Sequential Processes. In the case of sequential processes, this information indicates explicitly the possible transactions from a state.

Transcription

1 Sequental Processes Each state of a system can be represented by a process expresson that carres nformaton ab both the behavor and the structure of the system. In the case of sequental processes, ths nformaton ndcates explctly the possble transactons from a state. The set P seq of sequental process expressons s defned by the followng syntax: P :: A a,,a n Σ I α.p where I s any fnte ndexng set. We use P, Q, P, to stand for process expressons. 53

2 Notatons We defne processes lke procedures n standard programmng languages. We presuppose an nfnte set of process dentfers and we use A, B,, or helpful names lke er. We use α, β, to range over Σ. We consder processes to be parametrc on names. For example, we wrte A a,b,c to mean the process A wth name parameters a, b, c. We wrte a for a sequence a,, a n of names. 54

3 Name Substtuton If a and b are sequences of names of length n wth all names n a are parwse dstnct and P s a process expresson, then {b/a}p denotes the result of replacng a by b n P where n. The set of names, whch occur n the process expresson P s denoted fn(p) the set of free names of P. 55

4 Defnng Equatons We assume that every process dentfer A has a defnng equaton of the form A( a) de f P where P A s a summaton, and the names a a,..., (all dstnct) nclude all the free names fn(p A ) of P A. If b s any sequence on n names, not necessarly dstnct, then the process express A b should mean the same as {b/a}p. A a n 56

5 Summaton If I {, 2, 3}, then a summaton Σ I α.p s wrtten as α.p + α 2.P 2 + α 3.P 3 The order of the sub-terms s nsgnfcant. If I, then Σ I α.p s the empty sum, wrtten. 57

6 Structural Congruence Two sequental process expressons P and Q are structurally congruent, wrtten P Q, f we can transform one nto the other by replacng occurrences of A b by {b/a}p, or vce versa, for arbtrary A defned by A( a) de f P. For example: A(a,b) a.a a,b + b.b a,a B(c,d) c.d. Then we the structural equvalences B a,a a.a. and A a,b a.(a.a a,b + b.b a,a ) + b.a.a. A 58

7 LTS of Sequental Processes The labeled transton system of sequental processes over Σ s defned to have states P seq, and transtons as as follows: If P Σ I α.p then for each I, P. α P 59

8 Boolean er n n A buffer process wth capacty two, (ntally empty). The process receves boolean values (the recept of s the acton n ) and transmts them n the same order (e.g. the transmsson of s the acton ). Whle storng the sequence s, where s {,,,,, }, the process s n the state s. 6

9 6 Boolean er Equatons ),,,n (n. n.. n. {,} {,} +

10 Boolean er States n n n n n n 62

11 63 Process Equatons {,} {,} {,}.. n.... n.. n. + +

12 Scheduler A set of agents P, n, s to be scheduled to perform a certan task repeatedly. More precsely, P wshes to perform the task repeatedly, and a scheduler s requred to ensure that the agents ntate the task n cyclc order, begnnng wth P. Each agent sgnals ts completon of the task to the scheduler. The dfferent task-performances may overlap each other n tme, but the scheduler must ensure that each P fnshes one performance before t starts another. 64

13 Scheduler Specfcaton We suppose that P requests to start the task by pressng the button a on the scheduler, and sgnals completon of the task by pressng b. a a 2 a n b b 2 b n Specfcaton:. The scheduler must requre a,, a n to occur cyclcally, begnnng wth a, 2. For each, the scheduler must requre a and b to occur alternately, begnnng wth a, 3. The scheduler must permt any of ts buttons to be pressed at any tme provded () and are not volated. 65

14 Scheduler States We use Sched,X to denote a state of the scheduler, wth {,,n} and X {,, n}. The parameter ndcates that t s P s turn to ntate the task next; the parameter X represents the set of agents currently performng the task. In state Sched,X : Only P can ntate the task (provded that X), Any P ( X) can complete. Scheduler Sched,X Sched X X, b.sched b.sched,x - +,X - a.sched +,X ( X) ( X) + s nterpreted modulo n 66

15 Counter A counter able to hold natural numbers s smple process wth nfntely many states. nc zero dec Count Count n+ nc.count nc.count + n+ 2 zero.count + dec.count n 67