notes16 Undecidability and Unrecognizability

notes16 Undecidability and Unrecognizability - CS 373:...

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Unformatted text preview: CS 373: Theory of Computation Gul Agha Mahesh Viswanathan Fall 2010 1 1 Undecidability 1.1 Recap Decision Problems and Languages • A decision problem requires checking if an input (string) has some property. Thus, a decision problem is a function from strings to boolean . • A decision problem is represented as a formal language consisting of those strings (inputs) on which the answer is “yes”. Recursive Enumerability • A Turing Machine on an input w either (halts and) accepts, or (halts and) rejects, or never halts. • The language of a Turing Machine M , denoted as L ( M ), is the set of all strings w on which M accepts. • A language L is recursively enumerable/Turing recognizable if there is a Turing Machine M such that L ( M ) = L . Decidability • A language L is decidable if there is a Turing machine M such that L ( M ) = L and M halts on every input. • Thus, if L is decidable then L is recursively enumerable. Undecidability Definition 1. A language L is undecidable if L is not decidable. Thus, there is no Turing machine M that halts on every input and L ( M ) = L . • This means that either L is not recursively enumerable. That is there is no turing machine M such that L ( M ) = L , or • L is recursively enumerable but not decidable. That is, any Turing machine M such that L ( M ) = L , M does not halt on some inputs. Big Picture 2 Regular L n 1 n Decidable Recursively Enumerable Languages Figure 1: Relationship between classes of Languages Machines as Strings • For the rest of this lecture, let us fix the input alphabet to be { , 1 } ; a string over any alphabet can be encoded in binary. • Any Turing Machine/program M can itself be encoded as a binary string. Moreover every binary string can be thought of as encoding a TM/program. (If not the correct format, considered to be the encoding of a default TM.) • We will consider decision problems (language) whose inputs are Turing Machine (encoded as...
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This note was uploaded on 10/04/2011 for the course CS 373 taught by Professor Viswanathan,m during the Fall '08 term at University of Illinois, Urbana Champaign.

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notes16 Undecidability and Unrecognizability - CS 373:...

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