Computational Theory Assignment Help

The theory of computation, also known as automata theory is a computer science branch that is concerned with designing abstract and self-propelled computing devices. These devices should automatically follow a predetermined sequence of operations. A finite automaton is a kind of automaton which has a finite number of states. This article will introduce you to the fundamental concepts of finite automata, regular languages, pushdown automata, and Turing machines and decidability.

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What is Automata?

Automata means “self-acting” in Greek. From this, we can define an automaton (the singular of automata) as an abstract self-propelled computing device that automatically follows a sequence of operations that have been predetermined. A Finite State Machine or Finite Automaton is an automaton with a finite number of states.

Definition of related terminologies

  • Alphabet – It refers to any finite set of symbols
  • String – refers to a finite sequence of symbols taken from an alphabet set
  • Length of a string – This is the number of symbols present in a string
  • Kleene star – This is the unary operator on a set of symbols or strings. It gives the infinite set of all possible strings of all possible lengths.
  • Language – This is the subset of Kleene star for some alphabet. It can be infinite or finite.

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There are two types of finite automaton:

  • Deterministic Finite Automaton (DFA)

In this type of finite automaton, one can determine the state to which the machine will move for each input symbol. It is for this reason why it is called a deterministic automaton. Since it has a finite number of states, hence, such a machine is called deterministic finite automaton or deterministic finite machine.

Graphical representation of a DFA

State diagrams, a type of digraphs are used to represent a DFA.

  • The states represent the vertices
  • The transitions are shown by the arcs labeled with an input alphabet
  • An empty single incoming arc denotes the initial state
  • The double circles indicate the final state

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  • Non-deterministic Finite Automaton

A machine in non-deterministic finite automaton can move to any combination of the states in the machine for a particular input symbol. In simple terms, you cannot determine the exact state to which the machine moves. Since the machine has a finite number of states, it is called a non-deterministic Finite Automaton or non-deterministic Finite Machine.

A non-deterministic finite automaton is obtained if unpredictability is introduced in a model. This can be represented as shown in the figure below. This figure represents one of the simplest examples of non-determinism in theoretical computer science.

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DFA vs NDFA

Our computational theory project help professionals have outlined the differences between DFA and NDFA in the table below

DFA NDFA
It is called DFA because, for each input symbol, the transition from a state is to a particular next state. It is called NDFA because, for each input symbol, the transition from a state can be to multiple next states
In DFA, you cannot use empty string transitionsEmpty string transitions are permitted in NDFA
DFA allows backtrackingBacktracking is not possible in NDFA
DFA requires more spaceLess space is required in NDFA
DFA accepts strings that transit to a final stateNDFA accepts strings which at least one of all possible transitions ends in a final state

Acceptors, Classifiers, and Transducers

Acceptor (recognizer)

An acceptor is an automaton that handles or computes a Boolean function. An acceptor can either accept or reject  the inputs given to it.

Classifier

A classifier gives a single output when it terminates and has more than two final states.

Transducer

A transducer is an automaton whose output is based on the current input and/or previous state. They can be of two types:

  • Mealy Machine – The output of this type of transducer depends on both the current input and current state.
  • Moore Machine – The out of a Moore machine depends only on the current state.

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How do you convert NDFA to DFA?

Here is the algorithm that converts an NDFA to its equivalent DFA

Algorithm

Input is an NDFA

The output should be an equivalent DFA

  1. Create a state table from the given NDFA
  2. Next, you should create a blank state table under possible input alphabets for the equivalent DFA
  3. You should mark the start state of the DFA by q0. The same should be done for DFA
  4. For each possible input alphabet, find out the combination of states
  5. Apply step 4 again every time you generate a new DFA state under the input alphabet columns. Otherwise, go to step six.
  6. The final states of the equivalent DFA are the states which contain any of the final states of the NDFA.

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Using Myhill-Nerode theorem in DFA Minimization

Algorithm

Input should be the DFA

The output should be the Minimized DFA

  1. You should draw a table for all pairs of states (Qi, Qj). They should not necessarily be directly connected. Initially, they are all unmarked.
  2. Take into account every state pair in the DFA and mark them. Set the final states to F.
  3. Repeat step two until you cannot mark any more states. Suppose there is an unmarked pair, mark it if the pair is marked for some input alphabet.
  4. Lastly, you should combine all the unmarked pairs (Qi, Qj) and make them a single state in the reduced DFA.

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DFA Minimization using the Equivalence Theorem

Suppose we have two states, X and Y in a DFA, we can combine them into {X, Y} if they are not distinguishable. Two states are said to be distinguishable if there is one string such that one is accepting state is accepting and the other one is not. For this reason, we can only say a DFA is minimal if and only if all the states are distinguishable.

Algorithm

  1. Divide all states into two partitions, final states, and non-final states. They should also be denoted by P0. You should remember that all states in a partition are 0th After that, you should take a counter k and initialize it with 0.
  2. Next, you should increment k with 1. Divide the states in PK into two partitions if they are k distinguishables, for each partition in PK. We can say that two states in the partition, X and Y are distinguishable if they have any input S.
  3. Repeat step 2 if PK is not equal to PK-1. Otherwise, you should proceed to step 4.
  4. Lastly, combine the Kth equivalent sets and make them the new states of the reduced DFA.

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