1
vote
1answer
62 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 ...
-2
votes
0answers
37 views

Pumping Lemma & Regular Language

For each regular language L, we have an integer k such that: ...
-1
votes
1answer
35 views

$L \in RE$ Question in Computation [closed]

Let L be a language. Suppose a TM exists that halts on all words in L. Which of the following statements is true? a) if L is r.e we have such TM. b) if L is r.e and complement of L is r.e then we ...
9
votes
2answers
169 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
0answers
42 views

predicate logic with assumption NP $\neq$ CO-NP?

Anyone could describe why: Set of All Tautology in propositional logic with assumption NP $\neq$ CO-NP is CO-NP Complete. Thanks. I ask it here before: Is the language of tautologies NP-complete? ...
0
votes
1answer
25 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 ...
6
votes
1answer
76 views

Who first discovered that some R.E. sets are not recursive?

Who first discovered that some recursively enumerable sets are not recursive, or equivalently that some semidecidable sets are undecidable? And in what context? Was the earliest formulation of this ...
-2
votes
1answer
73 views

D={ $ deg_T (A) | A \subseteq N$} Problem [closed]

Dear friends I wanted to ask the question that already asked 2 times but it's on-hold and after few days deleted, but I didn't get any answer. I try to solve it but confused. I don't know anything and ...
1
vote
1answer
61 views

set theory, Incompleteness and axiomatic systems

Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms ...
2
votes
1answer
77 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
75 views

Why is $x\mapsto x$-th prime number a partial recursive function?

I think that partial recursive functions correspond to all computable functions. Thus, if we can write a computer program to represent a function, the function is partial recursive. In computability ...
-1
votes
1answer
64 views

Undecidability of First Order Logic [closed]

friends! I read in Ebraham's Outline of Logic that first order logic is undecidable because it lacks an algorithmic procedure which reliably detects invalidity in every case. It is undecidable ...
-1
votes
1answer
69 views

The range of an increasing computable function relation is recursive [closed]

I read this post: A is recursive iff A is the range of an increasing function which is recursive, and also this tutorial on recursion theory. In the latter, I saw the following sentence: $A$ is ...
0
votes
0answers
140 views

range of one increasing computation function?

We know that that the range of any recursive partial function is recursively enumerable. Also we know the fact: Set A is recursive if and only if it is range of some increasing section partial ...
0
votes
1answer
40 views

Problems On Many-one Reducible [closed]

In computability theory and computational complexity theory, a many-one reduction is a reduction which converts instances of one decision problem into instances of a second decision problem. ...
-1
votes
1answer
60 views

Recursively enumerable language [closed]

In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable or Turing-acceptable) if it is a recursively ...
-1
votes
1answer
35 views

Set of Logical Result Problem [closed]

If we have a set of predicate formulas $A$, and there is an algorithm such that for every predicate formula $X$, (with input $X$), output YES iff $X \in A$. My question is about set of logical result ...
-4
votes
1answer
157 views

Primitive Recursive Predicate Problem [closed]

i get trouble with 2011 midterm exam question. if P(x) and Q(x) be a primitive recursive predicate. which of the following is not a primitive recursive? anyone could describe it for me? 1) $P(x) ...
-1
votes
1answer
106 views

Many-one Reducibility Understanding Problem [closed]

We know for every set $B$, that be r.e have: $$B\leq_mK$$ (The set $B$ is many-one reducible, or m-reducible, to the set $K$) we know $K$ is r.e and define: $$K=\{ e:e\in W_e\}$$ my challenge is: ...
-2
votes
1answer
78 views

Computation & R.E Set Problem [closed]

i ran into a old-midterm question recently, without any definition and tutorial Suppose A is a subset of Natural Numbers that includes all numbers except some finite numbers. why A ...
-1
votes
1answer
55 views

Computable Set & Function

we know that i read this sentence are true? can anyone say an example for following sentence? there are a non computable set A such that
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
88 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?
1
vote
1answer
30 views

What techniques are there to search for first order sentence equivalence?

Suppose we have a first order sentences $\phi$, $\psi$, and $\chi$ such that: $\phi$ $\longleftrightarrow$ ($\psi$ $\land$ $\chi$) And $\phi$ and $\chi$ are known or fixed. How can we search for a ...
1
vote
0answers
34 views

Meaning of Biinterpretability.

I'm reading this paper: http://www.math.cornell.edu/~shore/papers/pdf/hyp9.pdf and I am struggling with the meaning of Biintereptability, to quote the paper A degree structure $D$ is ...
0
votes
1answer
56 views

Function Combination on Computer Science

I read some material on Computational Function, every one could describe the result of following combination? suppose $g_1(x)=3x$, $g_2(x)=4x$, $f(x,y)=x+y$, how we compute combination of $f$ with ...
0
votes
0answers
71 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
46 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
73 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)
0
votes
1answer
49 views

Complexity & Computation & Logic Problem [closed]

As i study for prepare to CS Final exam, i have some challenges. can i say all of following statements are true? 1) each infinite recursive set, is union of two disjoint infinite recursive set? 2) ...
0
votes
0answers
44 views

Proof: All recursive functions are arithmetic (logic)

So I'm trying to understand the proof of the following statement: > All recursive functions are arithmetic The proof begins with: "It is sufficient to show that all arithmetic functions satisfy ...
5
votes
1answer
53 views

Nonstandard models of PA with a decidable order relation.

There this exercise in Models of Peano Arithmetic (Kaye 1991, p.157), which asks to define a recursive binary relation on $\mathbb{N}^2$, such that $M \upharpoonright < $ is isomorphic to ...
2
votes
1answer
64 views

Question about the definition of Diophantine sets

I am currently reading "A Course in Mathematical Logic for Mathematicians" by Manin. The book defines Diophantine sets as follows: The projections of the level sets of a special kind of primitive ...
4
votes
3answers
113 views

Diophantine equations and Hilbert's 10th Problem, how did MRDP do it?

I'm having a bit of trouble understanding the Wiki explanation of MRDP's (Matiyasevich, Robinson, Davis, Putnam)'s Theorem, which explains that Hilbert's 10th problem is unsolvable. The MRDP ...
1
vote
0answers
93 views

Range/Image of a Non-Decreasing Total Recursive Function is Recursive

How do I show that the range of a non-decreasing, total-recursive function is recursive? I've made reference to this question, but the method used there is not clear to me. My attempt: Let $f$ be ...
3
votes
2answers
86 views

How does undecidability of 'theoremhood' imply that human ingenuity is necessary in mathematics?

In Robert Stoll's "Set Theory and Logic", there is the following passage on effectiveness of theorems (p. 375) : Mathematical logicians have shown that for many interesting axiomatic theories ...
2
votes
2answers
142 views

How can the Gödel sentence be Pi_1

The Gödel sentence must be provable or unprovable by itself - you have to resolve all definitions until it only uses the elementary symbols of Peano arithmetic. What is the correct way to resolve ...
2
votes
1answer
104 views

$\textbf{Q}$ fails to prove some correct $\forall$-rudimentary sentence

Show that the existence of a semirecursive set that is not recursive implies that any consistent, axiomatizable extension of Q fails to prove some correct $\forall$-rudimentary sentence. I ...
1
vote
1answer
56 views

A Question About Tennebaum's Theorem?

Tennenbaum's theorem proves there are no countable recursive nonstandard models of Peano arithmetic. It is a proof by contradiction. If our countable, nonstandard model is recursive, then, given a ...
2
votes
0answers
86 views

Ackermann function is not primitive recursive

The function of the Ackermann function is defined as $$ A_{0}(y)= y+1$$ $$ A_{x+1}(0)= A_{x}(1)$$ $$ A_{x+1}(y +1)= A_{x}(A_{x+1}(y))$$ I want to show that the function of ackermann is primitive ...
0
votes
0answers
86 views

A is recursive iff A is the range of an increasing function which is recursive

Working a problem stated in Enderton, but stated better and apparently stronger in Soare. All citations hereon are for Soare (1987). Would appreciate help on the proof. I know there has to be a more ...
1
vote
1answer
37 views

Kolmogorov (Kolmogoroff- ) Complexity of infinite sequences, Request for Proof

Let $\xi \in X^{\omega}$ be an infinite sequence and denote by $\xi[1\ldots n]$ its length $n$ initial segment. Then (due to Martin-Löf) the following holds: For every $\xi \in X^{\omega}$ there ...
1
vote
1answer
30 views

Direct proof that $K \leq_\mathrm{T} Rec$

Soare's Recursively Enumerable Sets and Degrees (1987) shows that $Rec = \left\{ e : W_e \text{ is recursive} \right\}$ is $\Sigma^0_3$-complete via its relationship to other index sets, namely $Cof$ ...
0
votes
1answer
36 views

Kolmogorov (Kolmogoroff-) Complexity, Contradiction with Invariance Theorem.

Fix some programming languages $S$ which is rich enough such that one can write interpreters for $S$ in $S$. Define $$ K(w) := \mbox{length of a shortest program producing $w$}. $$ Now fix some ...
7
votes
1answer
108 views

Proof-theoretic characterization of the primitive recursive functions?

The total recursive functions are exactly those number-theoretic functions that can be represented by a $\Sigma_1$ formula of first-order arithmetic. Is there a similar characterization of the ...
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 ...
1
vote
1answer
76 views

Problem from Cutland's Computability: 3.2. problem 3

The problem goes as follows. Let f: N --> N, such that f is partial, N is the natural numbers, and let m $\in$ N. Construct a non-computable function g such that g(x) = f(x) for x$\le$m. Proof: By ...
0
votes
1answer
70 views

Complexity of Recursively Inseparable Sets

I am interested in examples of recursively inseparable sets. A standard example is the set of positive integers encoding a Turing machine that halts in an odd number of steps on blank input versus ...
1
vote
1answer
49 views

Non-computable c.e. sets are Kurtz random

I'm trying to directly show that non-computable c.e. sets are Kurtz random, without using the concept of genericity, but to little success. I assume by way of contradiction that $\emptyset'$ (for ...
14
votes
3answers
295 views

Mathematical Notation and its importance

You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...