Concept expressions:

Note: conjunction and disjunction are n-ary, n>1

• Lukasiewicz T-norm: (and x y) = max(0, x+y -1)
• Lukasiewicz T-conorm: (or x y) = min(1, x+y)
• Lukasiewicz implication: x => y is 1 if x ≤ y, else is max(0, 1 - x + y)
• Gödel T-norm: (and x y) = min(x, y)
• Gödel T-conorm: (or x y) = max(x, y)
• Gödel implication: x => y is 1 if x ≤ y, else is y
• Kleene-Dienes implication: x => y is max(1-x,y)

• By default, the conjuction (and C1 C2) and disjunction (or C1 C2) are interpreted according the selected semantics (Classical Logic, Zadeh Semantics, Lukasiewicz Logic)
• Negation is interpreted always as Lukasiewicz negation: (not x) = 1- x
• Zadeh semantics uses: Lukasiewicz negation, Gödel T-norm, Gödel T-conorm, Kleene-Dienes implication in the "(all R C)" and "(kd-implies C1 C2)" concepts. For general concept inclusion axioms, (implies C1 C2) is the same as: for all individuals d, the degree C1(d) is less or equal than C2(d). This is the same as stating (l-implies C1 C2 1). Similarly for 'concept inclusion'
• Lukasiewicz Logic uses: Lukasiewicz negation, conjunction and disjunction and implication.

Logic selection:

Definition of the fuzzy logic is not done in the Config file, but in the KB with the statement:

 (define-fuzzy-logic [lukasiewicz | zadeh | classical])

Indiscernibility Relation Declarations:

Definitions a role R as an indescirnibility relation

(define-fuzzy-similarity R)

R is a reflexive and symmetric

(define-fuzzy-equivalence R)

R is a reflexive, symmetric, and transitive

Truth constants:

(define-truth-constant constant n)

Defines a name for a truth constant n, where n is a rational in [0,1].

Example:

(define-truth-constant totally 1.0)
(define-truth-constant very 0.8)
(define-truth-constant quite 0.7)
(define-truth-constant moderately 0.6)
(define-truth-constant enough 0.4)
(define-truth-constant little 0.2)

We then may use them in the expressions as, e.g.

(instance jim Tall totally)
(instance maria Tall little)

Concept modifier:

(define-modifier CM linear-modifier(b))

Defines a name for a modifier as a linear hedge. b>0. E.g. (define-modifier very linear-modifier(3))

(define-modifier CM triangular-modifier(a,b,c))

Defines a name for a triangular modifier, where at 0 the value is a, at b the value is 1, while at 1 the value is c. E.g. (define-modifier more-or-less triangular-modifier(0,0.8,0.7))

Concrete Fuzzy Concepts:

Defines a name for a fuzzy set with explicit fuzzy membership function.

(define-fuzzy-concept CFC crisp(k1,k2,a,b))

(define-fuzzy-concept CFC left-shoulder(k1,k2,a,b)) with a < b

(define-fuzzy-concept CFC right-shoulder(k1,k2,a,b)) with a < b

(define-fuzzy-concept CFC triangular(k1,k2,a,b,c)) with a < b < c

(define-fuzzy-concept CFC trapezoidal(k1,k2,a,b,c,d)) with a < b < c < d

(define-fuzzy-concept CFC linear(k1,k2,a,b))

(define-fuzzy-concept CFC modified(mod,F))

E.g. (define-fuzzy-concept Young left-shoulder(0,100,10,30))

Fuzzy Numbers:

Fuzzy numbers, such as "about 3", "about 4.5", can be represented as concrete fuzzy concepts having triangular membership function. A fuzzy number can be defined using the syntax

(define-fuzzy-number-range k1 k2)

which defines the range of all fuzzy numbers used. Then any individual fuzzy number is defined by means of

(define-fuzzy-number name FN)

where FN is a fuzzy number expression and defines the fuzzy number name as defined by the expression FN. A fuzzy number correponds to a triangular fuzzy membership function, with range k1, k2, i.e, triangular(k1, k2, a, b, c).

No recursive definition is allowed.

Any rational number n is a fuzzy number with triangular membership function, where a=b=c.

On fuzzy numbers, we may perform arithmetic operations, such as addition, substraction, multiplication and division. Let n be rational number , n+ be a non zero rational and

(define-fuzzy-number fn1 a1, b1, c1)

(define-fuzzy-number fn2 a2, b2, c2)

The following are fuzzy number expressions with parameters as shown in the last column

Note:

• f+ and f* may also have L arguments, e.g, (f+ fn1 fn2 ... fnL)

Example: Three exams of a student have been evaluated qualitatively using the degree scale VeryPoor, Poor, Fair, Good, VeryGood, defined as fuzzy numbers below

(define-fuzzy-number-range -100 100)

(define-fuzzy-number VeryPoor 0, 0, 2)

(define-fuzzy-number Poor 1, 2.5, 4)

(define-fuzzy-number Fair 3, 5, 7)

(define-fuzzy-number Good 6, 7.5, 9)

(define-fuzzy-number VeryGood 8, 10, 10)

The overall evaluation can be defined as:

(define-concept Exam1 (some hasScore VeryPoor))

(define-concept Exam2 (some hasScore Fair))

(define-concept Exam3 (some hasScore Good))

(define-concept StudentScore (w-sum (0.3 Exam1) (0.2 Exam2) (0.5 Exam3)) )

where the weights reflects the importance of the exams.

Datatype Restrictions:

Defines a name for a functional datatype attribute (also called concrete feature) that then can be used to form concepts according to the syntax below. Concrete features are defined using two instructions, functionality and range:

 (functional featureName)

 (range featureName *integer* k1 k2) | (range F *real* k1 k2) | (range F *string*) | (range F *boolean*)

Boolean values are: true, false

E.g.,

(functional hasAge)

(range hasAge *integer* 0 150),

defines the attribute hasAge as integer valued. The values should be between 0 and 150.

Datatype restrictions are concepts having the following syntax:

For instance, given the definition

(functional hasAge)

(range hasAge *integer* 0 150)

the concept

(and Person (>= hasAge 18))

denotes the set of people whose age is at least 18.

Another example si the following:

%----

(functional totalPrice)

(range totalPrice *real* 0 1000)

(functional netPrice)

(range netPrice *real* 0 1000)

(instance a (= netPrice 50.21))

(instance a (= totalPrice (netPrice + (0.2*netPrice) + -20) ) )

(show-concrete-fillers-for a netPrice totalPrice)

(min-instance? a (>= totalPrice 30))

%-----

General Concept Inclusions Axioms:

The degree d may be either a rational number in [0,1] or a variable (string). Variables may be used when some special constraints on the values are requested, by using Inequetions (see below)

Role Axioms:

NOTE: (define-fuzzy-similarity RN) is used together "upper" and "lower approximations concepts in Fuzzy Rough Set reasoning. For instance,

(define-fuzzy-similarity S)

(instance a (ua S C) 0.8)

(instance a (la S D) 0.7)

(related a b S 0.9)

(min-instance? b D)

#(min-instance? a (some S (and C D)))

Weighted Concept:

(w-sum (n1 C1) ... (nk Ck))

Represents a concept, which is the weighted sum of the concepts Ci, i.e. n1*C1 + ... + nk*Ck. We assume, ni in [0,1] and n1+ ..+nk=1.

Example of usage:

(define-concept Dinner (and Person (w-sum (0.2 Bottle) (0.3 Plate) (0.2 Fork) (0.2 Spoon) (0.1 Knife))))

"A dinner is something in which there is a Person and possibly, a bottle, a plate, a fork, a spoon, a knife".

(w-sum-zero (n1 C1) ... (nk Ck))

Represents a concept that is as the weighted sum, except that if an argument is 0, the expression is 0.

Example of usage:

(define-concept Dinner (and Person (w-sum-zero (0.2 Bottle) (0.3 Plate) (0.2 Fork) (0.2 Spoon) (0.1 Knife))))

"A dinner is something in which there is a Person and possibly, a bottle, a plate, a fork, a spoon, a knife. If one of the arguments is 0, i.e missing, then we have no dinner".

(n C)

For n in [0,1], this represents a concept such that for any individual d, the degree of being individual d an instance of (n C), i.e., (n C)(d), is given by n times the degree of being d an instance of C, i.e., n * C(d). As a note, (w-sum (n1 C1) ... (nk Ck)) is the same as (l-or (n1 C1) ... (nk Ck)).

Other aggregation methods are via OWA, quantifier-based OWA or Fuzzy Integrals.

Threshold Concept:

([>= d] C)

([<= d] C)

Represents a concept, where d in [0,1] or d is a variable and C is a concept. E.g., the degree of being individual a an instance of ([>= d] C), i.e., ([>= n] C)(a), is 0 if a is an instance of C to degree less than d, otherwise the degree is C(a). Similarly, the degree of being individual a an instance of ( [<= d] C), i.e., ([<= d] C)(a), is 0 if a is an instance of C to degree greather than d, otherwise the degree is C(a).

Example of usage:

(implies ([>= 0.8] Dinner) GoodDinner))

"A a dinner with degree greater or equal than 0.8 is a good dinner"

As such ([>= 0.8] Dinner) acts as thresholded concept with threshold 0.8.

Assertions:

The degree d may be either a rational number in [0,1] or a variable (string). Variables may be used when some special constraints on the values are requested, by using Inequetions (see below)

Linear Inequation Axioms:

Allow to specify constraints on the truth values.

where

Example 1.

(instance a C x1)

(instance a D x2)

(constraints (x1 + x2 = 0.5))

Example 2.

Tom is taller than Tim.

(instance tim (not Tall) x1)

(instance tom Tall x)

(constraints (x1 + x = 1))

Crisp Declarations:

Declares concepts Ai and roles Rj as crisp.

(crisp-concept A1 ... An)

(crisp-role R1 ... Rn)

Query Expressions:

Multiple queries can be added to the file.

A simple example of maximization problem is the following:

(instance a Beta xBeta)

(instance a Alpha xAlpha)

(constraints (xBeta<=0.5) (xAlpha + -0.3*y<=0) (BND y BV) (0.5*xBeta + 0.5*xAlpha + -1*x=0))

(min-var? x)

Show Expressions:

Statements have been added to show values in an optimal solution.

Example:

(functional hasPrice)

(define-fuzzy-concept AudiTTPrice right-shoulder(0,50000,30500,31500))

(define-concept AudiTT (and SportsCar (some hasPrice AudiTTPrice)))

(define-fuzzy-concept BuyerPrice left-shoulder(0,50000,30000,32000))

(define-concept BuyerQuery (and SportsCar (some hasPrice BuyerPrice)))

(max-sat? (and BuyerQuery AudiTT))

(show-concrete-fillers hasPrice)

"Ask about the maximal agreement among buyer and sold item AUdiTT and show the price of this agreement"