4 Types and Advantages of Theory of Computation

• Bhumika Dutta
• Jul 19, 2021
• Information Technology

Theory of Computation

Theory of Computation is a branch of computer science and Mathematics that focuses on the logic of computation and how different problems are solved using algorithms and the efficiency of the solutions.

It is a scientific control concerned with the investigation of computing features such as natural, artificial, and otherwise fictitious. Most importantly, it intends to become acquainted with the atmosphere of resourceful computation.

Theory of Computation is very important as it helps in writing efficient algorithms that operate on computer devices, research and development of programming languages and in compiler design and construction that is efficient. The primary goal of creating this theory was to broaden approaches for explaining and analysing the active performance of discrete systems.

The knowledge of Theory of Computation is critical for its applications, which include the construction of intelligent technology, cognitive psychology, and philosophy, as well as diverse models of computing such as algorithm, compiler, and VLSI design, etc.

(Must read: What is computational science?)

Types of Theory of Computation

This branch is divided into 4 parts, namely:

• Automata Theory

• Formal Language

• Computability theory

• Complexity theory.

Let us briefly discuss all of them in the article.

1. Automata Theory

Automata Theory is a theoretical branch of Computer Science and mathematics and deals with the study of complex computational problems and abstract machines. The word Automata is derived from the word “Automaton” which is closely related to the word “Automation”.

Automata are machines that accept a string as input and process it through a finite number of states before reaching the end state. The primary objective for creating automata theory was to create tools for describing and analysing the dynamic behaviour of discrete systems.

The basic terms frequently used in Automata Theory are:

• Symbols:  These are either individual objects or separate entities. These can be any letter, alphabet or any picture.

• Strings: These are a finite collection of symbols from the alphabet, and are denoted by w.

• Language: A collection of appropriate strings is called a language. A language can be Finite or Infinite.

An automaton with a finite number of states is called a Finite Automaton (FA) or Finite State Machine (FSM). An elaborate discussion about Finite Automata is given below

(Also read: 7 branches of discrete mathematics)

Finite Automata

Finite Automata, also known as the Finite State Machine, is a simple machine that is able to recognize patterns. It is an abstract machine with five components or tuples. It contains a set of states and rules for going from one state to the next, but it is dependent on the input symbol used. It is essentially an abstract representation of a digital computer.

Finite automata has two states: Accept State or Reject State.

Types of Finite Automata

Two types of Finite Automata are there:

• Deterministic Finite Automata (DFA): In DFA, the computer only travels to one state for each input character. Here, the uniqueness of the calculation is referred to as deterministic.  The null move is not accepted by DFA.

• Non-deterministic Finite Automata (NFA): NFA is used to send any number of states for a certain input. It is capable of accepting the null move.

1. Formal Language Theory

Formal Language Theory is a branch of Computer Science and Mathematics dealing with representing languages as a collection of operations on an alphabet. Automata is used to generate and recognize different formal languages, hence they are closely related.

It was first initiated by Noam Chomsky in the 1950s. The field of formal language studies is concerned with the syntax of languages and their internal structural patterns. Due to the rise of linguistics in the field of formal language, the syntactic regularities of natural languages now have a means for comprehension.

In computer science, formal languages are used to define the grammar of programming languages as well as formalised versions of subsets of natural languages in which the words of the language represent concepts with specific meanings or semantics. As discussing natural language, read more about Introduction to Natural Language Processing)

The Chomsky Hierarchy

A Formal language is a set of sequences or strings over some finite vocabulary identified with words, morphemes or sounds. There are four types of languages in the Chomsky Hierarchy:

• Computably Enumerable Languages: The languages that can be defined by any formal grammar are known as Computably Enumerable Languages. Any formal or algorithmic procedure can be expressed by grammar. For eg- derivation of logic, rules of chess etc. All computably enumerable languages are semi-decidable.

• Context-Sensitive languages: Grammars where the left hand side of each rule is longer than the right hand side are called Context-sensitive languages. The specification of this class of grammars assures that a decision method for the membership problem is instantly established. Eg- set of all prime numbers, set of all square numbers etc.

• Context-free languages: These are languages defined by context-free grammar. In this case, the non-terminals can be read as syntactic category names, and the arrow ‘ ' can be interpreted as ‘consists of.' As a result, the derivation of a string x in such a language enforces an implicit hierarchical structure of x into ever bigger sub-strings. For this particular reason, context free languages are often referred to as phrase structure languages.

• Regular languages: The languages that are defined by regular grammar are known as regular languages. Regular grammars are also context-free where the non-terminals can be seen as category symbols and the arrow as consists of. Non-terminals, according to another natural meaning, are the names of an automaton's states. (source)

1. Computability Theory

Computability theory, also known as Recursion Theory, is a branch of Mathematics and Computer Science that is primarily concerned with the extent to which an issue may be solved by a computer. It originated in the 1930s with the study of computable functions and Turing degrees.

The statement that a Turing computer cannot solve the halting issue is one of the most significant conclusions in computability theory because it is an example of a concrete problem that is both straightforward to define and impossible to solve with a Turing machine. The halting issue result serves as the foundation for most of computability theory.

(Recommended read: What is Compliance Testing?)

Turing Computability

A set of natural numbers is said to be a computable set if, given a number n, a Turing machine stops with output 1 if n is in the set and stops with output 0 if n is not in the set. A function f from natural numbers to natural numbers is a computable or recursive function if a Turing machine stops and returns output f on input n. (n).

1. Computational Complexity Theory

Computational Complexity Theory is that branch of Theory of Computation that classifies computational problems  according to their resource usage. These computational problems are solved by different algorithms.

An issue is considered inherently complex if its solution necessitates considerable resources, regardless of the method utilised. This intuition is formalised by the theory, which introduces mathematical models of computing to analyse these issues and measure their computational complexity, i.e., the amount of resources required to solve them, such as time and storage.

There are major aspects of Computational Complexities:

• Time Complexity: The amount of time or the number of steps taken by the computation to be performed.

• Space Complexity: The amount of memory required to perform the computation.

Computer scientists describe the time or space necessary to solve the issue as a function of the size of the input problem in order to assess how much time and space a specific method takes.

(Suggested blog: What is Group Theory? Properties (Axioms) and Applications)

There are many advantages of the Theory of Computation. Some of them are listed below:

1. Theory of Computation deals with how efficiently any algorithm would solve any computational problem. Also, abstract machines are introduced in the Computational theory, which are defined mathematically. Hence, the algorithms would not need to change every time any physical hardware gets enhanced.

2. There is a massive amount of work that has been made possible in the portion of NLP (Natural Language Processing) that involves the construction of FSMs (Finite State Machines), also known as FSA (Finite State Automata).

3. Theory of Computation has helped in many fields such as Cryptography, Design and Analysis of Algorithm, Quantum Calculation, Logic within Computer Science, Computational Difficulty, Randomness within Calculation and Correcting Errors in Codes

4. A computational model can cope with complexity in ways that verbal arguments cannot, resulting in satisfactory answers for what would otherwise be ambiguous hand-wavy arguments. Furthermore, computational models can manage complexity at several levels of analysis, allowing data from various levels to be integrated and connected.

(Read also: What is System Analysis and Design?)

Conclusion

Theory of Computation and Automata are simple subjects of Computer Science and Engineering that attempt a deep understanding of computational problems and analysis. It makes us understand the establishment of formal mathematical models of computing that mirror the computer's real-world. It also helps us to understand different algorithms, hardwares and softwares. Hence it is one of the most essential and core areas to study for computer engineers or aspirants.

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