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Expert Systems. Linguistic variables: a quintuple (x,T(x),U,G, ) X is the name of the variable;

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expert systems
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.
approximate reasoning
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 B’s in the implication and the conclusion.

Premise x is A’

Implication If x is A then y is B

Conclusion y is B’

Zadeh’s 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.
fuzzy implications
Fuzzy Implications
  • Ordering of fuzzy implications: Fig. 11.2.
selection of fuzzy implications
Selection of Fuzzy Implications
  • It must satisfy the following formula:
multiconditional approximate reasoning
Multiconditional Approximate Reasoning
  • Disjunctive if-then rules
  • Conjunctive if-then rules









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.
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.
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.
  • 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


RO:occurrence relation S x D

RC:confirmability relation S x D

RS:occurrence 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)

=> draw different types of diagnostic conclusions.

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.

  • Membership function:
  • Linguistic terms: always, often, unspecific, seldom, never
  • Modifier: very (concentration operation u2)
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:.5< max [ uR1 , uR2])

d1 and d2 are suitable hypotheses for p1 and p2.

d2 is the acceptable hypothesis for p3

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


  • Wi are fuzzy weight factors, Ri are fuzzy relation matrices.















Deductive Inference Mechanism
  • Bi (y) = Ai o Ri (y) ( max-min composition)
  • rank(B) =

(rank all rules)

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)]}.





x , y

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 πk 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

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


Procedure (GOAL): Determine Aggregated Possibility of Goals


in which πk = aggregated possibility of goal Yk; πik = possibility of goal Yk with respect to past case Ci; NCB = number of cases in the case base; and i = 1, 2, …, NCB.

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.
In general, the rules of FLC systems are not extracted from the human expert through the system but formulated explicitly by the FLC designer.
  • Finally, because of their purpose, their inputs are normally observations of technological systems and their outputs control statements.
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.
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.
fuzzy automata
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 =
    • 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
fuzzy automata1
Fuzzy Automata



Fuzzy Relations

R and S


Ct = Et+1



fuzzy automata4
Fuzzy Automata
  • Deterministic fuzzy automaton (p. 352)
  • “o”  max-min; other schemes.
  • Probabilistic automaton of the Moore type