In the theory of computation, a Moore machine is a finite-state machine whose
output values are determined solely by its current state. This is in contrast
to a Mealy machine, whose 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 consisting of the following:
a finite set of states a start state which is an element of
a finite set called the input alphabet a finite set called the output alphabet
a transition function mapping a state and the input alphabet to the next state
an output function mapping each state to the output alphabet
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 output producer Relationship with Mealy machines
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, as opposed to just the current state. When represented
as a state diagram, for a Moore machine, each node is
labeled with an output value; for a Mealy machine, each arc is labeled
with an output value. Every Moore machine is equivalent to
the Mealy machine with the same states and transitions and the output function
, which takes each state-input pair and yields , where is 's output 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 from state to state . The input causing the transition labels the
edge . The output corresponding to that input, is the label of state . 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 as output for the transition .
Examples Types according to number of
inputs/outputs. = Simple=
Simple Moore machines have one input and one output:
edge detector using XOR binary adding machine
clocked sequential systems 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. 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, until the Moore machine changes state again.
= 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 automata are defined as having states, input
symbols and output symbols. Nine theorems are proved about the structure
of , and experiments with . Later, " 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 machine , such that every two of its states are
distinguishable from one another, then there exists an experiment of length
which determines the state of 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 is an machine, such that
every two of its states are distinguishable from one another, then
there exists a branched experiment of length at most through which one may
determine the state of at the end of the experiment.
Theorem B. There exists an 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 . 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, 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.
See also Synchronous circuit
Mealy machine Algorithmic state machine
References Further reading
Conway, J.H.. Regular algebra and finite machines. London: Chapman and Hall. ISBN
0-412-10620-5. Zbl 0231.94041. Moore E. F. Gedanken-experiments on
Sequential Machines. Automata Studies, Annals of Mathematical Studies, 34,
129–153. Princeton University Press, Princeton, N.J.(1956).
Karatsuba A. A. Solution of one problem from the theory of finite automata. Usp.
Mat. Nauk, 15:3, 157–159. Karacuba A. A. Experimente mit Automaten
Elektron. Informationsverarb. Kybernetik, 11, 611–612.
Karatsuba A. A. List of research works its length is n+1/2.