Work with Indeterminateness

NEST works with weights that express as the indeterminateness in the knowlage base as the indeterminateness in answers of users. While weights of rules are set by one value, the weights during a consultation are represented by intervals. Weights of the rules w (and answers of users that have only one value) are converted to the interval [w;w].

7 functions are defined for a work with indeterminateness (NEG, CONJ, DISJ, CTR, GLOB, NORM, IMP). These functions are defined on the interval [-1;1] (function NORM is defined on the set of real numbers). If the knowlage base uses another weight range, the weights`ll be regulated to this interval.

4 types of the work with indeterminateness are implemented in NEST:

• standard;
• logical;
• neuronal;
• hybrid.

5 functions (NEG, CONJ, DISJ, NORM, IMP) are defined equally for all types, in others two, there are some differences.

Function NEG

Function NEG is used for the calculation of a negation of a proposition. It`s defined as following:

NEG(w) = -w

or for the work with interval [w1;w2]:

NEG([w1;w2]) = [-w2;-w1]

Function CONJ

Function CONJ is used for the calculation of a conjunction of two propositions. It`s defined as following:

CONJ(v,w) = min(v,w)

or for the work with intervals [v1,v2],[w1;w2]:

CONJ([v1;v2],[w1;w2]) = [min(v1,w1); min(v2,w2)]

Function DISJ

Function DISJ is used for the calculation of a disjunction of two forms. It`s defined as following:

DISJ(v,w) = max(v,w)

or for the work with intervals [v1,v2],[w1;w2]:

DISJ([v1;v2],[w1;w2]) = [max(v1,w1); max(v2,w2)]

Function CTR

Function CTR is used for the calculation of a contribution of a rule (or for put together a weight of context and a weight of condition). It`s defined as following:

• standard, neuronal and hybrid type of undeterminateness:

  CTR(a,w) = 0       ; a ≤ 0
  CTR(a,w) = a * w   ; a > 0

• logical type of undeterminateness:

  CTR(a,w) = 0                              ; a < 0
  CTR(a,w) = sign(w) * max(0,a + |w| - 1)   ; a ≥ 0

For the work with intervals [a1;a2],[w1,w2], it is used the expression
CTR([a1;a2],[w1,w2]) = [CTR(a1,w1);CTR(a2,w2)]

Function GLOB

Function GLOB is used fo put together more contributions of rules. It`s defined as following:

• standard type of undeterminateness:

  GLOB(v,w) = 0                    ; (v = 1, w = -1) or (v = -1, w = 1) 
  GLOB(v,w) = (v + w)/(1 + v * w)  ; other  

• logical and hybrid type of undeterminateness:

  GLOB(w1...wn) = min(Σw>0w ,1)  +  max(Σw<0w,-1)

• neuronal type of undeterminateness:

  GLOB(w1...wn) = min(max(sΣw, -1), 1)

  where s = 1

For the work with intervals [a1;a2],[w1,w2], it is used the expression
GLOB([a1;a2],[w1,w2]) = [GLOB(a1,w1);GLOB(a2,w2)]

Function NORM

This function is defined in the real numbers. It regulates intervals to the interval [-1;1]. It`s used after the calculation of compositional rule`s contributions with by funciton CTR and after the function GLOB during the calculation of a weight of proposition`s condition. It`s defined as following:

NORM(w) := min(max(w,-range), range)

or for the work with interval [w1;w2]:

NORM([w1;w2]) = [min(max(w1,-range), range); min(max(w2,-range), range)]

where range = 0.999 (or another count of „9“ depends on the number of decimal places)

Function IMP

Function IMP is used for the calculation of a final weight of integrity restriction that decides if the integrity restriction was breaked and how much. It`s defined as following:

IMP(w,v) = 0                  ; w <= 0   
IMP(w,v) = max(0, min(w - v)) ; w > 0

or for the work with interval [w1;w2]:

IMP([w1;w2],[v1,v2]) = [max(0, w1 - v1); max(0, w2 - v2)]

where w is the weight of condition (that can be modified by a context by the function CTR) and v is the weight of conclusion.