Fault-based refinement-testing for CSP

Abstract

The process algebra CSP has been studied as a notation for model-based testing. Theoretical and practical work has been developed using its trace and failure semantics, and their refinement notions as conformance relations. Two sets of tests have been defined and proved to be exhaustive, in the sense that they can identify any SUT that is non-conforming with respect to the relevant refinement relation. However, these sets are usually infinite, and in this case, it is obviously not possible to apply them to verify the conformity of an SUT. Some classical selection criteria based on models have been studied. In this paper, we propose a procedure for online test generation for selection of finite test sets for traces refinement from CSP models. It is based on the notion of fault domains, focusing on the set of faulty implementations of interest. We investigate scenarios where the verdict of a test campaign can be reached after a finite number of test executions. We illustrate the usage of the procedure with some case studies.

Appendix: Refinement laws

Law

altI Alternation introduction

w : [pre, post]

$\sqsubseteq altI$

if □ i ∙ g_i & w : [g_i ∧ pre, post] fi

provided $\mathit {pre} \Rightarrow (\bigvee i \bullet g_{i})$

Syntactic restrictions:

• Each g_i is a well-scoped predicate.

• No g_i has free dashed variables.

• {i ∙ g_i} is non-empty.

Law

assigI Assignment introduction

w, VCL : [pre, post]

$\sqsubseteq \mathit {assigI}$

VCL := el

provided $\mathit {pre} \Rightarrow \mathit {post}[el/vl^{\prime }][\_/^{\prime }]$

Syntactic restrictions:

• vl contains no duplicated variables.

• vl and el have the same length.

• el is well-scoped and well-typed.

• el has no free dashed variables.

• The corresponding variables of vl and expressions of el have the same type.

Law

cfR Contract frame

w, x : [pre, post]

$\sqsubseteq ~cfR$

x : [pre, post[w/w^′]]

Syntactic restrictions The variables of w are not in x.

Law

fassigI Following assignment introduction

w, VCL : [pre, post]

$\sqsubseteq \mathit {fassigI}$

w, VCL : [pre, post[el[w^′,vl^′/w, VCL]/VC^′]]; VCL := el

Syntactic restrictions:

• vl contains no duplicated variables.

• vl and el have the same length.

• el is well-scoped and well-typed.

• el has no free dashed variables.

• The corresponding variables of vl and expressions of el have the same type.

Law

itI Iteration introduction

$w :[inv,inv[w^{\prime }/w] \land \lnot (\bigvee i \bullet g_{i}[w^{\prime }/w])]$

$\sqsubseteq \mathit {itI}$

do □i ∙ g_i&w : [inv ∧ g_i,inv[w^′/w] ∧ 0 ≤ vrt[w^′/w] < vrt] od

Syntactic restrictions:

• vrt is a well-scoped and well-typed integer.

• Each g_i and vrt have no free dashed variables. expression.

Law

vrbI Variable introduction

w : [pre, post]

= vrbI

|[vardvl ∙VCL, w : [pre, post]]|

wheredvl declares the variables of vl.

Syntactic restrictions:

• dvl is well-scoped and well-typed.

• The variables of vl and vl^′ are not free in w : [pre, post] and are not dashed.

Law

seqcI Sequential composition introduction

w, x : [pre, post]

$\sqsubseteq \mathit {seqcI}$

w : [pre, mid[w^′/w]]; w, x : [mid, post]

Syntactic restrictions:

• mid is well-scoped and well-typed.

• mid has no free dashed variables.

• No free variable of post is in w.

Law

seqcI Sequential composition introduction

w, x, y!, z! : [pre, post]

$\sqsubseteq \mathit {seqcI}$

$|[ \textbf {con} \mathit {dcl} \bullet w,y!:[\mathit {pre},\mathit {mid}];\vspace *{2pt} w,x,y!,z! :[\mathit {mid}[cl/w] [\_/^{\prime }],\mathit {post}[cl/w] ] ]|$

wheredcl declares the constants of cl.

Syntactic restrictions:

• mid is well-scoped and well-typed.

• The names of cl and cl^′ are not free in mid and w, x, y!,z! : [pre, post].

• cl and w have the same length.

• The constants of cl have the same type as the corresponding variables of w.

Law

sP Strengthen postcondition

w : [pre, post]

$\sqsubseteq \mathit {sP}$

w : [pre, npost]

providedpre ∧npost ⇒post

Syntactic restrictionsnpost is well-scoped and well-typed.