# Questions tagged [decidability]

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### Why we can reduce $A_{TM}$ to $ALL_{CFG}$, but we can not reduce $A_{TM}$ to $E_{CFG}$

If a $PDA$ can be constructed to check whether a string is not a computation history for a Turing Machine. Like in the proof of $ALL_{CFG}$ is not decidable. Then we can construct a $PDA$ that accepts ...
1 vote
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### Reduction from $ALL$ to $DECIDE$

Let $DECIDE=${$<M> :\ M\ halts\ on \ all \ inputs$} and I wish to show its unrecognizable using a reduction from $ALL=${$<M> :L(M)=\Sigma ^*$} using a deterministic turing machine $R$ ...
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1 vote
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### How much is decidability compromised within this restriction of the fixpoint combinator?

Though purely functional programming languages, such as Haskell, is commonly thought to have no side-effects, there is a caveat: Recursive calls may hang. I considered this to be undesirable, and ...
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1 vote
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### Turing-reducibility for guaranteed decider

The following exercise is taken from Theoretical Computer Science by Atiba. Use Rice's theorem to demonstrate that every decidable language is Turing reducible to some language that is already ...
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1 vote
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### A decidable language that can't be decided by a circuit ensemble of linear size

Let Size(O(n)) be the set of languages the can be decided by a circuit ensemble (a sequence of circuits C_i for every natural i s.t input size is i) such that every circuit's size is linear (in input ...
1 vote
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### Mapping Reduction from HALT?

I've been given a task to determine whether L={〈M〉|M is a TM that loops on the input c (a constant)} is decidable. I can prove co-L is recognizable so I figured a reduction from HALT to co-L would ...
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### A program that solves the Halting Problem for programs with N states

My question relates to the conclusions drawn from the Halting Problem. I understand that the Halting Problem proves that no program H(P,i) exists that determines if P(i) halts or not for P in general. ...
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### Reduce A ∶= {x ∈ N ∣ x < 10} to Halting Problem on empty tape

I am preparing for an exam in computability and still learning about the idea of reductions. I found an interesting problem to start with and am curious if my approach is correct: Let H0 be the ...
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### Proving a language is recursively enumerable

Prove that the following language is recursively enumerable: L = {<M,x> | Turing machine M enters the same configuration twice on input x} I have tried to construct a TM that maintains the ...
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### How can decidability/semi-decidability/undecidability have impact on REAL life applications/examples?

How decidability/semi-decidability/undecidability has impact on REAL life applications/examples? I understand that it can be used for implementation of various algorithms but what else?
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### Prove that the problem of REGEX producing strings with 111 as substring is decidable

I have been given the following problem and was wondering if my solution is correct (taken from the textbook exercise in the book Introduction to the Theory of Computation by Martin Sipser): Given <...
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### Loop in control flow graph of a program versus checking whether program goes into infinite loop or not

Which of these is true or false. 1)If Control Flow graph(CFG) doesn't contain any loop, the program will never go into infinite loop. 2)If CFG contains loop, the program may or may not go into ...
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### Prove that $L = \{ \langle M \rangle | \text{ M is a PDA, L(M) contains at least 1 string w that } |w| \leq n \}$ is recursive?
Description Similar to the encoding of a Turing Machine, we can encode a Push-Down Automata. Denote $\langle M \rangle$ as the encoding of PDA M, and a natural number n, is language \$L = \{ \langle M \...