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On Implementing Usual Values

In many cases commonsense knowledge consists of knowledge of what is usual. In this paper we develop a system for reasoning with usual information. This system is based upon the fact that these pieces of commonsense information involve both a probabilistic aspect and a granular aspect. We implement this system with the aid of possibility-probability granules.


Introduction
An ability to handle commonsense reasoning is a crucial need in the development of the artificial intelligence [1] .
A number of variants of so-called non-monotonic logics have been introduced as aids in developing these commonsense reasoning mechanisms [2-41. Recently L. A. Zadeh has suggested that a central role in a theory of com,nonsense must be played by a concept of "usuality". This concept reflects the fact that in many cases commonsense reasoning involves reasoning with usual values for variables. For example, the statement "a cup of coffee costs fifty cents," is more precisely reflected by the statement "a cup of coffee usually costs fifty cents. " In a number of recent presentations Zadeh [5-7] has suggested some properties which must be present in any theory of usuality. In this paper we introduce a formal mechanism for representing and manipulating usual values. This formalism allows for both logical and arithmetic manipulations of usual values. The formal structure used to represent these usual values are Possibility-Probability granules <Poss-Prob granules>. These structures have been studied by Yager [8][9][10] and are based upon a combination of the linguistic variables introduced by L. A.
Zadeh in his theory of approximate reasoning (11] and the evidential structures introduced by 6. Shafer [12] .
As we shall see, the idea of usuality implies some random or probabilistic phenomenon at play as well as some idea of granularity. The idea of granularity necessitates the use of set theoretic constructs in the form of possibility distributions which enable us to account for the fact that humans conceptualize in terms of gross concepts. The introduction of the probabilistic aspect also provides a departure from the approach taken by the non-monotonic logicians.
We further note that these usual values play a central role as default values in frames and other similar type structures [13][14][15] .
Since much of the information in knowledge-based systems involves the use of commonsense knowledge. This theory of usuality will greatly impinge in· this area.

On Possibility-Probability Granules
In preparation for presenting our representation scheme for usual values we briefly review the idea of possibility-probability granules. For more details on these structures see (8- In particular we associate with each statement V is Ak a probability pk, which indicates the probability that V is Ak is the appropriate proposition expressing our knowledge about V.
We note that their exists no restriction on the relationship between the Ak's,other than they be non-null fuzzy subsets of X, however the pk's must sum to one.
Care should be taken to understand that pk is not the probability that the value of an experiment on X results in an element in Ak but more in the following vain. There exists some other space Y= <y&, Yzt ••• ••• ,y"} in which we perform a random experiment in which pk is the probability that Yk is the outcome. If as a result of this experiment yk occurs then we say that V is Ak is the canonical statement restricting V.
In [10] Yager has shown that the structure captured by these Poss-Prob granules is similar to that of a Dempster-Shafer belief structure in which the Ak's are the focal elements and the pk's are the weights. However, we note in this Poss-Prob framework the Ak's can be fuzzy subsets.
Because of the similarity with the Dempster-Shafer belief structure we shall denote the knowledge that a variable's value is controlled by a Poss-Prob granule where m is a basic probability assignment function <bpa> with focal elements {Ak> and weights m<Ak> = P k • Two important concepts associated with Dempster-Shafer are the measures of plausibility and belief defined on subsets of X. Because of the allowance for fuzzy sets as focal elements we must provide more general definitions for these measures in the framework of Poss-Prob granules. In particular, if we denote Prob<B> ti indicate the probability that the value for V lies in the set B then The use of Poss-Prob granules provides a very powerful mechanism for representing various different types of knowledge about the variable V in a unified structure.
One particular type of knowledge we shall find useful in this paper is the case in which we know that Prob<B> is "at least a. " Formally this can be stated as a � Prob<B> S 1.
Thus in this case we require that Bel<B> =a and Pl<B> = 1. It can easily be shown that the least restrictive Poss-Prob granule which can represent this information is a bpa m on X such that m<B> = a and m<X> = 1-a.

On Usual Values and Their Representation
As we noted in a number of recent papers L. A. Zadeh has introduced the concept of usuality and discussed its central role in any theory of commonsense reasoning.
In particular Zadeh argues that in many cases the types of knowledge which constitutes commonsense involves the knowledge about the usual value of some variable. Furthermore we understand Zadeh to suggest that in many cases these usual values are vague and imprecise of the type best represented by a linguistic value and the associated ideas of a fuzzy subset and a possibility distribution. This imprecision is due to the granular nature of human conceptualization.
If V is a variable taking its value in the set X and A is a fuzzy subset of X representing the usual value of V, we could then say, Examples of the above of linguistic structure would be "usually basketball players are tall, " "usually birds fly" "usually Mary comes home at about 8 o'clock.
In many instances of natural language discourse we suppress the word usually and simply say, for example, "birds fly" rather than "usually birds fly. " If one doesn't recognize this shorthand many difficulties follow.
The statement "usually Mary comes home at 8 o'clock, " as formalized by Usually V is A embodies a number of different forms of uncertainty. We see that the statement U<V> is A implies some probabilistic phenomenon in our knowledge about the variable V.
We shall now provide a formal framework for representing this type of knowledge.
According to Zadeh the statement Usually V is A should be interpreted as indicating that the probability that the event A occurs as the value for the variable V is "at least a", where a is some number close to one. The usual the occurrence of A the closer a is to one.
As we have indicated in the previous section this type of information on the variable V can be represented as a Poss-Prob granule.
In particular the knowledge that usually V is A can be represented as the Poss-Prob where m is a bpa on X, the frame of V, such that m<A> = a and m<X> = 1-a.
Thus we see that the effect of the statement usually V is A is to say that a portion of the time the value of V is determined by the proposition V is A and that for 1-a of the time V is unknown.
The form V is m shall constitute a canonical type of representation for usual information.
In the next section we shall provide for the translation of various linguistic structures involving usual values into these structures.
Before preceding we note that a non-probabilistic assertion such as "John is about 30 years old" can be written in this formation as V is m1 where V is the attribute John's age and m�, is a bpa such that m�.<B> = 1 where B is "a bout thirty. "

Translation of Compound Statements
In this section we shall provide some procedures for trans lating compound linguistic statements involving usual values into formal structures in terms of Poss-Prob granules.
Our purpose here is to put these complex linguistic statements into forms which enable us to us e the sophisticated mechanisms available _for combining these structures as necessary in the course of the reasoning process.
The approach here is based upon a generalization of the approach used  Based upon the work of Yager [10] we can extend this operation to apply to Poss-Prob granules. Assume V1 is m1 and Vz is mz are two Poss-Prob defined over the sets X and Y respectively, the rule developed by Yager states that <V1 is m1> � <Vz is mz) translates into <V1, Vz> is m1 � mz where m = m1 � mz is a bpa defined on X x Y such that for all the focal elements Ak of m1 and B� of mz m<Ak � 8�) = m1<Ak> * mz<B�>.
Equivalently we can define m such that for any where the summation is taken over all Ak,B� such that Ak�B�=A.
We note in the special case where V1 = Vz = V defined on X then <V is m1> � V is mz translates into V is m where m is a bpa on X defined as in the abovee Dempster's rule is a special case when � = 0.
Let us use this rule to translate some linguistic statements which involve an embedded usuality qualification.
Consider the statement "if V1 is A then usually Vz is B. " This statement can be seen to be of the form "if V1 is m1 then vz is m:z," where m1 is a bpa on X such that m1<A> = 1 and mz is a bpa on Y such that mz<B> = a and mz<Y> = 1-a.
In the case we are interested in with A � B = D and A � Y = X x Y we get <V1, Vz> is m• where m• is a bpa on X x Y such that m*<D> = a and m*<X x Y> = 1-a.
We note in this case D<x, y> =M in <1, 1-A<x> + B <y>>a Parenthetically we note that this is the same translation as the statement "usually if V1 is A then Vz is B. " Consider next the statement, "if usually V1 is A then V:z is B. " This can be translated into "if V1 is m1 then Vz is mz" where m1 is on X such that m1 <A> =a and m1<X> = 1-a. mz is on Y such that mz<B> = 1.
Using our translation rule we get <V1,Vz> is m•. m• is a bpa on X such that m*<D> = a and m*<H> = 1-a in which D<x, y> =M in [1, 1-A<x> + B<y>J and H <x, y> = B <y>. Next consider the statement "if usually V1 is A then usually Vz is B. " This can be first translated as "if V1 is m1 then Vz is mz" where m1 is a bpa on X such that m1<A> =a and m1<X> = 1-a and mz is a bpa on Y such that mz<B> = a and mz<X> = 1-a. Consider next the proposition "usually V1 is A and usually Vz is Be" Formally this becomes V1 is m1 and Vz is mz where m1 is a bpa on X such that m1<A> =a and m1<X> = 1-a. mz is a bpa on Y where mz<B> =a and m 2 <Y > = 1-a. This can be seen to be equivalent to <V1, Vz) is m• where m is on X x Y such that m*<D1> = a2, m*<Dz) = a(1-a>, m*<D3> = a<1-a> and m* (D4) = <1-a)2• In this structure D1<x,y> = Min<A<x>, B<y>>, Dz<x, y> = A<x>, �<x, y> = B<y> and D4<x, y> = 1.

Reasoning With Usual Values
In this section we shall look at the structure of some examples of reasoning with usual values.
Consider the following two propositions P1: Usually [if V1 is A then Vz is Bl P:z: Vs. is C.
In the above we are assuming that A and C are fuzzy subsets of X, the base set of Vs. and B is a fuzzy of Y, the base set of V:z. These two pieces of data can be written as Poss-Prob granules, Ps.: <Vs., V:z> is m1 and P:z: <V1> is m:z.
In the above m1 is a bpa on X x Y such that m1<H> =a and m1<X In many cases we may have to perform mathematical or arithmetic operations on these usual values.
In this section we develop the calculus necessary to perform these operations. The ability to handle both logical and arithmetic manipulations provides this approach with a very sophisticated mechanism for building expert systems.
Assume V1 and Vz are two variables taking their values in the real line R.
Let V1 is m1 and Vz is mz be two Poss-Prob granules in which m1 and mz are basic probability assignments on R. Let {Ak} be the focal elements of m1 and let {B�} be the focal elements of mz.
We note that both the A's and B's are fuzzy subsets of R.
Let "V = V1 .J. Vz" where .1. is any arithmetic operation, (addition +, subtraction -, multiplication *, division 1, or exponentiation >. In [101 Yager has shown that in this situation "V is m" where m is a bpa on R such that for any A C R m<A> = E m1<A1> * mz<B�>, where the summation is taken over all As and B� such that As.J.B�=A. In '< Jrder to evaluate the above we must use fuzzy arithmetic [171. In particular if E and F are two fuzzy numbers, fuzzy subsets of the real line, then E .1. F = 6, where 6 is a fuzzy subset of R such that 6 = LJv.ac" {E(y) A F(z)/y .J. z}.
Let us look at the situation for various forms of m1 and mz.
Consider the case in which our knowledge is P1: Usually V1 is A and Pz: Vz is B in which A and B are fuzzy subsets of R.
First we see that these two pieces of data can be represented in terms of granules in the following way. P1: V1 is m1 in which m1 is a bpa on R defined by m1<A> =a and m1<R> = 1-a. Pz: Vz is mz where mz is also a bpa on R defined by mz<B> =