8/26/2021»»Thursday

## Sequence Detector Using Moore Machine

8/26/2021

Design a sequence detector implementing a Moore state machine using three always blocks. The Moore state machine has two inputs (a in 1:0) and one output (y out). The output y out begins as 0 and remains a constant value unless one of the following input sequences occurs: (i) The input sequence a in 1:0 = 01, 00 causes the output to. This Verilog project is to present a full Verilog code for Sequence Detector using Moore FSM. A Verilog Testbench for the Moore FSM sequence detector is also provided for simulation. The Moore FSM keeps detecting a binary sequence from a digital input and the output of the FSM goes high only when a '1011' sequence is detected.

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${displaystyle (Q,q_{0},Sigma ,O,delta ,G)}$ consisting of the following:

• A finite set of states${displaystyle Q}$
• A start state (also called initial state) ${displaystyle q_{0}}$ which is an element of ${displaystyle Q}$
• A finite set called the input alphabet${displaystyle Sigma }$
• A finite set called the output alphabet${displaystyle O}$
• A transition function${displaystyle delta :Qtimes Sigma rightarrow Q}$ mapping a state and the input alphabet to the next state
• An output function ${displaystyle G:Qrightarrow O}$ 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 (${displaystyle Stimes Sigma }$ 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 ${displaystyle G(s,sigma )rightarrow G_{M}(delta _{M}(s,sigma ))}$, which takes each state-input pair ${displaystyle (s,sigma )}$ and yields ${displaystyle G_{M}(delta _{M}(s,sigma ))}$, where ${displaystyle G_{M}}$ is ${displaystyle M}$'s output function and ${displaystyle delta _{M}}$ is ${displaystyle M}$'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 ${displaystyle s_{i}rightarrow s_{j}}$ 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 ${displaystyle s_{i}}$. 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:

• edge detector using XOR
• clocked sequential systems (a restricted form of Moore machine where the state changes only when the global clock signal changes)

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.

Example moore machine

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', 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.

### Moore Machine Sequence Detector

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.

## References

1. ^ abMoore, 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. ISBN9780557708574. Retrieved 1 July 2014.CS1 maint: discouraged parameter (link)
3. ^Karatsuba, A. A. (1960). 'Solution of one problem from the theory of finite automata'. Uspekhi Mat. Nauk (15:3): 157–159.

### Sequence Detector Using Moore Machine Design

• Conway, J.H. (1971). Regular algebra and finite machines. London: Chapman and Hall. ISBN0-412-10620-5. Zbl0231.94041.CS1 maint: discouraged parameter (link)
• 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.

• Media related to Moore machine at Wikimedia Commons

This example shows how to use Mealy and Moore machines for a sequence recognition application in signal processing. For more information, see Overview of Mealy and Moore Machines.

In this model, two Stateflow® charts use a different set of semantics to find the sequence 1,2,1,3 in the input signal from a Signal Editor (Simulink) block. Each chart contains an input data `u` and two output data:

• `seqFound`Digital cube clock. indicates when the chart finds the sequence. A value of `false` means that the chart is still searching for the sequence. A value of `true` means that the chart has found the sequence.

• `status` records the status of the sequence recognition. This value ranges from 0 to 4 and indicates the number of symbols detected by the chart.

In the Moore chart, the outputs depend only on the current state of the chart. This chart computes its output values in the state actions. At each time step, the chart executes the actions for the current state, evaluates the input `u`, and makes the transition to a new state.

In the Mealy chart, the outputs depend on the current state of the chart and the input. This chart computes its output values in the condition actions of its transitions. At each time step, the chart evaluates the input `u`, makes the transition to a new state, and executes the corresponding condition actions.

When you simulate the model, the `seqFound` scope shows that the output of the Moore chart lags one time step behind the output of the Mealy chart. The delay is a result of the Moore semantics, in which the output is based on the state of the chart at the start of each time step and not on the current input.

### Reference

Katz, Bruce F. Digital Design: From Gates to Intelligent Machines, 2006.