# Questions tagged [rice-theorem]

Rice's Theorem states that any (non-trivial) property of Turing machines is undecidable.

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### How to prove that the problem $\text{"If$L$is a context-free language, then, is$\overline{L}$also context-free?"}$ is undecidable?

Lately I came across a problem: $\text{"If$L$is a context-free language, then, is$\overline{L}$also context-free?"}$ And I need to comment on its decidability. Now I know that context free ...
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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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### Is there a connection between the Undecidability Theorem and "software complexity"?

I was reading Complexity: The Emerging Science at the Edge of Order and Chaos and a certain passage got me really intrigued. When discussing Chris Langton's explorations of artificial life algorithms,...
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### Halting (on empty input tape) for an infinite subset of all Turing machines

As is well known, there is no single procedure for deciding whether any given Turing machine halts on an empty input tape. This is easily shown, e. g., by applying Rice's theorem. But what if, instead ...
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### Determine if a language is Decidable or semi decidable

Consider the language $L = \{\langle M \rangle: \text{$M$accepts at most two single-letter words}\}$, where $\langle M\rangle$ is the encoding of Turing machine $M$. We need to determine, without ...
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### Modern vs. original formulations of Rice's theorem

Rice's theorem states that any nontrivial semantic property of Turing machines is undecidable. However, when I opened “Classes of recursively enumerable sets and their decision problems” by H. G. Rice,...
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### Is Rice's Theorem equivalent to the Halting problem?

As I understand it Rice's Theorem seems to imply the existence of the Halting problem. That is, with Rice's Theorem, we can prove that the Halting problem is undecidable. However, to me, it seems like ...
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### EVEN-CFL Decidable / Undecidable - Rice-Theorem

Let EVEN-CFL $=\left\{w | M_{w} \text { is a } \mathrm{TM}, \text { such that } L\left(M_{w} \right) \\ \text{ has only words with even length and is context free.}\right .\}$ Question : Is EVEN-CFL ...
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### How can I apply Rice's theorem?

I am learning for my computability and complexity exam in which there is always an exercise to decide whether some problem is decidable or not. In one of the past exams, there was the following ...
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### Why Rice theorem work for decidability?

Rice's theorem states: Every nontrivial property of recursively enumerable language is undecidable. I came across following problems, which Ullman's books say both are undecidable: Turing ...
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### Proving that Rice's theorem does not apply to a property

This is related to an assignment, but I would still appreciate help in formalising proof either through private message or on this topic. The question is about if Rice theorem applies to certain ...
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### Decidability of decision problems

Can somebody give intuition how to answer those questions? From one side I can say that most of them are undecidable because we can reduce the halting problem to them (or halting problem can appear ...
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### Rice Theorem - Problem to understand and apply it

I have struggle to understand the Rice Theorem. My understanding of Rice Theorem: The purpose of this Theorem is to proof that some given language L is undecidable iff the language has a non-trivial ...
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### How to determine if this problem is decidable?

I am currently stuck on the following problem: Given a WHILE-program P and the knowledge that all input variales are set to 0, is it decidable if a specific instruction is reached 1000 times? My ...
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### Rice's Theorem - usage on $DFA$ or $LBA$

I have read about Rice's Theorem on Sipser's book, and I think I understand it quite well. I understand that it can be used to show that a language is not decidable. However I am not sure about one ...
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### Can we enumerate finite sequences which have no halting continuation?

Note: this question has been cross-posted to Math.SE, after about a week here. I am trying to deepen my understanding of the relationship between the Halting Problem and Godel's Completeness Theorem (...
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### Rice's theorem applicable to the following language?

Let $L= \{\langle M \rangle \mid M \text{ halts on } \langle M \rangle \}$ be a language where $\langle M \rangle$ is the Code of the TM $M$. $L$ is undecidable. I've heard that I can't use Rice's ...
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### Language of TM is Undecidable

why is this Problem$$L = \{ \langle M\rangle \mid L(M) \text{ is undecidable}\}$$ undecidable? I thought if we know $L(M)$ the turingmaschine accepts all $x \in L(M)$, so $L(M)$ is in every case ...
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### proof of the rice's theorem

Let $P$ be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given TM’s language has property $P$ is undecidable. Proof:(This is from sipser'...
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### What is the Name of the Problem or Technique of Determining if a Line in a Program Will Execute

If I were to pose the question: "Given a program $P$ containing statement $X$, will $X$ be executed (given enough runs with all possible inputs)?" This strikes me of being a relative of the Halting ...
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### Trying to prove semidecidability of an undecidable language

I have been having a hard time understanding whether the set $S = \{ M \mid |L(M)| = 5 \}$ is semidecidable or not, where $M$ is a generic Turing Machine and $L(M)$ the language accepted by such TM, ...
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### Why is "accepted by Turing Machine with even number of states" a trivial property?

$$L = \left\{ \left< M \right>~\middle|~ \small{ \begin{array}{l} L(M)\text{ is recognized by a Turing Machine} \\ \text{having even number of states} \end{array} } \right\}.$$ Isn't $L$ same ...
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### Proving that a class of languages is a subset of RE for Rice Theorem

Consider language $L = \{<M> |L(M) \subseteq L(0(0\cup1)^*) \}$ where $<M>$ is a valid encoding of a turing machine. I know that the language is applicable for Rice Theorem. Now, I ...
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### The bounded halting problem is decidable. Why doesn't this conflict with Rice's theorem?

One statement of Rice's theorem is given on page 35 of "Computational Complexity: a Modern Approach" (Arora-Barak): A partial function from $\{0,1\}^*$ to $\{0,1\}^*$ is a function that is not ...
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### What is the definition of a property?

I have seen 2 answers in stackoverflow: A "trivial" property is one that holds either for all languages or for none. The property is trivial if it contains every TM, or if it is empty. My problem is:...
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### Is undecidability of TMs' properties a statistical statement?

We know (by Rice's theorem) that is it not possible to decide a non-trivial property of a given TM. We could say therefore that we cannot be sure at 100 percent that a given TM has a certain non-...
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### Rice's Theorem for Total Computable Functions

Fix a Gödel numbering, and write $\phi_n$ for the function coded by $n$. Rice's theorem states that if $P$ is the set of partial computable functions, and $A \subseteq P$, then the decision problem ...
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### Reduction without Rice's Theorem

How can I show that the following language is neither semi decidable nor co-semi decidable without using Rice's Theorem? Further for the following language, how would I show that it IS co-semi ...
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### RICE theorem applications

I am having some confusion in understanding RICE's theorem. It says every non trivial property of RE in undecidable. I need to understand when to apply RICE's theorem and when to not. Questions ...
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### How to determine enumerability after applying Rice's theorem?

To my knowledge, lots of languages can be classified as undecidable after applying Rice's theorem, for example {"M" | L(M) is regular}. But what I am not sure is, how to determine if a language is ...
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### Decide the set of all Turing machines with $L(M)=\left\{\langle M\rangle\right\}$

How can I prove that the language $L=\left\{\langle M\rangle\mid L(M)=\left\{\langle M\rangle\right\}\right\}$ is not decidable? When trying to use a diagonal argument, I cannot conclude from \$L(M)\...