PPT-Turing Machines Recursive and Recursively Enumerable Languages

Turing Machine 1 TuringMachine Theory The purpose of the theory of Turing machines is to prove that certain specific languages have no algorithm Start with a language about Turing machines themselves. They provide a precise formal de64257nition of what it means for a function to be computable Many other de64257nitions of computation have been proposed over the years for example one could try to formalize precisely what it means to run a program They provide a precise formal de64257nition of what it means for a function to be computable Many other de64257nitions of computation have been proposed over the years for example one could try to formalize precisely what it means to run a program (Chapter . 4.2). H. éctor Muñoz-Avila. Undecidable Languages. We are going to make 2 proofs:. An . existence. proof:. We show that a . language L . must exist that cannot be . decided/recognized with . Machines. Recursive and Recursively Enumerable Languages. Turing Machine. 1. Turing-Machine Theory. The purpose of the theory of Turing machines is to prove that certain specific languages have no algorithm.. Theory of Computation Lecture 16: A Universal Program VII. 1. Recursively Enumerable Sets. Definition:. The set B  N is called . recursively enumerable. and . Structural Induction. ICS 6D. Sandy . Irani. Recursive Definitions. A recursive definition defines a sequence or set in terms of smaller instances.. A . recursively defined sequence . (. recurrence relations. Machines. Recursive and Recursively Enumerable Languages. Turing Machine. 1. Turing-Machine Theory. The purpose of the theory of Turing machines is to prove that certain specific languages have no algorithm.. A language is called . Recursively Enumerable. if there is a Turing Machine that accepts on . any. . input within the language.. Reminder. : A language is called . Recursive. if there is a Turing Machine that accepts on . Universalities. Sources. Material in this lecture comes from, “Handbook of Natural Computing,” Editors . Grzegorz. Rosenberg, Thomas Back and . Joost. N. . Kok. , Springer 2014. . Combining functions. The Chomsky Hierarchy. Type. Language. Grammar. Automaton. 0. Recursively Enumerable. Unrestricted. DTM. - NTM. 1. Context Sensitive. Context. Sensitive. Linearly Bounded Automaton. 2. Context Free. Learning . Objectives. At the conclusion of the chapter, the student will be able to:. Explain and differentiate the concepts of computability and decidability. Define the Turing machine halting problem. Section 5.3. Section Summary. Recursively Defined Functions. Recursively Defined Sets and Structures. Structural Induction. Generalized Induction. Recursively Defined Functions. . Definition. : A . The Turing Test Minds & Machines Alan Turing British mathematician known for: Turing Machines (1936) Breaking German Enigma (WWII) Turing Test (1950) ? “I propose to consider the question, 'Can machines think?' This should begin with definitions of the meaning of the terms 'machine 'and 'think'. … [But] Instead of attempting such a definition I shall replace the question by another... The new form of the problem can be described in terms of a game which we call the 'imitation game'.“ Halting Problem. Your Questions?. Previous class days' material. Reading Assignments. HW . 15 . problems. Anything . else. Key ideas so far . Let . U. be . {. (. ,. . ). , . a. , . q. , . y.