Richer interface automata with optimistic and pessimistic compatibility

Abstract

Modal transition systems are a popular semantic underpinning of interface theories, such as Nyman et al.’s IOMTS and Bauer et al.’s MIO, which facilitate component-based reasoning for concurrent systems. Our interface theory MIA repaired a compositional flaw of IOMTS-refinement and introduced a conjunction operator. In this paper, we first modify MIA to properly deal with internal computations including internal must-transitions, which were largely ignored already in IOMTS. We then study a MIA variant that adopts MIO’s pessimistic—rather than IOMTS’ optimistic—view on component compatibility and define, for the first-time in a pessimistic, non-deterministic setting, conjunction and disjunction on interfaces. For both the optimistic and pessimistic MIA variant, we also discuss mechanisms for extending alphabets when refining interfaces, which is a desired feature for perspective-based specification. We illustrate our advancements via a small example.

Appendix: Proof of Lemma 4(b)–(e)

Proof of Part (b). We show by induction on $k$ that there exists a $\overline{P}_k$ such that . Part (a) implies the case $k=1$. Assume the claim holds for $k$. Now, there are two cases: if $p_{k+1} \notin \overline{P}_k$, then $\overline{P}_{k+1} = \overline{P}_k \subseteq (P'{\setminus }\{p_1, \ldots , p_{k+1}) \cup \,\bigcup _{i=1}^{k+1} P_i$. Otherwise, by Part (a). Hence, $\overline{P}_{k+1} \subseteq (((P'{\setminus }\{p_1, \ldots , p_k\}) \cup \,\bigcup _{i=1}^{k} P_i){\setminus }\{p_{k+1}\}) \cup P_{k+1} \subseteq (P'{\setminus }\{p_1, \ldots , p_{k+1}) \cup \,\bigcup _{i=1}^{k+1} P_i$. $\square$

Proof of Part (c). The proof proceeds by induction on the overall number of applications of Definition 2(a’). If this is $0$, then $\overline{P} =_{\text {df}}\bigcup _{i=1}^{n} P_i$. Otherwise, assume w.l.o.g. that , $p_1 \in P''_1$, $p_1 {\mathop {\longrightarrow }\limits ^{\tau }} P''$ and $P'_1 = (P''_1{\setminus }\{p_1\}) \cup P''$. By induction hypothesis, there exists a $\hat{P}$ such that . If $p_1 \notin \hat{P}$, then $\hat{P} \subseteq \,\bigcup _{i=1}^{n} P'_i$ and we are done. Otherwise, . Since $\hat{P}\subseteq P''_1 \cup \,\bigcup _{i=2}^{n} P'_i$ implies $\hat{P}{\setminus }\{p_1\} \subseteq (P''_1{\setminus }\{p_1\}) \cup \,\bigcup _{i=2}^{n} P'_i$, we obtain $\overline{P} \subseteq \bigcup _{i=1}^{n} P'_i$. $\square$

Proof of Part (d). The proof is by induction on the derivation of . For $P=P'$, choose $\overline{P} =_{\text {df}}P''$. Otherwise, assume , $p\in \hat{P}$, $p {\mathop {\longrightarrow }\limits ^{\tau }} \hat{P}'$ and $P' = (\hat{P}{\setminus }\{p\}) \cup \hat{P}'$. By induction hypothesis, there exists a $\overline{P}'$ such that . If $p \notin \overline{P}'$, then $\overline{P}'\subseteq P'$ and we are done. Otherwise, $\overline{P} =_{\text {df}}(\overline{P}'{\setminus }\{p\}) \cup \hat{P}'\subseteq P'$. $\square$

Proof of Part (e). For $1 \le i \le n$, we have such that $p_j^i {\mathop {\longrightarrow }\limits ^{o}} P_j^i$ for $1 \le j \le k_i$, and can derive from by repeated application of Definition 2(a), i.e., . By Part (d), we get for each $P_j^i$ a $P_j^{'i}$ such that .

When applying Part (b), we obtain some $\hat{P}$ such that . With Definition 2(b) we get , where $U$ is the union of some of the $P_j^i$. Taking these $P_j^i$ as the $P_i$ in Part (c) yields such that $\overline{P}$ is contained in the union of the resp. $P_j^{'i}$ and, thus, in $\bigcup _{i=1}^{n} P_i$. $\square$