1
vote
1answer
59 views

Countable Set & Formal Grammar

We know set A is countable if A is finite or in a one-to-one mapping to natural numbers. I try to summarize my though. I think the following proposition is true. suppose $\Sigma$ is arbitrary ...
0
votes
1answer
30 views

Formulating the Twin Prime Conjecture as a Language Recognition problem.

I'm trying to figure out how to formulate the Twin Prime Conjecture as a language recognition problem. I've got: A = {p: p is the largest prime such that p + 2 is prime} B = {p: p and p+2 are both ...
0
votes
1answer
23 views

Are there known patterns among minimal expressions?

Let $R = F[z_1, z_2, \dots]$ be the finite-degree polynomials in a countable number of variables. Let $\mathcal{E}(R)$ be the set of all expressions of polynomials. Note that there could be an ...
0
votes
0answers
45 views

My notes on $\Bbb{Z}/p\Bbb{Z}$-theoretic computational complexity

(Question at the very bottom) Def 1. Let $F = \Bbb{Z}_p$ be a finite field. Then an $F^k$-machine is a machine with $k$ input / output memory slots. All computations are done in the field $F$ and ...
1
vote
0answers
50 views

If $P=NP$, prove that $L' \in NP$

I think I'm overthinking this problem and need some hints in the right direction. The goal of this question is to show that if $P=NP$ then for every language $L \in NP$ via a polynomial time verifier ...
2
votes
1answer
62 views

Prove 2-HamiltonianCycle $\in \textbf{NP}$

Just want to verify that I have the right idea here with this hamiltonian cycle question. $HC$ = $\{\langle G \rangle$ | $G=(V,E)$ is an undirected graph such that there is a simple cycle (no vertex ...
2
votes
1answer
59 views

Prove that RE is closed under reduction

Prove that the class RE is closed under reduction. Definitions: A language $ A \subseteq \Sigma^*$ is called reducible to $ B \subseteq \Gamma^*$ ( denoted by $A \leq B$) if there is a computable ...
1
vote
0answers
32 views

Why every regular language is in $\text{TIME}(n)$?

How can I prove that every regular language $R$ has linear time complexity, i.e. every regular language satisfies $$R \in \text{TIME}(n)$$
7
votes
5answers
425 views

A computer's memory is finite, so how can there be languages more powerful than regular?

A computer has a finite memory. There are no computers with infinite memory. Therefore the only languages that a computer can process are those whose member strings are finite. As I recall, the ...
1
vote
2answers
68 views

Sequences and Languages

Let $U$ be the following language. A string $s$ is in $U$ if it can be written as: $s = 1^{a_1}01^{a_2}0 ... 1^{a_n}01^b$, where $a_1,..., a_n$ are positive integers such that there is a 0-1 ...
2
votes
1answer
188 views

Function problem vs. decision problem

Take the set $FP$ of number-theoretic functions that are computable in polynomial time. Let us restrict to those functions with range in $\{0,1\}$, $FP_{0,1}$. Is there any correspondence with ...
2
votes
1answer
516 views

Turing Machine Vs Linear Bounded Automata

Example of language accepted by Turing Machine but not by Linear Bounded Automata ? Is there any EXPSPACE language?
4
votes
1answer
133 views

Minimal DFA satisfying a finite view of a language.

Say one has a language $L \subseteq \Sigma^*$, but one doesn't know what strings are actually part of the language. All one has is a finite view of the language: a finite set of strings $A \subseteq ...
3
votes
1answer
195 views

Regular expressions which first disagree after an exponential length

Problem 8.24 of Sipser's Introduction to the Theory of Computation asks: For each $n$, exhibit two regular expressions $R$ and $S$ of length $poly(n)$ where $L(R)\not =L(S)$, but where the first ...
5
votes
3answers
301 views

How complicated is the set of tautologies?

Consider the set $\mathcal T$ of all tautologies in the propositional calculus in which the only operators allowed are $\to$ and $\neg$, and involving only the two variables $x$ and $y$. How ...
1
vote
1answer
138 views

Reduction over intersection of languages

Given two languages $L1$ and $L2$, such that $L2$ is NP-Hard under polytime (many-one or Turing) reduction. Let $L=L1\cap L2$. 1- Is it true that if $L2$ is polytime (many-one or Turing) reducible to ...