Discrete Time Process Algebra with Abstraction

ion in absolute timing. where all timing refers to an absolute clock. Here again. we only consider the two-phase version. In section 4. we have discrete time process algebra with parametric timing. where absolute and relative timing are integrated. For parametric timing. we introduce a model based on time spectrum sequences. An underlying viewpoint of the present paper is that for a given time free atomic action. there may be different timed versions. We mention some: fts(a) stands for action a in the fIrst time slice. followed by immediate termination. cts(a) is a in the current time slice with immediate termination. ats(a) is a in any time slice with immediate termination. atstau(a) = ats(a)·ats('t) is a in any time slice followed by silent termination in any subsequent slice. It turns out that for a an atomic action or 't, the interpretation of a as atstau(a) is the appropriate (homomorphic) embedding of time free process algebra into timed process algebra. There are many practical uses conceivable for timed process algebras. In particular. we mention the TOOLBus (see [BEK94. 95]). This TOOLBus contains a program notation called T which is syntactically sugared discrete time process algebra. Programs in T are called T-scripts. The runtime system is also described in terms of discrete time process algebra. By using randomised symbolic execution the TOOLB us implementation enacts that the axioms of process algebra can be viewed as correctness preserving transformations of T -scripts. A comparable part of disrete time process algebra that is used to describe T-scripts has also been used for the description of </lSDL, flat SOL. a subset of SOL that leaves out modularisation and concentrates on timing aspects (see [BEM95]). 2. DISCRETE TIME PROCESS ALGEBRA WITH RELATIVE TIMING. We start out from the relative discrete time process algebra of [BAB95]. First. we consider the theory with only nondelayable actions. next we add the delayable or time free actions. 2.1 BASIC PROCESS ALGEBRA WITHOUT TIME FREE ACTIONS. The signature of BPAct,:t has constants cts(a) (for a E A). denoting a in the current time slice, and cts(/», denoting a deadlock at the end of the current time slice. The superscript denotes that time free • atoms are not part of the signature. Also. we have the immediate deadlock constant /) introduced in [BAB95]. This constant denotes an immediate and catastrophic deadlock. Within a time slice. there is no explicit mention of the passage of time. we can see the passage to the next time slice as a clock tick. Thus. the cts(a) can be called nondelayable actions: the action must occur before the next clock tick. The operators are alternative and sequential composition. and the relative discrete time unit delay areI (the notation a taken from [HER90]). The process areI (x) will start X after one clock tick. i.e. in the next time slice. In addition. we add the auxiliary operator VreI. This operator disallows an initial time step, it gives the part of a process that starts with an action in the current time slice. It was called a time out at the end of the current time slice in [BAB92], there. the notation x» dt 1 was used for VreI (x). (The greek letter v sounds like "now"; this correspondence is even stronger in Dutch.) The axiom DRT I is the time factorization axiom: it says that the passage of time by itself cannot determine a choice. The addition of a silent step in strong bisimulation semantics now just amounts to Discrete time process algebra with abstraction 3 the presence of a new constant cts('t), with the same axioms as the cts(a) constants. We write At = Au{'t} , AS = Au{o} etc. The standard process algebra BPAS can be considered as an SRM specification (Subalgebra of Reduced Model, in the terminology of [BAB94]) of the present theory: consider the initial algebra of • BPAdrt + DCS, reduce the signature by omitting 0, Orel, Yrel, then BPAs is a complete axiomatisation of the reduced model, under the interpretation of a,o by cts(a), cts(o) (note that x + cts(o) = x for all • closed terms x except 0). X+Y=Y+X Al Orel(X) + Orel(Y) = Orel(X + y) DRTl (x + y) + z = x + (y + z) A2 Orel(X),Y = Orel(X'Y) DRTI • X+X=X A3 Orel(O) = cts(o) DRT3 (x+y)·z=x·z+y·z A4 cts(a) + cts(o) = cts(a) DRT4 • • (x·y)·z = x·(y·z) A5 Yrel(O) = 0 DCSI Yrel(cts(a)) = cts(a) DCS2 • x+o=x A6ID Yrel(Orel(X)) = cts(o) DCS3 • • o,x= 0 A7ID Yrel(X + y) = Yrel(X) + Yrel(Y) DCS4 Yrel(X'Y) = Yrel(X)'Y DCS5 TABLE I. BPAdrt + DCS. 2.2 STRUCTURED OPERATIONAL SEMANTICS. We give a semantics in terms of operational rules. We have the foIlowing relations on the set of closed process expressions P: action step s P x At x P, notation p ~ p' action termination s S x At, notation p ~ " time step ~ P x P, notation p ~ p' immediate deadlock s P, notation ID(p) (denotes action execution) (execution of a terminating action). (denotes passage to the next time slice) (immediate deadlock, holds only for process • expressions equal to 0). We enforce the time factorization axiom DRTI by phrasing the rules so that each process expression has at most one o-step: in a transition system, each node has at most one outgoing o-edge. This operational semantics uses predicates and negative premises. Still, using terminology and results of [VER94], the rules satisfy the panth format, and determine a unique transition relation on closed process expressions. Strong bisimulation is defined as usual, so a binary relation R on P is a strong bisimulation iff the following conditions hold: i. if R(p,q) and p ~ p' (u E AuJ), then there is q' such that q ~ q' and R(p',q') ii. if R(p,q) and q ~ q' (u E At,,), then there is p' such that p ~ p' and R(p',q') iii. if R(p,q) then p ~ " iff q ~" (a E At) and ID(p) iff ID(q). Two terms p,q are strong bisimulation equivalent, p H q, if there exists a strong bisimulation relating them. From [BAB95] we know that the axiomatisation in table I is sound and complete for the model of closed process expressions modulo strong bisimulation.