Questions tagged [decidability]
The decidability tag has no usage guidance.
45
questions
0
votes
1
answer
10
views
L is a recognizable undecidable language ,M is a Turing machine that recognizes L, does M reject or infinitely loop for s belonging to L-complement?
If $L$ is a decidable language, $M$ is a Turing machine that determines $L$. For $\forall s \in L$, M accepts, and for $\forall s \in \overline{L}$, M rejects
However, my question is that
If $L$ is a ...
1
vote
0
answers
35
views
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 ...
0
votes
1
answer
74
views
Is the infinite union of decidable languages decidable?
I am currently struggling with figuring out the following problem:
Given decidable languages L1, L2, L3, L4, ...
Is the infinite union of Languages L1, ...... decidable? I have an intution that it is ...
-1
votes
1
answer
29
views
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 \...
0
votes
1
answer
29
views
Show that Lu is m-reducible to the language L = {⟨M, x⟩ | M(x) terminates with an empty tape}
Question: Given a language L, L = {⟨M, x⟩ | M(x) terminates with an empty tape}, show that Lu is m-reducible to L by finding a computable function f: Σ* -> Σ*, where for every w, w ∈ Lu if and only ...
1
vote
1
answer
47
views
Is it possible to determine if a 0-arity function [a program with no input] will always terminate
The halting problem concerns programs which take input.
By framing the halting problem on the diagonal argument it is clear why this is so.
What about programs with no input, constant functions.
Can ...
1
vote
1
answer
62
views
Is it computable to find the cardinality of intersection of two recursively enumerable sets?
I am well aware that recursively enumerable sets (which are subsets of $\mathbb N$) are closed under intersection. What is more interesting is whether or not the cardinality of the intersection is ...
0
votes
1
answer
65
views
Are there any formal systems or programming languages in which its only possible to define functions that have inverses?
Consider an algorithm $f(x)$.
Are there formal systems or programming languages that only allow $f(x)$ to be defined if $f^-1(x)$ exists?
1
vote
1
answer
28
views
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 ...
1
vote
1
answer
150
views
Language of Turing machines that go through some configuration infinitely many times on empty input
I've been going through some questions on old homework. Here was a question that confused me somehow.
Question: Given a language $$L=\{\langle M\rangle\ |\ M \text{ is a Turing machine. } M \text{ ...
0
votes
1
answer
92
views
Decidability of a context free Grammar
Say that a Context Free Grammar is red when it accepts every word of length 3 that begins with a, and extremely red when it accepts every word that begins with a.
Is redness decidable? or Semi ...
1
vote
1
answer
105
views
A decidable language that can't be decided by a circuit ensemble of linear size
Let Size(O(n)) be the set of languages the can be decided by a circuit ensemble (a sequence of circuits C_i for every natural i s.t input size is i) such that every circuit's size is linear (in input ...
0
votes
0
answers
69
views
Prove that the problem of REGEX producing strings with 111 as substring is decidable
I have been given the following problem and was wondering if my solution is correct (taken from the textbook exercise in the book Introduction to the Theory of Computation by Martin Sipser):
Given <...
1
vote
1
answer
217
views
For any two languages A and B there exists J such that both A and B are Turing reducible to J
Here is the my attempt:
Proof : Suppose $J = \{aa' \mid a \in A\} \cup \{bb' \mid b \in B\}$ such that $a'$ and $b'$ are the symbols that do not belong to any $w \in A \cup B$ and $a,b$ are strings.
...
1
vote
0
answers
360
views
Prove that determining if a PDA has an infinite language is decidable
I have been given the following problem and was wondering if my solution is correct (taken from the textbook exercise in the book Introduction to the Theory of Computation by Martin Sipser):
Given $$\...
1
vote
0
answers
48
views
Prove that the problem of CFG producing epsylon 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 $$\...
0
votes
2
answers
472
views
Show that ALL DFA is decidable
I have been given the following problem and was wondering if my solution is correct (taken from the textbook exercise in the book Introduction to the Theory of Computation by Martin Sipser):
Given $$\...
2
votes
1
answer
94
views
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 ...
1
vote
0
answers
44
views
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 ...
-2
votes
1
answer
37
views
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?
2
votes
1
answer
459
views
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 ...
0
votes
1
answer
21
views
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∗ ...
0
votes
0
answers
39
views
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 $L_1 \leq_m L_2$, ...
0
votes
1
answer
43
views
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 ...
0
votes
0
answers
95
views
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, ...
0
votes
0
answers
59
views
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
1
answer
123
views
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 ...
-1
votes
1
answer
53
views
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 ...
0
votes
1
answer
146
views
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-...
0
votes
0
answers
35
views
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 ...
0
votes
1
answer
235
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 ...
0
votes
1
answer
163
views
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 ...
0
votes
2
answers
369
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 ...
0
votes
1
answer
153
views
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?
0
votes
1
answer
89
views
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 ...
0
votes
0
answers
35
views
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 ...
0
votes
0
answers
32
views
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 ...
-1
votes
1
answer
54
views
How to determine whether this language is regular?
I've encountered this question recently: Given $\Sigma=\{\sigma_1, \sigma_2, ..., \sigma_n\}$ and $n\ge 2$, determine whether the following language is regular or not:
$$
L_1=\{w\in\Sigma^*|for \ 1 \...
0
votes
1
answer
42
views
Prove decidable
L={⟨M⟩: M is a DFA and for each string in L(M) the number of 1s is more than or equal to the number of 0s }
T = "On input where M is encoded DFA"
...
0
votes
0
answers
57
views
The role of diagonalization - asymmetry between TM and Recursion Theory
This might be a slightly strange or irrelevant question. My apologies if it is. I'll try to formulate it the best I can.
First, here is an hypothesis: diagonalization is syatematically used to prove ...
3
votes
1
answer
66
views
Effectively decidable vs. noneffectively (or ineffectively) decidable
The introduction of https://www.sciencedirect.com/science/article/pii/0001870882900482 starts with the following sentence:
The word problem for commutative semigroups is effectively decidable.
I ...
1
vote
2
answers
333
views
Prove that { $\langle M \rangle$ : $M$ is a TM and $L(M)$ is decidable} is undecidable
So I want to prove that $$ \big\{\langle M \rangle : \text{ M is a TM and } L(M) \text{ is decidable} \big\}$$ is undecidable.
To do so I want to reduce it from$\ \overline{A_{TM}}$ with a function ...
1
vote
1
answer
61
views
Decidability of $\{⟨G⟩ \mid \text{$G$ is CFG and $L(G) ⊈ \Sigma^+$}\}$
I want to prove that the following language is decidable:
$$\mathit{SEQ}_{\mathit{CFG}} = \{⟨G⟩ \mid \text{$G$ is CFG and $L(G) ⊈ L$}\}, \text{ where } L = \Sigma^* - \{\epsilon\}$$
So, I think about ...
-9
votes
1
answer
547
views
Are the halting problem proofs refuted by software engineering?
This has been completely rewritten today 2022-09-16 to address all of the objections from thousands of reviews in the last 12 months.
Are the halting problem proofs refuted by software engineering ?
<...
4
votes
1
answer
1k
views
A language is Turing recognizable iff it is a projection of a decidable language
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 ...