# Moore machine

Last updated

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.” [1]

## Formal definition

A Moore machine can be defined as a 6-tuple ${\displaystyle (S,S_{0},\Sigma ,\Lambda ,T,G)}$ consisting of the following:

• A finite set of states ${\displaystyle S}$
• A start state (also called initial state) ${\displaystyle S_{0}}$ which is an element of ${\displaystyle S}$
• A finite set called the input alphabet ${\displaystyle \Sigma }$
• A finite set called the output alphabet ${\displaystyle \Lambda }$
• A transition function ${\displaystyle T:S\times \Sigma \rightarrow S}$ mapping a state and the input alphabet to the next state
• An output function ${\displaystyle G:S\rightarrow \Lambda }$ 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.[ clarification needed ]

### 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 (${\displaystyle S\times \Sigma }$ as the input to ${\displaystyle G}$), as opposed to just the current state (${\displaystyle S}$ as the input to ${\displaystyle G}$). When represented as a state diagram,

• for a Moore machine, each node (state) is labeled with an output value;
• for a Mealy machine, each arc (transition) is labeled with an output value.

Every Moore machine ${\displaystyle M}$ is equivalent to the Mealy machine with the same states and transitions and the output function ${\displaystyle G(s,\sigma )\rightarrow G_{M}(s)}$, which takes each state-input pair ${\displaystyle (s,\sigma )}$ and yields ${\displaystyle G_{M}(s)}$, where ${\displaystyle G_{M}}$ is ${\displaystyle M}$'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 ${\displaystyle s_{i}\rightarrow s_{j}}$ from state ${\displaystyle s_{i}}$ to state ${\displaystyle s_{j}}$. The input causing the transition ${\displaystyle s_{i}\rightarrow s_{j}}$ labels the edge ${\displaystyle (s_{i},s_{j})}$. The output corresponding to that input, is the label of state ${\displaystyle s_{i}}$. [2] 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 ${\displaystyle s_{j}}$ as output for the transition ${\displaystyle s_{i}\rightarrow s_{j}}$.

## 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:

Current stateInputNext stateOutput
A0D0
1B
B0E0
1C
C0F0
1C
D0G0
1E
E0H0
1F
F0I0
1F
G0G0
1H
H0H0
1I
I0I1
1I

### 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", [1] the ${\displaystyle (n;m;p)}$ automata (or machines) ${\displaystyle S}$ are defined as having ${\displaystyle n}$ states, ${\displaystyle m}$ input symbols and ${\displaystyle p}$ output symbols. Nine theorems are proved about the structure of ${\displaystyle S}$, and experiments with ${\displaystyle S}$. Later, "${\displaystyle S}$ 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 ${\displaystyle (n;m;p)}$ machine ${\displaystyle S}$, such that every two of its states are distinguishable from one another, then there exists an experiment of length ${\displaystyle {\tfrac {n(n-1)}{2}}}$ which determines the state of ${\displaystyle S}$ 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 ${\displaystyle S}$ is an ${\displaystyle (n;m;p)}$ machine, such that every two of its states are distinguishable from one another, then there exists a branched experiment of length at most ${\displaystyle {\tfrac {(n-1)(n-2)}{2}}+1}$ through which one may determine the state of ${\displaystyle S}$ at the end of the experiment.

Theorem B. There exists an ${\displaystyle (n;m;p)}$ 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 ${\displaystyle {\tfrac {(n-1)(n-2)}{2}}+1}$.

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. [3]

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.

## Related Research Articles

A finite-state machine (FSM) or finite-state automaton, finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of states at any given time. The FSM can change from one state to another in response to some external inputs and/or a condition is satisfied; the change from one state to another is called a transition. An FSM is defined by a list of its states, its initial state, and the conditions for each transition. Finite state machines are of two types – deterministic finite state machines and non-deterministic finite state machines. A deterministic finite-state machine can be constructed equivalent to any non-deterministic one.

In the theory of computation, a branch of theoretical computer science, a pushdown automaton (PDA) is a type of automaton that employs a stack.

In the theory of computation, a Mealy machine is a finite-state machine whose output values are determined both by its current state and the current inputs. This is in contrast to a Moore machine, whose (Moore) output values are determined solely by its current state. A Mealy machine is a deterministic finite-state transducer: for each state and input, at most one transition is possible.

In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite state machine (DFSM), or deterministic finite state automaton (DFSA)—is a finite-state machine that accepts or rejects a given string of symbols, by running through a state sequence uniquely determined by the string. Deterministic refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943.

In automata theory, a finite state machine is called a deterministic finite automaton (DFA), if

In computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The concept of an ATM was set forth by Chandra and Stockmeyer and independently by Kozen in 1976, with a joint journal publication in 1981.

In automata theory, an alternating finite automaton (AFA) is a nondeterministic finite automaton whose transitions are divided into existential and universal transitions. For example, let A be an alternating automaton.

A finite-state transducer (FST) is a finite-state machine with two memory tapes, following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton that maps between two sets of symbols. An FST is more general than a finite-state automaton (FSA). An FSA defines a formal language by defining a set of accepted strings, while an FST defines relations between sets of strings.

In theoretical computer science and mathematical logic a string rewriting system (SRS), historically called a semi-Thue system, is a rewriting system over strings from a alphabet. Given a binary relation between fixed strings over the alphabet, called rewrite rules, denoted by , an SRS extends the rewriting relation to all strings in which the left- and right-hand side of the rules appear as substrings, that is , where , , , and are strings.

In computer science, in particular in automata theory, a two-way finite automaton is a finite automaton that is allowed to re-read its input.

In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They are related to quantum computers in a similar fashion as finite automata are related to Turing machines. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata.

In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function.

DEVS abbreviating Discrete Event System Specification is a modular and hierarchical formalism for modeling and analyzing general systems that can be discrete event systems which might be described by state transition tables, and continuous state systems which might be described by differential equations, and hybrid continuous state and discrete event systems. DEVS is a timed event system.

A queue machine or queue automaton is a finite state machine with the ability to store and retrieve data from an infinite-memory queue. It is a model of computation equivalent to a Turing machine, and therefore it can process the same class of formal languages.

A read-only Turing machine or Two-way deterministic finite-state automaton (2DFA) is class of models of computability that behave like a standard Turing machine and can move in both directions across input, except cannot write to its input tape. The machine in its bare form is equivalent to a Deterministic finite automaton in computational power, and therefore can only parse a regular language.

An embedded pushdown automaton or EPDA is a computational model for parsing languages generated by tree-adjoining grammars (TAGs). It is similar to the context-free grammar-parsing pushdown automaton, except that instead of using a plain stack to store symbols, it has a stack of iterated stacks that store symbols, giving TAGs a generative capacity between context-free grammars and context-sensitive grammars, or a subset of the mildly context-sensitive grammars. Embedded pushdown automata should not be confused with nested stack automata which have more computational power.

In formal language theory, a grammar is a set of production rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form.

In computer science, more specifically in automata and formal language theory, nested words are a concept proposed by Alur and Madhusudan as a joint generalization of words, as traditionally used for modelling linearly ordered structures, and of ordered unranked trees, as traditionally used for modelling hierarchical structures. Finite-state acceptors for nested words, so-called nested word automata, then give a more expressive generalization of finite automata on words. The linear encodings of languages accepted by finite nested word automata gives the class of visibly pushdown languages. The latter language class lies properly between the regular languages and the deterministic context-free languages. Since their introduction in 2004, these concepts have triggered much research in that area.

In computer science and mathematical logic, an infinite-tree automaton is a state machine that deals with infinite tree structures. It can be seen as an extension of top-down finite-tree automata to infinite trees or as an extension of infinite-word automata to infinite trees.

In automata theory, a timed automaton is a finite automaton extended with a finite set of real-valued clocks. During a run of a timed automaton, clock values increase all with the same speed. Along the transitions of the automaton, clock values can be compared to integers. These comparisons form guards that may enable or disable transitions and by doing so constrain the possible behaviors of the automaton. Further, clocks can be reset. Timed automata are a sub-class of a type hybrid automata.

## References

1. Moore, Edward F (1956). "Gedanken-experiments on Sequential Machines". Automata Studies, Annals of Mathematical Studies. Princeton, N.J.: Princeton University Press (34): 129–153.
2. Lee, Edward Ashford; Seshia, Sanjit Arunkumar (2013). Introduction to Embedded Systems (1.08 ed.). UC Berkeley: Lulu.com. ISBN   9780557708574 . Retrieved 1 July 2014.
3. Karatsuba, A. A. (1960). "Solution of one problem from the theory of finite automata". Uspekhi Mat. Nauk (15:3): 157–159.
• Conway, J.H. (1971). 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 (1960).
• Karatsuba A. A. Experimente mit Automaten (German) Elektron. Informationsverarb. Kybernetik, 11, 611–612 (1975).
• Karatsuba A. A. List of research works .