Abstract
This paper presents an assume-guarantee specification theory (aka interface theory from [11]) for modular synthesis and verification of real-time processes with critical timing constraints. Four operations, i.e. conjunction, disjunction, parallel and quotient, are defined over specifications, drawing inspirations from classic specification theories like refinement calculus [4, 19]. We show that a congruence (or pre-congruence) characterised by a trace-based semantics [14] captures exactly the notion of substitutivity (or refinement) between specifications.
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Notes
- 1.
Note that the existence of incompatibility errors does not mean that the composed system is un-usable; an environment can still usefully exploit the system by only utilising the part of the system that is free of the incompatibility errors, as has been well explained in [11].
- 2.
It were Carroll Morgan and Joseph M. Morris who first added miracle to refinement calculus.
- 3.
Our timed framework originally appeared in [9]. However, the version presented here contains important technical extension as well as presentational improvements.
- 4.
Invariants and guards on output actions are constraints on the system (aka guarantees) whereas co-invariants and guards on input actions are constraints on the environment (aka assumptions).
- 5.
\(\mathcal {P}\) is time-additive providing \(p \xrightarrow {d_1+d_2} s'\) iff \(p \xrightarrow {d_1} s\) and \(s \xrightarrow {d_2} s'\) for some \(s \in S\).
- 6.
Note that invariant and co-invariant are downward-closed. Thus, the only way to violate them is to exceed their upper bounds.
- 7.
One further case missing above is that, for an action transition, there is possibility that its guard is respected but the invariant/co-invariant of its destination (say l) is violated. In such situation, a state (l, t) is treated (1) as \(\top \) if t violates the invariant in location l and (2) as \(\bot \) if t violates the co-invariant in l while the invariant holds.
- 8.
For \(i \in \{0,1\}\) and \(p_i=(l_i, t_i)\), \(p_0 \times p_1 = ((l_0,l_1),t_0 \uplus t_1)\) (t0 and \(t_1\) are clock-disjoint).
- 9.
Containment of \(\rightarrow _0 \cup \rightarrow _1\) is not required for parallel composition, but is necessary for conjunction and disjunction.
- 10.
The technique was inspired by a discussion with Roscoe on angelic choice in CSP.
- 11.
Note that the above definition exploits the fact that the addition or removal of false-guarded transitions to AT will not change the semantics of the automata.
- 12.
The modified determinisation procedure first appeared in the Definition 4.2 of [26], which is for the untimed case.
- 13.
We say an acyclic TIOTS is a tree if (1) there does not exist a pair of transitions in the form of \(p \xrightarrow {a} p''\) and \(p' \xrightarrow {d} p''\), (2) \(p \xrightarrow {a} p'' \wedge p' \xrightarrow {b} p''\) implies \(p=p'\) and \(a=b\) and (3) \(p \xrightarrow {d} p'' \wedge p' \xrightarrow {d} p''\) implies \(p=p'\).
- 14.
For the former, \(\mathcal {G}_s\) generates exactly a time interval (0, d] of delays from p, after which \(\mathcal {G}_s\) arrives at another plain state with a enabled. For the latter, an infinite time interval \((0,\infty )\) of delays are enabled at p. The delays either all lead to plain states or \((0,\infty )\) can be further partitioned into two intervals s.t. the delays in the first interval lead to plain states while those of the second lead to \(\top \) or \(\bot \).
- 15.
We choose not to call it a winning strategy as it serves additional purpose for our paper.
- 16.
Given a \(\top /\bot \) complete interface, we say a plain state p is an auto-\(\top \) iff \(p \xrightarrow {a} \top \) for some \(a \in I\); a plain state p is a semi-\(\top \) iff (1) all output transitions in p or any of its time-passing successors lead to the \(\top \) state, and (2) there exists \(d \in \mathbb {R}^{>0}\) s.t. \(p \xrightarrow {d} \top \).
- 17.
We omit the two operators in this paper due to space limitation.
- 18.
That is, they can distinguish the \(\bot \) state from the \(\bot \)-winning states by stopping time immediately.
- 19.
It is easy to verify that realisable specifications are closed under \(\parallel \) defined in Sect. 3 since \(\parallel \) preserves auto-\(\top \) and semi-\(\top \) freedom.
- 20.
With the extension, blocked synchronisation, i.e. an action being enabled on one process but not so on the other, becomes possible.
- 21.
- 22.
- 23.
The mirror-based definition of quotient (for the untimed case) was first presented by Verhoeff as his Factorisation Theorem [24].
- 24.
Composition of untimed specifications will not generated new unrealisable behaviours.
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Acknowledgments
We benefit from discussions with Prof. David Dill and Prof. Jeff Sanders on timed extension of trace theory and refinement calculus.
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Chilton, C., Kwiatkowska, M., Moller, F., Wang, X. (2017). A Specification Theory of Real-Time Processes. In: Gibson-Robinson, T., Hopcroft, P., Lazić, R. (eds) Concurrency, Security, and Puzzles. Lecture Notes in Computer Science(), vol 10160. Springer, Cham. https://doi.org/10.1007/978-3-319-51046-0_2
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