Questions tagged [decidability]

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If predicate P is partially-decidable, is ¬P decidable, partially decidable or undecidable?

I was learning about decidability when I thought of this question: If P is partially decidable, is ¬P decidable, partially decidable or undecidable? I think it is Undecidable since if ¬P holds then P ...
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A language of natural numbers is decidable iff it is either finite or the image of some strictly increasing computable function

Suppose $L \subseteq \mathbb N$ such that, for the purpose of Turing machine computation, numbers in $L$ are represented by strings over the alphabet $\{0, 1\}$ in the standard binary notation. Prove ...
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1 vote
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Does description method matter in Rice’s theorem?

If $\mathcal{p}$ is a nontrivial property of formal languages, then $L_{\mathcal{p}} = \{ \langle M \rangle \mid L(M) \in \mathcal{p} \}$ is undecidable by Rice’s theorem. What if we describe ...
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Decision problem

Prove the following theorem Let A and B be two languages on an alphabet Σ. If A ≤p B and B ∈ P, then A ∈ P. Could anyone be able to prove it?
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Useless states in a PDA

I am trying to solve a problem in Sipser's Introduction to the Theory of Computation book, which reads: 4.22 A useless state in a pushdown automaton is never entered on any input string. Consider the ...
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Show that $Y \subseteq A^*$ is decidable

Let A be a nonempty alphabet, $X ⊆ A^*$ a decidable set, and $Y ⊆ A^*$ be a semi-decidable set. We assume that $Y ⊆ X$ and that $X \setminus Y ⊆ A^*$ is semi-decidable. Show that then the set Y ⊆ A∗ ...
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If $L_1 \leq_m L_2$, and $L_2$ is decidable, is $L_1$ then decidable?

There is a lemma in our textbook that asks us to prove the following: If $L_1 \leq_m L_2$, and $L_2$ is decidable, then $L_1$ is decidable I tried proving this by saying that if $L_1 \leq_m L_2$, ...
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Can 3-SAT be recognized in less than exponential time?

Obviously it is an open question if $3$-SAT can be decided in a polynomial amount of time. But what results do we know about its recognizabilty? Can $3$-SAT be recognized in a polynomial amount of ...
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Decidability of the language of a regular expression being a subset of a given context free language

Let L be a language of pairs $\langle R,G\rangle$, with the first element being a regex and the second being a CFG. Is it decidable that G accepts whatever the regular expression does? In other words, ...
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Complement of a context free language

Consider the context-free language of balanced parentheses of three kinds: $$L = \{w \in \{ (, ), [,], \{, \} \}^∗ \mid \text{all parentheses in }w \text{ are properly balanced}\}$$ What will be the ...
1 vote
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Decidability for intersection of context free and regular languages

I am wondering if the following are decidable or undecidable and why. L is a CFL and R is a regular language. How does the complement of the context-free language change the decidability of the ...
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Is there any problems with equating Turing Machines with Algorithms and Language with Problems?

In a lot of the online explanation of complexity theory, the author proposes the following. "The definition associated with complexity theory (e.g., definition of NP) is phrased in terms of ...
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Does Turing machine move left on particular input?

We know that RE language is the collection of unrestricted grammar which is known as type-0 grammar that's why emptiness, finiteness of every RE languages is undecidable. My question is how I check ...
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Why is it undecidable to check the emptiness and finiteness of a context-sensitive grammar?

Context-sensitive languages have context-sensitive grammars, and context-free languages have context-free grammars. Using context-free grammars, we can decide the finiteness and emptiness of context-...
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Why linear bounded automata emptiness and finiteness isn't decidable? [duplicate]

We know that linear bounded automata has tape size based on input size which is limited. Context-sensitive languages are accepted by LBA. My question is that if LBA has limited tape size why it's ...
204 views

Why REC languages is undecidable under emptiness and finiteness?

Membership problem of Recursive languages are decidable. My approach: Let $L$ be a recursive language and $M$ be the Turing Machine that accepts it. For string $w,$ if $w ∈ L,$ then $M$ halts in ...
67 views

Whether $L(G)=L(R)$ is decidable for DCFL and CFL?

Let $G_1$ be the context free grammar and $R$ be regular language. Now I have to check whether $L(G_1)=L(R)$ is decidable or not? My approach $\overline{L(G_1)}=\overline{L(R)}$. Now $L(G_1)$ not ...
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Rice theorem could apply except RE language?

You know that Rice theorem is applicable to check decidability of RE language. Also we know that all regular, deterministic context free, context free, recursive languages are RE languages. $Q_1:$ So ...
220 views

Regularity of CFG and DCFL

I read that it is undecidable whether, given a CFG $G$, $L(G)$ is regular. And there exists no algorithm that, given a CFG $G$ such that $L(G)$ is regular, outputs a DFA that accepts $L(G)$. My ...
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Why finiteness problem of CFL is decidable?

We know that every $CFL$ has infinite configuration space. Due to this equality problem is undecidable. But why finiteness property is decidable inspite having infinite configuration space?
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Reduction from undecidability, decidability to decididabilty

If given any two language both $L_1$ and $L_2$ are decidable then why both $L_1\leq_m^\mathsf{}L_2$ and $L_2\leq_m^\mathsf{}L_1$ are false. Please provide easy explanation with any counterexample ...
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Is this language in RE?

Given the following language: $$L=\left \{ <M> | \exists L \in R \quad s.t \quad L(M)\subseteq L \right \}$$ I need to determine it's compuation class(R or RE). I used Rice Theorem as follows to ...
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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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Could H correctly decide that P never halts?

Because a halt decider must compute the mapping from its inputs to an accept or reject state on the basis of the actual behavior of its actual inputs, when H is a simulating halt decider H(P,P) would ...
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I was wondering how to prove that a language $C$ is Turing-recognizable iff a decidable language $D$ exists such that $C = \{x \mid \exists y \;(\langle x, y\rangle \in D)\}$. I do not know how to ...