Formal Languages and Automata Theory: Decidability and Recognizability, Slides of Theory of Automata

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Formal Languages and Automata Theory

Outline

• Examples for

– Decidable

– Undecidable but Recognizable

– Unrecognizable

Example 1

-- Is it decidable?

Proof: Decidable. Construct a TM M as follows: The shaded area: L(A) L(B)

Example 2

-- Is it decidable?

 Mw 

A decider

of target

problem

Construct

M

w The answer: NOT decidable. The difficulty: How to prove a language to be not decidable? accept reject The target problem is embedded. < M > w

How to construct M

w

M halts on

( )

w

L M is infinite

if and

only if

w

( )  w L M Our Target … Construction On any input string: s Simulate M on w ; If M halts, accept s , else reject s , end

Example 2

-- Is it decidable?  NO.

Construct

M

w

D

〈 M 〉 Accept,^ if L ( M )^ finite Reject, if L ( M ) NOT finite D 〈 Mw 〉 〈 M 〉 w

HALTTM = {(〈 M 〉, w ): M is a TM that halts on input

w } Accept, if M halts Reject, if M does not halt reject accept M^ halts on L ( Mw )^ is infinite if and only if w

L ( Mw )

How to construct M

w

M halts on

( ) w L M^ Contains two equal length strings

if and

only if

w

L ( Mw ){ a , b } Our Target … Construction On any input string: s Simulate M on w ; If M halts, accept if s=a or s=b else reject s , end

Example 4

-- Is it decidable?

Construct

M

w Proof: Suppose we have a TM D such that

D

〈 M 〉 Accept,^ if … D 〈 Mw 〉 〈 M 〉 w Accept, if M accept w A TM = {(〈 M 〉, w ): M is a TM that accepts w } Reject, if NOT … Reject, if not