1. CSE 555 Protocol Engineering
Dr. Mohammed H. Sqalli
Computer Engineering Department
King Fahd University of Petroleum & Minerals
Credits: Dr. Abdul Waheed (KFUPM)
Spring 2004 (Term 032)

2. Finite State Machines (FSMs)

3. Topics (Ch. 8) Finite state machines (FSMs) Petri nets
Informal and formal descriptions
Execution of machines
Minimization of machines
Combining machines
Extended FSMs
Generalization of machines
Petri nets
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4. Motivation for Using FSMs
A communication system exhibits:
Data flow
Deals with data input and its manipulation
Control flow
Deals with state changes
Can be expressed through finite state machine models
FSMs are used extensively in hardware to model sequential machines:
Inputs with well-defined next state
Output produced
Applicable model for protocols:
Interactions can be modeled as inputs/outputs
Protocol operations can be modeled as state changes
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5. Introduction A protocol is often described as a state machine
Helps understanding at a low level of abstraction
Design criteria can be expressed in terms of desirable vs. undesirable states
A state machine typically defines:
Actions that a process is allowed to take
Events that a process can expect to happen
Actions that a process will take in response to various events
A formal model of a communicating state machine is used for:
Formal validation of protocols
Protocol synthesis
Conformance testing
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6. Informal Description
Several informal techniques to describe a finite state machine:
Transition tables
Transition diagrams
A Turing machine
Communicating FSMs
Asynchronous coupling
Synchronous coupling
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7. Transition Table
A finite state machine is informally specified in the form of a transition table
Table specifies a set of transition rules for each control state
Each row  specifies one rule
Each state  usually specified by more than one rule
Example:
Consider four control states: q0, q1, q2, and q3
Each transition rule has four parts:
Current state
Input
Output
Next state
These parts correspond to columns in the transition table…
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8. Example
“-”: don’t care
In col#1: Rule applies to all states
In col#2: Rule applies to all possible values of input
“-”: no change
In col#3: Output signal does not change
In col#4: Control state remains unaffected
Conditions for transition rule to be executable (first two columns):
The current control state in which the machine must be; and
A condition on the environment of the machine
Example: value of an input signal
Effect of a transition (last two columns) specify:
How the environment changes due to transition
Example: value of an output signal
New state the machine attains after applying transition
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9. State Transitions
How many possible transition rules can be executable in a given state of the machine?
None
One
Multiple
Number of executable transition rules determines type of that state:
No executable transition rule  machine in an end state
Precisely one executable rule  deterministic state
More than one executable rules  non-deterministic choice to select a transition rule
Selection criteria is undefined
Without further information, all options are equally likely
Machines that can make such choices are called non-deterministic machines
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10. Example Two transition rules for the same state q1:
When input signal is 1, only the first rule is executable
When input signal is 0, both rules are executable
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11. Transition Diagram State transition diagram
Graphical representation of the state machine
Behavior more easily understood
Control states  circles
Transition rules  directed edges
Edge labels (c/e), where c: transition condition, and e: effect
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12. Turing Machines
Can be considered as a generalization of finite state model
For a truly finite state machine, environment must be of finite state
If finite state requirement is dropped, the machine is called a Turing machine
Environment for a Turing machine:
A tape of infinite length
Tape consists of a sequence of squares
Each square can store one of a finite set of symbols
All tape squares are initially blank
Machine can read or write one tape symbol at a time
Machine moves the tape left or right by one square at a time
State transitions:
Condition of transition depends on: current state and symbol
Effect of transition rule: output a new symbol onto current square, a possible left or right move of the tape, and a jump to a new control state
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13. Example Two output signals:
One to overwrite the current state on the tape
Other to move the tape left or right by one square
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14. Limitations of Turning Machine
Difficult to extend
To model interaction of multiple FSMs is hard
Infinite number of potential states of the environment:
Many problems will become computationally intractable
We need to explore other variants of the FSMs
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15. Communicating Finite State Machines
What happens when input and output signals are allowed to overlap
Example: feedback the output as an input
Assumption: signals have a finite range of possible values
Values can change only at precisely defined moments
Machine executes a two-step algorithm for ever:
Step # 1: inspect input signal values and select an executable transition rule
Step # 2: machine changes state wrt the rule and updates its output signals
A signal has a state much like a finite state machine
A signal can be interpreted as a variable that can be evaluated or assigned to only at precisely defined moments
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16. Example: Revisited Behavior of this state machine is now fully defined
Even if we assume a feedback from output to input signal
At each step, machine inspects the output value that was set in the previous transition
Machine loops through following states for ever: q0, q2, and q1
Elaborate systems of interacting machines can be built this way
By connecting output signals of one machine to the input signals of another
The machines must share a common “clock” for their two-step algorithm
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17. Modeling Distributed Systems using FSMs
FSMs are most useful if they can directly model distributed computer systems
Two ways to do this:
Asynchronous communication among systems
Synchronous communication
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18. Asynchronous Coupling
Asynchronous coupling model:
Machines are coupled via bounded first-in first-out (FIFO) message queues
Signals are now abstract objects called messages
Input signals are retrieved from input queues
Output signals are appended to output queues
Finiteness of the model:
All queues and sets of signals are still finite
How synchronization is achieved?
By defining both input and output signals to be conditionals on the state of the message queues
If an input queue is empty, no input signal is available from that queue  corresponding transition rule is unexecutable
If an output queue is full, no output signal can be generated for that queue  corresponding transition rule in unexecutable
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19. Asynchronous Coupling (Cont’d)
Assumption: restrict model to one synchronization event per transition rule
A single rule can specify either input or output but not both
This assumption simplifies the model
This assumption also models the real behavior of a distributed system process more closely
Execution of a transition rule is atomic
A single send/receive operation in most distributed systems is also atomic
Example: transition table model of simple version of alternating bit protocol
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20. Example: Alternating Bit Protocol
Possibility of a retransmission
Is not modeled in this table
Retransmissions are modeled
by adding these two rules
Last received message can
be marked as correct in states
q1 and q4
Table models the possibility of retransmission not their probability
This is okay as our analysis needs to be independent of any assumptions on the timing or speed of individual processes
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21. Example: State Transition Diagrams
Timeout option in the sender will produce an extra self loop on states q1 and q3
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22. Synchronous Coupling
For a transition, a signal has to be selected by precisely two machines simultaneously
In one machine as output signal, which is an input signal to the second
When such a match occurs, both machines make corresponding transitions simultaneously
As in case of asynchronous coupling, we allow only one synchronizing event per transition rule
Synchronous communication can be considered a special case of asynchronous communication with a queue capacity of zero slots
Consider an example of synchronously coupled state machines
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23. Example: Synchronously Coupled FSMs
First machine (user):
One input selection P in state q0
One output selection V in state q1
Second machine (server):
Same but inputs and output are swapped
Synchronous coupling:
We create two machines of the first type (User) and combine them with one machine of the second type (Server)
The two user machines cannot be both in state q1 simultaneously
Synchronous communication is binary: exactly two machines participate
One with a given input selection and other with same output selection
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24. Formal Description of FSMs
A communicating FSM can be defined as a daemon
It accepts input symbols
Symbols are defined as abstract objects without content
Generates output symbols
Changes its inner state in accordance with a pre-defined plan
FSM daemons communicate through FIFO queues
Output of one daemon is mapped onto the input of another
Formal definition of a message queue:
It is a triple (S, N, C) where:
S is a finite set called the queue vocabulary
N is an integer that defines the number of slots in the queue
C is the queue contents, which is an ordered set of elements from S
Elements of S and C are called messages
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25. Formal Description: Communicating FSMs
A communicating FSM is defined as:
A tuple of the form (Q, q0, M, T) where:
Q is a finite, non-empty set of states
q0 is an element of Q  the initial state
M is a set of message queues
T is a state transition relation
Relation T has two arguments: T(q, a):
q is the current state
a is an action from: inputs, outputs, and null action e
T defines a set of zero or more possible successor states in set Q for current state q
This contains precisely one state unless non-determinism is modeled
When T(q, a) is not explicitly defined T(q, a) = F
T(q, e) specifies a spontaneous transition: sufficient condition for such transition is that the machine is in state q
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26. Execution of Machines Consider a set of P FSMs
Assume asynchronous coupling only
Overlapping set of message queues whose union is M
This set of communicating FSMs is executed by applying following rules:
Set all machines in their initial states and initialize all message queues to empty
Select an arbitrary machine i and an arbitrary transition rule Ti with an action a such that and execute it
If no executable transition rule remains, the algorithm terminates
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27. Executability of the Action a
Action a can be:
An input action; output action; or null action
Let d(a) be destination queue of an action a
Let m(a) be the message that is sent or received
Let Ni represent the number of slots in message queue i
Following three rules can be used to determine if a is executable:
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28. Minimization Of Machines
Basic idea: equivalence of two FSMs
Two machines are equivalent if they can generate the same sequence of output symbols when offered with same sequence of input symbols
Machines can make non-deterministic choices
Possible to generate different output for same input even for two machines that are equal
Rule of equivalence:
Machines must have equivalent choices to be in equivalent states
States within a single machine are equivalent if:
Machine can be started in any one of these states; and
Generate same set of possible sequences of outputs when offered any given test sequence of inputs
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29. Example: Minimization
Equivalent
state transitions
Equivalent state
transition diagram
Both machines seem to behave similarly
One has three fewer states
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30. Example: Minimization
These two PROMELA processes are equivalent
Two sequences of messages: {q?a;q?b and q?a;q?c}
For non-deterministic communicating FSMs, processes A and B are not equivalent
The input sequence of q?a;q?b is always accepted by process B but may lead to an unspecified reception in process A
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31. Minimization: Approach
Set of control states of a communicating finite state machine can be minimized by:
Replacing every set of equivalent states with a single state
This does not change external behavior of the machine
Formally, equivalence relationship defines a partitioning of the states into a finite set of disjoint equivalence classes
The smallest machine equivalent to the given one will have as many states as the original machine has equivalence classes
We can now define an algorithm for minimization of an arbitrary finite state machine with |Q| states
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32. Minimization Algorithm
Step 1:
Define an array E of |Q|x|Q| boolean values
Initially all elements E[i,j] are set to the truth value of the following condition, for all actions a:
Two states are not equivalent unless the corresponding state transition relations are defined for the same actions
Step 2:
If machine contains only deterministic choices, T defines a unique successor state for all true entries of array E. Change the value of all those entries E[i,j] to the value of:
It means that states are not equivalent unless their successors are also equivalent
When T(i,a) and T(j,a) can have more than one element, the value of E[i,j] is set to false if either of the following conditions is false for any action a:
It means that states i and j are not equivalent unless for every possible successor state p of state i there is at least one equivalent successor state q of state j, and vice versa.
Repeat step 2 until the number of false entries in E can no longer increase
This procedure always terminates
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33. Example: Equivalence
Equivalence array E is obtained after applying steps 1 and 2
Following state pairs are equivalent: (q0, q3), (q1, q5), and (q2, q4)
Therefore, we can reduce the state table to 3-state FSM
Entries in array E are symmetric
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34. Conformance Testing Problem
Procedure of testing equivalence of states can also be applied to determine the equivalence of two machines
Determine that every state in one machine has an equivalent in the other machine
Machines need not be equal to be equivalent
A practical application:
A formal protocol specification in FSM form should be equivalent to an implementation of that specification
That is, implementation seen as black box should respond to input signals exactly as the reference machine would
Problem: find the right set of test sequences to establish the equivalence or non-equivalence of two machines
This problem is known as fault detection or conformance testing problem
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35. Combining Machines
Combining two FSMs into a single FSM reduces the complexity of formal validation based on FSM model
The problem is to find a tuple (Q, qo, M, T) for the combined machine, given two machines (Q1, q01, M1, T1) and (Q2, q02, M2, T2)
Algorithm:
Total states are: |Q1|x|Q2| and defined a combined set of states Q. The initial state q0 of the new machine is the combination q01q02
The set of message queues M is the union of two machines separate queues
For each state q1q2 in Q, define transition relation T for each action a as the non-deterministic choice of the corresponding relation of M1 and M2 separately when placed in individual states q1 and q2
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