Lecture

Recursive Enumerability: Turing Machines and Undecidable Languages

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Description

This lecture discusses the concept of recursively enumerable languages and the role of Turing machines in accepting these languages. The instructor begins by explaining the process of simulating a Turing machine on various input words, illustrating how all combinations can be explored without omission. The discussion progresses to the existence of languages that are not recursively enumerable, specifically constructing a language L0 that lacks an associated Turing machine. The instructor employs diagonalization to demonstrate that L0 cannot be accepted by any Turing machine. Furthermore, the lecture explores the complement of L0, showing that while it can be enumerated, it does not have a Turing machine that decides it. The instructor concludes by highlighting the implications of these findings on the understanding of decidable and undecidable problems in computation, particularly focusing on the halting problem and the universal language, which cannot be decided by any Turing machine.

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Ontological neighbourhood

Biology

Cell biology: Cell cycle

Theoretical computer science

Theory of computation: Computability theory

Mathematics

Mathematical logic: Set theory

Related concepts (32)

Recursively enumerable language

In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages.

Recursive language

In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a Turing machine that, when given a finite sequence of symbols as input, always halts and accepts it if it belongs to the language and halts and rejects it otherwise.

Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.

Undecidable problem

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run. A decision problem is a question which, for every input in some infinite set of inputs, answers "yes" or "no"..

Computability

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power.

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