Applications of Temporal Logic to the Specification and Verification of Reactive Systems: A Survey of Current Trends

As a generic c o m p u t a t i o n a l model we p resen t the model of fair transition systems (FTS) . T h e p r e s e n t a t i o n a n d discussion of th is model follows IMP1]. A fair t r ans i t i on sys t em consists of the following cons t i tuen t s : A (possibly infinite) set of states. T A finite set of transitions. Each t r ans i t i on v E T is a func t ion r: ~ ~ 2 ~:. O A set of initial states. 0 C_ E. J A Justice family. This is a family of sets J = ( J 1 , . . , J=}, where each J i ___ T presents a justice requ i rement . jr _ A Fairness family. This is a family of sets 7 = { F 1 , . . . ,Fk}, where each Fi _ T presents a fairness requ i rement . If 7(s) # 0 we say t h a t the t r a n s i t i o n 7 is enabled on t h e s t a t e s. If r ( s ) = O, t hen v is said to be disabled on s. Let S C T be a set of t rans i t ions . If for some v E S, r is enab led on s, we say t h a t the set S is enab led on s. Given an F T S P , we define a computation of P to be a finite or infini te sequence of s ta tes a n d t rans i t ions : v0 vl 1"2 0": 8 0 ~ 8 1 ~ 8 2 -~ " • " sat is fying the following requi rements : • Ini t ia l i ty: so E 0 • Consecut ion: For each i, s i+ l C r i (s i ) • Te rmina t ion : a is finite and t e r m i n a t e s in Sk only if Sk is terminal, i.e., for every ~ E T, ~(sk) --O. • Just ice: Let J i E J be a jus t ice set which is cont inuous ly enabled beyond some sk. Then , some t r ans i t i on in J i m u s t be taken at least once beyond sk, i.e., the re mus t be a j > k such t h a t ~-j E J i . Th i s implies, in fact , t h a t t r ans i t i ons in J i mus t be t aken infinitely m a n y t imes in a . • Fairness: Let F / E Jr be a fairness set which is enab led infinitely m a n y t imes in a . Then , t r ans i t ions in Fi mus t be taken infinitely m a n y t imes in a . Some o the r works, such as [GFMR] and [AOI, refer to our no t ion of jus t i ce as weak fairness and to our no t ion of fairness as strong fairness.

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