Moore machine

In the theory of computation, a Moore machine is a finite-state machine whose output values are determined only by its current state. This is in contrast to a Mealy machine, whose (Mealy) output values are determined both by its current state and by the values of its inputs. The Moore machine is named after Edward F. Moore, who presented the concept in a 1956 paper, “Gedanken-experiments on Sequential Machines.”
Formal definition.
A Moore machine can be defined as a 6-tuple formula_1 consisting of the following:
A Moore machine can be regarded as a restricted type of finite-state transducer.
Visual representation.
Table.
State transition table is a table showing relation between an input and a state.
Diagram.
The state diagram for a Moore machine or Moore diagram is a diagram that associates an output value with each state.
Moore machine is an output producer.
Relationship with Mealy machines.
As Moore and Mealy machines are both types of finite-state machines, they are equally expressive: either type can be used to parse a regular language.
The difference between Moore machines and Mealy machines is that in the latter, the output of a transition is determined by the combination of current state and current input (formula_9 as the input to formula_10), as opposed to just the current state (formula_11 as the input to formula_10). When represented as a state diagram,
Every Moore machine formula_13 is equivalent to the Mealy machine with the same states and transitions and the output function formula_14, which takes each state-input pair formula_15 and yields formula_16, where formula_17 is formula_13's output function and formula_19 is formula_13's transition function.
However, not every Mealy machine can be converted to an equivalent Moore machine. Some can be converted only to an "almost" equivalent Moore machine, with outputs shifted in time. This is due to the way that state labels are paired with transition labels to form the input/output pairs. Consider a transition formula_21 from state formula_22 to state formula_23. The input causing the transition formula_21 labels the edge formula_25. The output corresponding to that input, is the label of state formula_22. Notice that this is the source state of the transition. So for each input, the output is already fixed before the input is received, and depends solely on the present state. This is the original definition by E. Moore. It is a common mistake to use the label of state formula_23 as output for the transition formula_21.
Examples.
Types according to number of inputs/outputs.
Simple.
Simple Moore machines have one input and one output:
Most digital electronic systems are designed as clocked sequential systems. Clocked sequential systems are a restricted form of Moore machine where the state changes only when the global clock signal changes. Typically the current state is stored in flip-flops, and a global clock signal is connected to the "clock" input of the flip-flops. Clocked sequential systems are one way to solve metastability problems. A typical electronic Moore machine includes a combinational logic chain to decode the current state into the outputs (lambda). The instant the current state changes, those changes ripple through that chain, and almost instantaneously the output gets updated. There are design techniques to ensure that no glitches occur on the outputs during that brief period while those changes are rippling through the chain, but most systems are designed so that glitches during that brief transition time are ignored or are irrelevant. The outputs then stay the same indefinitely (LEDs stay bright, power stays connected to the motors, solenoids stay energized, etc.), until the Moore machine changes state again.
Worked Example.
A sequential network has one input and one output. The output becomes 1 and remains 1 thereafter when at least two 0's and two 1's have occurred as inputs.
A moore machine with nine states for the above description is shown on the right. The initial state is state A, and the final state is state I. The state table for this example is as follows:
Complex.
More complex Moore machines can have multiple inputs as well as multiple outputs.
Gedanken-experiments.
In Moore's paper "Gedanken-experiments on Sequential Machines", the formula_29 automata (or machines) formula_11 are defined as having formula_31 states, formula_32 input symbols and formula_33 output symbols. Nine theorems are proved about the structure of formula_11, and experiments with formula_11. Later, "formula_11 machines" became known as "Moore machines".
At the end of the paper, in Section "Further problems", the following task is stated:
Another directly following problem is the improvement of the bounds given at the theorems 8 and 9.
Moore's Theorem 8 is formulated as:
Given an arbitrary formula_29 machine formula_11, such that every two of its states are distinguishable from one another, then there exists an experiment of length formula_39 which determines the state of formula_11 at the end of the experiment.
In 1957, A. A. Karatsuba proved the following two theorems, which completely solved Moore's problem on the improvement of the bounds of the experiment length of his "Theorem 8".
Theorem A. If formula_11 is an formula_29 machine, such that every two of its states are distinguishable from one another, then there exists a branched experiment of length at most formula_43 through which one may determine the state of formula_11 at the end of the experiment.
Theorem B. There exists an formula_29 machine, every two states of which are distinguishable from one another, such that the length of the shortest experiments establishing the state of the machine at the end of the experiment is equal to formula_43.
Theorems A and B were used for the basis of the course work of a student of the fourth year, A. A. Karatsuba, "On a problem from the automata theory", which was distinguished by testimonial reference at the competition of student works of the faculty of mechanics and mathematics of Moscow Lomonosow State University in 1958. The paper by Karatsuba was given to the journal "Uspekhi Mat. Nauk" on 17 December 1958 and was published there in June 1960.
Until the present day (2011), Karatsuba's result on the length of experiments is the only exact nonlinear result, both in automata theory, and in similar problems of computational complexity theory.
Further reading.
Moore-and-Mealy-Machine