In [12], decision logic (DL) is proposed as a means to represent knowledge discovered from data tables. This logic is called decision logic because it is particularly useful for a decision table, which is a data table S = (U, A), whereAcan be partitioned into two sets,C(condition attributes) andD(de- cision attributes). Through data analysis, decision rules relating condition and decision attributes can be derived from the table. A rule is then represented as an implication between two formulas of DL.
Since DL can represent knowledge discovered from precise data tables, we generalize it to fuzzy decision logic (FDL) for rule representation in FDT. The basic alphabet of FDL consists of a finite set of attribute symbolsA=
{a1, a2, . . . , am} and, for 1 ≤ i≤ m, a finite set of linguistic terms Li. The
atomic formula of an FDL is a descriptor (ai, li), whereai ∈ Aand li ∈ Li.
The set of well-formed formulas (wff) of FDL is the smallest set containing the atomic formulas and closed under the Boolean connectives ¬,∧, and ∨. If ϕand ψ are wffs of FDL, then ϕ−→ ψ is a rule in FDL, whereϕ is the antecedent of the rule andψ is the consequent.
Each element in the universe of an FDT corresponds to an object and an atomic formula (i.e., an attribute-value pair) describes the value of an individual attribute of an object. Thus, atomic formulas (and wffs) can be verified or falsified in each object. This gives rise to a satisfaction relation between the universe and the set of wffs.
Many natural language terms are highly context-dependent. For example, the word “tall” in “a tall basketball player” has a quite different meaning than it has in “a tall child”. To model context-dependency, we associate a context with each FDL. The context determines the domain of values of each attribute and assigns an appropriate meaning to each linguistic term. Formally, given an FDT (U, A), a context associated with an FDL is a function,ct, that maps each linguistic termli ∈ Li toct(li)∈ P˜(Vi) for 1≤i≤m, whereVi is the
domain of values of attributefi. We assume each FDT has a fixed context.
Each linguistic term is interpreted as a fuzzy subset of attribute values, so an object may satisfy an atomic formula in FDL to some degree. Thus, the satisfaction between data records and wffs is a quantitative relation.
The semantics of FDL depend on how the fuzzy sets in the FDT and FDL contexts are interpreted. A fuzzy set can be interpreted disjunctively or conjunctively, and the difference between disjunctive and conjunctive inter- pretations corresponds to the bipolar representation of possibilistic logic [1].
3.1 Disjunctive Interpretation
In disjunctive interpretation, a fuzzy set is considered as constraints imposed by a linguistic term over the domain. LetV be the domain of possible values and X be a fuzzy set in ˜P(V); then, the membership degree of an element,
v, in X stipulates the possibility that the actual value is v. The disjunctive interpretation of fuzzy sets is appropriate for incomplete information systems. When we do not know the exact value of an attribute, we can encode the incomplete information by a fuzzy set. This is the interpretation adopted in the ordinary possibility theory by Zadeh [19].
Given a domain V, a possibility distribution onV is a function π: V →
[0,1]. A possibility distributionπis called normalized if supv∈V π(v) = 1. Two measures onV can be derived fromπ. They are called the possibility and the necessity measures and are denoted by Π and N respectively. Formally,Π
andN : 2V →[0,1] are defined as Π(A) = sup
v∈A π(v),
N(A) = 1−Π(A),
whereAis the complement of Awith respect toV.
These two measures correspond to our uncertainty about the crisp event
A when a piece of vague informationπ is available. They can be extended to measure the uncertainty of fuzzy events [4]. The extended measures, still denoted byΠ andN, are defined asΠ andN : ˜P(V)→[0,1],
Π(X) = sup
v∈V
µX(v)⊗π(v), N(X) = inf
v∈Vπ(v)→⊗µX(v),
whereµX is the membership function of a fuzzy eventX,⊗: [0,1]×[0,1]→
[0,1] is a t-norm,2and→
⊗: [0,1]×[0,1]→[0,1] is the residuated implication
function for⊗defined asa→⊗b= sup{x|x⊗a≤b}.
If an FDT S = (U, A) represents an incomplete information system, for eachx∈U andfi ∈A,fi(x) is considered as a possibility distribution over Vi. Thus, its membership function is equivalent to a possibility distribution πi,xover the domainVi. The wffs of FDL can then be evaluated in each data
record ofU according to the following evaluation function,E:
1. E(x,(ai, li)) = Ni,x(ct(li)), where Ni,x is the necessity measure corre-
sponding toπi,x
2. E(x,¬ϕ) = 1−E(x, ϕ)
2A binary operation⊗is a t-norm iff it is associative, commutative, and increasing in both places, and 1⊗a=aand 0⊗a= 0 for alla∈[0,1].
3. E(x, ϕ∧ψ) =E(x, ϕ)⊗E(x, ψ) 4. E(x, ϕ∨ψ) =E(x, ϕ)⊕E(x, ψ)
where⊕is a t-conorm defined bya⊕b= 1−(1−a)⊗(1−b).
In the disjunctive interpretation, fi(x) indicates the incomplete informa-
tion about the value of the attributefiof the objectx. To measure the extent
to whichxsatisfies the atomic formula (ai, li), we have to evaluate the truth
degree of the statement “for every possible valuevin the domain of attribute
fi, ifxhas the valuev, then vsatisfies the linguistic labelli”. In fuzzy logic,
the truth degree of such a formula is defined by inf
v∈Viµfi(x)(v)→⊗µct(li)(v) = infv∈Viπi,x(v)→⊗µct(li)(v),
which is equal to Ni,x(ct(li)). In other words, E(x,(ai, li)) is the necessity
degree of the fuzzy eventct(li) in accordance with the incomplete information fi(x). This explains the intuition behind the definition of the evaluation func-
tionE for the atomic formulas in the disjunctive case. The definition of the evaluation function for other wffs follows the standard approach in fuzzy logic.
3.2 Conjunctive Interpretation
In conjunctive interpretation, a fuzzy set represents positive knowledge in the domain, which is appropriate for the representation of multi-valued data. For example, in an FDT, the attribute “programming skills” may have a fuzzy set value (C + + : 1, Java : 0.8, Pascal : 0.6). By using conjunctive interpre- tation, the object with this attribute value has the programming ability of all three languages. If the fuzzy set were interpreted disjunctively, the object would only have the programming skills of one of these languages.
Mathematically, we can consider each linguistic label li as a possibility
distribution over domainVi. Thus, the membership function ofct(li) is equiv-
alent to a possibility distributionπli and its corresponding necessity measure
is denoted byNli.
If an FDT S= (U, A) represents multi-valued data, then the atomic wffs of FDL are evaluated in eachx∈U by
E(x,(ai, li)) =Nli(fi(x)),
and the evaluation function can be extended to any wffs in the same manner as in disjunctive interpretation.
In conjunctive interpretation, the fuzzy set fi(x) indicates all properties
that x has with respect to its attributefi. Therefore,x satisfies the atomic
formula (ai, li) if it possesses all properties stipulated by the linguistic label li. In fuzzy logic, to evaluate if xpossesses all properties stipulated by the
every possible value v in the domain of attributefi, ifv is stipulated by li,
thenv belongs tofi(x)”. The truth degree of such a formula is defined by
inf
v∈Viµct(li)(v)→⊗µfi(x)(v),
which is equal toNli(fi(x)). This explains the intuition behind the definition
of the evaluation function,E, for the atomic formulas in the conjunctive case.