States, state graphs and transition testing

Software Testing
Techniques
Compiled with reference from:
Software Testing Techniques: Boris Beizer

Contents
 SYNOPSIS
 MOTIVATIONAL OVERVIEW
 STATE GRAPHS
 GOOD STATE GRAPHS AND BAD
 STATE TESTING
 TESTABILITY TIPS

SYNOPSIS
 The state graph and its associated state table are useful models for describing software behavior.
 The finite-state machine is a functional testing tool and testable design programming tool

MOTIVATIONAL OVERVIEW
 The finite-state machine is as fundamental to software engineering as boolean algebra.
 State testing strategies are based on the use of finite-state machine models for software structure, software
behavior, or specifications of software behavior.
 Finite-state machines can also be implemented as table-driven software, in which case they are a powerful
design option. Independent testers are likeliest to use a finite-state machine model as a guide to the design
of functional tests—especially system tests.

STATE GRAPHS
 States
 Inputs and Transitions
 Outputs
 State Tables
 Time Versus Sequence
 Software Implementation
1. Implementation and Operation
2.Input Encoding and Input Alphabet
3.Output Encoding and Output Alphabet
4.State Codes and State-Symbol Products
5.Application Comments for Designers
6.Application Comments for Testers

STATE GRAPHS
States
 The word “state” is used in much the same way it’s used in ordinary English, as in “state of the union,” or “state of
health.”
 The Oxford English Dictionary defines “state” as: “A combination of circumstances or attributes belonging for the
time being to a person or thing.”
 A program that detects the character sequence “ZCZC” can be in the following states:
 1. Neither ZCZC nor any part of it has been detected.
 2. Z has been detected.
 3. ZC has been detected.
 4. ZCZ has been detected.
 5. ZCZC has been detected
Figure11.1

STATE GRAPHS
States(continued)
 A moving automobile whose engine is running can have the following states with respect to its
transmission:
1. Reverse gear
2. Neutral gear
3. First gear
4. Second gear
5. Third gear
6. Fourth gear
 Like,
 A person’s checkbook can have the what states with respect to the bank balance?
 A word processing program menu can be in the what states with respect to file manipulation?

STATE GRAPHS
Inputs and Transitions
 Result of those inputs, the state changes, or is said to have made a transition. Transitions are denoted by
links that join the states
 A finite-state machine is an abstract device that can be represented by a state graph having a finite
number of states and a finite number of transitions between states.
 The ZCZC detection example can have the following kinds of inputs:
 1. Z
 2. C
 3. Any character other than Z or C, which we’ll denote by A

STATE GRAPHS
Inputs and Transitions
 The state graph of Figure 11.1 is interpreted as follows:
 1. If the system is in the “NONE” state, any input other than a Z will keep it in that state.
 2. If a Z is received, the system transitions to the “Z” state.
 3. If the system is in the “Z” state and a Z is received, it will remain in the “Z” state. If a C is received, it will go to the
“ZC” state; if any other character is received, it will go back to the “NONE” state because the sequence has been broken.
 4. A Z received in the “ZC” state progresses to the “ZCZ” state, but any other character breaks the sequence and causes a
return to the “NONE” state.
 5. A C received in the “ZCZ” state completes the sequence and the system enters the “ZCZC” state. A Z breaks the
sequence and causes a transition back to the “Z” state; any other character causes a return to the “NONE” state.
 6. The system stays in the “ZCZC” state no matter what is received.

STATE GRAPHS
Outputs
 An output* can be associated with any link. Outputs are denoted by letters or words and are separated
from inputs by a slash as follows: “input/output.”
 The state graph is shown in Figure 11.2.
 As in the previous example, the inputs and actions have been simplified. There are only two kinds of
inputs (OK, ERROR) and four kinds of outputs (REWRITE, ERASE, NONE, OUT-OF-SERVICE).

STATE GRAPHS
State Tables
INPUT
STATE OKAY ERROR
1 1/NONE 2/REWRITE
2 1/NONE 4/REWRITE
3 1/NONE 2/REWRITE
4 3/NONE 5/ERASE
5 1/NONE 6/ERASE
6 1/NONE 7/OUT
7 . . . . . .
1. Each row of the table corresponds to a state.
2. Each column corresponds to an input condition.
3. The box at the intersection of a row and column specifies the next state (the transition) and the output,
if any

STATE GRAPHS
Time Versus Sequence
Time Sequence
1. State graphs don’t represent time 1.State graphs represent sequence.
A transition might take microseconds or centuries; a system
could be in one state for milliseconds and another for eons, or the
other way around;
the state graph would be the same because it has no notion of
time
Although the finite-state machine model can be elaborated to include notions of time in
addition to sequence, such as timed Petri nets

Software Implementation
1.Implementation and Operation
 There are four tables involved:
 1. A table or process that encodes the input values into a compact list (INPUT_CODE_TABLE).
 2. A table that specifies the next state for every combination of state and input code
(TRANSITION_TABLE).
 3. A table or case statement that specifies the output or output code, if any, associated with every state-
input combination (OUTPUT_TABLE).
 4. A table that stores the present state of every device or process that uses the same state table—e.g., one
entry per tape transport (DEVICE_TABLE).

 The routine operates as follows, where # means concatenation:
 BEGIN
 PRESENT_STATE := DEVICE_TABLE(DEVICE_NAME)
 ACCEPT INPUT_VALUE
 INPUT_CODE := INPUT_CODE_TABLE(INPUT_VALUE)
 POINTER := INPUT_CODE#PRESENT STATE
 NEW_STATE := TRANSITION_TABLE(POINTER)
 OUTPUT_CODE := OUTPUT_TABLE(POINTER)
 CALL OUTPUT_HANDLER(OUTPUT_CODE)
 DEVICE_TABLE(DEVICE_NAME) := NEW_STATE
 END

 1. The present state is fetched from memory.
 2. The present input value is fetched. If it is already numerical, it can be used directly; otherwise, it may
have to be encoded into a numerical value, say by use of a case statement, a table, or some other process.
 3. The present state and the input code are combined (e.g., concatenated) to yield a pointer (row and
column) of the transition table and its logical image (the output table).
 4. The output table, either directly or via a case statement, contains a pointer to the routine to be executed
(the output) for that state-input combination. The routine is invoked (possibly a trivial routine if no output
is required).
 5. The same pointer is used to fetch the new state value, which is then stored.

2.Input Encoding and Input Alphabet
 The alternative to input encoding is a huge state graph and table because there must be one outlink in
every state for every possible different input.
 Input encoding compresses the cases and therefore the state graph.
 Another advantage of input encoding is that we can run the machine from a mixture of otherwise
incompatible input events, such as characters, device response codes, thermostat settings, or gearshift
lever positions.
 The set of different encoded input values is called the input alphabet.

3.Output Encoding and Output Alphabet
 There are only a finite number of such distinct actions, which we can encode into a convenient output
alphabet.

4.State Codes and State-Symbol Products
 The term state-symbol product is used to mean the value obtained by any scheme used to convert the
combined state and input code into a pointer to a compact table without holes.
 “state codes” in the context of finite-state machines, we mean the (possibly) hypothetical integer used to
denote the state and not the actual form of the state code that could result from an encoding process.

5. Application Comments for Designers
 State-table implementation is advantageous when either the control function is likely to change in the
future or when the system has many similar, but slightly different, control functions.
 This technique can provide fast response time—one pass through the above program can be done in ten to
fifteen machine instruction execution times.
 It is not an effective technique for very small (four states or less) or big (256 states or more) state graphs.

6. Application Comments for Testers
 Independent testers are not usually concerned with either implementation details or the economics of this
approach but with how a state-table or state-graph representation of the behavior of a program or system
can help us to design effective tests.
 There is an interesting correlation, though: when a finite-state machine model is appropriate, so is a finite-
state machine implementation.

GOOD STATE GRAPHS AND BAD
1. General
 1. The total number of states is equal to the product of the possibilities of factors that make up the state.
 2. For every state and input there is exactly one transition specified to exactly one, possibly the same,
state.
 3. For every transition there is one output action specified. That output could be trivial, but at least one
output does something sensible.*
 4. For every state there is a sequence of inputs that will drive the system back to the same state.**

 Figure 11.3 shows examples of improper state graphs.

2.State Bugs
2.1Number of States
 The number of states in a state graph is the number of states we choose to recognize or model.
 In practice, the state is directly or indirectly recorded as a combination of values of variables that appear
in the data base.
 Find the number of states as follows:
 1. Identify all the component factors of the state.
 2. Identify all the allowable values for each factor.
 3. The number of states is the product of the number of allowable values of all the factors.

2.State Bugs
Impossible States
GEAR R, N, 1, 2, 3, 4 = 6 factors
DIRECTION Forward, reverse, stopped = 3 factors
ENGINE Running, stopped = 2 factors
TRANSMISSION Okay, broken = 2 factors
ENGINE Okay, broken = 2 factors
TOTAL = 144 states

2.State Bugs
Equivalent States
 Two states are equivalent if every sequence of inputs starting from one state produces exactly the same
sequence of outputs when started from the other state.

2.State Bugs
Equivalent States
 Equivalent states can be recognized by the following procedures:
 1. The rows corresponding to the two states are identical with respect to input/output/next state but the name of the next state
could differ. The two states are differentiated only by the input that distinguishes between them. This situation is shown in
Figure 11.6. Except for the a,b inputs, which distinguish between states A and B, the system’s behavior in the two states is
identical for every input sequence; they can be merged.
 2. There are two sets of rows which, except for the state names, have identical state graphs with respect to transitions and
outputs. The two sets can be merged (see Figure 11.7).

2.State Bugs
Equivalent States
 Equalent states

2.State Bugs
Equivalent States

 Unmergable procedure
2.State Bugs
Equivalent States

3.Transition Bugs
1. Unspecified and Contradictory Transitions
 Exactly one transition must be specified for every combination of input and
state.
 A program can’t have contradictions or ambiguities. Ambiguities are impossible because
the program will do something (right or wrong) for every input.

3.Transition Bugs
3.2An Example
 Specifications are one of the most common source of ambiguities and contradictions.
 the first statement of the specification:
 Rule 1: The program will maintain an error counter, which will be incremented whenever there’s an error.

3.Transition Bugs
3.2An Example
INPUT
STATE OKAY ERROR
0 0/none 1/
1 2/
2 3/
3 4/
4 5/
5 6/
6 7/
7 8/
There are only two input values, “okay” and “error.”

Rule 2: If there is an error, rewrite the block.
Rule 3: If there have been three successive errors, erase 10 centimeters of tape and then rewrite the block.
Rule 4: If there have been three successive erasures and another error occurs, put the unit out of service.
Rule 5: If the erasure was successful, return to the normal state and clear the error counter.
Rule 6: If the rewrite was unsuccessful, increment the error counter, advance the state, and try another rewrite.
Rule 7: If the rewrite was successful, decrement the error counter and return to the previous state.
3.Transition Bugs
3.2An Example

 Adding rule 2, we get
INPUT
STATE OKAY ERROR
0 0/NONE 1/REWRITE
1 2/REWRITE
2 3/REWRITE
3 4/REWRITE
4 5/REWRITE
5 6/REWRITE
6 7/REWRITE
7 8/REWRITE
3.Transition Bugs
3.2An Example

Rule 3: If there have been three successive errors, erase 10 centimeters of tape and then rewrite the block.
3.Transition Bugs
3.2An Example
INPUT
STATE OKAY ERROR
0 0/NONE 1/REWRITE
1 2/REWRITE
2 3/REWRITE, ERASE, REWRITE

Rule 4: If there have been three successive erasures and another error occurs, put the unit out of service.
3.Transition Bugs
3.2An Example
Rule 3, if followed blindly, causes an unnecessary rewrite. It’s a minor bug, so let it go for now, but it pays to
check such things. There might be an arcane security reason for rewriting, erasing, and then rewriting again.
INPUT
STATE OKAY ERROR
0 0/NONE 1/RW
1 2/RW
2 3/ER, RW
3 4/ER, RW
4 5/ER, RW
5 6/OUT
6
7

Rule 5: If the erasure was successful, return to the normal state and clear the counter.
3.Transition Bugs
3.2An Example
Rule 4 terminates our interest in this state graph so we can dispose of states beyond 6. The details of state 6 will not be covered by
this specification; presumably there is a way to get back to state 0. Also, we can credit the specifier with enough intelligence not to
have expected a useless rewrite and erase prior to going out of service.
INPUT
STATE OKAY ERROR
0 0/NONE 1/RW
1 2/RW
2 3/ER, RW
3 0/NONE 4/ER, RW
4 0/NONE 5/ER, RW
5 0/NONE 6/OUT
6

Rule 6: If the rewrite was unsuccessful, increment the error counter, advance the state, and try another rewrite
3.Transition Bugs
3.2An Example
Rule 7: If the rewrite was successful, decrement the error counter and return to the previous
state.
INPUT
STATE OKAY ERROR
0 0/NONE 1/RW
1 0/NONE 2/RW
2 1/NONE 3/ER, RW
3 0/NONE
2/NONE
4/ER, RW
4 0/NONE
3/NONE
5/ER, RW
5 0/NONE
4/NONE
6/OUT
6

Rule 7 got rid of the ambiguities but created contradictions. The specifier’s intention was probably:
Rule 7A: If there have been no erasures and the rewrite is successful, return to the previous state.
3.Transition Bugs
3.2An Example

3.Transition Bugs
3.3 unreachable state
 An unreachable state is like unreachable code—a state that no input sequence can reach.
 An unreachable state is not impossible, just as unreachable code is not impossible.

3.Transition Bugs
3.4 Dead States
 A dead state, (or set of dead states) is a state that once entered cannot be left.

4. Output Errors
 The states, the transitions, and the inputs could be correct, there could be no dead or unreachable states,
but the output for the transition could be incorrect.

5. Encoding Bugs
 The behavior of a finite-state machine is invariant under all encodings. That is, say that the states are
numbered 1 to n.

STATE TESTING
1.Impact of Bugs
 1. Wrong number of states.
 2. Wrong transition for a given state-input combination.
 3. Wrong output for a given transition.
 4. Pairs of states or sets of states that are inadvertently made equivalent (factor lost).
 5. States or sets of states that are split to create inequivalent duplicates.
 6. States or sets of states that have become dead.
 7. States or sets of states that have become unreachable.

STATE TESTING
2. Principles
 The starting point of state testing is:
 1. Define a set of covering input sequences that get back to the initial state when starting from the initial state.
 2. For each step in each input sequence, define the expected next state, the expected transition, and the expected output
code.
 A set of tests, then, consists of three sets of sequences:
 1. Input sequences.
 2. Corresponding transitions or next-state names.
 3. Output sequences.

STATE TESTING
3. Limitations and Extensions
 1. Simply identifying the factors that contribute to the state, calculating the total number of states, and
comparing this number to the designer’s notion catches some bugs.
 2. Insisting on a justification for all supposedly dead, unreachable, and impossible states and transitions
catches a few more bugs.
 Insisting on an explicit specification of the transition and output for every combination of input and state
catches many more bugs.
 4. A set of input sequences that provide coverage of all nodes and links is a mandatory minimum
requirement.
 5. In executing state tests, it is essential that means be provided (e.g., instrumentation software) to record
the sequence of states (e.g., transitions) resulting from the input sequence and not just the outputs that
result from the input sequence.

STATE TESTING
4.What to Model
 1. Any processing where the output is based on the occurrence of one or more sequences of events, such as detection of
specified input sequences, sequential format validation, parsing, and other situations in which the order of inputs is important.
 2. Most protocols between systems, between humans and machines, between components of a system (CHOI84, CHUN78,
SARI88).
 3. Device drivers such as for tapes and discs that have complicated retry and recovery procedures if the action depends on the
state.
 4. Transaction flows where the transactions are such that they can stay in the system indefinitely—for example, online users,
tasks in a multitasking system.
 5. High-level control functions within an operating system. Transitions between user states, supervisor’s states, and so on.
Security handling of records, permission for read/write/modify privileges, priority interrupts and transitions between interrupt
states and levels, recovery issues and the safety state of records and/or processes with respect to recording recovery data

 6. The behavior of the system with respect to resource management and what it will do when various
levels of resource utilization are reached. Any control function that involves responses to thresholds
where the system’s action depends not just on the threshold value, but also on the direction in which the
threshold is crossed. This is a normal approach to control functions
 7. A set of menus and ways that one can go from one to the other. The currently active menus are the
states, the input alphabet is the choices one can make, and the transitions are invocations of the next menu
in a menu tree. Many menu-driven software packages suffer from dead states—menus from which the
only way out is to reboot.
 8. Whenever a feature is directly and explicitly implemented as one or more state-transition tables.
STATE TESTING
4.What to Model

STATE TESTING
5. Getting the Data
 State testing, more than most functional test strategies, tends to have a labor-intensive data-gathering
phase and tends to need many more meetings to resolve issues.

STATE TESTING
6. Tools
 There are tools to do the same for hardware logic designs.
 That’s the good news.
 The bad news is that these systems and languages are proprietary, of the home-brew variety, internal,
and/or not applicable to the general use of software implementations of finite-state machines

TESTABILITY TIPS
1. A Balm for Programmers
 The key to testability design is easy: build explicit finite-state machines.

TESTABILITY TIPS
2. How Big, How Small?
 There are about eighty possible good and bad three-state machines, 2700 four-state machines, 275,000
five-state machines, and close to 100 million six-state machines, most of which are bad.

TESTABILITY TIPS
3. Switches, Flags, and Unachievable Paths
 Figure 11.9. Program with One Switch Variable.

Switches, Flags, and Unachievable Paths
(continued)
 Three-Switch-Veriable Program.

TESTABILITY TIPS
3.Switches, Flags, and Unachievable Paths
 Switches in a Loop.

TESTABILITY TIPS
4. Essential and Inessential Finite-State Behavior
 The simplest essential finite-state machine is a flip-flop. There is no logic that can implement it without
some kind of feedback. we cannot describe this behavior by a decision table or decision tree unless you
provide feedback into the table or call it recursively.

TESTABILITY TIPS
5. Design Guidelines
 1.. Start by designing the abstract machine. Verify that it is what you want to do. Do an explicit analysis, in the form of a state
graph or table, for anything with three states or more.
 2. Start with an explicit design—that is, input encoding, output encoding, state code assignment, transition table, output table,
state storage, and how you intend to form the state-symbol product. Do this at the PDL level. But be sure to document that
explicit design.
 3. Before you start taking shortcuts, see if it really matters. Neither the time nor the memory for the explicit implementation
usually matters

TESTABILITY TIPS
5. Design Guidelines
 4. Take shortcuts by making things implicit only as you must to make significant reductions in time or space and only if you can
show that such savings matter in the context of the whole system.
After all, doubling the speed of your implementation may mean nothing if all you’ve done is shaved 100
microseconds from a 500-millisecond process. The order in which you should make things implicit are: output encoding, input encoding,
state code, state-symbol product, output table, transition table, state storage. That’s the order from least to most dangerous.
 6. Consider a hierarchical design if you have more than a few dozen states.
 7. Build, buy, or implement tools and languages that implement finite-state machines as software if you’re doing more than a dozen
states routinely.
 8. Build in the means to initialize to any arbitrary state. Build in the transition verification instrumentation (the coverage analyzer).
These are much easier to do with an explicit machine.

Important
 1. The behavior of a finite state machine is invariant under all encodings. Justify? (16 M)**
 2. Write testers comments about state graphs (8 M)**
 3. What are the types of bugs that can cause state graphs? (8 M)*
 4. What are the principles of state testing. Discuss advantages and disadvantages. (8 M)
 5. Write the design guidelines for building finite state machine into code. (8 M)
 6. What are the software implementation issues in state testing? (8 M)
 7. Explain about good state and bad state graphs. (8 M)
 8. Explain with an example how to convert specification into state-graph. Also discuss how
contradictions can come out. (16 M)
 9. Write short notes on: (16 M)
 i. Transition Bugs
 ii. Dead States
 iii. State Bugs
 iv. Encoding Bugs

States, state graphs and transition testing

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