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Questions tagged [decidability]

Use this tag for questions about the existence of an algorithm that can and will return a correct true or false value to a decision problem.

In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a Boolean true or false value that is correct (instead of looping indefinitely, crashing, returning "don't know," or returning a wrong answer). Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership can exist for them.

307 questions
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Why does the unsolvability of the Halting Problem give a negative solution to the decision problem?

So Hilbert famously asked for a formalization of all of mathematics which was computationally decidable. Gödel is credited with shattering the idea that "all of mathematics" can be formalized. After all, one can't even formalize all of elementary…
Sebastian Oberhoff
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Show that $B$ is decidable

Let $A$ and $B$ be semi decidable languages. Moreover, $A \cup B$ and $A \cap B$ are decidable. I want to show that $B$ is decidable. Firstly, I would like to know if the following approach works: First, we can check if $w \in A \cup B$. Since this…
kklaw
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Is calculability calculable?

I m not at all a specialist so im gonna say how i understand the terms i use even if it's basic for some so we are in the same page. Notation: let $P$ be a turing machine $P(t)$ is the word writen after $t$ steps. let $M$ be a word $M[n]$ is the nth…
Guill Guill
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Show that the following language is decidable

$E_{cfg}=\{ \langle G \rangle \mid G \text{ is context-free grammar}, \ L(G)=\emptyset \}$ I think the following algorithm should work: Test whether or not the input $w$ codes a cfg. If not, reject. When $w$ codes a cfg, apply "output algorithm"…
Ettore
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Largest decidable subsystem of Robinson's arithmetic $\mathbb{Q}$

What is the largest decidable subsystem of $\mathbb{Q}$? Is there any nice axiomatization of it known? Edit: Given Noah Schweber's comment, let me rephrase the question, why is there no largest decidable subsystem of $\mathbb{Q}$?
Ilk
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Could the decidability of a theorem be undecidable?

Given some theorem $T$, could the question "is $T$ decidable?" be undecidable? I assume the answer is yes, and if it is, could the decidability of a theorem be undecidable even if the theorem itself is decidable?
Ty Jensen
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$\mathsf{DSPACE}(n) \subsetneq \mathbb{E}$: there is a decidable language no LBA can handle..

Use the following lemma and diagonalisation: Lemma: Let $M$ be a linear bounded automaton. Then, there is a lba $M'$, that accepts the same language and halts on every input. ($\mathsf{DSPACE}(n)$ is all languages accepted by some linear bounded…
Ettore
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