In this section, we consider about the transition fromIintoI4.We then consider about the transition of processes induced by this transition of interval truth values. Both generate together the congruence theorem.
We have noticed that interval truth value sets considered have two versions: Iand its subset I4. The latter is a sublattice of the previous one. So, one can consider one kind of homomorphisms fromIintoI4.Based on the operational se- mantics, we pay attention to the meet-homomorphisms, i.e., functions preserving finite meets ofI.
4.1 Quantizers of Interval Truth Values
Definition 4. A truth-quantizer from Iinto I4 is a function preserving finite meets with respect to the truth order relation ≺. Similarly, a vague-quantizer from Iinto I4 is a function preserving finite meets with respect to the vague order relation ⊂. A quantizer will mean a truth-quantizer or a vague-quantizer if no confusion occurs.ω, ω0 will rang over the set of all quantizers.
Letx= (x1, . . . , xn) be a vector of interval truth values. We defineω(x) =
(ω(x1), . . . , ω(xn)).
In the next, we will show that each centerζ induces a vague-quantizer and a numericν does a truth-quantizer. Thus, we have a family of such quantizers.
Now, let us take any centerζ= (a, b)∈Iζ.Then it divides the setIinto four
parts: ζU, ζC, ζFand ζT,where
– ζU={x= (xa, xb)∈I|xa≤a, b≤xb}
– ζC={x= (xa, xb)∈I|a≤xa, xb≤bbutx6= (a, b)}
– ζF={x= (xa, xb)∈I|xa < a, xb< b}and
Fuzzy Interval-valued Processes Algebra 53 Hence,I= ζU∪ ζC∪ ζT∪ ζF.
Similarly, for a numericν∈Iν, one can define νT, νF, νUand νC.
They are sets below, respectively.
– νT={x= (xa, xb)∈I|a≤xa, b≤xb}
– νF={x= (xa, xb)∈I|xa≤a, xb≤bbutx6= (a, b)}
– νU={x= (xa, xb)∈I|xa < a, b < xb}and
– νC={x= (xa, xb)∈I|a < xa, xb< b}.
We also have that I= νT∪ νF∪ νU∪ νC.
F U a b ζ ζC ζT ζF ζU C T ζ= [a, b],(a+b= 1) F U a b ν νF νC νU νT C T ν= [a, b],(a=b)
Based on the division ofIby a centerζand a numericν, we define two kinds of quantizers fromIinto I4.The specific definitions are below.
Definition 5. For a given centerζ and numericν, we define functions, denoted still byζ andν fromIintoI4respectively by
ζ(x) = U, if x∈ ζU C, if x∈ ζC T, if x∈ ζT F, if x∈ ζF ν(x) = T, if x∈ νT F, if x∈ νF U, if x∈ νU C, if x∈ νC.
Proposition 1. For a given center ζ, the transition function ζ is a vague- quantizer. Simultaneously, for a given numeric ν, the transition function ν is a truth-quantizer.
4.2 Omega Quantization of Processes
In this paragraph, we pay attention to setting up an omega quantization, denoted as qω, of processes from P intoP4 induced by a quantizer ω from Iinto I4. A congruence theorem will be given, which shows a reasoning rule as follows.
P αx −→P0 qω(P)
αω(x)
−→ qω(P0).
Definition 6. For a given quantizerω and any interval-valued processP ∈ P, theω-quantization ofP is inductively defined as follows:
54 Y. Chen, J. Zhou 1. IfP =0, thenqω(P) =0.
2. IfP is an interval-valued constantA, thenP =Ais defined by the equation ofAdef= Q, whereQis an interval-valued process, and thenqω(P) =qω(A)
def = qω(Q). 3. IfP =αx.Q, thenqω(P) =αω(x).qω(Q). 4. IfP =Pi∈IxiPi, thenqω(P) =Pi∈Iω(xi)qω(Pi). 5. IfP =x(P1|. . .|Pn), thenqω(P) =ω(x)(qω(P1)| · · · |qω(Pn)). 6. IfP =Q\L, then qω(P) =qω(Q)\L, and 7. IfP =Q[f], thenqω(P) =qω(Q)[f].
The operation of omega quantization, indeed, induces a map fromP, the set of all interval-valued processes, into P4, the set of all four value processes for a
given quantizerωfromIintoT4.Thisωquantization of processes is denoted as qω. The following theorem shows that both quantizerω and its induced omega
quantization generate the congruence theorem.
Theorem 1. For any pair of interval-valued processes P and P0 and a given quantizerω,P αx
−→P0 impliesqω(P) αω(x)
−→4qω(P0).
Proof. This proof is done by using structural induction on the process P and the transition rules fromActthrough Con.
Theorem 2. Suppose that P is an interval-valued process in P, Q is a four interval-valued process in P4,Y ∈I4, and ω is a quantizer function fromIinto I4. Then, qω(P)
αY
−→4 Q implies that there exist P0 ∈ Ψ and y ∈ I such that
P −→αy P0,qω(P0) =Q, andY =ω(y).
Proof. We also can render the proof by induction on the construction ofP.
P αx- P0 qω(P) - qω(P 0) 4 αω(x) ? ? qω ω qω ? Assumption @@R qω(P) αY -4 Q=qω(P0) P ? - αy P0 ? qω ω qω ? Assumption@ @ I Pictures of Theorem 1 and 2
Fuzzy Interval-valued Processes Algebra 55
5
Conclusion and Future Work
This paper has introduced the new model of process calculus called fuzzy interval- valued process algebra (FICCS, shortly) to resolve nondeterminism that arises in concurrent and communication systems based on the fuzzy interval value logic. This model is an extension of standard CCS of Milner’s. In FICCS, an interval truth value is assigned to either process or action and the interval truth valued vector to a summation process or a composition process, which are similar to the way Ying tackles with the additive probabilistic process algebra model [YING02]. When interval truth values become numerics, these interval-valued processes are those which probabilistic CCS considers about. But, they are complete distinct due to the distinct logic used. In our model, the logic is fuzzy interval logic. In probabilistic model, however, the logic is the probabilistic logic. As a result, the operators involved are very distinct too. In FICCS, the operators are lattice operators such as the meet and union. In probabilistic model, however, they are numeric operators such as the multiplication and plus.
This paper has established the syntax of FICCS and its two versions of opera- tional semantics due to existence of two kinds of order-relations between interval truth values in fuzzy interval logic. For special four interval truth values F,T, C and U representing for false, true, contradiction and unknown respectively, a corresponding four fuzzy interval-valued CCS (FI4CCS, for short) has been
established. Then, this paper mainly pays attention to quantization of processes from FICCS into FI4CCS. A congruence theorem has been given indicating that
this quantization transfers the operational semantics from FICCS into FI4CCS.
This paper is just at the beginning of study of the fuzzy process algebra. The future work will be to fulfil the theory of fuzzy process algebra and its application in the practice of computer technology.
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