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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0
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2answers
147 views

Termination of a Fast Exponentiation problem

Here's the problem I am stuck on. There exists a fast exponentiation program like the following: Given inputs a in the set of all Real numbers, b in the set of Natural numbers, initialize ...
0
votes
1answer
111 views

Is DFA (Deterministic Finite Automata) a kind of predicate?

When I read a book on computation theory, I found a interesting thing: A Language L was defined by a DFA(Deterministic Finite Automata) like this, L = {$\omega$ | the last input of $\omega$ causes ...
3
votes
0answers
154 views

Simplifying Relations in a Group

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked ...
1
vote
1answer
88 views

Ackermann function and primitive recursiveness

If we define $b_n(m) := a(n,m)$ for all $n$ and $m \in \mathbb{N}$. For which $n$ is the function $b_n$ primitive recursive and for which $n$ it is not a primitive recursive function? Can anyone ...
2
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1answer
75 views

Decidable language closed under complement

Why are decidable languages closed under complement? So if L is decidable why is the complement of L also decidable.
1
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1answer
89 views

Generating a context free grammar

How do I generate a context free grammar for a language $$\left\{a^ib^jc^k:i=j\text{ or }j=k,\text{ and }i,j,k\ge 0\right\}\;?$$ Thanks.
2
votes
1answer
172 views

Is the Church-Kleene Ordinal describable with Kleene's $O$?

Kleene's $O$ is an ordinal notation system that uses certain natural numbers to represent transfinite ordinals. It is a recursive notation system (although it's not decidable whether a number ...
1
vote
1answer
44 views

Is recursive join of a sequence of low sets also low?

A set $A$ is low when $\deg(A)'\leq 0'$. Suppose we have a sequence of low sets $(A_i)_{i\in\omega}$ such that for every $n\in\omega$ we have $$\deg(\bigoplus_{i<n}A_i)'\leq 0'$$ Let ...
4
votes
1answer
92 views

A function agreeing with a primitive recursive function at all but finitely many points is primitive recursive

Show that if $f:\mathbb{N} \rightarrow \mathbb{N}$ is primitive recursive, $A \subseteq \mathbb{N}$ is a finite set, and $g$ is a total function agreeing with $f$ at every point not in $A$, ...
4
votes
1answer
152 views

Is ordinal analysis a non-recursive project?

A recursive ordinal is an ordinal that is the order-type for some recursive relation (i.e. a recursive well-ordering). We can represent recursive ordinals as natural numbers using Kleene's $O$, an ...
4
votes
2answers
237 views

context free grammar problem

$L$ is the context free grammar over $\{a, b\}$ $S \rightarrow aSb \; | \;bR \; |\;Ra$ $R \rightarrow bR \;|\;aR\;|\;\epsilon$ Briefly describe this CFG with English sentences and prove your ...
0
votes
1answer
308 views

Simultaneous recursion

I have no idea how to even start proving the following theorem: If $f_0, f_1: \mathbb{N}^r \rightarrow \mathbb{N}$ and $g_0, g_1: \mathbb{N}^{r+3} \rightarrow \mathbb{N}$ are primitive recursive, ...
0
votes
2answers
74 views

context free grammar design

Design a context free grammar and PDA for the following language. $$\Sigma = \{0,1\},\qquad L = \left\{uv \mid u \in \sum^{*} \;v\in \sum^{*}1\sum^{*} \text{ with }|u| \geq |v| \right\}$$ I'm not ...
0
votes
1answer
71 views

Enumerating relations that are true infinitely often

Let us concentrate on relations on natural numbers. Is it possible to enumerate all computable unary relations that are true infinitely often? I would guess no but I can't see a direct way to prove ...
5
votes
0answers
313 views

How is the Kleene normal form theorem for $\Sigma^1_1$ relations proved in RCA0?

All of the following concerns Simpson's Subsystems of Second Order Arithmetic (2nd ed.). In the notes subsequent to lemmas VII.1.6 and VII.1.7 (pp. 245–246), Simpson remarks that both lemmas are ...
5
votes
3answers
1k views

Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: $$c_{\small d}x^{\small ...
2
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0answers
61 views

Why is positional number system natural?

In the theory of computation, one mainly deals with maps $\Sigma^*\rightarrow\Sigma^*$. To discuss computation on other sets $X$ than $\Sigma^*$, one fixes a representation ...
1
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1answer
224 views

Proof of undecidability of $FINITE_{\text{TM}}$ and $USELESS_{\text{TM}}$

I came across these 2 problems about proving of undecidability of languages: $1$. $FINITE_{\text{TM}} = \{\langle M \rangle | M \text{ is a Turing machine and } L(M) \text{ is a finite language} \}$. ...
4
votes
1answer
695 views

Why doesn't diagonalization prove that integers are not countable?

I understand how Cantor's diagonalization argument works with respect to disproving that a bijection between integers and real numbers can exist. What I don't get is why the same reasoning doesn't ...
0
votes
1answer
177 views

Arithmetic Hierarchy problems

Is $\Sigma^{0}_n$ closed under intersection? I think yes m I correct? Let $L_1$ be an $\Sigma^{0}_1$ complete language, and $L_2$ be a $\Pi^{0}_1$ complete language, such that $\emptyset \neq L= ...
3
votes
1answer
56 views

A diophantine program for a basic For Loop

By Matiyasevich, every computable function has a diophantine representation. I am wondering if there is a general way to represent a simple iterative for loop. Specifically: Let $f(x)$ be any ...
3
votes
1answer
61 views

Can this simple functional language be simplified further without losing any computational power?

Here is a definition of a very simplistic programming language (it is not Turing complete). Input to a function is any natural number. The following functions are primitive to the language: ...
4
votes
1answer
129 views

Decidable & Recursive predicates

Let $C$ be a decidable predicate in the language of arithmetic HA, that is $$ \vdash (\forall \underline x)\: C(x) \vee \neg C(x).$$ $C$ is recursive if there exists a computable characteristic ...
-4
votes
1answer
250 views

How does 3-sat work in laymen's terms?

I know only basic math like so: (+,-,x,\,). And I studied a little bit of programming up to the point of knowing a little bit about Boolean values.I desperately want to understand the 3-sat question ...
1
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1answer
61 views

Partially correct algorithm for a decidable problem

$D$ is decision problem whose inputs are the natural numbers. Suppose $A$ is an algorithm to solve $D$ in which we know that it is: partially correct for all inputs halts on all inputs >= 1000. ...
3
votes
1answer
113 views

Looking for counterexamples where the output of a computable function always has a computably checkable property, but PA cannot prove this

Suppose we have a computable function $f$, say over the naturals, and a decidable set $S$ of naturals, such that $f(x) \in S$ for all $x$. In this case, for any specific $x$, there is some specific ...
1
vote
1answer
87 views

Why SKI when SK is complete

Why people talk about SKI calculus when S and K combinators can be used to create any other combinator including I?
1
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0answers
56 views

A diophantine definition of the Kleene star

Let $f(x \, | \, y_1, \dots, y_n)$ be a Diophantine polynomial that generates the Diophantine set $F$. By Matiyasevich, the set $F^*$ (Kleene star of $F$) is also Diophantine. My question: how can ...
3
votes
1answer
124 views

Showing the Rec is $\Sigma_3^0$-complete

In Soare's Computability Theory and Applications, he gives a very quick proof that the following set is $\Sigma_3^0$-complete: $$\text{Rec} := \{e \mid W_e \text{ is recursive}\}$$ It's fairly ...
8
votes
1answer
137 views

Does an undecidable decision problem have a ZFC-independent instance?

Is it true that every undecidable decision problem has an instance whose solution is independent of ZFC? For example, let $G$ be a finitely-presented group with undecidable word problem. Does there ...
1
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2answers
1k views

Is the function, calculating square root of natural number — computable?

My question is about domain of the term "computable". Consider Turing machine, that calculates square roots of natural numbers. If it gets 4 then it prints out ...
21
votes
3answers
5k views

Are there any examples of non-computable real numbers?

Is this true, that if we can describe any (real) number somehow, then it is computable? For example, $\pi$ is computable although it is irrational, i.e. endless decimal fraction. It was just a luck, ...
1
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1answer
196 views

Non-computable numbers and surreals

Can non computable numbers be expressed with surreal numbers? Show the construction using Conway's definition of surreal.
1
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0answers
83 views

Systematic way of creating the complement of a regular grammar?

Regular languages are closed under complement. And any regular language can be generated using a regular grammar. Is there a systematic way to create the rewrite rules for the complement of a regular ...
15
votes
1answer
503 views

Explicit automorphisms of the field of algebraic numbers

The field $\overline {\bf {Q}} $ of algebraic numbers admits many automorphisms other than conjugation. This follows from Galois theory: the field $\overline {\bf {Q}}$ can be realized as the union ...
4
votes
1answer
180 views

A Turing machine for which halting is outside ZFC

If, given Turing machine T, "T halts" or "T doesn't halt" could be derived from axioms of ZFC, halting problem would be in R. As it isn't, there must exist a Turing machine for which truth or ...
9
votes
5answers
561 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 ...
3
votes
1answer
133 views

Why do complex grammars require powerful algorithms?

I am reading a fabulous book on Formal Languages and in the book it says: As the rewrite rules of a grammar become more complex, the algorithm for recognizing the associated language becomes ...
5
votes
2answers
147 views

Running programs in nonstandard models of PA

I came across the following problem in several places, to paraphrase: Let $T$ be a recursively axiomatizable, consistent extension of PA. Then there exists some $e$ such that the $e'$th program ...
2
votes
2answers
127 views

Theory vs. Deductive Theory

Looking through some notes on model and computability theory, I noticed that the definition of the term 'theory' changed between them; in particular, the model theory (based on Hodges' text) defined a ...
0
votes
2answers
156 views

Universal Turing Machine with an Oracle

(Note: For convenience, I'm phrasing this in terms of computer programs rather than Turing machines.) Consider computer program P which does the following: It asks the user to enter the code of a ...
2
votes
1answer
112 views

A few basic questions about the arithmetical hierarchy, mostly about terminology.

I was reading about the arithmetical hierarchy, and I have a few questions, mostly notational. For completeness, here's the definition given over at Wikipedia. The classifications $\Sigma_n$ and ...
1
vote
1answer
61 views

Computability text emphasizing the arithmetical point of view?

I learned here that A set is recursively enumerable if and only if it is at level $\Sigma^0_1$ of the arithmetical hierarchy. Is there an introductory text in computability theory that takes ...
2
votes
0answers
103 views

Acceptable numbering of partial computable functions required to be one variable?

Soare in a yet unpublished textbook (I happened to be in a class taught by one of his former graduate students where we were field-testing a rough draft of his new textbook) Computability Theory and ...
6
votes
1answer
120 views

Ackermann function and $f_\omega$

The Wikipedia page of Ackermann function states that Ackermann function is "roughly comparable" to $f_\omega$ in fast-growing hierarchy. Is there some standard way to make the "roughly comparable" ...
5
votes
2answers
238 views

Random reals and Martin-Löf randomness

My questions are about the relationship between the following notions of randomness: A real $r$ is random over the model $M$ if $r\notin B$ for every null Borel set $B$ coded in $M$. A real $r$ is ...
0
votes
1answer
133 views

Let F be a function from $N ^{n} \longrightarrow N$. Show that if F is computable/ recursive then its graph is computable

Let F be a function from $N ^{n} \longrightarrow N$. Show that if F is computable then its graph is computable. According to the definition of computable/recursive I am looking at, a relation is ...
2
votes
0answers
96 views

Prove that $\{(x,y): W_x\text{ and }W_y\text{ are recursively separable}\}$ is $\Sigma_3$-complete

Prove that $\{(x,y): W_x\text{ and }W_y\text{ are recursively separable}\}$ is $\Sigma_3$-complete This is a question from Soare's Recursively Enumerable Sets and Degrees. I have little idea how ...
3
votes
1answer
178 views

If P=NP, then NP = coNP. Why is this so?

I read that if we assume that P = NP, then NP = coNP. I am unable to understand why this is so.
1
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0answers
50 views

Regular Functional Algorithms

A language is regular if it is accepted by a read-only Turing machine. I am curious about applying this model to functional problems rather than decision problems. Definition: A functional read-only ...