Theory of Computation Basics

Study Notes

The theory of computation deals with three central areas: automata, computability, and complexity.
These areas are linked by the question: "What are the fundamental capabilities and limitations of computers?"

Automata theory deals with definitions and properties of mathematical models of computation.
Models of computation include finite automaton, used in text processing, compilers, and hardware design.
Context-free grammar is another model, used in programming languages and artificial intelligence.
Automata theory is a good starting point for studying the theory of computation as it introduces concepts relevant to other areas of computer science.

Computability theory classifies problems as solvable or not solvable by computers.
During the first half of the 20th century, mathematicians like Kurt G¨odel, Alan Turing, and Alonzo Church discovered that certain basic problems cannot be solved by computers.
One example is determining whether a mathematical statement is true or false, which seems like a natural task for solution by computer but cannot be performed by a computer algorithm.

Complexity theory classifies problems as easy or hard based on their computational difficulty.
Researchers have developed an elegant scheme for classifying problems according to their computational difficulty.
This scheme can be used to demonstrate that certain problems are computationally hard, even if it's unable to prove that they are.
Cryptography is an area that has been directly affected by complexity theory, as it requires hard computational problems to design secure codes.

Theories of computability and complexity are closely related, with computability theory introducing concepts used in complexity theory.

Deals with definitions and properties of mathematical models of computation.
Models, such as finite automata and context-free grammars, play a role in applied areas of computer science, including text processing, compilers, and hardware design.
Automata theory is an excellent place to begin the study of the theory of computation.

Classifies problems as easy or hard based on computational difficulty.
Researchers have discovered an elegant scheme for classifying problems according to their computational difficulty.
Complexity theory has pointed cryptographers in the direction of computationally hard problems around which they have designed revolutionary new codes.
One applied area directly affected by complexity theory is cryptography, which requires hard computational problems to ensure secure secret codes.

Developed by mathematicians such as Kurt Gödel, Alan Turing, and Alonzo Church, who discovered that certain basic problems cannot be solved by computers.
One example is determining whether a mathematical statement is true or false, which no computer algorithm can perform.
This theory led to the development of ideas concerning theoretical models of computers, eventually helping to construct actual computers.

The course focuses on three central areas of the theory of computation: automata, computability, and complexity.
These areas are linked by the question: "What are the fundamental capabilities and limitations of computers?"
Each area interprets this question differently, and the answers vary according to the interpretation.