theory of computation case study topics

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Theory of Computation (TOC) studies the fundamental strengths and limits of computation, how these strengths and limits interact with computer science and mathematics, and how they manifest themselves in society, biology, and the physical world.

At its core, TOC investigates tradeoffs among basic computational resources. These resources include computation time, space, communication, parallelization, randomness, quantum entanglement, and more. As computational systems come in many forms and the goals of computation are diverse, TOC studies the limits of computation in its many manifestations. These are determined by what access we have to the computation’s input: do we have access to it as a whole, or does it come as a stream; as samples from a distribution; in encrypted form; or in fragments? Limits are also determined by the environment within which the computation takes place. Beyond the architecture and connectivity of the computational environment determining where the data is produced and stored and where the computation takes place, we are interested in the presence of other forces, such as adversaries who might want to eavesdrop on the computation, or strategic parties who want to influence the computation to their benefit.

Moreover, computation takes place both in systems that are explicitly computational but also systems that are not explicitly computational, such as biological systems, the human brain, social networks, and physical systems. As such, TOC provides a scientific lens with which to study such systems, and the study of these systems motivates new models of computation and computational tradeoffs, to be studied in turn by TOC.

MIT’s TOC faculty research an unusually broad spectrum of both core TOC and interdisciplinary topics, including algorithms, optimization, complexity theory, parallel and distributed computing, cryptography, computational economics and game theory, computational algebra and number theory, computational geometry, quantum computation, computational biology, machine learning, statistics, and numerical computation.

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Superurop poster showcase, doctoral thesis: integrated hardware security for practical and low overhead protections.

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Introduction of Theory of Computation

Automata theory (also referred to as the Theory Of Computation) is a branch of Computer Science and Mathematics that studies how machines compute functions and solve problems. This field is mainly focused on mathematical structures called automata and is crucial for the purpose of studying processes occurring in discrete systems.

What is Automata Theory?

In automata theory, scientists and engineers can predict the behavior of computing systems thereby improving problem-solving approaches. Originally developed to describe and explain the dynamics of systems, automata theory is the theoretical base of the formal languages theory, grammar , and computational complexity.

Basic Terminologies of Theory of Computation

Now, let’s understand the basic terminologies, which are important and frequently used in the Theory of Computation.

A symbol (often also called a character ) is the smallest building block, which can be any alphabet, letter, or picture.

Alphabets (Σ)

Alphabets are a set of symbols, which are always finite .

String

A string is a finite sequence of symbols from some alphabet. A string is generally denoted as w and the length of a string is denoted as |w| .

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The Theory of Computation explores automata, languages, and complexity. If you want to dive deeper into this subject for GATE, the GATE CS Self-Paced Course covers it extensively.

Closure Representation in TOC

L + : It is a Positive Closure that represents a set of all strings except Null or ε-strings.

L * : It is “ Kleene Closure “, that represents the occurrence of certain alphabets for given language alphabets from zero to the infinite number of times. In which ε-string is also included.

From the above two statements, it can be concluded that:

Note: Σ* is a set of all possible strings(often power set(need not be unique here or we can say multiset) of string) So this implies that language is a subset of Σ*.This is also called a “Kleene Star” .

Kleene Star is also called a “Kleene Operator” or “Kleene Closure” . Engineers and IT professionals make use of Kleene Star to achieve all set of strings which is to be included from a given set of characters or symbols. It is one kind of Unary operator. In Kleene Star methodology all individual elements of a given string must be present but additional elements or combinations of these alphabets can be included to any extent.

A language is a set of strings , chosen from some Σ* or we can say- ‘ A language is a subset of Σ* ‘ . A language that can be formed over ‘ Σ ‘ can be Finite or Infinite .

It is an important branch of computation that is concerned with formal languages, and automata theory in particular. It provides a basis for other courses such as Turing machines and computational complexity that are very important in computer science.

Introduction of Theory of Computation – FAQs

What is the relevance of the automata theory in computer science.

Automata theory is used in modeling computational problems hence enhancing the understanding and design of systems such as compilers, interpreters among others.

what is the purpose of using Kleene Star in the study of formal languages?

The Kleene Star extends symbols from a given alphabet where one is able to create infinite strings from it or even the null string.

Is it possible to implement automata theory into real life?

Of course, automata theory has found its use in certain areas like compiler design, artificial intelligence, network security and natural language processing.