Questions about Turing computability and recursion theory, including the halting problem and other unsolvable problems. Questions about the resources required to solving particular problems should be tagged (computational-complexity).

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3
votes
1answer
208 views

Can a polynomial size CFG over large alphabet describe any of these languages:

Can a polynomial size CFG over large alphabet describe any of these languages: Each terminal appears 0 or 2 times Word repetition {www* | w \in \Sigma *} (word repetition of an arbitrary word w) ...
2
votes
1answer
246 views

Restricted read twice BDDs and context free grammars

Several papers give poly-time algorithms for constrained paths on labelled graphs, e.g. [1] Quote: Given an alphabet Σ, a (directed) graph G whose edges are weighted and Σ-labeled, and a formal ...
5
votes
5answers
686 views

Can a polynomial size CFG over large alphabet describe a language, where each terminal appears even number of times?

Can a CFG over large alphabet describe a language, where each terminal appears even number of times? If yes, would the Chomsky Normal Form be polynomial in |Σ| ? EDIT: What about a language where ...
3
votes
1answer
349 views

Diophantine equation and Turing Machine

If a Diophantine equation is: exists v, p(x,v) = 0 (where v is a vector of finitely many integers) for some polynomial p, is there a Turing machine which prints out all values of x?
2
votes
1answer
2k views

Quotient of a regular language

According to wikipedia the right quotient of a regular language with ANY other language is regular. I have not been able to find a proof of this fact. All the sources talk about quotient with another ...
3
votes
3answers
244 views

Are there problems that can't be expressed as languages?

OK, so I was told in CSTheory that I should be asking here. So my question is the following: I've taken my first course on Language Theory and we saw the "standard" classification of languages. We ...
6
votes
1answer
335 views

Is there an infinite set of strings whose Kolmogorov complexities are computable?

Is there an infinite set of strings whose Kolmogorov complexities are computable?
9
votes
2answers
661 views

Can we reduce the number of states of a Turing Machine?

My friend claims that one could reduce the number of states of a given turning machine by somehow blowing up the tape alphabet. He does not have any algorithm though. He only has the intuition. But ...
2
votes
1answer
179 views

Hint on Kleene's O

I figured out this order on $\omega^2$: elements of different columns are ordered by their column number; within column $m$, the order is $0 \gt 1 \gt 2 \gt \cdots \gt n$, where $n$ is the first such ...
4
votes
1answer
249 views

(Semi-)Decidability of Turing-completeness of cellular automata

This is a follow-up to the question Undecidability in Conway’s Game of Life I posted at mathoverflow. For some cellular automata it can be proven that they can simulate a Turing machine, normally by ...
2
votes
2answers
240 views

Does this proof for the undecidability of the halting problem violate the axiom of regularity?

One proof of the halting problem goes by contradiction like this : Assume there is a Turing machine $H$ that can decide the halting problem, then construct a Turing machine $Q$ that takes as input a ...
11
votes
7answers
1k views

“Proof” that ZFC is inconsistent using Turing machines

I came across the following "proof" for the inconsistency of ZFC and can't find the flaw in it (if there is one...): Construct a Turing machine A which sequentially runs on all proofs in ZFC and ...
7
votes
4answers
359 views

Are the computable reals finitary?

In the comment thread of an answer, I said: The computable numbers are based on the intuitionistic continuum, and are not finitary. To which T.. replied: Computable numbers are not based on ...
7
votes
3answers
420 views

Why are $\Delta_1$ sentences of arithmetic called recursive?

The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ ...