Institut für Technische Informatik, Technische Universität Wien, Treitlstraße 3/182/1, A-1040 Wien, Austria
Embedded systems are often used in safety-critical
environments. Thus, thorough testing of them is
mandatory. To achieve a required structural code-coverage
criteria it is beneficial to derive the test data
at a higher program-representation level than machine
code. Higher program-representation levels include, beside
the source-code level, languages of domain-specific
modeling environments with automatic code generation.
For a testing framework with automatic generation
of test data this will enable high retargetability of
the framework. In this article we address the challenge of ensuring
that the structural code coverage achieved at a higher
program representation level is preserved during the
code generations and code transformations down to machine
code. We define the formal properties that have
to be fullfilled by a code transformation to guarantee
preservation of structural code coverage. Based on
these properties we discuss how to preserve code coverage
achieved at source-code level. Additionally, we discuss
how structural code coverage at model level could
be preserved. The results presented in this article are
aimed toward the integration of support for preserving
structural code coverage into compilers and code generators.
1. Introduction
Testing is a mandatory process to assess the correct behavior of safety-critical systems. Even the increasing use of formal verification cannot make testing obsolete, as there is always a gap between the formal model and the real system with all the issues of integration.
The use of formal (=executable) models increasingly pervades the software engineering process. Formal models are used as part of the specification, as high-level software descriptions with automatic code generation, or as a tool for formal verification and model-based testing [1].
When generating test data it is beneficial to operate at the same representation level where the software is developed, which may be at the source-code level or at a domain-specific modeling environment like ezRealtime [2], MATLAB/Simulink [3, 4], Statemate [5], or Scade [6]. The advantage of test-data generation at this high-level program representation is on the one side reduced complexity and availability of explicit knowledge of the program behavior that might get lost during code generation and compilation. On the other side, a test-data generator operating on such a high-level program representation could be easily retargeted to different platforms. Beside conventional testing, the support for retargetability is also of high interest for hybrid timing analysis, that is, an approach to determine the timing behavior of a program based on the combination of execution-time measurements and program analyses [7, 8].
Structural code-coverage criteria are metrics to analyze and quantify the control-flow coverage that is achieved for a given set of test data. Using a model-based or source-based test-data generator raises the challenge of ensuring that adequate structural code-coverage has been achieved at machine-code level [9]. Code generators and compilers perform many transformations on the program or model given as input. Some of these code transformations can compromise structural code-coverage by copying, reordering, or moving conditions inside the program or even creating new conditions and decisions. For example, optimizations like loop unrolling, loop inversion, reverse if-conversion, and condition reordering [10] can disrupt full structural code-coverage. In general, full structural code-coverage cannot be guaranteed in this case without taking the burden of analyzing the machine code.
We propose an approach toward the preservation of structural code coverage when transforming the program. To achieve this, we introduce in Section 3 a notation to formally define structural code-coverage criteria. In Section 4 we present coverage preservation criteria for the different variants of structural code coverage. As described in Section 5, these criteria can help to extend a compiler with the ability of preserving coverage achieved at source level. The code coverage is preserved by prohibiting all code transformations that can disrupt the concrete structural code coverage metric. If full coverage preservation is not strictly required, the compiler may be used in a special mode where all available code transformations are allowed but a warning is emitted if structural code coverage may be compromised by an applied code optimization. Issues of preserving model coverage by code generators are discussed in Section 6.
2. Related Work
Structural coverage criteria are used as a supplementary criterion to monitor the progress of testing [11]. The DO178b document introduces the modified condition-decision coverage (MCDC) as a supplementary criterion for testing systems of safety-criticality level A [12]. Vilkomir proposes solutions to overcome some weaknesses of MCDC [13]. Vilkomir and Bowen have formally modeled structural code-coverage criteria using the Z notation [14]. The formalization we present in this article is basically equivalent, with the difference that we also support hidden-control flow [15], which is necessary to model code coverage for languages like ANSI C or ADA. Further, our notation is more compact, which has shown to be helpful for developing coverage-preservation criteria.
Model-based development aims to use high-level system representations within the system engineering process. For example, the Object Management Group proposes the Model-Driven Architecture, which explicitly differentiates between platform-independent and platform-specific models [16]. Models can be used to automatically generate source code. Another model-based approach is model-based testing where abstract models are used to guide the generation of test data [1, 17]. Using models to verify the correctness of the system requires evidence of the model's correctness [18].
Directly related to our work is the relationship of achieved model coverage and the resulting code coverage. Baresel et al. analyzed this relationship empirically, finding some correlation between the degree of model coverage and the resulting degree of code coverage [19]. Rajan et al. have shown for MCDC that the correlation of the degree of model coverage and the degree of code coverage depends on the code generation patterns [20]. To test safety-critical systems we want to do better than relying on accidental coverage correlations.
Elbaum et al. empirically studied the preservation of code coverage for software evolution with different change levels. They concluded that even relatively small modifications in the software may impact the coverage in a way that is hard to predict [21]. Their results also motivate our work for the preservation of code coverage.
A method complementary to our approach is described by Harman et al. Testability transformation results in a transformed program to be used by a test-data generator to improve its ability of generating test data for the original program [22].
3. Basic Definitions
In this section we give a list of basic definitions. These definitions are used to describe properties of structural code coverage and to preserve structural code coverage.
Program . Denotes the program before () and after () the transformations for which we want to preserve structural code coverage.
Control-Flow Graph (CFG). Is used to model the control flow of a program [23]. A CFG consists of a set of nodes representing basic blocks (see below), a set of edges representing the control flow (also called control-flow edges), a unique entry node , and a unique end node .
Program Scope of A Program . Is a fragment of with well-defined interfaces for entry and exit. We denote the set of program scopes in a program as . The concrete partitioning of a program into scopes is application-specific. For example, in [24] a program partitioning is used that allows to trade the number of required test data against the number of instrumentation points. Another feature of scopes is that nested scopes can be used to hide details. This feature allows to reduce the program complexity of the surrounding scope.
Scoped Path. Of a program scope is a sequence of control-flow edges from an entry point of the scope to an exit point of the scope. In case of nested program scopes, the whole inner program scope is a single block in the paths of the outer program scope. A scoped path of a program scope is uniquely represented by its starting basic block and the necessary TRUE/FALSE evaluation result of all conditions along the scoped path. We denote the set of scoped paths in a program scope as . The paths within a program , that is, the scoped paths where the program scope subsumes the whole program, is denoted as .
Basic Block. Of a program is a code sequence of maximal length with a single entry point at the beginning and with the only allowed occurrence of a control-flow statement at its end. We denote the set of basic blocks in a program as . The set of all basic blocks along a scoped path is denoted as . Note that in cases of program paths with cycles, will contain multiple instances of the basic blocks in the program code. If a scoped path goes through a nested program scope, all the basic blocks from the nested program scope are hidden for this path. The starting basic block of a scoped path is denoted as .
Decision. Is a Boolean expression composed of conditions that are combined by Boolean operators. If a condition occurs more than once in the decision, each occurrence is a distinct condition [25]. However, the input of a decision is the set of its conditions without duplicates. A decision is composed of one or more basic blocks. We denote the set of decisions of a program as .
There are source languages, where decisions are hidden by an implicit control flow. For example, in ANSI C due to the short-circuit evaluation the following statement a = (b && c); contains the decision (b && c). The short-circuit evaluation of ANSI C states that the second argument of the operators && and is not evaluated if the result of the operator is already determined by the first argument. See Section 5.4 for further details. The correct identification of hidden control flow is important, for example, to analyze decision coverage.
Condition. Is a Boolean expression. We consider only lowest-level conditions, that is, conditions that do not contain operators with Boolean arguments [25]. A condition is composed of one or more basic blocks. We denote the set of conditions of a decision as . The set of all conditions within a program is denoted as .
The set of all conditions that directly control edges along a scoped path is denoted as . Note that in cases of program paths with cycles, will contain multiple instances of the conditions in the program code. If a scoped path goes through a nested program scope, all the conditions from the nested program scope are hidden for this path. To follow a certain path, it is also important whether a condition evaluates to TRUE/FALSE. Whether a condition has to be evaluated as TRUE or as FALSE is given by the syntactical structure of a program. For a given scoped path we denote by all the conditions that have to be evaluated as TRUE and by all the conditions that have to be evaluated as FALSE to follow . It holds that and .
Input Data . Defines the set of all possible valuations of the input variables of a program. (Valuation of a variable means the assignment of concrete values to it. The valuation of an expression means the assignment of concrete values to all variables within the expression.)
Test Data . Defines the set of valuations of the input variables that have been generated with structural code coverage analysis done at source-code level. Since exhaustive testing is intractable in practice, is assumed as a true subset of the program's input data space : . If we would consider exhaustive testing () there would be no challenge of structural code-coverage preservation.
Reachability Valuation Defines the set of valuations of the input variables that trigger the execution of expression , where can be a condition, decision, or a basic block.
Satisfiability Valuation , Defines the sets of valuations of the input variables that trigger the execution of the condition/decision with a certain result of : is the input-dataset, where
refers to TRUE and is the set, where
refers to FALSE. The following properties always hold for , :
Consider the following example of C code to get an intuition about the meaning of the satisfiability valuations:
void f (int a,b)
if (a==3 && b==2)
return 1;
return 0;
.
For this code fragment we assume
It follows that
(and not the larger set
due to the hidden control flow caused by the short-circuit evaluation of ANSI C; see Section
5.4). It follows that
Only those input data that trigger the execution of condition
b==2 and evaluate it to TRUE are within
(
b==2). With
the conditions
a==3 and
b==2 are both executed and evaluated to TRUE. Further, it holds that
The definition of , , and depends on whether the programming language has hidden control flow (see Section 5.4). Above definitions allow to formally describe structural code coverage criteria. We will also use them to describe requirements to preserve structural code coverage.
3.1. Structural Code-Coverage Criteria
Structural code-coverage criteria are metrics to analyze and quantify the control-flow coverage that is achieved for a given set of test data. Execution traces are used to collect the coverage information. In general, the satisfaction of a structural code-coverage criterion is not the primary test-case generation strategy in functional testing. Instead, structural code-coverage achieved during testing is analyzed as a supplementary measure to decide whether the implemented functionality has been sufficiently tested and does not contain any unintended functionality. However, there are also rare testing scenarios where the satisfaction of a certain code-coverage is the primary directive for test-data generation. For example, in measurement-based timing analysis an estimation of the worst-case execution time (WCET) is derived by systematic measurements [24].
In the following we review the properties of several structural code-coverage criteria.
Line Coverage. Is not a serious code coverage criterion, as without strict coding guidelines there is an ambiguous mapping from source lines to statements. In the extreme case one could write the whole program within one source line. Historically, line coverage was used as an easy hack when tools for analyzing statement coverage were missing. Thus, we do not discuss preservation of line coverage in this work.
Statement Coverage (SC). Requires that every statement of a program is executed at least once. Statement coverage alone is quite weak for functional testing [26] and should best be considered as a minimal requirement. Using our above definitions, we can formally define SC as follows:
Note that the boundary recognition of basic blocks can be tricky due to hidden control-flow. A statement in a high level language like ANSI C can consist of more than one basi cblock. For example, the ANSI C statement f=(a==3 && b==2); consists of multiple basic blocks due to the short-circuit evaluation order of ANSI C expressions.
Decision coverage (DC). Requires that each decision of a program has been tested at least once with each possible outcome. Decision coverage is also known as branch coverage or edge coverage. Decision coverage implies statement coverage:
Condition Coverage (CC). Requires that each condition of the program has been tested at least once with each possible outcome. It is important to mention that CC does not imply DC. A formal definition of CC is given in (8)
Note that our definition requires in case of short-circuit operators that each condition is really executed. This is achieved by the semantics of . However, often definitions are used that do not explicitly consider short-circuit operators (e.g., [27]), thus having in case of short-circuit operators only a “virtual” coverage since they do not guarantee that the short-circuit condition is really executed for the evaluation to TRUE as well as for the evaluation to FALSE.
Condition/Decision Coverage (CDC). Requires that both, condition coverage and decision coverage are achieved.
Modified Condition/Decision Coverage (MCDC). Requires to show that each condition can independently affect the outcome of the decision [12]. Thus, having conditions in a decision, test cases are required to achieve MCDC. Note that MCDC implies DC and CC. A formal definition of MCDC is given in (9) based on the set of input test data . It requires that for each condition of a decision there exists two test vectors such that the predicate symbol holds, which ensures that the two test vectors show different outcomes for as well as but the same outcomes for all other conditions within . This is exactly how MCDC is described above
The predicate symbol tests whether one of the test data , is a member of the input dataset and the other one a member of the input data set . If this predicate symbol is TRUE it is guaranteed that the expression evaluates to both, TRUE and FALSE:
The predicate symbol tests whether the input-data set provides a constant outcome for the evaluation of . Actually, the predicate symbol is used to test whether there exists a test-data subset for a given condition, such that the results of all other conditions remain unchanged. Thus, this predicate symbol is used to ensure that each condition can independently control the output of the decision:
The above definition of MCDC is the original definition given in the RTCA/DO178b document [12]. However, this definition is rather strict, so that people thought of some less restrictive definitions. For example, it is not possible with the original definition to cover a decision with strongly coupled conditions. (Two conditions are strongly coupled, iff they have the same input data partitioning for their satisfiability valuation, i.e., .) As described in [25], there exist at least three definitions of MCDC:
(i)Unique-Cause MCDC: this is the original definition given in [12]. (ii)Unique-Cause + Masking MCDC: this definition of MCDC is less restrictive as it requires in case of strongly coupled conditions to test only that one of them covers the decision (masking) [15]. (iii)Masking MCDC: this is less restrictive than the two above, as it does not require the Unique-Cause. A condition is masked if its value cannot influence the outcome of a decision due to the overruling values of other conditions. For Masking MCDC it is sufficient to show that each condition can affect the outcome of the decision without being masked. However, Masking MCDC is not required to test whether conditions do independently cover the decision. It focuses more on testing the correct implementation of subexpressions within a Boolean expression. According to Chilenski the metric Masking MCDC should be the preferred form of MCDC as it provides the same error detection probability but allows for more different test sets and thus the generation of test data more is cost-effective [25].
Within this article we focus on the original definition of MCDC. Extending above formal definition of MCDC to Unique-Cause + Masking MCDC or Masking MCDC is straight-forward. One has to exchange the predicate by another predicate that formalizes the semantics of the alternative MCDC criterion.
Multiple Condition Coverage (MCC). Requires besides DC and CC that each possible combination of outcomes of the conditions of each decision is executed at least once. MCC demands a rather high number of test cases: to achieve full MCC of a decision with conditions tests are necessary. MCC is desired in theory, but MCC tends to be infeasible for industrial code, because there are too many conditions per decision [27]. Thus, in this work we do not address MCC.
Path Coverage (PC). Requires that each path of a program has been tested at least once. Since the number of paths within a program typically grows exponentially with the program size (PC is even stronger than MCC), we do not address PC.
Scoped Path Coverage (SPC). Is a coverage metric recently introduced by the authors. We use this type of code coverage for measurement-based timing analysis [7]. In this article we formalize this coverage metric to reason about necessary properties of a compilation profile for preserving SPC. The basic idea of SPC is to partition the program into program scopes and cover all possible paths within each program scope. Actually, PC is just the special case of using SPC in combination with only one program scope covering the whole program .
The appropriate partitioning of a program into program scopes depends on the concrete testing goal. For example, in case of our research on measurement-based timing analysis [7] we use the partitioning of the program into scopes to achieve a compromise between precision of measurement results (the larger the segments the more precise) and number of necessary measurements.
SPC requires that each path within a program scope is tested at least once. Thus, there must be a test datum that covers all basic blocks along the path. Using our above definitions, we can formally define SPC as follows:
Note, that the condition “” of (13) ensures that in the pathological case of having a program scope that is completely free of conditions, coverage of the only single path in the program scope is guaranteed.
Whether SPC is feasible in practice, depends on the program complexity itself and also on the application-specific partitioning of a program into program scopes.
Examples of test vectors sufficient for full coverage according to the different coverage metrics are given in Table 1. The ANSI C code example is a decision including three conditions. Note that in C the operators and && influence the control flow of the program due to the short-circuit evaluation in ANSI C. This small example is meant to support the definition of different variants of coverage metric. It is not meant to show the relative costs of the different variants of structural coverage metric.
Table 1: Example: Sufficient test vectors per coverage metric.
The condition coverage (CC) needs a relative high number of test vectors. This is because of test vectors that enforce the entering of a program decision do not necessarily enforce the execution of a specific condition within the decision. Multiple condition coverage (MCC) has a relative high cost for testing a single decision. However, when looking at the whole program, then path coverage (PC) is typically much more complex, and depending on the definition of program scopes, scoped path coverage (SPC) requires significantly less test vectors than PC.
4. Preservation of Structural Code Coverage
The challenge of structural code-coverage preservation is to ensure for a given structural code coverage of a program that this code coverage is preserved while the program is transformed into another program . This scenario is shown in Figure 1. Of course if a program will be transformed, also the sets of basic blocks , the set of program decisions , or program scopes may get changed. As shown in Figure 1, the interesting question is whether a concrete code transformation preserves the structural code coverage of interest.
Figure 1: Coverage-preserving program transformation.
When transforming a program, we are interested in the program properties that must be maintained by the code transformation such that a structural code coverage of the original program by the test-data set is preserved to the transformed program. Based on these properties one can adjust a source-to-source transformer or a compiler to use only those optimizations that preserve the intended structural code coverage.
These coverage-preservation properties to be maintained have to ensure that whenever the code coverage is fulfilled at the original program by some test data then this coverage is also fulfilled at the transformed program with the same test data:
The code coverage preservation can be applied on any type of code transformation, for example, on a source-to-source transformer or a compiler.
In the first step, we have to determine for each code transformation of the code transformer whether it preserves a given structural code coverage. We call this the coverage profile of a code transformation. The determination of the coverage profile is shown in Figure 2. The structural code coverage metrics of interest have to be formalized and based on that the coverage preservation criteria have to be determined. The coverage preservation criteria together with description of a code optimization are used to calculate the coverage profile of that optimization. The construction of a formal model of the code optimization in Figure 2 is an intermediate step that is necessary if one wants to use formal verification to determine the coverage profile. In case the coverage profile is determined manually, such a formal model of the code optimization is not needed.
Figure 2: Determination of a coverage profile.
In the second step, the coverage preservation has to be integrated into the code transformer. As an example we assume the code transformer is a compiler, as shown in Figure 3. This coverage-preserving compiler will have an input parameter to set the code coverage metric to be preserved. The coverage-preserving compiler can have two operation modes.
Figure 3: Application of a coverage profile.
Safe Mode
In this mode the coverage-preserving compiler will apply only those code optimizations that preserve the given code coverage metric. With this operation mode we assure coverage preservation at the cost of a potential degradation of performance.
Full-Optimization Mode
In this mode the coverage-preserving compiler will apply all code transformations but it will emit a warning whenever a code transformation has been used that does not ensure the preservation of the given coverage metrics. The warning message should be as specific as possible to support the user in determining additional test data to regain code coverage for the optimized code.
The determination of the coverage profile for a given code transformation and the realization of a coverage-preserving compiler are not the focus of this article. Within this article we present the foundation for such a coverage preservation framework and discuss issues that challenge its applicability.
In the following we present coverage preservation criteria for several variants of structural code-coverage metrics. The important aspect is that these preservation criteria are independent of the concrete test data that achieve the structural code coverage at the original program.
4.1. Preserving Statement Coverage (SC)
Equation (15) of Theorem 4.1 provides a coverage preservation criterion for statement coverage. Equation (15) essentially says that for each basic block of the transformed program there exists a basic block of the original program such that reaching with a given test vector implies that also is reached with the same test vector.
Theorem 4.1 (Preservation of SC). Assuming that a set of test data achieves statement coverage on a given program , then (15) provides a sufficient—and without further knowledge about the program and the test data (there is now knowledge about the test data or the program assumed), also necessary—criterion for guaranteeing preservation of statement coverage on a transformed program .
Proof. Preservation of SC: Part 1, showing sufficiency: Since is assumed to achieve SC on , it holds for each that . Since (15) states that it follows that for each we also have . Thus, SC is preserved at .
Part 2, showing necessity by indirect proof: Assuming there exists a basic block of such that for all basic blocks of it holds that , then each contains at least one input that is not in . If consists of exactly those inputs, then is never reached although SC holds in , which implies that SC is not preserved.
4.2. Preserving Condition Coverage (CC)
To define a coverage preservation criterion for CC (Theorem 4.2) we use the auxiliary predicate given in (16).
The predicate is only TRUE if the set of input data includes at least the true-satisfiability valuation or the false-satisfiability valuation of expression , where is either a condition or a decision. The predicate is used for the coverage preservation criterion of CC (and also DC) to test whether the evaluation of any expression of the original program to both, TRUE and FALSE, implies that the test data include at least one element of , needed for the coverage of an expression in the transformed program
Equation (17) states that for each condition of the transformed program there exists at least one condition of the original program whose coverage implies that evaluates to TRUE and there exists at least one condition of the original program whose coverage implies that evaluates to FALSE.
Theorem 4.2 (Preservation of CC). Assuming that a set of test data achieves condition coverage on a given program , then (17) provides a sufficient—and without further knowledge about the program and the test data, also necessary—criterion for guaranteeing preservation of condition coverage on a transformed program :
Proof. Preservation of CC: Part 1, showing sufficiency: Since is assumed to achieve CC on , it holds for each that and . Since (17) states that for each it holds that
it follows that for each we also have
Thus, CC is preserved at .
Part 2, showing necessity by indirect proof: Assuming there exists a condition of program such that for all conditions of program it either holds that (a)(b)
then it is possible that(a): (b):
which in both cases violates the preservation of CC.
Simplification of the CC Preservation Criteria
The goal of defining the coverage preservation criterion is to decide for a set of code transformations whether they could potentially disrupt the structural code coverage achieved on the original program. Typically, when checking the preservation of structural code coverage, one would simplify (17) by just checking whether each condition is kept equal or simply is inverted. This would result in the simpler criterion given in (20)
Working with the simple constraint of (20) may be sufficient in practice when analyzing the effect of concrete code transformations, since many transformations do not modify the conditions within a decision, but only their grouping into decisions. The simplified criterion is sufficient to allow only such code transformations that do not introduce new conditions with new unique satisfiability by the test data. Further, some transformations just invert a condition, which can be checked also with this simplified criterion.
4.3. Preserving Decision Coverage (DC)
To define a coverage preservation criterion for DC (Theorem 4.3) we use the auxiliary predicate given in (16), which is also used for preserving CC.
Equation (21) of Theorem 4.3 provides a coverage preservation criterion for decision coverage. Equation (21) essentially says that for each decision of the transformed program there exists at least one decision of the original program whose coverage implies that evaluates to TRUE and there exists at least one decision of the original program whose coverage implies that evaluates to FALSE.
Theorem 4.3 (Preservation of DC). Assuming that a set of test data achieves decision coverage on a given program , then (21) provides a sufficient—and without further knowledge about the program and the test data, also necessary—criterion for guaranteeing preservation of decision coverage on a transformed program
Proof. Preservation of DC: Part 1, showing sufficiency: since is assumed to achieve DC on , it holds for each that and . Since (21) states that for each (1)(2)
it follows that for each we also have and . Thus, DC is preserved at .
Part 2, showing necessity by indirect proof: assuming there exists a decision such that for all conditions it either holds that(a)(b)
then it is possible that(a) or(b)
which in both cases violates the preservation of DC.
Guaranteeing Decision Coverage
Guaranteeing the preservation of a structural code coverage criterion that depends on the coverage of decisions of a program is challenging, since there are many ways to re-group conditions into hierarchies of decisions without changing the program semantics.
The criterion given in (21) imposes quite strong restrictions on the performed code transformations, since it requires that for each decision there is an adequate decision of the original program such that decision coverage is preserved. For example, consider the following code transformation:if (a==3)
if (a==3&&b==2) if (b==2)
c();c();
inlined style
noninlined style
Such a transformation is quite typical when source-code is transformed into assembly code. Actually, the only decision in the original code is (a==3 && b==2). Having decision coverage on the original code, there are numerous code transformations possible that do not preserve decision coverage.
Thus, it would be useful to have another criterion to guarantee decision coverage at the transformed program. Equation (22) provides a sufficient criterion for guaranteeing decision coverage on the transformed program, assuming that condition coverage is fulfilled on the original program
The new criterion requires a different, but not stronger, structural code coverage at the original code to guarantee decision coverage at the transformed code. This criterion is typically more flexible when generating assembly code (which typically does not have control-flow statements with complex decisions). Further, in case that condition decision coverage (CDC) is fulfilled at the original program, one may chose between the criteria of (21) and (22) to guarantee decision coverage at the transformed program.
4.4. Preserving MCDC
Preserving MCDC coverage on a transformed program is especially challenging, since the code transformation may produce arbitrary groupings of conditions into decisions. Especially the requirement that each condition can independently influence the outcome of its conditions, is rather complex to check.
As the MCDC coverage preservation criterion is rather complex, we derive them in two steps. First, we describe a rather naive criterion that is relatively ease to understand. This criterion is sufficient but not necessary (too strict). Second, we describe a “realistic” (more detailed) criterion that is sufficient and necessary.
A Naive Coverage Preservation Criterion
A sufficient but not necessary coverage preservation criterion for MCDC is given in (23). The predicate symbol is used in the same way as the real criterion: it is used to express that only input data that fulfill MCDC at the original program have to be considered for coverage preservation
This naive criterion is not necessary since it requires the coverage preservation of the conditions in the transformed program by a single condition from the original program .
Another drawback of this naive criterion is that it is based on a concrete set of test data that are used to achieve MCDC at the original program. To ensure coverage preservation in general, it would be necessary to ensure that the criterion holds for all possible sets of test data that achieve MCDC at the original program, which tends to be intractable in practice.
A Realistic Coverage Preservation Criterion
To define an easier testable (but more complicated) coverage preservation criterion for MCDC (Theorem 4.4) we use the auxiliary predicate given in (24). The predicate is similar to the predicate symbol , with the difference that it performs the control check on all members of two sets of input data. The predicate is used for the coverage preservation of MCDC to test whether the condition of the original program refers to TRUE for one input data set or and refers to FALSE for the other. Besides , also the predicate (10) is used to describe the preservation criterion for MCDC coverage
The criterion given in equ_preserve_mcdc states that for each condition of a decision of the transformed program there exist two sets of input data and whose members achieve the criterion needed for MCDC coverage. Further, there has to be a condition of the original program such that the is a subset of either the true-satisfiability valuation or the false-satisfiability valuation (tested with the predicate ). the same requirement as .
Theorem 4.4 (Preservation of MCDC). Assuming that a set of test data achieves MCDC coverage on a given program , then (25) provides a sufficient—and without further knowledge about the program and the test data, also necessary—criterion for guaranteeing preservation of MCDC coverage on a transformed program Proof. Preservation of MCDC: Part 1, showing sufficiency: Since is assumed to achieve MCDC on , it holds for each and for each that there exist at least two test vectors such that . Since as defined in (10) for each condition is the formal definition of MCDC it directly follows that
is a sufficient criterion to ensure that MCDC is preserved at program .
Part 2, showing necessity by indirect proof: Assuming there exists a decision with a condition such that for all input-data subsets it either holds that
then it is possible that
(for all conditions in the original program condition coverage is not fulfilled; this case is already excluded by assumption of having MCDC coverage at )
(there is no MCDC coverage at the original program ; this case is already excluded by assumption of having MCDC coverage at )
(the test data do not provide MCDC coverage at the transformed program ) which in each case violates the preservation of MCDC: Case (a) and (b) violate the preservation of MCDC since they are in contradiction with the requirement that MCDC is achieved at the original program. Case (c) states that there exists a condition in the transformed program for which there are no test data to achieve unique cause coverage, which is required for MCDC.
4.5. Preserving Scoped Path Coverage (SPC)
To define a coverage preservation criterion for SPC (Theorem 4.5) we use the auxiliary predicate given in (31).
The predicate is only TRUE if there is at least one condition from the set of conditions whose true-satisfiability valuation is a subset of the input data or there is at least one condition from the set of conditions whose false-satisfiability valuation is a subset of the input data . The predicate is used for the coverage preservation criterion of SPC to test whether for a condition in the transformed program with true/false-satisfiability valuation there exist two conditions in the original program whose true/false coverage are a subset of
As stated in Theorem 4.5, (32) provides a coverage preservation criterion for SPC. Equation (32) says that for each scoped path of the transformed program there exists a scoped path such that the reachability of the first basic block of implies the reachability of the first basic block of . Further, Equation (32) states that for each condition of that has to be evaluated to TRUE, there exists a condition of a scoped path in the original program that will imply the True evaluation of (by predicate ). Finally, Equation (32) states that for each condition of that has to be evaluated to FALSE, there exists a condition of a scoped path in the original program that will imply the FALSE evaluation of (by predicate ).
Theorem 4.5 (Preservation of SPC). Assuming that a set of test data achieves scoped path coverage on a given program , then (32) provides a sufficient—and without further knowledge about the program and the test data, also necessary—criterion for guaranteeing preservation of scoped path coverage on a transformed program
Proof. Preservation of SPC: Part 1, showing sufficiency: Since is assumed to achieve SPC on , it holds for each and each with that there exists test data with
Since (32) states that
it follows that
As (32) also states that
it follows that
Finally, as (32) states that
it follows that
Thus, SPC is preserved at .
Part 2, showing necessity by indirect proof: Assuming there exists a scoped program path such that for all scoped program paths