Reduced Forms of Disjuncts

Fuzzy Multi-dimensional Analysis	Alexandr A. Savinov
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4. Reduced Forms of Disjunctions

Let us consider the following example. Three disjunctions

{1, 0, 1, 1}	{0.5, 1}
{1, 0.5, 1, 1}	{0, 1}
{1, 0.5, 1, 1}	{0.2, 1}

are semantically equivalent, i.e., they represent the same distribution over the universe of discourse. Thus in general case disjunctions represent semantics not uniquely, i.e., several different on the form disjunctions can represent the same on the meaning proposition about the universe. The uniqueness of representation takes place only for disjunctions with the constant equal to 0, when there is at least one element from the universe with the distribution value 0. If the disjunction constant (the minimal value of the corresponding distribution) is not equal to 0, then its representation is not unique because the components which are between 0 and the disjunction constant may be changed in this interval (provided that this does not change the constant itself). So it is clear that in the disjunction

{1, 0, 1, 1}

{0.5, 1}

with the constant 0.5 the second component of the first section may be changed between 0 and 0.5, e.g.,

{1, 0.27, 1, 1}

{0.5, 1}

To overcome this non-uniqueness of representation let us introduce a so called reduced forms of disjunction. The disjunction d is said to be in k-th reduced form iff the constants of all its sections d_i except for the k-th section d_k are equal to 0 constant(d_i) = min(d_ij) = 0

and all the rest of components are exactly greater than constant(d_k):

In other words, disjunction in k-th reduced form may not contain components in the interval (0,constant(d_k)] (i.e., any component is either equal to 0 or is greater than constant(d_k), and in addition, each proposition must include at least one component equal to 0 except for the k-th section. For example, the disjunction

{1, 0.2, 0.9, 1} {1, 0}

is in 1st reduced form, whereas the disjunctions

{1, 0.2, 0.9, 1}	{1, 0.2}
{1, 0.2, 0.9, 1}	{1, 0.1}
{1, 0.2, 0.9, 1}	{0.2, 0.2}

are not reduced.

This definition does not say how to reduce disjunctions. Now we will propose a procedure for reducing disjunctions which is based on operations of subtraction/addition of the value p from/to the section d_i. This operations result in a new local distribution d_i-p/d_i+p such that

d_i-p = d_i, if d_i > p
d_i-p = 0, otherwise

and

d_i+p = d_i, if d_i > p
d_i+p = p, otherwise

Thus to compute d_i-p (d_i+p) we have to change onto 0 (p) all the components which are less than or equal to p (Fig. 5).

Fig. 5. Operation of subtraction/addition of the value from/to the section.

The whole procedure for reducing disjunctions is the following:

find the disjunction constant:

_1j

_nj

subtract the disjunction constant from all non-k-th propositions
add the disjunction constant to the k-th proposition

For example, the constant of the disjunction

{0, 0.2, 0.3, 1}

{0.3, 1}

is equal to 0.3, therefore its 1st reduced form is the following:

{0.3, 0.3, 0.3, 1}

{0, 1}

According to this approach if a disjunction is in a reduced form then it involves a section which is responsible for storing the disjunction constant value. We can transfer the constant from one section to another but such a section will always exist and thus we have n different reduced forms.

Another approach [S93b, S93c, S96a] consists in that we introduce one special component which is responsible for storing the disjunction constant value and is said to be also the disjunction constant (or constant proposition in logical terms). For the sake of simplicity such a representation is not used in this paper but it is really useful in many situations (e.g., when representing disjunctions in knowledge base) since the reduced form is defined uniquely and the disjunction constant is represented explicitly.

Fuzzy Multi-dimensional Analysis	Alexandr A. Savinov
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