AutoTest for software engineering for automated testing

272: Software Engineering
Fall 2008
Instructor: Tevfik Bultan
Lecture 10: Testing, Automated Testing

Finding Errors in Software
• We discussed various approaches to finding errors in programs
– Static analysis techniques and tools such as
• automated theorem proving, ESC/Java
• model checking, Java pathfinder
– Dynamic monitoring of assertions and contracts
• JContractor
• JML runtime assertion checker
• Although these are interesting and promising research areas the
most common way of looking for software errors in industry is testing
– Testing: Checking correctness of software by executing the
software on some inputs (test cases)

Software Testing
• Goal of testing
– finding faults in the software
– demonstrating that there are no faults in the software (for the test
cases that has been used during testing)
• It is not possible to prove that there are no faults in the software using
testing
• Testing should help locate errors, not just detect their presence
– a “yes/no” answer to the question “is the program correct?” is not
very helpful
• Testing should be repeatable
– could be difficult for distributed or concurrent software
– effect of the environment, uninitialized variables

Testing Software is Hard
• If you are testing a bridge’s ability to sustain weight, and you test it
with 1000 tons you can infer that it will sustain weight  1000 tons
• This kind of reasoning does not work for software systems
– software systems are not linear nor continuous
• Exhaustively testing all possible input/output combinations is too
expensive
– the number of test cases increase exponentially with the number
of input/output variables

Some Definitions
• Let P be a program and let D denote its input domain
• A test case d is an element of input domain d  D
– a test case gives a valuation for all the input variables of the
program
• A test set T is a finite set of test cases, i.e., a subset of D, T  D
• The basic difficulty in testing is finding a test set that will uncover the
faults in the program
• Exhaustive testing corresponds to setting T = D

Exhaustive Testing is Hard
• Number of possible test cases
(assuming 32 bit integers)
– 232
 232
= 264
• Do bigger test sets help?
– Test set
{(x=3,y=2), (x=2,y=3)}
will detect the error
– Test set
{(x=3,y=2),(x=4,y=3),(x=5,y=1)}
will not detect the error although
it has more test cases
• It is not the number of test cases
• But, if T1  T2, then T1 will detect
every fault detected by T2
int max(int x, int y)
{
if (x > y)
return x;
else
return x;
}

Exhaustive Testing
• Assume that the input for the max procedure was an integer array of
size n
– Number of test cases: 232 n
• Assume that the size of the input array is not bounded
– Number of test cases: 
• The point is, naive exhaustive testing is pretty hopeless

Random Testing
• Use a random number generator to generate test cases
• Derive estimates for the reliability of the software using some
probabilistic analysis
• Coverage is a problem

Generating Test Cases Randomly
• If we pick test cases randomly it is
unlikely that we will pick a case where
x and y have the same value
• If x and y can take 232
different values,
there are 264
possible test cases. In 232
of them x and y are equal
– probability of picking a case where
x is equal to y is 2-32
• It is not a good idea to pick the test
cases randomly (with uniform
distribution) in this case
• So, naive random testing is pretty
hopeless too
bool isEqual(int x, int y)
{
if (x = y)
z := false;
else
z := false;
return z;
}

Types of Testing
• Functional (Black box) vs. Structural (White box) testing
– Functional testing: Generating test cases based on the
functionality of the software
– Structural testing: Generating test cases based on the structure of
the program
– Black box testing and white box testing are synonyms for
functional and structural testing, respectively.
• In black box testing the internal structure of the program is
hidden from the testing process
• In white box testing internal structure of the program is taken
into account
• Module vs. Integration testing
– Module testing: Testing the modules of a program in isolation
– Integration testing: Testing an integrated set of modules

Functional Testing, Black-Box Testing
• Functional testing:
– identify the the functions which software is expected to perform
– create test data which will check whether these functions are
performed by the software
– no consideration is given how the program performs these
functions, program is treated as a black-box: black-box testing
– need an oracle: oracle states precisely what the outcome of a
program execution will be for a particular test case. This may not
always be possible, oracle may give a range of plausible values
• A systematic approach to functional testing: requirements based
testing
– driving test cases automatically from a formal specification of the
functional requirements

Domain Testing
• Partition the input domain to equivalence classes
• For some requirements specifications it is possible to define
equivalence classes in the input domain
• Here is an example: A factorial function specification:
– If the input value n is less than 0 then an appropriate error
message must be printed. If 0  n < 20, then the exact value n!
must be printed. If 20  n  200, then an approximate value of n!
must be printed in floating point format using some approximate
numerical method. The admissible error is 0.1% of the exact
value. Finally, if n > 200, the input can be rejected by printing an
appropriate error message.
• Possible equivalence classes: D1 = {n<0}, D2 = {0  n < 20}, D3 = {20
 n  200}, D4 = {n > 200}
• Choose one test case per equivalence class to test

Equivalence Classes
• If the equivalence classes are disjoint, then they define a partition of
the input domain
• If the equivalence classes are not disjoint, then we can try to
minimize the number of test cases while choosing representatives
from different equivalence classes
• Example: D1 = {x is even}, D2 = {x is odd}, D3 = {x  0}, D4={x > 0}
– Test set {x=48, x= –23} covers all the equivalence classes
• On one extreme we can make each equivalence class have only one
element which turns into exhaustive testing
• The other extreme is choosing the whole input domain D as an
equivalence class which would mean that we will use only one test
case

Testing Boundary Conditions
• For each range [R1, R2] listed in either the input or output
specifications, choose five cases:
– Values less than R1
– Values equal to R1
– Values greater than R1 but less than R2
– Values equal to R2
– Values greater than R2
• For unordered sets select two values
– 1) in, 2) not in
• For equality select 2 values
– 1) equal, 2) not equal
• For sets, lists select two cases
– 1) empty, 2) not empty
R1 R2

Testing Boundary Conditions
• For the factorial example, ranges for variable n are:
– [, 0], [0,20], [20,200], [200, ]
– A possible test set:
• {n = -5, n=0, n=11, n=20, n= 25, n=200, n= 3000}
– If we know the maximum and minimum values that n can take we
can also add those n=MIN, n=MAX to the test set.

Structural Testing, White-Box Testing
• Structural Testing
– the test data is derived from the structure of the software
– white-box testing: the internal structure of the software is taken
into account to derive the test cases
• One of the basic questions in testing:
– when should we stop adding new test cases to our test set?
– Coverage metrics are used to address this question

Coverage Metrics
• Coverage metrics
– Statement coverage: all statements in the programs should be
executed at least once
– Branch coverage: all branches in the program should be
executed at least once
– Path coverage: all execution paths in the program should be
executed at lest once
• The best case would be to execute all paths through the code, but
there are some problems with this:
– the number of paths increases fast with the number of branches in
the program
– the number of executions of a loop may depend on the input
variables and hence may not be possible to determine
– most of the paths can be infeasible

Statement Coverage
• Choose a test set T such
that by executing program
P for each test case in T,
each basic statement of P
is executed at least once
• Executing a statement once
and observing that it
behaves correctly is not a
guarantee for correctness,
but it is an heuristic
– this goes for all testing
efforts since in general
checking correctness is
undecidable
bool isEqual(int x, int y)
{
if (x = y)
z := false;
else
z := false;
return z;
}
int max(int x, int y)
{
if (x > y)
return x;
else
return x;
}

Statement Coverage
areTheyPositive(int x, int y)
{
if (x >= 0)
print(“x is positive”);
else
print(“x is negative”);
if (y >= 0)
print(“y is positive”);
else
print(“y is negative”);
}
Following test set will give us statement
coverage:
T1 = {(x=12,y=5), (x= 1,y=35),
(x=115,y=13),(x=91,y= 2)}
There are smaller test cases which will
give us statement coverage too:
T2 = {(x=12,y=  5), (x= 1,y=35)}
There is a difference between these two
test sets though

Statement vs. Branch Coverage
assignAbsolute(int x)
{
if (x < 0)
x := -x;
z := x;
}
Consider this program segment, the test set
T = {x=1} will give statement coverage,
however not branch coverage
(x < 0)
x := -x
z := x
B0
B1
B2
Test set {x=1} does not
execute this edge, hence, it
does not give branch coverage
true false
Control Flow Graph:

Control Flow Graphs (CFGs)
• Nodes in the control flow graph are basic blocks
– A basic block is a sequence of statements always entered at the
beginning of the block and exited at the end
• Edges in the control flow graph represent the control flow
if (x < y) {
x = 5 * y;
x = x + 3;
}
else
y = 5;
x = x+y;
(x < y)
x = 5 * y
x = x + 3
y = 5
x = x+y
B1 B2
B0
B3
• Each block has a sequence of statements
• No jump from or to the middle of the block
• Once a block starts executing, it will execute till the end
Y N

Branch Coverage
• Construct the control flow graph
• Select a test set T such that by executing program P for each test
case d in T, each edge of P’s control flow graph is traversed at least
once
(x < 0)
x := -x
z := x
B0
B1
B2
Test set {x=1} does not
execute this edge, hence, it
does not give branch coverage
Test set {x= 1, x=2}gives
both statement and branch
coverage
true false

Path Coverage
• Select a test set T such that by executing program P for each test
case d in T, all paths leading from the initial to the final node of P’s
control flow graph are traversed

Path Coverage
{
if (x >= 0)
else
if (y >= 0)
else
}
(x >= 0)
B0
B1
print(“x is p”)
B2
print(“x is n”)
(y >= 0)
B3
B4
print(“y is p”)
B5
print(“y is n”)
return
B6
Test set:
T2 = {(x=12,y=  5), (x= 1,y=35)}
gives both branch and statement
coverage but it does not give path coverage
Set of all execution paths: {(B0,B1,B3,B4,B6), (B0,B1,B3,B5,B6), (B0,B2,B3,B4,B6),
(B0,B2,B3,B5,B6)}
Test set T2 executes only paths: (B0,B1,B3,B5,B6) and (B0,B2,B3,B4,B6)
true false
true false

Path Coverage
{
if (x >= 0)
else
if (y >= 0)
else
}
(x >= 0)
B0
B1
print(“x is p”)
B2
print(“x is n”)
(y >= 0)
B3
B4
print(“y is p”)
B5
print(“y is n”)
return
B6
Test set:
T1 = {(x=12,y=5), (x= 1,y=35),
(x=115,y=13),(x=91,y= 2)}
gives both branch, statement and path
coverage
true false
true false

Path Coverage
• Number of paths is exponential in the number of conditional
branches
– testing cost may be expensive
• Note that every path in the control flow graphs may not be executable
– It is possible that there are paths which will never be executed
due to dependencies between branch conditions
• In the presence of cycles in the control flow graph (for example loops)
we need to clarify what we mean by path coverage
– Given a cycle in the control flow graph we can go over the cycle
arbitrary number of times, which will create an infinite set of paths
– Redefine path coverage as: each cycle must be executed 0, 1, ...,
k times where k is a constant (k could be 1 or 2)

Condition Coverage
• In the branch coverage we make sure that we execute every branch at
least once
– For conditional branches, this means that, we execute the TRUE
branch at least once and the FALSE branch at least once
• Conditions for conditional branches can be compound boolean
expressions
– A compound boolean expression consists of a combination of
boolean terms combined with logical connectives AND, OR, and
NOT
• Condition coverage:
– Select a test set T such that by executing program P for each test
case d in T, (1) each edge of P’s control flow graph is traversed at
least once and (2) each boolean term that appears in a branch
condition takes the value TRUE at least once and the value FALSE
at least once
• Condition coverage is a refinement of branch coverage (part (1) is
same as the branch coverage)

Condition Coverage
something(int x)
{
if (x < 0 || y < x)
{
y := -y;
x := -x;
}
z := x;
}
T = {(x=1, y=1), (x=1, y=1)} will achieve
statement, branch and path coverage,
however T will not achieve condition
coverage because the boolean term (y <
x) never evaluates to true. This test set
satisfies part (1) but does not satisfy part (2).
(x < 0 || y < x)
y := -y;
x := -x;
z := x
B0
B1
B2
true false
Control Flow Graph
T = {(x=1, y=1), (x=1, y=0)}
will not achieve condition coverage
either. This test set satisfies part (2)
but does not satisfy part (1). It does
not achieve branch coverage since
both test cases take the true branch,
and, hence, it does not achieve
condition coverage by definition.
T = {(x=1, y=2), {(x=1, y=1)}
achieves condition coverage.

Multiple Condition Coverage
• Multiple Condition Coverage requires that all possible combination of truth
assignments for the boolean terms in each branch condition should
happen at least once
• For example for the previous example we had:
x < 0 && y < x
• Test set {(x=1, y=2), (x=1, y=1)}, achieves condition coverage:
– test case (x=1, y=2) makes term1=true, term2=true, and the whole
expression evaluates to true (i.e., we take the true branch)
– test case (x=1, y=1) makes term1=false, term2=false, and the whole
expression evaluates to false (i.e., we take the false branch)
• However, test set {(x=1, y= 2), (x=1, y=1)} does not achieve multiple
condition coverage since we did not observe the following truth
assignments
– term1=true, term2=false
– term1=false, term2=true
term1 term2

Types of Testing
• Unit (Module) testing
– testing of a single module in an isolated environment
• Integration testing
– testing parts of the system by combining the modules
• System testing
– testing of the system as a whole after the integration phase
• Acceptance testing
– testing the system as a whole to find out if it satisfies the
requirements specifications

Unit Testing
• Involves testing a single isolated module
• Note that unit testing allows us to isolate the errors to a single module
– we know that if we find an error during unit testing it is in the
module we are testing
• Modules in a program are not isolated, they interact with each other.
Possible interactions:
– calling procedures in other modules
– receiving procedure calls from other modules
– sharing variables
• For unit testing we need to isolate the module we want to test, we do
this using two things
– drivers and stubs

Drivers and Stubs
• Driver: A program that calls the interface procedures of the module
being tested and reports the results
– A driver simulates a module that calls the module currently being
tested
• Stub: A program that has the same interface as a module that is
being used by the module being tested, but is simpler.
– A stub simulates a module called by the module currently being
tested

Drivers and Stubs
Driver
Module
Under Test
Stub
procedure
call
procedure
call
access to global
variables
• Driver and Stub should have the same interface as the modules they replace
• Driver and Stub should be simpler than the modules they replace

Integration Testing
• Integration testing: Integrated collection of modules tested as a group
or partial system
• Integration plan specifies the order in which to combine modules into
partial systems
• Different approaches to integration testing
– Bottom-up
– Top-down
– Big-bang
– Sandwich

Module Structure
A
C
D
E F G
H
• A uses C and D; B uses D; C uses E and F; D uses F, G, H and I; H uses I
• Modules A and B are at level 3; Module D is at level 2
Modules C and H are at level 1; Modules E, F, G, I are at level 0
• level 0 components do not use any other components
• level i components use at least one component on level i-1 and no
component at a level higher than i-1
I
B
• We assume that
the uses hierarchy is
a directed acyclic graph.
• If there are cycles merge
them to a single module
level 0
level 1

Bottom-Up Integration
• Only terminal modules (i.e., the modules that do not call other
modules) are tested in isolation
• Modules at lower levels are tested using the previously tested higher
level modules
• Non-terminal modules are not tested in isolation
• Requires a module driver for each module to feed the test case input
to the interface of the module being tested
– However, stubs are not needed since we are starting with the
terminal modules and use already tested modules when testing
modules in the lower levels

Bottom-up Integration
A
C
D
E F G
H
I
B

Top-down Integration
• Only modules tested in isolation are the modules which are at the
highest level
• After a module is tested, the modules directly called by that module
are merged with the already tested module and the combination is
tested
• Requires stub modules to simulate the functions of the missing
modules that may be called
– However, drivers are not needed since we are starting with the
modules which is not used by any other module and use already
tested modules when testing modules in the higher levels

Top-down Integration
A
C
D
E F G
H
I
B

Other Approaches to Integration
• Sandwich Integration
– Compromise between bottom-up and top-down testing
– Simultaneously begin bottom-up and top-down testing and meet
at a predetermined point in the middle
• Big Bang Integration
– Every module is unit tested in isolation
– After all of the modules are tested they are all integrated together
at once and tested
– No driver or stub is needed
– However, in this approach, it may be hard to isolate the bugs!

System Testing, Acceptance Testing
• System and Acceptance testing follows the integration phase
– testing the system as a whole
• Test cases can be constructed based on the the requirements
specifications
– main purpose is to assure that the system meets its requirements
• Manual testing
– Somebody uses the software on a bunch of scenarios and
records the results
– Use cases and use case scenarios in the requirements
specification would be very helpful here
– manual testing is sometimes unavoidable: usability testing

System Testing, Acceptance Testing
• Alpha testing is performed within the development organization
• Beta testing is performed by a select group of friendly customers
• Stress testing
– push system to extreme situations and see if it fails
– large number of data, high input rate, low input rate, etc.

Regression testing
• You should preserve all the test cases for a program
• During the maintenance phase, when a change is made to the
program, the test cases that have been saved are used to do
regression testing
– figuring out if a change made to the program introduced any faults
• Regression testing is crucial during maintenance
– It is a good idea to automate regression testing so that all test
cases are run after each modification to the software
• When you find a bug in your program you should write a test case
that exhibits the bug
– Then using regression testing you can make sure that the old
bugs do not reappear

Test Plan
• Testing is a complicated task
– it is a good idea to have a test plan
• A test plan should specify
– Unit tests
– Integration plan
– System tests
– Regression tests

Mutation Analysis
• Mutation analysis is used to figure out the quality of a test set
• Mutation analysis creates mutants of a program by making changes
to the program (change a condition, change an assignment, etc.)
• Each mutant program and the original program are executed using
the test set
• If a mutant and the original program give different results for a test
case then the test set detected that the mutant is different from the
original program, hence the mutant is said to be dead
• If test set does not detect the difference between the original program
and some mutants, these mutants are said to be live
• We want the test set to kill as many mutants as possible
– Mutant programs can be equivalent to the original program, hence
no test set can kill them

Automated Testing
• Automated testing refers to the techniques which generate the test sets
automatically
• I will talk about two tools on automated testing
– TestEra, Korat
• TestEra is a specification-based functional (black-box) testing tool
– Requires the user to write input/output specifications
• Korat is also a kind of functional (black-box) testing tool
– Requires the user to write a specification as a method in the class that is
being testing
• Both tools are used for unit testing
– Testing of complex data structures
• Both tools automatically generate test cases from specifications
– They exhaustively generate all non-isomorphic test cases within a given
scope

TestEra and Korat
• The references are:
– ``TestEra: Specification-based Testing of Java Programs Using
SAT.'' S. Khurshid and D. Marinov. Automated Software
Engineering Journal, Volume 11, Number 4. October 2004.
– ``Korat: Automated Testing Based on Java Predicates.'' C.
Boyapati, S. Khurshid and D. Marinov. ACM/SIGSOFT
International Symposium on Software Testing and Analysis
(ISSTA 2002), Rome, Italy. Jul 2002.

TestEra Framework
• TestEra is a framework for automated testing of Java programs
• TestEra’s main focus is unit testing of complex data structures
– Examples: Red-black trees, linked lists etc.
• TestEra has also been used in analyzing larger Java programs
– Examples: Alloy Analyzer, Intentional Naming System

TestEra Framework
• TestEra automatically generates all non-isomorphic test cases within
a given input size
– the input size corresponds to the scope in Alloy specifications
• TestEra evaluates the correctness criteria for the automatically
generated test cases
• TestEra uses Alloy and Alloy Analyzer to generate the test cases and
to evaluate the correctness criteria
• TestEra produces concrete Java inputs as counterexamples to
violated correctness criteria

TestEra Framework
• TestEra framework requires the following:
– A specification of inputs to a Java program written in Alloy
• precondition
– A correctness criterion written in Alloy
• Class invariant and post-condition
– An concretization function
• which maps instances of Alloy specifications to concrete Java
objects
– An abstraction function
• which maps the concrete Java objects to instances of Alloy
specifications

TestEra Framework
• TestEra generates all the non-isomorphic input instances using Alloy
Analyzer
– Two instances are isomorphic if there is a one to one mapping
between the atoms of the two instances which preserve all the
relations
• TestEra uses the concretization function to translate the instances of
the Alloy specification to Java inputs
– These inputs form the test set
• TestEra runs the program on the test set
• TestEra maps the output produced by the program to Alloy using the
abstraction function
• Finally, TestEra uses the Alloy Analyzer to check the input and the
output against the correctness criteria

TestEra Framework
TestEra
spec
Alloy
input spec
Java
tester
Alloy I/O
spec
Alloy
Analyzer
Concretization Run Code Abstraction
Alloy
instances
Alloy
input
Java input
Java
output
Alloy
output
counter
example
Check
correctness
fail
pass

Steps of the TestEra Framework
• Identify a sequence of method calls to analyze
• Create an Alloy specification for the inputs of these methods
• Create an Alloy specification of the correctness criteria by relating the
inputs to the outputs of these methods
• Define a concretization translation a2j from an Alloy instance of the
input specification to a Java input to these methods
• Define an abstraction translation j2a from a Java output to an Alloy
instance of the specification of the correctness cretiria that relates
inputs to outputs

TestEra Spec
• A TestEra specification is a combination of Alloy and Java code
– This specification is split to three files
• Alloy input specification
• Java code for
– translating input instances from Alloy to Java
– running the sequence of Java methods to test
– translating the Java output back to Alloy
• Alloy specification for the correctness criteria which relates the
input values to the output values

TestEra Analysis
• TestEra uses Alloy Analyzer to generate all non-isomorphic instances
of the Alloy input specification
• Each instance is translated to Java input using concretization
– this forms the test case for the sequence of Java methods to be
tested
• The sequence of methods are run on this input
• The produced output is translated using abstraction back to Alloy
• The input and output instances are checked against the correctness
criterion
• If the check fails a counter-example is reported
– otherwise next Alloy instance is used for testing

An Example
• A recursive method for performing merge sort on acyclic singly linked
lists
Java:
class List {
int elem;
List next;
static List mergeSort(List l) { ... }
}
Alloy:
module list
import integer
sig List {
elem: Integer,
next: lone List }
Signature declaration introduces the List type
with functions:
elem: List  Integer
next: List  List
next is a partial function which is indicated
by the keyword lone

Input Specification
module list
import integer
sig List {
elem: Integer,
next: lone List }
fun Acyclic(l: List) {
all n: l.*next | lone n.~next // at most one parent
no l.~next } // head has no parent
one sig Input in List {}
fact GenerateInputs {
Acyclic(Input) }
Subsignature Input is a
subset of List and it has
exactly one atom which is
indicated by the keyword
one

Correctness Criteria
fun Sorted(l: List) {
all n: l.*next | some n.next => n.elem <= n.next.elem }
funPerm(l1: List, l2:List)
all e: Integer | #(e.~elem & l1.*next) =
#(e.~elem & l2.*next) }
fun MergeSortOK(i:List, o:List) {
Acyclic(o)
Sorted(o)
Perm(i,o) }
one sig Output in List {}
fact OutputOK {
MergeSortOk(Input, Output) }
# denotes cardinality
of sets

Counter-Examples
• If an error is inserted in the method for merging where
(l1.elem <= l2.elem) is changed to (l1.elem >= l2.elem)
• Then TestEra generates a counter example
Counterexample found:
Input List: 1 -> 1 -> 3 -> 2
Output List: 3 -> 2 -> 1 -> 1

Abstraction and Concretization Translations
• An abstraction function: j2a
– translate Java instance to Alloy instance
• A concretization function: a2j
– translate Alloy instance to Java instance
• These functions are written in Java by the user

Concretization
• Concretization is implemented in two stages
1. Create a Java object for each atom in Alloy specification and
store this correspondence in a map
2. Establish the relationships among the Java objects created in the
first step

TestEra Case studies
• Red-Black trees
– Tested the implementation of Red-Black trees in java.util.TreeMap
– They introduced some bugs and showed that they can catch them
with TestEra framework
• Intentional Naming System
– A naming architecture for resource discovery and service location
in dynamic networks
– Found some bugs
• Alloy Analyzer
– Found some bugs in the Alloy Analyzer using TestEra framework

Korat
• Another automated testing tool
– Similar to TestEra but does not require extra Alloy specifications
– Application domain is again unit testing of complex data structures
• It uses Java predicates to generate the test cases
– These are Java methods which return a boolean value
– For example pre and post-conditions of methods
• Korat generates the test cases from pre and postconditions of
methods
• There is no need to write an extra specification if the class contract is
written as Java predicates (like the JContractor approach)

Korat
• Korat uses the method precondition to automatically generate all
nonisomorphic test cases up to a give small size
– Given a predicate and a bound on the size of its inputs Korat
generates all nonisomorphic inputs for which the predicate returns
true
• Korat then executes the method on each test case and uses the
method postcondition as a test oracle to check the correctness of
output
– Korat exhaustively explores the bounded input space of the
predicate but does so efficiently by monitoring the predicate’s
executions and pruning large portions of the search space

An Example: Binary Tree
import java.util.*;
class BinaryTree {
private Node root;
private int size;
static class Node {
private Node left;
private Node right;
}
public boolean repOk() {
// this method checks the class invariant:
// checks that empty tree has size zero
// checks that the tree has no cycle
// checks that the number of nodes in the tree is
// equal to its size
}

Finitization
• Korat uses a finitization description to specify the finite bounds on the
inputs (scope)
public static Finitization finBinaryTree(int NUM_node){
Finitization f = new Finitization(BinaryTree.class);
ObjSet nodes = f.createObjects(“Node”, NUM_node);
nodes.add(null);
f.set(“root”, nodes);
f.set(“size”, NUM_node);
f.set(“Node.left”, nodes);
f.set(“Node.right”, nodes);
return f;
}
• Korat automatically generates a finitization skeleton based on the type
declarations in the Java code
– Developers can restrict or extend this default finitization
Creates a set of objects of
Type “Node” with
NUM_node objects in
the set
The value of size
is set to NUM_node

Non-isomorphic Instances for finBinaryTree(3)
N0
N1
N2
N0
N1
N2
N0
N1
N2
N0
N1
N2
N0
N1 N2
right
right
right
left
left
right
left
left
right
left
Korat automatically generates non-isomorphic instances within a given bound
Each of the above trees correspond to 6 isomorphic trees. Korat only generates
one tree representing the 6 isomorphic trees.
For finBinaryTree(7) Korat generates 429 non-isomorphic trees in less than a second

Isomorphic Instances
N0
N1 N2
right
left
N0
N2 N1
right
left
N1
N2 N0
right
left
N1
N0 N2
right
left
N2
N1 N0
right
left
N2
N0 N1
right
left

Generating test cases
• The crucial component of Korat is the test case generation algorithms
• Consider the binary tree example with scope 3
– There are three fields: root, left, right
– Each of these fields can be assigned a node instance or null
– There is one root field and there is one left and one right field for
each node instance
– Given n node instances, the state space (the set of all possible
test cases) for the binary tree example is:
(n+1)2n + 1
– Most of these structures are not valid binary trees
• They do not satisfy the class invariant
– Most of these structures are isomorphic (they are equivalent if we
ignore the object identities)

Generating test cases
• There are two techniques Korat uses to generate the test cases
efficiently
1. Korat only generates non-isomorphic test cases
2. Korat prunes the state space by eliminating sets of test cases
which do not satisfy the class invariant

Isomorphism
• The isomorphism definition used in Korat is the following
– O1, O2, ..., On are sets objects from n classes
– O = O1  O2  ...  On
– P: the set of consisting of null and all values of primitive types that
the fields of objects in O can contain
– r O is a special root object
– Given a test case C, OC is the set of objects reachable from r in C
• Two test cases C and C’ are isomorphic iff there is a permutation 
on O, mapping objects from Oi to objects from Oi for all 1  i  n,
such that
o, o’  OC . f  fields(o) .p  P .
o.f == o’ in C iff (o).f == (o’) in C’ and
o.f == p in C iff (o).f == p in C’

Isomorphism
• In Korat isomorphism is defined with respect to a root object
– for example this
• Two test cases are defined to be isomorphic if the parts of their object
graphs reachable from the root object are isomorphic
• The isomorphism definition partitions the state space (i.e. the input
domain) to a set of isomorphism partitions
– Korat generates only one test case for each partition class

Generating Test Cases
• Korat only generates the test cases which satisfies the input
predicate: class invariant and the precondition
• Korat explores the state space efficiently using backtracking
– It does not generate all instances one by one and check the input
predicate
– It prunes its search of the state space based on the evaluation of
the input predicate
• If the method that checks the input predicate returns false
without checking a field then there is no need to generate test
cases which assign different values to that field
– In order to exploit this, Korat keeps track of the fields that
are accessed before the predicate returns false
– For this to work well, predicate method should return false as
soon as it detects a violation

• Korat orders all the elements in every class domain and every field
domain
• Each test case is represented as a vector of indices into the
corresponding field domains
N0
N1 N2
right
left
size=3
[1,0,2,3,0,0,0,0,]
Test Case Corresponding Vector
For the Binary Tree example assume that
The class domain is ordered as N0 < N1 < N2
The field domains for root, left and right are ordered as null < N0 < N1 < N2
The size domain has one element which is 3
[1,0,2,3,0,0,0,0,]
fields of the
Binary tree object
fields of the Node object N0
fields of the
Node object N1

• Search starts with a candidate vector set to all zeros.
• For each candidate vectors, Korat sets fields in the objects according
to the values in the vector.
• Korat then invokes repOk (i.e., class invariant) to check the validity of
the current candidate.
– During the execution of repOk, Korat monitors the fields that
reopOK accesses, it stores the indices of the fields that are
accessed by the repOK (field ordering)
– For example, for the binary tree example, if the repOK accesses
root, N0.left and N0.right, then the field ordering is 0, 2, 3
• Korat generates the next candidate vector by backtracking on the
fields accessed by repOk.
– First increments the field domain index for the field that is last in
the field-ordering
– If the field index exceeds the domain size, then Korat resets that
index to zero and increments the domain index of the previous
field in the field ordering

• Korat achieves non-isomorphic test case generation using the
ordering of field domains and the vector representation
• While generating the test cases, Korat ensures that the indices of the
objects that belong to the same class domain are listed in
nondecreasing order in the generated candidate vectors
• This means that during backtracking, Korat looks for fields
– that precede the field that is accessed last and
– that have an object from the same class domain as the field that is
accessed last
– and makes sure that the object assigned to the field that is
accessed last is higher in the ordering then those objects

Using Contracts in Testing
• Korat checks the contracts written in JML on the generated instances
//@ public invariant repOk(); //class invariant
/*@ requires has(n) // precondition
@ ensures !has(n) // postcondition
@*/
public void remove(Node n) {
...
}
• Korat uses JML tool-set to translate JML annotations into runtime
Java assertions

Using Contracts in Testing
• Given a finitization, Korat generates all non-isomorphic test cases
within the given scope (defined by the finitization) that satisfy the
class invariant and the pre-condition
• The post-conditions and class invariants provide a test oracle for
testing
– For each generated test case, execute the method that is being
tested and check the class invariant and the post-condition
• Korat uses JML tool-set to automatically generate test oracles from
method post-conditions written as annotations in JML

Korat Performance
• Checking a BinaryTree implementation with scope 8 takes 1/53
seconds, with scope 11 takes 56.21 seconds, with scope 12 takes
233.59 seconds.
• Test case generation with Korat is more efficient than the test case
generation with Alloy Analyzer.

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