UNIT-3

TRANSACTION FLOW, DATA FLOW AND DOMAIN TESTING

3.1 TRANSACTION FLOWS

INTRODUCTION:

 A transaction is a unit of work seen from a system user's point of view.

 A transaction consists of a sequence of operations, some of which are
performed by a system, persons or devices that are outside of the system.
 Transaction begin with Birth-that is they are created as a result of some
external act.

Example of a transaction: A transaction for an online information retrieval system

might consist of the following steps or tasks:

 Accept input (tentative birth)

 Validate input (birth)
 Transmit acknowledgement to requester
 Do input processing
 Search file
 Request directions from user
 Accept input
 Validate input
 Process request
 Update file
 Transmit output
 Record transaction in log and clean up (death)

TRANSACTION FLOW GRAPHS:

 Transaction flows are introduced as a representation of a system's

processing.
 The methods that were applied to control flow graphs are then used for
functional testing.
 Transaction flows and transaction flow testing are to the independent
system tester what control flows are path testing are to the programmer.
 The transaction flow graph is to create a behavioral model of the program
that leads to functional testing.
 The transaction flowgraph is a model of the structure of the system's
behavior (functionality).
 An example of a Transaction Flow is as follows:
 Figure 3.1: An Example of a Transaction Flow

Usage:

 Transaction flows are indispensable for specifying requirements of

complicated systems, especially online systems.
 A big system such as an air traffic control or airline reservation system,
has not hundreds, but thousands of different transaction flows.
 The flows are represented by relatively simple flowgraphs, many of
which have a single straight-through path.
 Loops are infrequent compared to control flowgraphs.
 The most common loop is used to request a retry after user input
errors. An ATM system, for example, allows the user to try, say three
times, and will take the card away the fourth time
 Implementation

Figure 4.2 A transaction flow and its implementation

The implementation of a transaction flow is usually implicit in the design

of the system’s control structure and associated database.
A transaction flow is a representation of a path taken by a transaction through a
succession of processing modules. Each transaction is represented by a token.
Figure 4.2 shows another transaction flow and the corresponding implementation
of a program that creates that flow. This transaction goes through input processing,
which classifies it as to type, and then passes transaction to pass back to process
A,followed by B.
Figure 4.2b shows is a diagrammatic representation of a software architecture
that might implement this and many other transactions. The system is controlled by an
executive/Scheduler/Dispatcher/Operating system.
In this diagram the boxes represents the processes and the links represents the
processing queues. The scheduler contains tables or codes that routes the transaction to
next process.
Figure 4.2c shows a possible implementation of this transaction processing
system. Let’s say that A,B,C,D,E transactions are performed depending on the priority of
a transaction .
First the scheduler invokes processing of module B, which cleans up all the
transactions and then disc reads are initiated and then scheduler starts to C module
Complications:

 In simple cases, the transactions have a unique identity from the time
they're created to the time they're completed.
 In many systems the transactions can give birth to others, and transactions
can also merge.
 Births: There are three different possible interpretations of the decision
symbol, or nodes with two or more out links. It can be a Decision, Biosis
or a Mitosis.

1. Decision: Here the transaction will take one alternative or the

other alternative but not both. (See Figure 3.2 (a))
2. Biosis: Here the incoming transaction gives birth to a new
transaction, and both transaction continue on their separate
paths, and the parent retains it identity. (See Figure 3.2 (b))
3. Mitosis: Here the parent transaction is destroyed and two new
transactions are created.(See Figure 3.2 (c))

Figure 3.2: Nodes with multiple outlinks

Mergers: Transaction flow junction points are potentially as troublesome as transaction flow
splits. There are three types of junctions: (1) Ordinary Junction (2) Absorption (3) Conjugation

1. Ordinary Junction: An ordinary junction which is similar to the junction in a

control flow graph. A transaction can arrive either on one link or the other. (See
Figure 3.3 (a))
2. Absorption: In absorption case, the predator transaction absorbs prey transaction.
The prey gone but the predator retains its identity. (See Figure 3.3 (b))
3. Conjugation: In conjugation case, the two parent transactions merge to form a
new daughter. In keeping with the biological flavor this case is called as
conjugation.(See Figure 3.3 (c))
 Figure 3.3: Transaction Flow Junctions and Mergers

 Births, absorptions, and conjugations are as problematic for the software designer as
they are for the software modeler and the test designer; as a consequence, such points
have more than their share of bugs. The common problems are: lost daughters,
wrongful deaths, and illegitimate births.

3.2 TRANSACTION FLOW TESTING TECHNIQUES:

Get The Transactions Flows:

 Complicated systems that process a lot of different, complicated transactions should have
explicit representations of the transactions flows, or the equivalent.
 Transaction flows are like control flow graphs, and consequently we should expect to have
them in increasing levels of detail.
 The system's design documentation should contain an overview section that details the main
transaction flows.
 Detailed transaction flows are a mandatory prerequisite to the rational design of a system's
functional test.
Inspections, Reviews and Walkthroughs:
 Transaction flows are natural agenda for system reviews or inspections.
In conducting the walkthroughs, you should:
 Discuss enough transaction types to account for 98%-99% of the transaction the system is
expected to process.
 Discuss paths through flows in functional rather than technical terms.
 Ask the designers to relate every flow to the specification and to show how that transaction,
directly or indirectly, follows from the requirements.
 Make transaction flow testing the cornerstone of system functional testing just as path
testing is the cornerstone of unit testing.
 Select additional flow paths for loops, extreme values, and domain boundaries.
 Design more test cases to validate all births and deaths.
 Publish and distribute the selected test paths through the transaction flows as early as
possible so that they will exert the maximum beneficial effect on the project.
Path Selection:
 Select a set of covering paths (c1+c2) using the analogous criteria that were used for
structural path testing.
 Select a covering set of paths based on functionally sensible transactions, as you would for
control flow graphs.
 Try to find the most tortuous, longest, strangest path from the entry to the exit of the
transaction flow.
Path Sensitization:
 Add your content...Most of the normal paths are very easy to sensitize-80% - 95%
transaction flow coverage (c1+c2) is usually easy to achieve.
 The remaining small percentage is often very difficult.
 Sensitization is the act of defining the transaction. If there are sensitization problems on the
easy paths, then bet on either a bug in transaction flows or a design bug.

Path Instrumentation:

 Instrumentation plays a bigger role in transaction flow testing than in unit path testing.
 The information of the path taken for a given transaction must be kept with that transaction
and can be recorded by a central transaction dispatcher or by the individual processing
modules.
 In some systems, such traces are provided by the operating systems or a running log.

3.3 DATA FLOW TESTING:

 Data flow testing is the name given to a family of test strategies based on selecting paths
through the program's control flow in order to explore sequences of events related to the
status of data objects.
 For example, pick enough paths to assure that every data object has been initialized prior
to use or that all defined objects have been used for something.

 The source code consists of data declaration statements-that is statements that define
data structures, individual objects, initial or default rules and attributes

Data Flow Machines:

There are two types of data flow machines with different architectures. (1) Von Neumann
machines (2) Multi-instruction, multi-data machines (MIMD).

Von Neumann Machine Architecture:

 Most computers today are Von-Neumann machines.
 This architecture features interchangeable storage of instructions and data in the same
memory units.
 The Von Neumann machine Architecture executes one instruction at a time in the
following, micro instruction sequence:
1. Fetch instruction from memory
2. Interpret instruction
 3. Fetch operands
4. Process or Execute
5. Store result
6. Increment program counter
7. GOTO 1
Multi-instruction, Multi-data machines (MIMD) Architecture:
 These machines can fetch several instructions and objects in parallel.
 They can also do arithmetic and logical operations simultaneously on different data objects.
 The decision of how to sequence them depends on the compiler.
Bug Assumption:
 The bug assumption for data-flow testing strategies is that control flow is generally correct
and that something has gone wrong with the software so that data objects are not available
when they should be, or silly things are being done to data objects.
 Also, if there is a control-flow problem, we expect it to have symptoms that can be detected
by data-flow analysis.
 Although we'll be doing data-flow testing, we won't be using data flow graphs as such.
Rather, we'll use an ordinary control flow graph annotated to show what happens to the data
objects of interest at the moment.
Data Flow Graphs:
 The data flow graph is a graph consisting of nodes and directed links.

 We will use a control graph to show what happens to data objects of interest at that moment.
 Our objective is to expose deviations between the data flows we have and the data flows we
want.
Data Object State and Usage:
 Data Objects can be created, killed and used.
 They can be used in two distinct ways: (1) In a Calculation (2) As a part of a Control Flow
Predicate.
 The following symbols denote these possibilities:
 Defined: d - defined, created, initialized etc
 Killed or undefined: k - killed, undefined, released etc
Usage: u - used for something (c - used in Calculations, p - used in a predicate)
Defined (d):
 An object is defined explicitly when it appears in a data declaration.
 Or implicitly when it appears on the left hand side of the assignment.
 It is also to be used to mean that a file has been opened.
 A dynamically allocated object has been allocated.
 Something is pushed on to the stack.
 A record written.
Killed or Undefined (k):
 An object is killed on undefined when it is released or otherwise made unavailable.
 When its contents are no longer known with certitude (with absolute certainty / perfectness).
 Release of dynamically allocated objects back to the availability pool.
 Return of records.
 The old top of the stack after it is popped.
 An assignment statement can kill and redefine immediately. For example, if A had been
previously defined and we do a new assignment such as A : = 17, we have killed A's
previous value and redefined A
Usage (u):
 A variable is used for computation (c) when it appears on the right hand side of an
assignment statement.
 A file record is read or written.
 It is used in a Predicate (p) when it appears directly in a predicate.
Data Flow Anomalies:
 An anomaly is denoted by a two-character sequence of actions.
 For example, ku means that the object is killed and then used, where as dd means that the
object is defined twice without an intervening usage.
 What is an anomaly is depend on the application.
 There are nine possible two-letter combinations for d, k and u. some are bugs, some are
suspicious, and some are okay.
1. dd :- probably harmless but suspicious. Why define the object twice without an
intervening usage?
2. dk :- probably a bug. Why define the object without using it?
3. du :- the normal case. The object is defined and then used.
4. kd :- normal situation. An object is killed and then redefined.
5. kk :- harmless but probably buggy. Did you want to be sure it was really killed?
6. ku :- a bug. the object doesnot exist.
7. ud :- usually not a bug because the language permits reassignment at almost any time.
8. uk :- normal situation.
9. uu :- normal situation.
 In addition to the two letter situations, there are six single letter situations.
 We will use a leading dash to mean that nothing of interest (d,k,u) occurs prior to the action
noted along the entry-exit path of interest.
 A trailing dash to mean that nothing happens after the point of interest to the exit.
 They possible anomalies are:
1. -k :- possibly anomalous because from the entrance to this point on the path, the
variable had not been defined. We are killing a variable that does not exist.
2. -d :- okay. This is just the first definition along this path.
3. -u :- possibly anomalous. Not anomalous if the variable is global and has been
previously defined.
4. k- :- not anomalous. The last thing done on this path was to kill the variable.
5. d- :- possibly anomalous. The variable was defined and not used on this path. But this
could be a global definition.
6. u- :- not anomalous. The variable was used but not killed on this path. Although this
sequence is not anomalous, it signals a frequent kind of bug. If d and k mean dynamic
storage allocation and return respectively, this could be an instance in which a
dynamically allocated object was not returned to the pool after use.
Data Flow Anomaly State Graph:
 Data flow anomaly model prescribes that an object can be in one of four distinct states:
o K :- undefined, previously killed, does not exist
o D :- defined but not yet used for anything
o U :- has been used for computation or in predicate
o A :- anomalous
 These capital letters (K, D, U, A) denote the state of the variable and should not be confused
with the program action, denoted by lower case letters.
 Unforgiving Data - Flow Anomaly Flow Graph: Unforgiving model, in which once a
variable becomes anomalous it can never return to a state of grace.

Assume that the variable starts in the K state - that is, it has not been defined or does not
exist. If an attempt is made to use it or to kill it (e.g., say that we're talking about opening,
closing, and using files and that 'killing' means closing), the object's state becomes anomalous
(state A) and, once it is anomalous, no action can return the variable to a working state.
 If it is defined (d), it goes into the D, or defined but not yet used, state. If it has been
defined (D) and redefined (d) or killed without use (k), it becomes anomalous, while usage (u)
brings it to the U state. If in U, redefinition (d) brings it to D, u keeps it in U, and k kills it.

Forgiving Data - Flow Anomaly Flow Graph

Forgiving model is an alternate model where redemption (recover) from the anomalous
state is possible.

This graph has three normal and three anomalous states and he considers the kk
sequence not to be anomalous. The difference between this state graph and Figure 3.5 is that
redemption is possible. A proper action from any of the three anomalous states returns the
variable to a useful working state.
The point of showing you this alternative anomaly state graph is to demonstrate that the
specifics of an anomaly depends on such things as language, application, context, or even your
frame of mind. In principle, you must create a new definition of data flow anomaly (e.g., a new
state graph) in each situation. You must at least verify that the anomaly definition behind the
theory or imbedded in a data flow anomaly test tool is appropriate to your situation.

Static Vs Dynamic Anomaly Detection:

 Static analysis is analysis done on source code without actually executing it. For example:
source code syntax error detection is the static analysis result.
 Dynamic analysis is done on the fly as the program is being executed and is based on
intermediate values that result from the program's execution. For example: a division by
zero warning is the dynamic result.
 If a problem, such as a data flow anomaly, can be detected by static analysis methods, then
it doesn’t belong in testing - it belongs in the language processor.
 There is actually a lot more static analysis for data flow analysis for data flow anomalies
going on in current language processors.
 For example, language processors which force variable declarations can detect (-u) and (ku)
anomalies.
 But still there are many things for which current notions of static analysis are Inadequate
they are
 1. Dead variables: Although it is often possible proved that a variable is dead or
alive at the given point in the program.
2. Arrays: Arrays are problematic in that array is defined or killed as a single
object. Arrays are dynamically calculated so there is no way to do static analysis
to validate the pointer value.
3. Records and pointers : In many applications we create the files and their names
dynamically and there is no way to determine without execution
4. Dynamic subroutine or function names in a call : A subroutine or function name
is a dynamic variable in a call
5. False Anomalies
6. Concurrency, Interrupts and system issues
Data Flow Model:
 The data flow model is based on the program's control flow graph
 Here we annotate each link with symbols (for example, d, k, u, c, p) or sequences of
symbols (for example, dd, du, ddd) that denote the sequence of data operations on that link
with respect to the variable of interest. Such annotations are called link weights.
 The control flow graph structure is same for every variable: it is the weights that change.

Components of the model:

The modeling rules are
1. To every statement there is a node and whose name or number is unique.
Every node has one outlink and atleast one inlink except exit nodes,
which do not have outlinks, and entry nodes,which do not have inlinks
2. Exit nodes are placed at the outgoing arrowheads of exit statements(Ex:
END, RETURN) to complete the graph and Entry nodes are placed at the
entry statements (Ex: BEGIN)
3. The outlink of simple statements are weighted by proper sequence of data
flow actions of that statement
4. Predicate nodes (Ex: IF-THEN-ELSE, DO WHILE,CASE) are weighted
with p-use
5. Every sequence of simple statements can be replaced by pair of nodes that
has weights on the link between them

Putting it Together:
Example:
 Figure 3.8 shows that the control flow graph for the program and the node labels and
marked the decision nodes with the variables.
Figure 3.9 shows that this control flowgraph annotated for variables X and Y and there
there is a dcc on the first link (1,3)
Figure 3.10 shows all the predicate uses for the Z variable and Figure 3.11shows the
control flow graph for V data flow .
Data flow testing is a family of test strategies based on selecting paths through the
program's control flow in order to explore sequences of events related to the status of variables
or data objects. Dataflow Testing focuses on the points at which variables receive values and the
points at which these values are used.

3.4 DATA FLOW TESTING STRATEGIES

 Data Flow Testing Strategies are structural strategies.
 In contrast to the path-testing strategies, data-flow strategies take into account what
happens to data objects on the links in addition to the raw connectivity of the graph.
 In other words, data flow strategies require data-flow link weights (d,k,u,c,p).
 Data Flow Testing Strategies are based on selecting test path segments (also
called sub paths) that satisfy some characteristic of data flows for all data objects.
  For example, all subpaths that contain a d (or u, k, du, dk).
 A strategy X is stronger than another strategy Y if all test cases produced under Y
are included in those produced under X - conversely for weaker.

Terminology:

Definition-Clear Path Segment, with respect to variable X, is a connected

sequence of links such that X is (possibly) defined on the first link and not redefined
or killed on any subsequent link of that path segment. ll paths in Figure 3.9 are
definition clear because variables X and Y are defined only on the first link (1,3) and
not thereafter. In Figure 3.10, we have a more complicated situation. The following
path segments are definition-clear: (1,3,4), (1,3,5), (5,6,7,4), (7,8,9,6,7), (7,8,9,10),
(7,8,10), (7,8,10,11). Subpath (1,3,4,5) is not definition-clear because the variable is
defined on (1,3) and again on (4,5). For practice, try finding all the definition-clear
subpaths for this routine (i.e., for all variables).

Loop-Free Path Segment is a path segment for which every node in it is

visited atmost once. For Example, path (4,5,6,7,8,10) in Figure 3.10 is loop free, but
path (10,11,4,5,6,7,8,10,11,12) is not because nodes 10 and 11 are each visited twice.

Simple path segment is a path segment in which at most one node is visited
twice. For example, in Figure 3.10, (7,4,5,6,7) is a simple path segment. A simple path
segment is either loop-free or if there is a loop, only one node is involved.

A du path from node i to k is a path segment such that if the last link has a
computational use of X, then the path is simple and definition-clear; if the penultimate
(last but one) node is j - that is, the path is (i,p,q,...,r,s,t,j,k) and link (j,k) has a
predicate use - then the path from i to j is both loop-free and definition-clear.

The Strategies

The structural test strategies discussed below are based on the program's control
flowgraph. They differ in the extent to which predicate uses and/or computational uses of
variables are included in the test set. Various types of data flow testing strategies in
decreasing order of their effectiveness are:

All - du Paths (ADUP): The all-du-paths (ADUP) strategy is the strongest data-
flow testing strategy discussed here. It requires that every du path from every
definition of every variable to every use of that definition be exercised under some
test.

For variable X and Y:In Figure 3.9, because variables X and Y are used only on link
(1,3), any test that starts at the entry satisfies this criterion (for variables X and Y, but
not for all variables as required by the strategy).

For variable Z: The situation for variable Z (Figure 3.10) is more complicated
because the variable is redefined in many places. For the definition on link (1,3) we
must exercise paths that include subpaths (1,3,4) and (1,3,5). The definition on link
(4,5) is covered by any path that includes (5,6), such as subpath (1,3,4,5,6, ...). The
(5,6) definition requires paths that include subpaths (5,6,7,4) and (5,6,7,8).

For variable V: Variable V (Figure 3.11) is defined only once on link (1,3). Because
V has a predicate use at node 12 and the subsequent path to the end must be forced for
both directions at node 12, the all-du-paths strategy for this variable requires that we
exercise all loop-free entry/exit paths and at least one path that includes the loop
caused by (11,4). Note that we must test paths that include both subpaths (3,4,5) and
(3,5) even though neither of these has V definitions. They must be included because
they provide alternate du paths to the V use on link (5,6). Although (7,4) is not used in
the test set for variable V, it will be included in the test set that covers the predicate
uses of array variable V() and U.

The all-du-paths strategy is a strong criterion, but it does not take as many tests as it
might seem at first because any one test simultaneously satisfies the criterion for
several definitions and uses of several different variables.

All Uses Startegy (AU):The all uses strategy is that at least one definition clear path
from every definition of every variable to every use of that definition be exercised
under some test. Just as we reduced our ambitions by stepping down from all paths (P)
to branch coverage (C2), say, we can reduce the number of test cases by asking that
the test set should include at least one path segment from every definition to every use
that can be reached by that definition.

For variable V: In Figure 3.11, ADUP requires that we include subpaths

(3,4,5) and (3,5) in some test because subsequent uses of V, such as on link (5,6), can
be reached by either alternative. In AU either (3,4,5) or (3,5) can be used to start paths,
but we don't have to use both. Similarly, we can skip the (8,10) link if we've included
the (8,9,10) subpath. Note the hole. We must include (8,9,10) in some test cases
because that's the only way to reach the c use at link (9,10) - but suppose our bug for
variable V is on link (8,10) after all? Find a covering set of paths under AU for Figure
3.11.

All p-uses/some c-uses strategy (APU+C) : For every variable and every definition
of that variable, include at least one definition free path from the definition to every
predicate use; if there are definitions of the variables that are not covered by the above
prescription, then add computational use test cases as required to cover every
definition.

For variable Z:In Figure 3.10, for APU+C we can select paths that all take the upper
link (12,13) and therefore we do not cover the c-use of Z: but that's okay according to
the strategy's definition because every definition is covered. Links (1,3), (4,5), (5,6),
and (7,8) must be included because they contain definitions for variable Z. Links (3,4),
(3,5), (8,9), (8,10), (9,6), and (9,10) must be included because they contain predicate
uses of Z. Find a covering set of test cases under APU+C for all variables in this
example - it only takes two tests.

For variable V:In Figure 3.11, APU+C is achieved for V by

(1,3,5,6,7,8,10,11,4,5,6,7,8,10,11,12[upper], 13,2) and (1,3,5,6,7,8,10,11,12[lower],
13,2). Note that the c-use at (9,10) need not be included under the APU+C criterion.

All c-uses/some p-uses strategy (ACU+P) : The all c-uses/some p-uses strategy
(ACU+P) is to first ensure coverage by computational use cases and if any definition
is not covered by the previously selected paths, add such predicate use cases as are
needed to assure that every definition is included in some test.

For variable Z: In Figure 3.10, ACU+P coverage is achieved for Z by path

(1,3,4,5,6,7,8,10, 11,12,13[lower], 2), but the predicate uses of several definitions are
not covered. Specifically, the (1,3) definition is not covered for the (3,5) p-use, the
(7,8) definition is not covered for the (8,9), (9,6) and (9, 10) p-uses.

The above examples imply that APU+C is stronger than branch coverage but
ACU+P may be weaker than, or incomparable to, branch coverage.

All Definitions Strategy (AD) : The all definitions strategy asks only every
definition of every variable be covered by atleast one use of that variable, be that use a
computational use or a predicate use.

For variable Z: Path (1,3,4,5,6,7,8, . . .) satisfies this criterion for variable Z, whereas
any entry/exit path satisfies it for variable V. From the definition of this strategy we
would expect it to be weaker than both ACU+P and APU+C.

All Predicate Uses (APU), All Computational Uses (ACU) Strategies : The all
predicate uses strategy is derived from APU+C strategy by dropping the requirement
that we include a c-use for the variable if there are no p-uses for the variable. The all
computational uses strategy is derived from ACU+P strategy by dropping the
requirement that we include a p-use for the variable if there are no c-uses for the
variable. It is intuitively obvious that ACU should be weaker than ACU+P and that
APU should be weaker than APU+C.

Ordering The Strategies:

Figure 3.12 compares path-flow and data-flow testing strategies. The arrows
denote that the strategy at the arrow's tail is stronger than the strategy at the arrow's
head.
 Figure 3.12 Relative Strength Of Structural Test Strategies

Slicing And Dicing:

 A (static) program slice is a part of a program (e.g., a selected set of statements)

defined with respect to a given variable X (where X is a simple variable or a data
vector) and a statement i: it is the set of all statements that could (potentially, under
static analysis) affect the value of X at statement i - where the influence of a faulty
statement could result from an improper computational use or predicate use of some
other variables at prior statements.
 If X is incorrect at statement i, it follows that the bug must be in the program slice for
X with respect to i
 A program dice is a part of a slice in which all statements which are known to be
correct have been removed.
 In other words, a dice is obtained from a slice by incorporating information obtained
through testing or experiment (e.g., debugging).
 The debugger first limits her scope to those prior statements that could have caused
the faulty value at statement i (the slice) and then eliminates from further
consideration those statements that testing has shown to be correct.
 Debugging can be modeled as an iterative procedure in which slices are further
refined by dicing, where the dicing information is obtained from ad hoc tests aimed
primarily at eliminating possibilities. Debugging ends when the dice has been
reduced to the one faulty statement.
 Dynamic slicing is a refinement of static slicing in which only statements on
achievable paths to the statement in question are included.
3.5APPLICATIONS,TOOLS AND EFFECTIVENESS

Comparison Random Testing, P2, AU - by Ntafos

• AU detects more bugs than

• P2 with more test cases
• RT with less # of test cases

Comparison of P2, AU - by Sneed

 AU detects more bugs with 90% Data Coverage Requirement.

Comparison of number of test cases for ACU, APU, AU & ADUP by Weyuker using
ASSET testing system


Test Cases Normalized. t = a + b * d d = # binary
decisions
 At most d+1 Test Cases for P2 loop-free
 No of Test Cases / Decision
ADUP > AU > APU > ACU > revised-APU

Comparison of # test cases for ACU, APU, AU & ADUP by Shimeall & Levenson

Test Cases Normalized. t = a + b * d (d = # binary decisions)

At most d+1 Test Cases for P2 loop-free
# Test Cases / Decision
ADUP ~ ½ APU*
AP ~ AC
 The data flow testing strategies have found it expedient and cost effective to
create supporting tools

Commercial tools :

 Can possibly do Better than Commercial Tools

 Easier Integration into a Compiler
 Efficient Testing
3.6DOMAINS AND PATHS

 Domain: In mathematics, domain is a set of possible values of an independant

variable or the variables of a function.
 Programs as input data classifiers: domain testing attempts to determine whether
the classification is or is not correct.
 Domain testing can be based on specifications or equivalent implementation
information.
 If domain testing is based on specifications, it is a functional test technique.
 If domain testing is based implementation details, it is a structural test technique.

The Model:

The following figure is a schematic representation of domain testing.

Figure 4.1: Schematic Representation of Domain Testing.

 Before doing whatever it does, a routine must classify the input and set
it moving on the right path
 An invalid input (e.g., value too big) is just a special processing case
called 'reject'.
 The input then passses to a hypothetical subroutine rather than on
calculations.
 In domain testing, we focus on the classification aspect of the routine
rather than on the calculations.
 Structural knowledge is not needed for this model - only a consistent,
complete specification of input values for each case.
 We can infer that for each case there must be atleast one path to
process that case.

A Domain Is A Set:

 An input domain is a set.

  If the source language supports set definitions (E.g. PASCAL set types and
C enumerated types) less testing is needed because the compiler does
much of it for us.
 Domain testing does not work well with arbitrary discrete sets of data
objects.
 Domain for a loop-free program corresponds to a set of numbers defined
over the input vector.

Domains, Paths And Predicates:

 In domain testing, predicates are assumed to be interpreted in terms of

input vector variables.
 If domain testing is applied to structure, then predicate interpretation must
be based on actual paths through the routine - that is, based on the
implementation control flowgraph.
 Conversely, if domain testing is applied to specifications, interpretation is
based on a specified data flowgraph for the routine; but usually, as is the
nature of specifications, no interpretation is needed because the domains
are specified directly.
 For every domain, there is at least one path through the routine.
 There may be more than one path if the domain consists of disconnected
parts or if the domain is defined by the union of two or more domains.
 Domains are defined their boundaries. Domain boundaries are also where
most domain bugs occur.
 For every boundary there is at least one predicate that specifies what
numbers belong to the domain and what numbers don't.

 For example, in the statement IF x>0 THEN ALPHA ELSE BETA

we know that numbers greater than zero belong to ALPHA
processing domain(s) while zero and smaller numbers belong to
BETA domain(s).

 A domain may have one or more boundaries - no matter how many

variables define it.

For example, if the predicate is x2 + y2 < 16, the domain is the inside of a
circle of radius 4 about the origin. Similarly, we could define a spherical
domain with one boundary but in three variables.

 Domains are usually defined by many boundary segments and therefore by

many predicates. i.e. the set of interpreted predicates traversed on that path
(i.e., the path's predicate expression) defines the domain's boundaries.

A Domain Closure:

 A domain boundary is closed with respect to a domain if the points on the

boundary belong to the domain.
  If the boundary points belong to some other domain, the boundary is said
to be open.
 Figure 4.2 shows three situations for a one-dimensional domain - i.e., a
domain defined over one input variable; call it x
 The importance of domain closure is that incorrect closure bugs are
frequent domain bugs. For example, x >= 0 when x > 0 was intended.

 Figure 4.2: Open and Closed Domains.

Domain Dimensionality:

 Every input variable adds one dimension to the domain.

 One variable defines domains on a number line.
 Two variables define planar domains.
 Three variables define solid domains.
 Every new predicate slices through previously defined domains and cuts
them in half.
 Every boundary slices through the input vector space with a
dimensionality which is less than the dimensionality of the space.
 Thus, planes are cut by lines and points, volumes by planes, lines and
points and n-spaces by hyperplanes.

Bug Assumption:

 The bug assumption for the domain testing is that processing is okay but
the domain definition is wrong.
 An incorrectly implemented domain means that boundaries are wrong,
which may in turn mean that control flow predicates are wrong.
 Many different bugs can result in domain errors. Some of them are:
 1. Double Zero Representation :In computers or Languages that
have a distinct positive and negative zero, boundary errors for
negative zero are common.
2. Floating point zero check:A floating point number can equal
zero only if the previous definition of that number set it to zero
or if it is subtracted from it self or multiplied by zero. So the
floating point zero check to be done against a epsilon value.
3. Contradictory domains:An implemented domain can never be
ambiguous or contradictory, but a specified domain can. A
contradictory domain specification means that at least two
supposedly distinct domains overlap.
4. Ambiguous domains:Ambiguous domains means that union of
the domains is incomplete. That is there are missing domains or
holes in the specified domains. Not specifying what happens to
points on the domain boundary is a common ambiguity.
5. Overspecified Domains:he domain can be overloaded with so
many conditions that the result is a null domain. Another way to
put it is to say that the domain's path is unachievable.
6. Boundary Errors:Errors caused in and around the boundary of
a domain. Example, boundary closure bug, shifted, tilted,
missing, extra boundary.
7. Closure Reversal:A common bug. The predicate is defined in
terms of >=. The programmer chooses to implement the logical
complement and incorrectly uses <= for the new predicate; i.e.,
x >= 0 is incorrectly negated as x <= 0, thereby shifting
boundary values to adjacent domains.
8. Faulty Logic:Compound predicates (especially) are subject to
faulty logic transformations and improper simplification. If the
predicates define domain boundaries, all kinds of domain bugs
can result from faulty logic manipulations.

Restrictions To Domain Testing:

Domain testing has restrictions, as do other testing techniques. Some of them include:

1. Co-incidental Correctness:
Domain testing isn't good at finding bugs for which the outcome is correct
for the wrong reasons. If we're plagued by coincidental correctness we may
misjudge an incorrect boundary. Note that this implies weakness for domain
testing when dealing with routines that have binary outcomes (i.e.,
TRUE/FALSE)
2. Representative Outcome:
Domain testing is an example of partition testing. Partition-testing
strategies divide the program's input space into domains such that all inputs within
a domain are equivalent (not equal, but equivalent) in the sense that any input
represents all inputs in that domain.
 If the selected input is shown to be correct by a test, then processing is
presumed correct, and therefore all inputs within that domain are expected
(perhaps unjustifiably) to be correct. Most test techniques, functional or structural,
fall under partition testing and therefore make this representative outcome
assumption. For example, x2 and 2x are equal for x = 2, but the functions are
different. The functional differences between adjacent domains are usually simple,
such as x + 7 versus x + 9, rather than x2 versus 2x.

3. Simple Domain Boundaries and Compound Predicates:

Compound predicates in which each part of the predicate specifies a

different boundary are not a problem: for example, x >= 0 AND x < 17, just
specifies two domain boundaries by one compound predicate. As an example of a
compound predicate that specifies one boundary, consider: x = 0 AND y >= 7
AND y <= 14.

This predicate specifies one boundary equation (x = 0) but alternates

closure, putting it in one or the other domain depending on whether y < 7 or y >
14. Treat compound predicates with respect because they’re more complicated
than they seem.

4. Functional Homogeneity of Bugs:

Whatever the bug is, it will not change the functional form of the boundary
predicate. For example, if the predicate is ax >= b, the bug will be in the value of
a or b but it will not change the predicate to ax >= b, say.

5. Linear Vector Space:

Most papers on domain testing, assume linear boundaries - not a bad

assumption because in practice most boundary predicates are linear.

6. Loop Free Software:

Loops are problematic for domain testing. The trouble with loops is that
each iteration can result in a different predicate expression (after interpretation),
which means a possible domain boundary change.
3.7 NICE AND UGLY DOMAINS:

NICE DOMAINS:

Where do these domains come from?

Domains are and will be defined by an imperfect iterative process aimed at achieving
(user, buyer, voter) satisfaction.

 Implemented domains can't be incomplete or inconsistent. Every input will be processed

(rejection is a process), possibly forever. Inconsistent domains will be made consistent.

 Conversely, specified domains can be incomplete and/or inconsistent. Incomplete in this

context means that there are input vectors for which no path is specified, and
inconsistent means that there are at least two contradictory specifications over the same
segment of the input space.

 Some important properties of nice domains are: Linear, Complete, Systematic, And
Orthogonal, Consistently closed, Convex and simply connected.

 To the extent that domains have these properties domain testing is easy as testing gets.

 The bug frequency is lesser for nice domain than for ugly domains.

Figure 4.3: Nice Two-Dimensional Domains.

Linear And Non Linear Boundaries:

 Nice domain boundaries are defined by linear inequalities or equations.

  The impact on testing stems from the fact that it takes only two points to determine a
straight line and three points to determine a plane and in general n+ 1 point to determine
an n-dimensional hyper plane.

 In practice more than 99.99% of all boundary predicates are either linear or can be
linearized by simple variable transformations.

Complete Boundaries:

 Nice domain boundaries are complete in that they span the number space from plus to
minus infinity in all dimensions.

 Figure 4.4 shows some incomplete boundaries. Boundaries A and E have gaps.

 Such boundaries can come about because the path that hypothetically corresponds to
them is unachievable, because inputs are constrained in such a way that such values can't
exist, because of compound predicates that define a single boundary, or because
redundant predicates convert such boundary values into a null set.

 The advantage of complete boundaries is that one set of tests is needed to confirm the
boundary no matter how many domains it bounds.

 If the boundary is chopped up and has holes in it, then every segment of that boundary
must be tested for every domain it bounds.

Figure 4.4: Incomplete Domain Boundaries.

Systematic Boundaries:
  Systematic boundary means that boundary inequalities related by a simple function such
as a constant.
In Figure 4.3 for example, the domain boundaries for u and v differ only by a constant.

where fi is an arbitrary linear function, X is the input vector, ki and c are constants, and g(i,c) is
a decent function over i and c that yields a constant, such as k + ic.

 The first example is a set of parallel lines, and the second example is a set of
systematically (e.g., equally) spaced parallel lines (such as the spokes of a wheel, if
equally spaced in angles, systematic).

 If the boundaries are systematic and if you have one tied down and generate tests for it,
the tests for the rest of the boundaries in that set can be automatically generated.

Orthogonal Boundaries:

 Two boundary sets U and V (See Figure 4.3) are said to be orthogonal if every
inequality in V is perpendicular to every inequality in U.

 If two boundary sets are orthogonal, then they can be tested independently

 In Figure 4.3 we have six boundaries in U and four in V. We can confirm the boundary
properties in a number of tests proportional to 6 + 4 = 10 (O(n)). If we tilt the boundaries
to get Figure 4.5,

 we must now test the intersections. We've gone from a linear number of cases to a
quadratic: from O(n) to O(n2).
 Figure 4.5: Tilted Boundaries.

Figure 4.6: Linear, Non-orthogonal Domain Boundaries.

Actually, there are two different but related orthogonality conditions. Sets of boundaries can be
orthogonal to one another but not orthogonal to the coordinate axes (condition 1), or boundaries
can be orthogonal to the coordinate axes (condition 2).

Closure Consistency:

 Figure 4.6 shows another desirable domain property: boundary closures are consistent
and systematic.

 The shaded areas on the boundary denote that the boundary belongs to the domain in
which the shading lies - e.g., the boundary lines belong to the domains on the right.

 Consistent closure means that there is a simple pattern to the closures - for example,
using the same relational operator for all boundaries of a set of parallel boundaries.

CONVEX:

 A geometric figure (in any number of dimensions) is convex if you can take two
arbitrary points on any two different boundaries, join them by a line and all points on
that line lie within the figure.

 Nice domains are convex; dirty domains aren't.

  You can smell a suspected concavity when you see phrases such as: ". . . except if . . .,"
"However . . .," ". . . but not. . . ." In programming, it's often the buts in the specification
that kill you.

Simply Connected:

 Nice domains are simply connected; that is, they are in one piece rather than pieces all
over the place interspersed with other domains.

 Simple connectivity is a weaker requirement than convexity; if a domain is convex it is

simply connected, but not vice versa.

 Consider domain boundaries defined by a compound predicate of the (Boolean) form

ABC. Say that the input space is divided into two domains, one defined by ABC and,
therefore, the other defined by its negation.

 For example, suppose we define valid numbers as those lying between 10 and 17
inclusive. The invalid numbers are the disconnected domain consisting of numbers less
than 10 and greater than 17.

 Simple connectivity, especially for default cases, may be impossible.

UGLY DOMAINS:

 Some domains are born ugly and some are uglified by bad specifications.

 Every simplification of ugly domains by programmers can be either good or bad.

 Programmers in search of nice solutions will "simplify" essential complexity out of

existence. Testers in search of brilliant insights will be blind to essential complexity and
therefore miss important cases.

 If the ugliness results from bad specifications and the programmer's simplification is
harmless, then the programmer has made ugly good.
 But if the domain's complexity is essential (e.g., the income tax code), such
"simplifications" constitute bugs.

 Nonlinear boundaries are so rare in ordinary programming that there's no information on

how programmers might "correct" such boundaries if they're essential.

Ambiguities And Contradictions:

 Domain ambiguities are holes in the input space.

  The holes may lie within the domains or in cracks between domains.

 Two kinds of contradictions are possible: overlapped domain specifications and

overlapped closure specifications

 Figure 4.7c shows overlapped domains and Figure 4.7d shows dual closure assignment.

Figure 4.7: Domain Ambiguities and Contradictions

Simplifying The Topology:

 The programmer's and tester's reaction to complex domains is the same - simplify

 There are three generic cases: concavities, holes and disconnected pieces.

 Programmers introduce bugs and testers misdesign test cases by: smoothing out
concavities (Figure 4.8a), filling in holes (Figure 4.8b), and joining disconnected pieces
(Figure 4.8c).
Figure 4.8: Simplifying the topology.

RECTIFYING BOUNDARY CLOSURES:

 If domain boundaries are parallel but have closures that go every which way (left, right,
left . . .) the natural reaction is to make closures go the same way (see Figure 4.9).
 If the main processing concerns about one or two domains and the spaces between them
are rejected, the likely treatment of closures is to make all boundaries point the same
way

Figure 4.9: Forcing Closure Consistency.

3.8 DOMAIN TESTING:

DOMAIN TESTING STRATEGY: The domain-testing strategy is simple, although possibly tedious
(slow).

1. Domains are defined by their boundaries; therefore, domain testing concentrates test
points on or near boundaries.
 2. Classify what can go wrong with boundaries, then define a test strategy for each
case. Pick enough points to test for all recognized kinds of boundary errors.
3. Because every boundary serves at least two different domains, test points used to
check one domain can also be used to check adjacent domains. Remove redundant
test points.
4. Run the tests and by posttest analysis (the tedious part) determine if any boundaries
are faulty and if so, how.
5. Run enough tests to verify every boundary of every domain.

Domain Bugs And How To Test For Them:

 An interior point (Figure 4.10) is a point in the domain such that all points within an
arbitrarily small distance (called an epsilon neighborhood) are also in the domain.
 A boundary point is one such that within an epsilon neighborhood there are points
both in the domain and not in the domain.
 An extreme point is a point that does not lie between any two other arbitrary but
distinct points of a (convex) domain.

Figure 4.10: Interior, Boundary and Extreme points.

 An on point is a point on the boundary.

 If the domain boundary is closed, an off point is a point near the boundary but in the adjacent
domain.
 If the boundary is open, an off point is a point near the boundary but in the domain being tested;
see Figure 4.11. You can remember this by the acronym COOOOI: Closed Off Outside, Open
Off Inside.
 Figure 4.11: On points and Off points.

 Figure 4.12 shows generic domain bugs: closure bug, shifted boundaries, tilted
boundaries, extra boundary, missing boundary.

Figure 4.12: Generic Domain Bugs.

Testing One Dimensional Domains:

 Figure 4.13 shows possible domain bugs for a one-dimensional open

domain boundary.
 The closure can be wrong (i.e., assigned to the wrong domain) or the
boundary (a point in this case) can be shifted one way or the other, we can
be missing a boundary, or we can have an extra boundary.
 Figure 4.13: One Dimensional Domain Bugs, Open Boundaries.

 In Figure 4.13a we assumed that the boundary was to be open for A. The bug we're
looking for is a closure error, which converts > to >= or < to <= (Figure 4.13b). One
test (marked x) on the boundary point detects this bug because processing for that
point will go to domain A rather than B.
 In Figure 4.13c we've suffered a boundary shift to the left. The test point we used for
closure detects this bug because the bug forces the point from the B domain, where it
should be, to A processing. Note that we can't distinguish between a shift and a
closure error, but we do know that we have a bug.
 Figure 4.13d shows a shift the other way. The on point doesn't tell us anything
because the boundary shift doesn't change the fact that the test point will be
processed in B. To detect this shift we need a point close to the boundary but within
A. The boundary is open, therefore by definition, the off point is in A (Open Off
Inside).
 The same open off point also suffices to detect a missing boundary because what
should have been processed in A is now processed in B.
 To detect an extra boundary we have to look at two domain boundaries. In this
context an extra boundary means that A has been split in two. The two off points that
we selected before (one for each boundary) does the job. If point C had been a closed
boundary, the on test point at C would do it.
 For closed domains look at Figure 4.14. As for the open boundary, a test point on the
boundary detects the closure bug. The rest of the cases are similar to the open
boundary, except now the strategy requires off points just outside the domain.
 Figure 4.14: One Dimensional Domain Bugs, Closed Boundaries.

Testing Two Dimensional Domains:

 Figure 4.15 shows possible domain boundary bugs for a two-dimensional domain.
 A and B are adjacent domains and the boundary is closed with respect to A, which
means that it is open with respect to B.
For Closed Boundaries:

1. Closure Bug: Figure 4.15a shows a faulty closure, such as might be caused by using
a wrong operator (for example, x >= k when x > k was intended, or vice versa). The
two on points detect this bug because those values will get B rather than A
processing.
2. Shifted Boundary: In Figure 4.15b the bug is a shift up, which converts part of
domain B into A processing, denoted by A'. This result is caused by an incorrect
constant in a predicate, such as x + y >= 17 when x + y >= 7 was intended. The off
point (closed off outside) catches this bug. Figure 4.15c shows a shift down that is
caught by the two on points.
3. Tilted Boundary: A tilted boundary occurs when coefficients in the boundary
inequality are wrong. For example, 3x + 7y > 17 when 7x + 3y > 17 was intended.
Figure 4.15d has a tilted boundary, which creates erroneous domain segments A' and
B'. In this example the bug is caught by the left on point.
4. Extra Boundary: An extra boundary is created by an extra predicate. An extra
boundary will slice through many different domains and will therefore cause many
test failures for the same bug. The extra boundary in Figure 4.15e is caught by two
on points, and depending on which way the extra boundary goes, possibly by the off
point also.
5. Missing Boundary: A missing boundary is created by leaving a boundary predicate
out. A missing boundary will merge different domains and will cause many test
failures although there is only one bug. A missing boundary, shown in Figure 4.15f,
is caught by the two on points because the processing for A and B is the same - either
A or B processing.

Figure 6.16 : Domain testing strategy

  The above figure 6.16 summarizes domain testing for two dimensional domains
it shows a domain all but one of whose boundaries are closed.
 There are two on points (closed circles) for each segment and one off point(open
circle)
 The on points for two adjacent boundary segments are shared if both segments are
open or if both are closed.

Equality & Inequality Predicates

 Equality predicates such as x+y=17 define lower-level dimensions

 We get two test points for equality predicates by considering adjacent domains as
shown in above figure
 There are three domains A,B and C . A and B are planar while c,defined by the
equality boundary predicate between A and B
 The test point for A is a and for B is b and on C there are c c—

Testing n-Dimensional domains

 The domains defined over an n-dimensional input space with p boundary

segments, the domain testing strategy generalize to require atmost (n+1)p
strategy.
 Equality predicates defined over m dimensions(m<n) create a subspace of n-
m dimensions, which is then tested the same way as n dimension

Procedure For Testing: The procedure is conceptually is straight forward. It can be done by
hand for two dimensions and for a few domains and practically impossible for more than two
variables.

1. Identify input variables.

2. Identify variable which appear in domain defining predicates, such as control flow
predicates.
3. Interpret all domain predicates in terms of input variables.
 4. For p binary predicates, there are at most 2p combinations of TRUE-FALSE values
and therefore, at most 2p domains. Find the set of all non null domains. The result is a
boolean expression in the predicates consisting a set of AND terms joined by OR's.
For example ABC+DEF+GHI ...... Where the capital letters denote predicates. Each
product term is a set of linear inequality that defines a domain or a part of a multiply
connected domains.
5. Solve these inequalities to find all the extreme points of each domain using any of
the linear programming methods.

Variations, Tools Effectiveness

 The basic domain testing strategy is N*1 strategy because it uses N on points and
one off point
 Clarke strategy uses N on and N off points per boundary segment(N*N
strategy) and strategies based on the vertex
 Richardson and Clarke used the generalization strategy i.e, partition and analysis
which includes verification, both structural and functional information

3.9 DOMAIN AND INTERFACE TESTING

Introduction:

 We defined integration testing as testing the correctness of the interface

between two otherwise correct components.
 Components A and B have been demonstrated to satisfy their component
tests, and as part of the act of integrating them we want to investigate
possible inconsistencies across their interface.
 Interface between any two components is considered as a subroutine call.
 We're looking for bugs in that "call" when we do interface testing.
 Let's assume that the call sequence is correct and that there are no type
incompatibilities.
 For a single variable, the domain span is the set of numbers between (and
including) the smallest value and the largest value. For every input
variable we want (at least): compatible domain spans and compatible
closures (Compatible but need not be Equal).

Domains And Range:

 The set of output values produced by a function is called the range of the
function, in contrast with the domain, which is the set of input values over
which the function is defined.
 For most testing, our aim has been to specify input values and to predict
and/or confirm output values that result from those inputs.
 Interface testing requires that we select the output values of the calling
routine i.e. caller's range must be compatible with the called routine's
domain.
  An interface test consists of exploring the correctness of the following
mappings:

caller domain --> caller range(caller unit test)

caller range --> called domain(integration test)
domain --> called range (called unit test)

Closure Compatibility:

 Assume that the caller's range and the called domain spans the same
numbers - for example, 0 to 17.
 Figure 4.16 shows the four ways in which the caller's range closure and
the called's domain closure can agree.
 The thick line means closed and the thin line means open. Figure 4.16
shows the four cases consisting of domains that are closed both on top
(17) and bottom (0), open top and closed bottom, closed top and open
bottom, and open top and bottom.

Figure 4.16: Range / Domain Closure Compatibility.

 Figure 4.17 shows the twelve different ways the caller and the called can
disagree about closure. Not all of them are necessarily bugs. The four
cases in which a caller boundary is open and the called is closed (marked
with a "?") are probably not buggy. It means that the caller will not supply
such values but the called can accept them.
 Figure 4.17: Equal-Span Range / Domain Compatibility
Bugs.

Span Compatibility:

 Figure 4.18 shows three possibly harmless span incompatibilities.

Figure 4.18: Harmless Range / Domain Span incompatibility bug

(Caller Span is smaller than Called).
 In all cases, the caller's range is a subset of the called's domain. That's not
necessarily a bug.
 The routine is used by many callers; some require values inside a range
and some don't. This kind of span incompatibility is a bug only if the
caller expects the called routine to validate the called number for the
caller.
 Figure 4.19a shows the opposite situation, in which the called routine's
domain has a smaller span than the caller expects. All of these examples
are buggy.
 Figure 4.19: Buggy Range / Domain Mismatches

 In Figure 4.19b the ranges and domains don't line up; hence good values
are rejected, bad values are accepted, and if the called routine isn't robust
enough, we have crashes.
 Figure 4.19c combines these notions to show various ways we can have
holes in the domain: these are all probably buggy.

Interface Range / Domain Compatibility Testing:

 For interface testing, bugs are more likely to concern single variables
rather than peculiar combinations of two or more variables.
 Test every input variable independently of other input variables to confirm
compatibility of the caller's range and the called routine's domain span and
closure of every domain defined for that variable.
 There are two boundaries to test and it's a one-dimensional domain;
therefore, it requires one on and one off point per boundary or a total of
two on points and two off points for the domain - pick the off points
appropriate to the closure (COOOOI).
 Start with the called routine's domains and generate test points in
accordance to the domain-testing strategy used for that routine in
component testing.

3.10 DOMAINS AND TESTABILITY

Linearizing Transformations

 We often convert non linear boundaries to equivalent linear boundaries this is

done by applying linear transformations
 The methods are
1. Polynomials: A boundary is specified by a polynomial or
multinomial in several variables for example each term x,x 2,x3 is
replaced by y1=x,y2=x2
2. Logarithmic transformations: products such as xyz can be
linearized by substituting u=log(x),v=log(y),z=log(y)
3. More general transforms are by using taylor series expansion we
can linearize

Coordinate transformations

 Nice boundaries are parallel sets .Parallel boundaries sets are sets of linearly
related boundaries they differ only by constant
 It is an O(n2) procedure for n boundary equations
  Generally we have n equalities and m variables and convert the inequalities into
equalities
 Select the equation and apply a procedure called Gram-schmidt orthogonalization
which transforms original set of variables x into a new

A canonical program form

Figure 4.1: Schematic Representation of Domain Testing.

The input variable is going to be linearized and orthogonalizes so that inequalities can be
removed

The routine structure will be

1. Input the data

2. Apply linearizing transforms to as many predicates as possible
3. Transform to an orthogonal coordinate syste
4. For each set of parallel hyperplanes in the orthogonal space determine the case
5. Test the remaining inequalities to determine the required database
6. Now direct the program to the correct case process routine by a control flow
predicate

Testing is clearly divided into the following:

 Testing the predicate

 Testing the coordinate systems
 Testing the individual case selections
 Testing the control flow
 Testing the case processing
Great Insights

 A good coordinate systems can break the back of many tough problems. Make the
domain design as explicit
 Look the transformations to new, orthogonal, coordinate sets therefore it is easy
to test
 Testers will test the domains based on the specifications to a minimal set