implement this type of inference yet. ANS may
  hold promise for this type of inference since
  they can extrapolate from their training rather
  than just provide a conditioned response or
  interpolation. That is, a neural net will always
  give its best guess for a solution.
• Heuristics. Rules of thumb based on experience.

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• Generate and Test. Trial and error. Often used
  with planning for efficiency.
• Abduction. Reasoning back from a true conclusion
  to the premises that may have caused the
  conclusion.
• Default. In the absence of specific knowledge,
  assume general or common knowledge by default.
• Autoepistemic. Self-knowledge
• Nonmonotonic. Previous knowledge may be incorrect
  when new evidence is obtained.
• Analogy. Inferring a conclusion based on the
  similarities to another situation.

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• Reasons for the use of fuzzy set theory in expert
  systems
• user-machine input/output description,
• imprecise knowledge
• uncertainty management.
• Fuzzy production rules condition/ conclusion
  parts contain linguistic variables.
• Fuzzy frames
• allowing slots to contain fuzzy sets as values,
• allowing partial inheritance through is-a slots.
• Fuzzy semantic nets.
• Fuzzy inference.

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• Medicine
• Sanchez
• Fuzzy set A the symptoms observed in the
  patient.
• Fuzzy relation R the medical knowledge that
  relates the symptoms to the
  diseases.
• Fuzzy set B the possible diseases of the patient
• B A o R

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• CADIAG-2
• ROoccurrence relation S x D
• RCconfirmability relation S x D
• RSoccurrence relation P x S
• R1 RS o RO (occurrence indication on P x D)
• R2 RS o RC (confirmability indication on P x D)
• R3 RS o (1-RO) (nonoccurrence indication on P x
  D)
• R4 (1-RS) o RO (nonsymptom indication on P x
  D)
• gt draw different types of diagnostic conclusions.

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• s1 occurs very seldom in patients with d1.
• S1 often occurs in patients with d2 but seldom
  confirms the presence of disease d2.
• S2 always occurs with d1 and always confirms the
  presence of d1 s2 never occurs with d2 and its
  presence never confirms d2.
• S3 very often occurs with d2 and often confirms
  the presence of d2.
• S3 seldom occurs in patients with disease d1.

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• Example
• Membership function

• Linguistic terms always, often, unspecific,
  seldom, never
• Modifier very (concentration operation u2)

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d1 is strongly confirmed for p2 (R2) (confirmed
diagnosis uR2(p,d) 1) d1 is excluded as a
possible diagnosis for p3 (excluded diagnosis
uR3(p,d) or uR4(p,d) 1)
(diagnostic hypotheses.5lt max uR1 , uR2) d1
and d2 are suitable hypotheses for p1 and p2. d2
is the acceptable hypothesis for p3
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• Building Expert Systems by Embedding Analogical
  Reasoning into Deductive Reasoning Mechanism
• Rule knowledge base
• IF X1 with (W1,R1) AND
• X2 with (W2,R2) AND
• Xm with (Wm,Rm) THEN
• Y.
• Wi are fuzzy weight factors, Ri are fuzzy
  relation matrices.

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PCPILE
TURBO PROLOG
MS-DOS
PC/AT or COMPATIBLE
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EXPERT
USER
INTERFACE
ANALOGICAL INFERENCE MECHANISM
DEDUCTIVE INFERENCE MECHANISM
CASE KNOWLEDGE BASE
RULE KNOWLEDGE BASE
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• Deductive Inference Mechanism
• Bi (y) Ai o Ri (y) ( max-min composition)

1. rank(B)

(rank all rules)
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• Case Knowledge Base
• Case base structure
• Similarity relation matrix expresses the
  relaxation of truth value of an attribute.

(2) Very very similar (Sij)1/4 very similar
(Sij)½ middle similar (Sij)1 less
similar (Sij)2 less less similar
(Sij)4 (3) Similarity degree S A o S o B
max min (x), S(x,y), (y).
AVT(0,0,0,.5,1) BMT(0,0,.5,1,.5) SMS S0.75
x , y
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• Analogical inference mechanism
• Do Procedure (CASE) for all cases
• Do Procedure (QTTRIBUTE) for all attributes
• End DO Procedure ( ATTRIBUTE).
• End Do Procedure (CASE).
• Do Procedure (GOAL) for all goals.
• End Do Procedure (GOAL).
• Return the sort of goals with its similarity
  possibility value pk as approximate solution.
• Procedure (ATTRIBUTE) Determine Similarity
  Degrees of Attributes
• Using the definition of similarity degree of
  fuzzy truth value, i.e., Eq. 7, we can draw the
  similarity degree Sij of an attribute Xi between
  the diagnosed case and past case Ci by the
  following equation

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(9)
In which Aj the fuzzy truth value of attribute
Xj of the diagnosed case Aij fuzzy truth value
of attribute Xj of past case Ci uaj the
mambership function of Aj and uaij the
membership function of Aij. Procedure (CASE)
Determine Similarity Degree of Cases For each
test case, Ci, we obtain a similarity degree, Si
with respect to the diagnosed case by aggregating
all the similarity degrees of attributes with the
weighted average method
(10)
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Procedure (GOAL) Determine Aggregated
Possibility of Goals
(11)
in which pk aggregated possibility of goal Yk
pik possibility of goal Yk with respect to past
case Ci NCB number of cases in the case base
and i 1, 2, , NCB.
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• Differences between expert systems and fuzzy
  logic control
• The existing FLC systems originated in control
  engineering rather than in AI.
• FLC models are all rule-based systems.
• By contrast to expert systems FLC serves almost
  exclusively the control of production systems
  such as electrical power plants, kiln cemen
  plants, chemical plants, etc., that is, their
  domains are even narrower than than those of
  expert systems.

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1. In general, the rules of FLC systems are not
   extracted from the human expert through the
   system but formulated explicitly by the FLC
   designer.
2. Finally, because of their purpose, their inputs
   are normally observations of technological
   systems and their outputs control statements.

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• Essential design problems in FLC
• Define input and control variables, that is ,
  determine which states of the process shall be
  observed and which control actions are to be
  considered.
• Engine-boiler combination state variables (steam
  pressure in boiler, speed of engine)
• Control actions (heat-input to boiler, throttle
  opening at the input of the engine cylinder)
• Define the condition interface, that is, fix the
  way in which observations of the process are
  expressed as fuzzy sets.

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• 3. Design the rule base, that is, determine
  which rules are to be applied under which
  conditions.
• Design the computational unit, that is, supply
  algorithms to perform fuzzy computations. Those
  generally lead to fuzzy outputs.
• Determine rules according to which fuzzy control
  statements can be transformed into crisp control
  actions.

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Defuzzification Methods Center of Area Method
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Defuzzification Methods Center of Maxima Method
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Defuzzification Methods Mean of Maxima Method
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Defuzzification Methods parameterized class
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Fuzzy Automata

• An automaton is called a fuzzy automaton when its
  states are characterized by fuzzy sets, and the
  production of responses and next states is
  facilitated by appropriate fuzzy relations
• A ltX,Y,Z,R,Sgt
• X is a nonempty finite set of input states
  (stimuli)
• Y is a nonempty finite set of output states
  (stimuli)
• Z is a nonempty finite set of internal states
  (stimuli)
• R is a fuzzy relation on Z x Y
• S is a fuzzy relation on X x Z x Y

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Fuzzy Automata
At
Ct
Fuzzy Relations R and S
Storage Ct Et1
Et
Bt
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Fuzzy Automata
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Fuzzy Automata
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Fuzzy Automata

• Deterministic fuzzy automaton (p. 352)
• o ? max-min other schemes.
• Probabilistic automaton of the Moore type