Finite automata with output, often referred to as Moore and Mealy machines, are extensions of traditional finite automata that include the ability to produce output in addition to recognizing input strings. These machines are used in various applications such as digital circuit design, control systems, and natural language processing.
- Moore Machine:
- In a Moore machine, output is associated with states rather than transitions.
- Formally, a Moore machine is defined as a 6-tuple (Q, Σ, Δ, q0, λ, ρ), where:
- Q is a finite set of states.
- Σ is a finite input alphabet.
- Δ: Q × Σ → Q is the transition function, specifying the next state given the current state and input.
- q0 ∈ Q is the initial state.
- λ: Q → Λ is the output function, mapping each state to an output symbol.
- Λ is a finite output alphabet.
- Mealy Machine:
- In a Mealy machine, output is associated with transitions.
- Formally, a Mealy machine is defined as a 6-tuple (Q, Σ, Δ, q0, λ, δ), where:
- Q is a finite set of states.
- Σ is a finite input alphabet.
- Δ: Q × Σ → Q is the transition function, specifying the next state given the current state and input.
- q0 ∈ Q is the initial state.
- λ: Q × Σ → Λ is the output function, mapping each state-input pair to an output symbol.
- Λ is a finite output alphabet.
In both types of machines:
- Inputs are consumed one symbol at a time, causing state transitions.
- Outputs are produced based on the current state (in Moore machines) or the current state and input symbol (in Mealy machines).
Differences between Moore and Mealy machines:
- In Moore machines, outputs are associated only with states and do not depend on the input symbols that caused the transitions.
- In Mealy machines, outputs are associated with transitions and can depend on both the current state and the input symbol.
Both Moore and Mealy machines have equivalent computational power, meaning any problem that can be solved by one type of machine can also be solved by the other. The choice between Moore and Mealy machines depends on the specific requirements of the application and the ease of modeling the problem.