- Automata Theory Tutorial
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- Automata Theory Introduction
- Deterministic Finite Automaton
- Non-deterministic Finite Automaton
- NDFA to DFA Conversion
- DFA Minimization
- Moore & Mealy Machines

- Classification of Grammars
- Introduction to Grammars
- Language Generated by Grammars
- Chomsky Grammar Classification

- Regular Grammar
- Regular Expressions
- Regular Sets
- Arden's Theorem
- Constructing FA from RE
- Pumping Lemma for Regular Grammar
- DFA Complement

- Context-Free Grammars
- Context-Free Grammar Introduction
- Ambiguity in Grammar
- CFL Closure Properties
- CFG Simplification
- Chomsky Normal Form
- Greibach Normal Form
- Pumping Lemma for CFG

- Pushdown Automata
- Pushdown Automata Introduction
- Pushdown Automata Acceptance
- PDA & Context Free Grammar
- PDA & Parsing

- Turing Machine
- Turing Machine Introduction
- Accepted & Decided Language
- Multi-tape Turing Machine
- Multi-Track Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- Linear Bounded Automata

- Decidability
- Language Decidability
- Undecidable Language
- Turing Machine Halting Problem
- Rice Theorem
- Post Correspondence Problem

- Automata Theory Useful Resources
- Automata Theory - Quick Guide
- Automata Theory - Useful Resources
- Automata Theory - Discussion

Finite automata may have outputs corresponding to each transition. There are two types of finite state machines that generate output −

- Mealy Machine
- Moore machine

A Mealy Machine is an FSM whose output depends on the present state as well as the present input.

It can be described by a 6 tuple (Q, ∑, O, δ, X, q_{0}) where −

**Q**is a finite set of states.**∑**is a finite set of symbols called the input alphabet.**O**is a finite set of symbols called the output alphabet.**δ**is the input transition function where δ: Q × ∑ → Q**X**is the output transition function where X: Q × ∑ → O**q**is the initial state from where any input is processed (q_{0}_{0}∈ Q).

The state table of a Mealy Machine is shown below −

Present state | Next state | |||
---|---|---|---|---|

input = 0 | input = 1 | |||

State | Output | State | Output | |

→ a | b | x_{1} |
c | x_{1} |

b | b | x_{2} |
d | x_{3} |

c | d | x_{3} |
c | x_{1} |

d | d | x_{3} |
d | x_{2} |

The state diagram of the above Mealy Machine is −

Moore machine is an FSM whose outputs depend on only the present state.

A Moore machine can be described by a 6 tuple (Q, ∑, O, δ, X, q_{0}) where −

**Q**is a finite set of states.**∑**is a finite set of symbols called the input alphabet.**O**is a finite set of symbols called the output alphabet.**δ**is the input transition function where δ: Q × ∑ → Q**X**is the output transition function where X: Q → O**q**is the initial state from where any input is processed (q_{0}_{0}∈ Q).

The state table of a Moore Machine is shown below −

Present state | Next State | Output | |
---|---|---|---|

Input = 0 | Input = 1 | ||

→ a | b | c | x_{2} |

b | b | d | x_{1} |

c | c | d | x_{2} |

d | d | d | x_{3} |

The state diagram of the above Moore Machine is −

The following table highlights the points that differentiate a Mealy Machine from a Moore Machine.

Mealy Machine | Moore Machine |
---|---|

Output depends both upon the present state and the present input | Output depends only upon the present state. |

Generally, it has fewer states than Moore Machine. | Generally, it has more states than Mealy Machine. |

The value of the output function is a function of the transitions and the changes, when the input logic on the present state is done. | The value of the output function is a function of the current state and the changes at the clock edges, whenever state changes occur. |

Mealy machines react faster to inputs. They generally react in the same clock cycle. | In Moore machines, more logic is required to decode the outputs resulting in more circuit delays. They generally react one clock cycle later. |

**Input** − Moore Machine

**Output** − Mealy Machine

**Step 1** − Take a blank Mealy Machine transition table format.

**Step 2** − Copy all the Moore Machine transition states into this table format.

**Step 3** − Check the present states and their corresponding outputs in the Moore Machine state table; if for a state Q_{i} output is m, copy it into the output columns of the Mealy Machine state table wherever Q_{i} appears in the next state.

Let us consider the following Moore machine −

Present State | Next State | Output | |
---|---|---|---|

a = 0 | a = 1 | ||

→ a | d | b | 1 |

b | a | d | 0 |

c | c | c | 0 |

d | b | a | 1 |

Now we apply Algorithm 4 to convert it to Mealy Machine.

**Step 1 & 2** −

Present State | Next State | |||
---|---|---|---|---|

a = 0 | a = 1 | |||

State | Output | State | Output | |

→ a | d | b | ||

b | a | d | ||

c | c | c | ||

d | b | a |

**Step 3** −

Present State | Next State | |||
---|---|---|---|---|

a = 0 | a = 1 | |||

State | Output | State | Output | |

=> a | d | 1 | b | 0 |

b | a | 1 | d | 1 |

c | c | 0 | c | 0 |

d | b | 0 | a | 1 |

**Input** − Mealy Machine

**Output** − Moore Machine

**Step 1** − Calculate the number of different outputs for each state (Q_{i}) that are available in the state table of the Mealy machine.

**Step 2** − If all the outputs of Qi are same, copy state Q_{i}. If it has n distinct outputs, break Q_{i} into n states as Q_{in} where **n** = 0, 1, 2.......

**Step 3** − If the output of the initial state is 1, insert a new initial state at the beginning which gives 0 output.

Let us consider the following Mealy Machine −

Present State | Next State | |||
---|---|---|---|---|

a = 0 | a = 1 | |||

Next State | Output | Next State | Output | |

→ a | d | 0 | b | 1 |

b | a | 1 | d | 0 |

c | c | 1 | c | 0 |

d | b | 0 | a | 1 |

Here, states ‘a’ and ‘d’ give only 1 and 0 outputs respectively, so we retain states ‘a’ and ‘d’. But states ‘b’ and ‘c’ produce different outputs (1 and 0). So, we divide **b** into **b _{0}, b_{1}** and

Present State | Next State | Output | |
---|---|---|---|

a = 0 | a = 1 | ||

→ a | d | b_{1} |
1 |

b_{0} |
a | d | 0 |

b_{1} |
a | d | 1 |

c_{0} |
c_{1} |
C_{0} |
0 |

c_{1} |
c_{1} |
C_{0} |
1 |

d | b_{0} |
a | 0 |

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