Stable Fuzzy Predicates And Predicate Converter

This paper is to discuss the program designing semantic on the base of the latticetopology. And the main research content is the Domain theory. Scott first initiates thedomain theory, the aim is to provide a mathematical model for the program seman-tic. Domain is a directed complete partial order set, called as dcpo. All the set of Scottopen sets is a Scott topology. Because of the different emphasitions and the mathe-matical tools when we research the program designing languages, the formal semanticscan be divided into four classes: operational semantics; denotational semantics; alge-bral semantics and axiomal semantics. The denotational semantics is thinked of asthe criteria semantic. Domain is the base and core of denotational semantics. And, thetopology knowleges have been used in the program semantics.Predicate transformer semantic is an important content in the axiomal semantics.The Hoare logic is the triple {P}S{Q}, P is the precondition, S is the program, Qis the post-condition. On the base of Hoare logic, Dijkstra first intruduces the defi-nation of the weakest precondition and presentes the predicate transformers, called asthe weakest predicate transformers, defined as wp, which maps a postcondition to theweakest precondition. So, this predicate transformer provide a semantical model forprogram language. Later, in order to provide a semantical model for the program withprobability choice, many researchers research the probabilistic predicate transformers.Then in order to account for many non-determinaton and dispremise phenomenons inreal world, professer Chen yixiang and professer Jung combines the Zadeh's fuzzy settheory with the Scott domain, intruducing the fuzzy predicate which is a Scott continuefunction from a domain (dcpo) to the unit interval [0, 1] and the fuzzy predicate trans-formers. At the same time, professer Chen yixiang and professer Plotkin presents thehealthy fuzzy predicate transformers in order to show the relation between the fuzzyprdecate transformers and the probability predicate transformers. In rencently research, fuzzy predicate transformer can provide the fuzzy logic senmantic for the probabilistccomputation.In this paper, the author will present a new fuzzy predicate, called as the stablefuzzy predicate on the base of the fuzzy predicate, which is a stable function from dcpo D with the consistent meets into the unit interval [0, 1]. The aim is to discuss thestable property of fuzzy predicate. In this paper, the author mainly will prove the setof all stable fuzzy predicates on dcpo D is a fuzzyξ—semitopology on the pointwiseorder. The author will give an example to show the linear add of stable fuzzy predicatesunnecessarily is the stable, but some special stable fuzzy predicates own this property.At the same time, the author prove the product of a number in the unit interval [0, 1]and a stable fuzzy predicate is stable. Also, the author show the set of all the consistentopen sets on D and the set of all the stabl fuzzy predicates is homomorphic. Thenin order to depict the semantic, the author will set up the complete consistent stablefuzzy predicate transformers, showing the relation with the state transformers. At thesame time, the author is going to prove the healthy fuzzy predicate transformers is acpo on the pointwise order and the linear properties.