Yixiang Chen and Jie Zhou
School of Mathematics, Physics and Informatics Shanghai Normal University
Shanghai 200234, People’s Republic of China
Abstract. In this paper, the authors propose a new model called as fuzzy interval-valued CCS (FICCS for short) to solve the problem of nondeterministic choices arises in concurrency and communication sys- tems. This model is designed in such a way that an interval value is assigned to the action of a prefix and an interval value distribution to components of a parallel composition and a summation based on the meta logic of fuzzy interval logic. Following the study of standard CCS, the authors give the syntax of FICCS and its two versions of operational semantics due to existence of two kinds of order-relations between in- terval truth values in fuzzy interval logic. We also introduce four fuzzy interval-valued CCS (FI4CCS, for short). Then, we consider the notion
of 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.
1
Introduction
As what Hoare stated in the preface of Milner’s book named with “ Commu- nication and Concurrency” [MIL89]. Milner’s CCS (Calculus of Communicating Systems)[MIL89] is the model of representing not only interactive concurrent systems, such as communications protocols, but also much of what is familiar in traditional computation e.g. data structures and storage regimes. A similar model, CSP (Communicating Sequential Processes) was independently received by Hoare[HOA85]. Both models have inspired a school of researchers throughout the world, who have contributed to its refinement, development and application. Among them, probabilistic CCS is one of important developments. One of its feature is that each command of a process summation expression is guarded by a probability and the sum of these probabilities is 1, i.e., it obeys the stochastic condition [GSST]. For example, summation processp1P1+p2P2,whereP1andP2
are sub-processes, however, p1 andp2 are non-negative real numbers satisfying
the stochastic conditionp1+p2= 1 and being designed to sub-processesP1and
P2as the probability of execution respectively. This model is called as stochastic
?This work was partly supported by National Key Research and Development Pro- gram (Grant NO. 2002CB312004), National Foundation of Natural Sciences of China (Grant No: 60273052), and the key Project of Educational Commission of Shanghai (Grant No: 02DZ46).
Fuzzy Interval-valued Processes Algebra 45 probabilistic CCS. Recently, Ying introduced a new model of probabilistic pro- cesses, called additive probabilistic process algebra (APPA, for short) [YING02]. In APPA, the probability satisfies the sub-stochastic condition, i.e., the sum of probabilities is less than or equal to 1. APPA is called as sub-stochastic prob- abilistic CCS. In whether stochastic or sub-stochastic probabilistic CCS, the probabilities have the same feature, which is that if subprocess P is executed with the probabilitypthen it is not executed with the probability 1−p. In other words, for each subprocess in a summation process, the sum of its probabilities of execution and non-execution is just 1.
One can consider about other tow cases. One is that the sum is less than 1, and the other that the sum is larger than 1. The first case depicts that there are some things that are unknown, since the sum of probabilities of execution and non-execution for a process is less than 1. The second one says that there are some things that are contradictive due to the sum of probabilities being larger than 1. Both cases can appear in a distributed systems.
In this paper, we employ the interval value [a, b] in fuzzy interval logic to represent them in such a way thatais the degree (instead of the probability) of subprocess’s execution, and 1−b the degree of its non-execution. A little more general, one proposed the notion of interval truth value, which is
[a, b] (a, b∈[0,1]),
whereacan be interpreted as a degree of affirmative assertion for a given state- ment, 1−bas a degree of negative assertion for the same statement.
The purpose of this paper is to propose a new model to deal with nondeter- ministic choice, based on the interval truth value and the fuzzy interval logic, that we refer to as fuzzy interval-valued process algebra (FICCS, for short). FICCS is an interval extension of CCS with interval values instead of probabil- ities. It borrows some ideas of probabilistic CCS’s. Its underlying (meta) logic is the fuzzy interval logic proposed by Mukaidono and Kikuchi [MUKK90]. So, our FICCS model is completely distinct from the probabilistic model.
As well-known, stochastic and vagueness are at least two forms of nondeter- minism and imprecision’s appearance in the real world. The toss of a fair coin is a classic stochastic example, and “young” and “old” are well-known concepts of vagueness. The situation in the computer world is similar, e.g., stochastic processes, and imprecise inputs. As what we have pointed above, in order to deal with stochastic phenomena, one uses methods from probability theory and statistics. Probabilistic processing is one of resolving stochastic phenomena in the computer world [Jones [JON90], Larsen and Skou [LS94], McIver and Mor- gan [MM01], Seidel [SEI95], van Glabbeek et al. [GSS95], Ying [YING02]]. For the denotational semantics of probabilistic processing, one proposed probabilistic powerdomain [Alvarez-Manilla et al. [AMJK03], Jones and Plotkin [JP89],Jung and Tix [JT98],Moshier and Jung [MJ02]], which comes from the classic pow- erdomain. Probabilistic predicate transformers is the third method to deal with stochastic phenomena [He et al. [HSM97], Kozen [KOZ81], McIver and Morgan [MM01], Morgan et al.[MMS96], Ying [YING03]].
46 Y. Chen, J. Zhou
These fuzzy analogues, however, have not been well considered, although as earlier as in 1968, Zadeh pioneers the notion of fuzzy algorithm. In 2003, Chen and Zhou in [CZ03] proposed a kind of fuzzy processing system, called interval CCS. The interval CCS is of the additive model, which is followed from Ying’s work. The most recently, Chen and Jung explored the notion of fuzzy predicate transformers [CJ04]. Chen and Plotkin proposed the notion of fuzzy healthy condition for fuzzy predicate transformers [CP04]. This paper aims at the proposal on fuzzy interval-valued process algebra, a non-additive model of CCS, based on the interval truth value in fuzzy interval logic. Its syntax and semantics are given.
The paper is organized as follows. Section 2 recalls some basic notions in fuzzy interval logic. Four particular interval truth values, T = [1,1] for true, F= [0,0] for false,U= [0,1] for unknown andC= [1,0] for contradiction, are considered. The notationI4will denote the set of these four interval truth values. Two kinds of ordering relation between interval truth values, called truth order and vague order respectively, are introduced. In section 3, the authors define the syntax of FICCS based on theIand one of FI4CCS based on theI4. With corresponding these two syntaxes, two kinds of operational semantics of FICCS and FI4CCS are introduced. In Section 4, the authors consider about the notion
of quantizers of interval truth values, which is indeed a function fromIintoI4 preserving the finite meets. Based on the quantizer, the authors introduce the notion of omega quantization of processes, which is a mapping from the setP of interval-valued processes of FICCS into the setP4of four interval-valued process
of FI4CCS. A congruence theorem indicating that omega quantization preserves
the operational semantics is given.
2
Fuzzy Interval Logic
Fuzzy interval logic was proposed by Mukaidono and Kikuchi[MUKK90]. In this fuzzy interval logic, propositions can take interval truth values, which can be viewed as special cases of linguistic truth values [ZAD75] or the general cases of numerical truth values.
2.1 Interval Truth Value
An interval truth value is expressed as
[a, b] (a, b∈[0,1]),
whereais interpreted as a degree of affirmative assertion for a given statement, 1−bas a degree of negative assertion for the same statement.
The set consisting of all interval truth values is expressed as I={[a, b]|a, b∈[0,1]}.
Four interval truth values [0,0],[1,1],[1,0] and [0,1] are very special. For [0,0], the affirmative degree is 0, but the negative degree is the largest 1. So, it
Fuzzy Interval-valued Processes Algebra 47 stands for false, which is denoted by F. For [1,1], the affirmative degree is 1, its negative degree, however, is 0. So, it is used to stand for truth, denoted by T. In the case of [1,0], both affirmative degree and negative degree are 1. So, it presents the greatest contradictive. We us it to stand for the contradiction, denoted byC. Finally, it is clear that both affirmative and negative degrees of [0,1] are 0. So, it says nothing. We us it to represent the unknown, denoted by U. The set of these four interval truth values is denoted byI4.That is,
I4={F,T,C,U}.
Notice thata and b are not asked to satisfy the condition ofa less than b. In other word, b is possibly less than a. So, there are three cases of relations betweenaandb:a=b, a < banda > b.
Ifa =b then a+ (1−b) = 1. That means that the sum of the affirmative degree and the negative degree is 1. It is what the probabilistic case considers about. These interval truth values will be called as centers. Iζ will denote the
set of all centers, andζ, ζ0, . . .will rang over centers.
If a < b then a+ (1−b) = 1−(b−a) < 1. That means that the sum of affirmative degree and negative degree is less than 1. So, there is some event not known. For example, if we take a= 1/3, b= 2/3, then both affirmative and negative degrees are 1/3. Their sum is 2/3. So, the remain 1/3 is of the unknown. Ifa > b thena+ (1−b) = 1 + (a−b)>1. So, the sum of affirmative and negative degrees is larger than 1. For example, we take a = 2/3 and b = 1/4, thena+ (1−b) = 2/3 + 3/4 = 17/12.Thus, 1−(a+ (1−b)) =b−a= 5/12.The 5/12 is of contradiction, for it can be of the affirmation and also of the negation. So, in this case, contradiction occurs.
Another interesting case is thata+b= 1. Thus, 1−b=a. So, both degrees of affirmation and negation area. These interval truth values will be called as numerics. The notationIν will denote the set of all numerics, andν, ν0, . . . will
rang over numerics.
We have seen that degrees of true (affirmation) and false (negation) for a statement are given independently, which is just of the feature of fuzzy logic, and completely different from the probability logic [NIL86]. But, the latter can be thought of as a special case of fuzzy interval truth value.
2.2 Truth Order and Vagueness Order
Two kinds of partial order relations≺and⊂are defined in the setIas follows:
Definition 1. [MUK94] Letx= [xa, xb] andy= [ya, yb] be elements ofI. Define
the truth order relation≺as
y≺xif and only ifya ≤ xa andyb ≤ xb.
From the definition, one can find out that ifxis larger than y in the truth order, then the affirmative degree of x0s is larger than one of y0s, the negative degree is, however, exactly oppositive. That is, the negative degree of x0s is
48 Y. Chen, J. Zhou
smaller than oney0s. This is the reason that we call such relation as the truth order relation. Under the truth order, the smallest element isF, and the greatest one is T.
Definition 2. [MUK94] Let x = [xa, xb] and y = [ya, yb] be elements of I.
Define the vague order relation⊂as
y⊂xif and only if xa ≤ ya andyb ≤ xb.
Notice that if they is a subset of thexthen y is the less than xunder the vague relation. But, the inverse is not true in general. In fact, the contradiction truth valueC=[1,0] is the least element under the vague order relation. Surely it is not a subset of any interval value truth except itself. The unknown truth valueU=[0,1] is, however, the greatest one.
Both induced orders onI4are shown in the following graphs.
@ @ @ @ ¡¡ ¡¡ F • U•¡ • C ¡¡ ¡ @ @ @ @ • T (I4,≺,∨,∧) @ @ @ @ ¡¡ ¡¡ C • F•¡ • T ¡¡ ¡ @ @ @ @ • U (I4,⊂,∪,∩) 2.3 Lattice Operators
Both truth order and vague order makeIandI4be lattices. From the viewpoint of logic, however, we have logical operations such as truth OR(∨), truth AND(∧), vague OR(∪), and vague AND(∩), which are defined as follows.
For elementsx= [xa, xb] andy= [ya, yb] of I, we define
– x∨y= [max(xa, ya),max(xb, yb)]
– x∧y= [min(xa, ya),min(xb, yb)]
– x∪y= [min(xa, ya),max(xb, yb)]
– x∩y= [max(xa, ya),min(xb, yb)].
Both (I4,∨,∧) and (I4,∪,∩) are sublattices ofI.
• x∧y x • • x∨y • y (I,≺,∨,∧) • x x∪y • • y • x∩y (I,⊂,∪,∩)
Fuzzy Interval-valued Processes Algebra 49 We will use<andt,uto represent any one of the two partial order relations
and its induced lattice operators as a general notation.