Replication, refinement & reachability: complexity in dynamic condition-response graphs

Abstract

We explore the complexity of reachability and run-time refinement under safety and liveness constraints in event-based process models. Our study is framed in the DCR$^\star$ process language, which supports modular specification through a compositional operational semantics. DCR$^\star$ encompasses the “Dynamic Condition Response (DCR) graphs” declarative process model for analysis, execution and safe run-time refinement of process-aware information systems; including replication of sub-processes. We prove that event-reachability and refinement are np-hard for DCR$^\star$ processes without replication, and that these finite state processes recognise exactly the languages that are the union of a regular and an $\omega$-regular language. Moreover, we prove that event-reachability and refinement are undecidable in general for DCR$^\star$ processes with replication and local events, and we provide a tractable approximation for refinement. A prototype implementation of the DCR$^\star$ language is available at http://dcr.tools/acta16.

Appendix: Proof of Proposition 6

We recap the original graph-based formalisation of DCR graph [10, 15, 17, 31, 41].

Definition 44

(DCR graph) A DCR graph is a tuple $(\mathsf {E}, \mathsf {R}, \mathsf {M})$ where

• $\mathsf {E}$ is a finite set of (labelled) events, the nodes of the graph.

• $\mathsf {R}$ is the edges of the graph. Edges are partitioned into four kinds, named and drawn as follows: the conditions $(\mathrel {\rightarrow \bullet })$, responses $(\mathrel {\bullet \rightarrow })$, inclusions $(\mathrel {\rightarrow +})$, and exclusions $(\mathrel {\rightarrow \%})$.

• $\mathsf {M}$ is the marking of the graph. This is a triple $(\mathsf {Ex},\mathsf {Re},\mathsf {In})$ of sets of events, respectively the previously executed $(\mathsf {Ex})$, the currently pending $(\mathsf {Re})$, and the currently included $(\mathsf {In})$ events.

When G is a DCR graph, we write, e.g., $\mathsf {E}(G)$ for the set of events of G, $\mathsf {Ex}(G)$ for the executed events in the marking of G, etc. We write $\mathord {(\mathrel {\rightarrow \bullet }e)}$ for the set $\{e^{\prime }\in \mathsf {E}\mid e^{\prime }\mathrel {\rightarrow \bullet }e\}$, write $\mathord {(e\mathrel {\bullet \rightarrow })}$ for the set $\{e^{\prime }\in \mathsf {E}\mid e \mathrel {\bullet \rightarrow }e^{\prime }\}$ and similarly for $\mathord {(e\mathrel {\rightarrow +})}$ and $\mathord {(e\mathrel {\rightarrow \%})}$.

Definition 45

(Enabled events) Let $G=(\mathsf {E}, \mathsf {R}, \mathsf {M})$ be a DCR graph, with marking $\mathsf {M}= (\mathsf {Ex},\mathsf {Re},\mathsf {In})$. An event $e \in \mathsf {E}$ is enabled, written $e\in {\mathsf {enabled}(G)}$, iff (a) $e \in \mathsf {In}{}$ and (b) $\mathsf {In}{}\cap \mathord {(\mathrel {\rightarrow \bullet }e)} \subseteq \mathsf {Ex}$.

Definition 46

(Execution) Let $G=(\mathsf {E}, \mathsf {R}, \mathsf {M})$ be a DCR graph with marking $\mathsf {M}=(\mathsf {Ex},\mathsf {Re},\mathsf {In})$. Suppose $e\in {\mathsf {enabled}(G)}$. We may execute e obtaining the resulting DCR graph $(\mathsf {E}, \mathsf {R}, \mathsf {M}^{\prime })$ with $\mathsf {M}^{\prime } =(\mathsf {Ex}^{\prime },\mathsf {Re}^{\prime },\mathsf {In}^{\prime })$ defined as follows.

• 1. $\mathsf {Ex}^{\prime } = \mathsf {Ex}\cup \{e\}$

• 2. $\mathsf {Re}^{\prime } = (\mathsf {Re}\backslash \{e\}) \cup \mathord {(e\mathrel {\bullet \rightarrow })}$

• 3. $\mathsf {In}^{\prime } = (\mathsf {In}\backslash \mathord {(e\mathrel {\rightarrow \%})}) \cup \mathord {(e\mathrel {\rightarrow +})}$

Definition 47

(Transitions) Let G be a DCR graph. If $e\in {\mathsf {enabled}(G)}$ and executing e in G yields H, we say that G has transition on e to H and write ${G}\xrightarrow {e}{H}$. A run of G is a (finite or infinite) sequence of DCR graphs $G_i$ and events $e_i$ such that: $G = {G_0}\xrightarrow {e_0}{{G_1}\xrightarrow {e_1}{\ldots }}$. A trace of G is a sequence of labels of events $e_i$ associated with a run of G. We write $\mathop {\mathsf {runs}}(G)$ and $\mathop {\mathsf {traces}}(G)$ for the set of runs and traces of G, respectively

Definition 48

(Acceptance) A run ${G_0}\xrightarrow {e_0}{{G_1}\xrightarrow {e_1}{\ldots }}$ is accepting iff for all n with $e \in \mathsf {In}(G_n)\cap \mathsf {Re}(G_n)$ there exists $m\ge n$ s.t. either $e_m = e$, or $e\not \in \mathsf {In}(G_{m})$. A trace is accepting iff it has an underlying run which is.

Definition 49

(Language) The language of a DCR graph G is the set of its accepting traces. We write $\mathop {\mathsf {lang}}({G})$ for the language of G.

This concludes our recap of DCR graphs (as opposed to DCR processes). Now, the proof.

Definition 50

Let T, U be terms. Define $T\cong U$ by taking for $\mathcal R$ ranging over the four relations

$$\begin{aligned} { e\, \mathcal{R} \,f }\, \parallel e \,\mathcal{R} \,f \cong e \,\mathcal{R}\, f \end{aligned}$$

(8)

and closing under monoid laws for $- \parallel -$ and $0$.

Lemma 51

If $T\cong U$ then ${T}\xrightarrow {e}{T^{\prime }}$ and ${U}\xrightarrow {e}{U^{\prime }}$ implies $T^{\prime }\cong U^{\prime }$.

Proof

It is straightforward to verify that the monoid laws and the idempotency rule (8) defining $\cong$ preserves transition in the above sense; the desiderata follows.

Definition 52

Observe that by the monoid laws, every term T can be read as a multi-set of relations; by the idempotency rule (8), it can be read as a set of relations. Write $\bar{T}$ for this set. For a process $P=[{M}]\;{T}$, take $\bar{P} = [{M}]\;{\bar{T}}$.

Definition 53

Let $P=[{M}]\;{T}$ be a process and $G=(\mathsf {E}, \mathsf {R}, \mathsf {M})$ a graph. Define $P\approx G$ iff

• 1. $\mathsf {E}= \mathsf {dom}(M)$

• 2. for all $e\in \mathsf {E}$ we have $M(e)=(h,i,r)$ iff $h=(e\in \mathsf {Ex})$ and $i=(e\in \mathsf {In})$ and $r=(e\in \mathsf {Re})$.

• 3. $\bar{T} = \mathsf {R}$.

Lemma 54

Let P be a process and G a graph s.t. $P \approx G$. For all e, if ${P}\xrightarrow {e}{P^{\prime }}$ then for some $G^{\prime }$ we have ${G}\xrightarrow {e}{G^{\prime }}$ with $P^{\prime }\approx G^{\prime }$; and vice versa.

Proof

By Lemma 51 it is sufficient to prove this result for $\bar{P}$. Assume $\bar{P} =[{M}]\;{\bar{T}}$ and $G=(\mathsf {E}, \mathsf {R}, \mathsf {M})$. We proceed by induction on the number of relations k in $\bar{T}=\mathsf {R}$.

For $k=0$, clearly $P^{\prime }=[{{e}\cdot {M}}]\;{T}$ and $G^{\prime }= (\mathsf {Ex}\cup \{e\},\mathsf {In},\mathsf {Re})$; the result follows.

For $k=1$, straightforward verification by cases on the single relation.

For $k>1$, we must have $T=\bar{T}_1 \parallel \bar{T}_2$ and $\mathsf {R}=\bar{T_1}\cup \bar{T_2}$ such that the number of relations in $\bar{T}_j$ are both smaller than k. Define $G_j= (\mathsf {E}, \bar{T_j},\mathsf {M})$. Note that if ${P}\xrightarrow {e}{P'}$ then we must have

$$\begin{aligned} \frac{ {[{M}]\;{\bar{T}_1}}\xrightarrow {e:\delta _1}{\bar{T}_1} \qquad {[{M}]\;{\bar{T}_2}}\xrightarrow {e:\delta _2}{\bar{T}_2} }{ {[{M}]\;{\bar{T}_1 \parallel \bar{T}_2}}\xrightarrow {e : \delta _1 \oplus \delta _2}{\bar{T}_1 \parallel \bar{T}_2} } \end{aligned}$$

(9)

In this case we have by induction that $G_j={(\mathsf {E}, \bar{T_j},\mathsf {M})}\xrightarrow {e}{(}{\mathsf {E}, \bar{T_j},\mathsf {M}^{\prime }_j}) = G_j^{\prime }$ for $j\in \{1,2\}$.

We show first that e is enabled in P iff it is in Q. Suppose e is enabled in P. By (9), e is then enabled in $[{M}]\;{T_1}$ and $[{M}]\;{T_2}$. By induction, it is then enabled in $G_1,G_2$, and by calculation on Definition 45 it is also enabled in G. Now suppose instead e is enabled in G. By Definition 45, e is also enabled in $G_1,G_2$. By induction, e is enabled in $[{M}]\;{\bar{T}_1}$ and $[{M}]\;{\bar{T}_2}$. It follows by the [par]-rule that e is enabled in P.

Now suppose we have transitions ${P}\xrightarrow {e}{P^{\prime }}$ and ${G}\xrightarrow {e}{G'}$. As noted we must have (9); by [effect] we have transitions

$$\begin{aligned} {[{M}]\;{\bar{T}_1}}\xrightarrow {e}{[{{e:\delta }\cdot {M}}]\;{\bar{T}_1}} \quad \text {and} \quad {[{M}]\;{\bar{T}_2}}\xrightarrow {e}{[{{e:\delta }\cdot {M}}]\;{\bar{T}_2}} \end{aligned}$$

(10)

By induction we must have transitions

$$\begin{aligned} {G_1}\xrightarrow {e}{G_1^{\prime }} \quad \text {and}\quad {G_2}\xrightarrow {e}{G_2^{\prime }} \end{aligned}$$

(11)

We prove that $P^{\prime }\approx G^{\prime }$. Items (1) and (3) of Definition 53 are immediate, so it is sufficient to show Item (2). Let $M_h(x) = h$ when $M(x) = (h,i,r)$; similarly for i, r. It is straightforward to verify using (10) and (11) that each of the following leads to a contradiction:

• 1. $M_h(f)\not =(f\in \mathsf {Ex}^{\prime })$, or

• 2. $M_i(f)\not =(f\in \mathsf {In}^{\prime })$, or

• 3. $M_r(f)\not =(f\in \mathsf {Re}^{\prime })$.

Theorem 55

The relation $\approx$ is a bisimulation.

Proof

Immediate from Lemma 54.

Corollary 56

Let P be a process and G a graph. Then $P\approx G$ implies $\mathop {\mathsf {lang}}({P}) = \mathop {\mathsf {lang}}({G})$.

Proof

Using Lemma 54, it is clear that a run of P gives rise to a run of G and vice versa. It is sufficient to show that such a runs is accepting for P iff it is for G; this is immediate by inspection of Definitions 2 and 48.

Proof of Proposition 6

Immediate from Theorem 55 and Corollary 56.