Prove whether the set of all proofs within any formal system is decidable, recognizable, or unrecognizable

Solutions

Expert Solution

A Turing machine M is said to recognize a language L if L = L(M).
A Turing machine M is said to decide a language L if L = L(M)
and M halts on every input.
L is said to be Turing-recognizable (Recursively Enumerable (R.E.)
or simply recognizable) if there exists a TM M which recognizes L.
L is said to be Turing-decidable (Recursive or simply decidable) if
there exists a TM M which decides L.

But not all languages are decidable! We will show:
Atm = {<M,w >|M is a TM and M accepts w } is undecidable.
However Atm is Turing-recognizable!

the answer has been provided below:

venereology answered 5 months ago

Next > < Previous

Related Solutions

PROOFS: 1. State the prove The Density Theorem for Rational Numbers 2. Prove that irrational numbers are dense in the set of real numbers

PROOFS: 1. State the prove The Density Theorem for Rational Numbers 2. Prove that irrational numbers are dense in the set of real numbers 3. Prove that rational numbers are countable 4. Prove that real numbers are uncountable 5. Prove that square root of 2 is irrational

Prove or Disprove The set of all finite strings is undecidable. The set of all finite...

Prove or Disprove The set of all finite strings is undecidable. The set of all finite strings is recognizable

Show epsilon arguments for any limit proofs: 1)Prove: If limf(x)x->a=L and c∈R, then limx->acf(x)=cL. 2) Prove:...

Show epsilon arguments for any limit proofs: 1)Prove: If limf(x)x->a=L and c∈R, then limx->acf(x)=cL. 2) Prove: If lim f(x)x->a = L and lim g(x)x->a = M, then lim(f(x)+g(x))x->a = L+M. 3) Find a counterexample to the converse of #2. 4) Prove: If lim f(x)x->a = L and lim g(x)x->a = M, then lim(f(x)g(x)) x->a= LM. 5) Find a counterexample to the converse of #4.

Prove that finding whether there are any states which will never be entered on any given...

Prove that finding whether there are any states which will never be entered on any given input is decidable in the case of a PDA, and, undecidable in case of a Turing machine.

Prove that isomorphism is an equivalent relation on the set of all groups.

Outline a set of ten (10) specific competencies for any formal position on a management negotiation...

Outline a set of ten (10) specific competencies for any formal position on a management negotiation team. Please describe in one sentence the specific competency and provide two specific & concrete examples of how the person would use the competency during the collective bargaining process.

Please use any of the methods to prove whether each of the following arguments is valid...

Please use any of the methods to prove whether each of the following arguments is valid or invalid. For each problem, please identify the method that you have decided to employ and make sure to show your work. 1. It is obvious that nuclear energy is needed. Nuclear energy is needed if and only if solar energy cannot be harnessed. And it is also true both that solar energy can be harnessed only if funds to do so are available,...

Prove that if A is an enumerable set all of whose members are also enumerable sets,...

Prove that if A is an enumerable set all of whose members are also enumerable sets, then UA is also enumerable.

DISCRETE MATH 1.Prove that the set of all integers that are not multiples of three is...

DISCRETE MATH 1.Prove that the set of all integers that are not multiples of three is countable.

Prove for the following: a. Theorem: (Cantor-Schroder-Bernstein in the 1800s) For any set S, |S| <...

Prove for the following: a. Theorem: (Cantor-Schroder-Bernstein in the 1800s) For any set S, |S| < |P(S)|. b. Proposition N×N is countable. c. Theorem: (Cantor 1873) Q is countable. (Hint: Similar. Prove for positive rationals first. Then just a union.)

Subjects