2. Introduction to automata theory
Automata theory is a branch of computer science and mathematics that
deals with abstract machines and computational models. It explores the
capabilities and limitations of computing devices through formal
languages and automata, which are mathematical representations of
machines.
Automata theory is foundational in understanding the concept of
computation, helping to analyze algorithms, design compilers, and study
the complexity of problems. It has applications in various fields such as
computer science, linguistics, artificial intelligence, and robotics
3. Regular Language
A regular language is a type of language that can be described using regular
expressions or can be recognized by a finite automaton. In simpler terms, it's a
pattern or set of strings that can be generated or identified by straightforward, finite
rules.
Regular languages have certain defining characteristics:
• Finite Automata: They can be recognized by finite automata, which are abstract
machines that process inputs and transition between states based on those inputs.
• Closure Properties: Regular languages are closed under several operations, such
as union, concatenation, and Kleene star (repetition). This means that combining
regular languages using these operations results in another regular language.
• Simple Structure: Regular languages are relatively simple compared to more
complex language types like context-free languages or recursively enumerable
languages.
4. • Moore Machine:
• Definition: A Moore machine is a
finite state machine where each
state produces an output, and the
output depends only on the current
state.
• Output Function: In a Moore
machine, the output is associated
directly with each state. When the
machine transitions from one state
to another, it emits an output
corresponding to the current state
• Z(t)=λ(q(t))
• Mealy Machine:
• Definition: A Mealy machine is a
finite state machine where each
transition is associated with both an
output and a state change, and the
output depends on both the current
state and the input symbol.
• Output Function: In a Mealy
machine, the output is a function of
the current state and the input
symbol that triggers the transition.
• Z(t)=λ(q(t),x(t))
5. MEALY MACHINE
• A Mealy machine is a machine in which output
symbol depends upon the present input symbol and
present state of the machine.
• The output is represented with each input symbol
for each state separated by /. The Mealy machine
can be described by 6 tuples (Q, q0, ∑, O, δ, λ’).
1.Q: finite set of states
2.q0: initial state of machine
3.∑: finite set of input alphabet
4.O: output alphabet
5.δ: transition function where Q × ∑ → Q
6.λ': output function where Q × ∑ →O
6. EXAMPLE :
Design a Mealy machine for a binary
input sequence such that if it has a
substring 101, the machine output A, if the
input has substring 110, it outputs B
otherwise it outputs C.
Solution: For designing such a machine,
we will check two conditions, and those
are 101 and 110. If we get 101, the output
will be A. If we recognize 110, the output
will be B. For other strings the output will
be C.
The partial diagram will be:
7. Now we will insert the possibilities of 0's and 1's for each state. Thus the Mealy
machine becomes:
8. TRANSITION
DIAGRAM FOR
MEALY MACHINE
• In a Mealy machine, the
transition between states is
again determined by the
current state and the
inputs, but in addition to
that, the outputs are also
determined by these inputs
and the current state.
9. MOORE MACHINE
• Moore machine is a finite state machine in which the next
state is decided by the current state and current input
symbol. The output symbol at a given time depends only
on the present state of the machine. Moore machine can be
described by 6 tuples (Q, q0, ∑, O, δ, λ) where,
1.Q: finite set of states
2.q0: initial state of machine
3.∑: finite set of input symbols
4.O: output alphabet
5.δ: transition function where Q × ∑ → Q
6.λ: output function where Q → O
10. EXAMPLE:
Design a Moore machine for a binary input
sequence such that if it has a substring 101, the
machine output A, if the input has substring 110, it
outputs B otherwise it outputs C.
Solution: For designing such a machine, we will
check two conditions, and those are 101 and 110.
If we get 101, the output will be A, and if we
recognize 110, the output will be B. For other
strings, the output will be C.
11. Now we will insert the possibilities of 0's and 1's for each state.
Thus the Moore machine becomes:
12. TRANSITION
DIAGRAM FOR
MOORE MACHINE
• In a Moore machine, the
transition between states is
determined solely by the
current state and the
inputs. The outputs,
however, are associated
with the states themselves,
not the transitions.
13. DIFFERENC
E
Moore Machine
• Output depends only upon
the present state.
• More states are required.
• There is less hardware
requirement for circuit
implementation.
• They react slower to
inputs(One clock cycle
later).
• Synchronous output and
state generation.
• Output is placed on states.
Mealy machine
• Output depends on the
present state as well as
present input.
• Less number of states are
required.
• There is more hardware
requirement for circuit
implementation.
• They react faster to inputs.
• Asynchronous output
generation.
• Output is placed on
transitions.
14. ADVANTAGES
Advantages of Moore Machines:
1.Simplicity: Moore machines are generally simpler to
design and understand because the outputs are directly
associated with states, making the relationship between
states and outputs clear.
2.Stable Outputs: Since the outputs are associated with
states, they remain stable for the duration of each state.
This can be advantageous in applications where output
stability is critical.
3.Less Sensitive to Input Timing: Moore machines are
less sensitive to variations in input timing because the
outputs are determined solely by the current state, not by
the inputs.
4.Better for Control Applications: Moore machines are
often preferred in control applications where the primary
concern is to produce stable outputs based on the current
state.
15. • Advantages of Mealy Machines:
1.Flexibility: Mealy machines offer more flexibility in output
generation because outputs can depend on both the current
state and the inputs. This allows for more complex behavior
and potentially more efficient designs.
2.Reduced Output Delay: Mealy machines can produce
outputs more quickly because the outputs can change
asynchronously with state transitions, allowing for shorter
output delay compared to Moore machines.
3.Resource Efficiency: In some cases, Mealy machines can be
more resource-efficient than Moore machines because they
can produce the same output for multiple states, reducing the
number of required states and transitions.
4.Compact Design: Mealy machines can sometimes lead to
more compact designs, especially when dealing with systems
that require complex behavior or have limited resources.
16. LIMITATIONS
Limitations of Moore Machines:
• Limited Flexibility: Moore machines have limited
flexibility in output generation since the outputs
are solely determined by the states.
• Output Delay: In Moore machines, output
changes only occur at state transitions.
• Sensitive to State Transitions: Moore machines
are sensitive to changes in state transitions.
• Increased State Count: Moore machines may
require more states compared to Mealy machines
to achieve the same functionality, especially for
systems with complex output behavior.
17. Limitations of Mealy Machines:
• Complex Design: Mealy machines can have a
more complex design compared to Moore
machines, especially for systems with complex
input-output relationships.
• Output Glitches: Since the outputs in Mealy
machines can change asynchronously with
state transitions there is a possibility of output
glitches or transient states during transitions.
• Difficult State Encoding: The increased
flexibility of Mealy machines can sometimes
make it difficult to encode the states efficiently,
especially in systems with a large number of
inputs and outputs.
18. APPLICATIONS
• Sequential Circuits: Both Moore and Mealy machines
are widely used in the design of sequential circuits,
such as counters, shift registers, and memory units.
• Automata Theory: Moore and Mealy machines are
central to automata theory, a branch of theoretical
computer science. They are used to model and analyze
the behavior of finite-state systems, such as formal
languages, regular expressions, and parsing algorithms.
• Hardware Description Languages (HDLs): Moore
and Mealy machines are used in hardware description
languages like VHDL and Verilog to specify and
simulate digital circuits.
• Testing and Verification: Moore and Mealy machines
are utilized in the testing and verification of digital
circuits and systems.
19. CONCLUSION
• In conclusion, Moore and Mealy machines are
foundational concepts in digital logic design and
sequential circuitry.
• Moore machines offer simplicity and stability, with
outputs dependent solely on the current state.
• On the other hand, Mealy machines provide greater
flexibility and efficiency, as outputs depend on
both the current state and the inputs.
• Both Moore and Mealy machines find applications
across various domains, including sequential
circuits, communication systems, embedded
systems, automata theory, and digital control.
• As digital technology continues to evolve, Moore
and Mealy machines will remain indispensable
tools for digital system design, adapting to
emerging technologies and driving innovation in
computing and engineering fields.
