Possibility and Necessity
8.5 RESOLVING CONTRADICTIONS USING POSSIBILITY IN A NECESSITY-BASED SYSTEM
An example of the use of possibility in a necessity-based system to resolve contradictions is given by example program echo.par. In this program, we wish to determine the anatomical significance of regions in an echocardiogram, noisy cross-sectional ultrasound images of a beating heart. To do this, the major steps are first, fuzzify the input data; next, determine preliminary classification using sup-porting data and necessities; last, rule out contradictory preliminary classifications using refuting data and calculated possibilities.
8.5.1 Input Data: Useful Ambiguities
Our critical input data are numerical values of the area, x- and y-centroid of each image region. Since the image is noisy, these values are somewhat uncertain, and are represented by fuzzy numbers. These values are fuzzified into three fuzzy sets; size (TINY SMALL MEDIUM LARGE HUGE), xpos (FAR-LEFT LEFT CENTER RIGHT FAR-RIGHT), and ypos (VERY-HIGH HIGH MIDDLE LOW VERY-LOW). Usually, more than one member of each of these fuzzy sets will have non-zero truth values; since the members are not mutually exclusive, ambigu-ities are almost certain to be present. Such ambiguambigu-ities are a very good thing; they lend robustness to the reasoning process.
In addition, we have a simple string attribute “border”, which has value “YES” if a region touches the border of the image, and “NO” if the region does not touch the border.
8.5.2 Output Data: Contradictions
Our output data, the region classifications, are represented by fuzzy set class fARTIFACT LUNG LA (left atrium) LV (left ventricle) LAþLV (merged left atrium and ventricle) RA (right atrium) RV (right ventricle) RAþRV (merged right atrium and ventricle) LAþLVþRAþRV (all four chambers artifactually merged)g. These members are mutually exclusive; if more that one member has a non-zero truth value, we have a contradiction that will have to be resolved.
8.5.3 Reaching Preliminary Classifications Using Supporting Evidence and Necessities
Rules for determining preliminary classifications all employ supporting data, and hence employ necessities, the default truth value for the FLOPS language. An example is IF (in Region area is LARGE AND xpos is RIGHT AND ypos is MIDDLE AND border = "NO")
THEN in 1 class is LV;
(In echo.par the preliminary classification rules are created automatically from a classification definition database; this permits modification of the classification defi-nitions without rewriting the program. However, this makes the rules more difficult to access; the “prule” command must be used to inspect them.)
Since the rules for classification as LV and LAþLV (and RV and RAþRV) have identical antecedents, we can be sure that there will be contradictory prelimi-nary classifications.
Suppose there are more than one rule whose consequent is “class is LV”. Since the default reasoning is (upward) monotonic, the aggregation procedure will store the largest of the existing and new truth values as the grade of membership of LV in fuzzy set class.
8.5 RESOLVING CONTRADICTIONS USING POSSIBILITY 147
8.5.4 Resolving Contradictions Using Refuting Evidence and Possibilities
Here, we will consider the rules for resolving the contradiction when a region has been classified both as LV and as LAþLV. In this case, if another region in the same image frame has been classified as LA, we can be sure that the LAþLV classification is wrong, and the LV classification is right; if in that frame no other region has been classi-fied as LA, then we are sure that the LV classification is wrong and the LAþLV classi-fication is right. The refuting data for LAþLV is then the presence of an LA region; the refuting data for LV is the absence of an LA region. Our rule antecedents will then read
rule (goal Refute LA+LV)
IF (in Region frame =<FR>AND class is LA) (8:21) (in Region frame = <FR>AND class is LAþLV)
THEN ...
rule (goal Refute LV)
IF (in Region frame =<FR>AND NOT class is LA) (8:22) (in Region frame = <FR>AND class is LV)
THEN ...
We must now think about how to formulate the consequent. Then antecedent is a complex proposition, say Q, that refutes a proposition; the antecedent will then assign a (possibly) new truth value(s) to that proposition. In a necessity-based system, the new truth value will be the proposition’s necessity.
The new truth value itself can be obtained from (8.17). This requires using the complement of the truth value of the antecedent. We must now employ non-mono-tonic reasoning; this can be done by storing truth values directly. In FLOPS, the com-bined truth value of antecedent and rule is available as the system-furnished variable
<pconf>. Since the truth value of the rule is 1.0 (1000 in FLOPS), the truth value of :Q for a particular rule instance is given by
tv(:Q) = (1<pconf>) Rules (8.21) and (8.22) then become:
rule (goal Refute LA+LV)
IF (in Region frame =<FR>AND class is LA) (8:23) (in Region frame =<FR>AND class is LAþLV)
THEN class:LA+LV = (min(class:LAþLV,(1 -<pconf>));...
rule (goal Refute LV)
IF (in Region frame =<FR>AND NOT class is LA) (8:24) (in Region frame =<FR>AND class is LV)
THEN class:LV = (min(class:LV,(1 -<pconf>)));...
The truth value of the antecedent represents how sure we are that the LV or LAVþLV classification should be refuted; its complement represents the possibility of the LV or LAVþLV classification.
8.6 SUMMARY
Necessity represents the degree to which the evidence considered to date supports the truth of a proposition or datum; possibility represents the extent to which the evi-dence considered to date fails to refute a proposition or datum. Initially, in the lack of any data supporting or refuting a datum, its necessity is 0 and its possibility is 1. As more and more supporting data are considered the necessity of a datum tends to increase monotonically, always subject to the restriction that the necessity of a datum must be equal to or less than its possibility; as more and more refuting data are considered, the possibility of a datum tends to decrease monotonically.
In systems that maintain two truth values, possibility and necessity, a datum of which we have no knowledge at all has necessity 0 and possibility 1. A datum that is known to be completely false has both necessity and possibility 0. A datum that is known to be completely true has both necessity and possibility 1.
In systems that maintain a single truth value, necessity, we cannot distinguish between a datum about which nothing is known from a datum known to be false, since both have necessity 0. In such systems, we consider a datum to be false until supporting evidence is found.
Even though necessity-based systems do not maintain the possibility of data, it is possible to calculate the possibility of a datum from refuting evidence; if the calcu-lated possibility is less than its existing necessity, the necessity must be reduced to obey the restriction that necessity is less than or equal to possibility. However, sup-porting evidence considered in subsequent firing of block of rules could increase the reduced necessity. It is then advisable first to fire blocks of rules to arrive at prelimi-nary conclusions, which may be ambiguous or contradictory. We then resolve any contradictions by considering refuting evidence in rule blocks fired later in the reasoning process.
8.7 QUESTIONS
8.1 We have said that if there is no supporting or refuting evidence for a datum, its necessity is 0 and its possibility is 1. Why is this?
8.2 We have postulated the relationship Nec(A)<= Pos(A)
Suppose that at the end of a rule-firing step, we have calculated Nec(A)¼0.6, and Pos(A)¼0.4. To maintain that relationship, we can either decrease Nec(A) to 0.4 or can increase Pos(A) to 0.6. Which should we do? Why?
8.7 QUESTIONS 149
8.3 Why should we consider supporting evidence before considering refuting evidence?
8.4 What is the basis for rejecting the conventional possibility axiom that in theory, when considering several propositions A, B, C,. . ., that A, B, and C are nested so that (A)B)C )?
8.5 What is the basis for rejecting the conventional possibility axiom that A AND NOT A¼min(A, 12A)?