3
votes
0answers
109 views

indices set and halting problem in computation course

I ran into a multiple choice question that confused me with this notation. anyone could help me? this is adapted from an old class quiz in Calgary. Suppose A is be indices (i think index set) of type ...
6
votes
0answers
79 views

Challenge on Some Definition on Formal Language & Recursive & Automata

We know set A is countable if A is finite or in a one-to-one mapping to natural numbers. Suppose $\Sigma$ be an arbitrary finite alphabet. I summarize my inference: a) Each arbitrary Language on ...
9
votes
2answers
186 views

The mother of all undecidable problems

It is usual to show that a problem P is undecidable by showing that the halting problem reduces to P. Is it the case that the halting problem is the mother of all undecidable problems in the sense ...
0
votes
1answer
31 views

Computable Function and Predicate Question

I See on Our Lecture note on Theory of Computation Course that: .... The basic characteristic of a computable function is that there must be a finite procedure (an algorithm) telling how to compute ...
2
votes
1answer
84 views

Many to one Reducible & Polynomial time

we know that If $A \le_p B$, then $A$ can be reduced to $B$ in polynomial time. we know that If $A \le_m B$, then $A$ is many to one reduction to $B$ . can we deduce that: if $A \le_m B$ then $A ...
0
votes
1answer
52 views

Logic & Computability Problem

i read this sentence in one exam that be false. anyone could say why? if predicate H(x) become false when a program with code r(x) halt on input l(x), then H be a computable predicate.
0
votes
1answer
90 views

Turing & Computability & Computation

We know if we have: we can show (T=t= Turin Redu.) but i have no idea why this relation be correct? any idea?
0
votes
0answers
67 views

TAUTOLOGIES NP-Complete Condition

The decision problem TAUTOLOGIES is, Given $\forall x_1 \forall x_2 ... \forall x_n$ $\phi(x_1, x_2, ... x_n)$ a set of universally quantified Boolean variables and a Boolean formula ...
0
votes
0answers
72 views

Primitive Recursive Predicate Challenge

I'm an Computer scientist, and I recently ran into a challenge. If we have primitive recursive predicate $P(x), Q(x)$, I think that all of following 4 expressions can be primitive recursive. Any hint ...
0
votes
0answers
48 views

Recursive Set and Complement Problem

if we have $$A=\{x:|W_x\ne\phi\}$$ can we say always my tight listed below is true? $A$ is recursive , $A$ is r.e, complement of $A$ is r.e, complement of $A$ is not recursive?
0
votes
1answer
78 views

Recursive Set Challenge

we knoe also we know for example if A be any arbitrary r.e set. can we always Necessarily the following is TRUE ? (always) any description is good. (bar sign means complement)
1
vote
3answers
70 views

How does one generally use partial function in logical statements?

How does one generally use partial function in logical statements? How it's done in practice? Specifically, let $M$ by a Turing machine, $f_M:\{0,1\}^*\to\{0,1\}$ the characteristic function which ...
2
votes
1answer
46 views

Undecidability of REGULAR_TM

In case you have Sipser's Introduction to the Theory of Computation 3rd edition, I am asking specifically about the proof of theorem 5.3, how the language REGULAR_TM is undecidable. \begin{equation} ...
4
votes
2answers
336 views

Why is propositional logic not Turing complete?

According to 1 (probably not the most relevant source), propositional logic is not Turing complete. Aren't all computations in computers performed using logic gates, which can be represented as ...
4
votes
1answer
112 views

How many digits of Chaitin's $\Omega$ constant would we know if we had a $\Sigma_1$-Oracle?

According to Wikipedia (and it seems intuitive from the definition itself), $\Omega$ is Turing equivalent to the halting problem and thus at level $\Delta_2^0$ of the arithmetical hierarchy. Do this ...
7
votes
4answers
459 views

Is the set of PA theorems the same as the set of solvable halting problems?

I am not sure if this is a trivial question. By Post's theorem we know that every PA (first order logic) theorem is equivalent to stating that a given input C in a given Turing machine halts or ...
3
votes
1answer
121 views

Question about $\Sigma_n$-soundness

According to wikipedia (http://en.wikipedia.org/wiki/%CE%A9-consistent_theory#Definition): "$\Sigma_n$-soundness has the following computational interpretation: if the theory proves that a program C ...
1
vote
2answers
178 views

I do not understand why the Turing computable sets of N are exactly the sets at level $\Delta_1^0$ of the arithmetical hierarchy

The reason I don't understand it is this. Take for example the twin primes conjecture, which is $\Pi_2^0$. The set of twin primes is computable right? (there is a Turing machine that enumerates all of ...
1
vote
1answer
74 views

Seeking Alternate Proof Regarding Closure Of Recursively Enumerable Languages Under Shrink

So I would like to show that the class of Recursively Enumerable languages are closed under the shrink operation. In other words, $\mathrm{shrink}_a(L) = \{x \mid x=\mathrm{shrink}_a(w), w\in L\}$ and ...