2
votes
2answers
79 views

Background for Kierstead terms

I was looking at some slides of John Longley's here, where he mentions "the Kierstead functional" $$\lambda f.f(\lambda x.f(\lambda y.x)) \ ,$$ (where $f$ should be of type $2$, and $x,y$ of ground ...
1
vote
1answer
91 views

Total conditional complexity

By $C(|)$ denote conditional complexity. By $CT(|)$ denote total conditional complexity. For every n there exist two strings $x$ and $y$ of length $n$ such that $C(x|y) = O(1)$ but $CT(x|y) \ge n $. ...
1
vote
1answer
35 views

How to select a subset of points from a universal to minimize the distance from outside to inside?

Here is the detailed problem. I have a set of N points in K-dimension space, called U, and I want select M points of them, called S. For each point p in U, we define the distance from p to S as $$ ...
15
votes
3answers
552 views

Which distributions can you sample if you can sample a Gaussian?

Explicitly: You have a computer that is able to pick a real number at random according to the normal distribution: $\mathcal{N}(0,1) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Which distributions can this ...
4
votes
1answer
132 views

Self-similarity in the theory of computability

Let $M = w_0w_1... \in \{0,1\}^*$. For any computable function $f$ define $M_f = w_{f(0)}w_{f(1)}...$ Let for any computable strictly increasing function $f$ there is continuous computable mapping ...
56
votes
2answers
2k views

How feasible is it to prove Kazhdan's property (T) by a computer?

Recently, I have proved that Kazhdan's property (T) is theoretically provable by computers (arXiv:1312.5431, explained below), but I'm quite lame with computers and have no idea what they actually ...
24
votes
10answers
2k views

Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not. Are there some large $n$ ...
4
votes
0answers
239 views

About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , ‎of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...
5
votes
2answers
483 views

Why is there no product type in simply typed lambda-calculus?

Consider simply typed $\lambda$-calculus that has only the unit type as primitive. We would like to encode the product and the sum types. An encoding of the product type in the untyped ...
1
vote
0answers
75 views

Is there any track for proving $D=NP$, besides showing that $D$ has polynomial-bounded universal quantifiers?

Background By the MRDP theorem, every for every recursively enumerable set $S$, there exists a Diophantine polynomial $p$ such that $$x \in S \iff \exists y_1, \dots, y_n \in \mathbb{N} \text{ such ...
18
votes
4answers
799 views

Kolmogorov complexity is the strongest noncomputable function

Yury I. Manin says that Kolmogorov complexity (in some nontrivial sense) is the strongest noncomputable function ("Колмогоровская сложность... невычислима... она во многих интересных смыслах ...
10
votes
2answers
375 views

What Turing-Complete models of computation carry a notion of time complexity that “agrees” with that of Turing Machines?

Certain models of computation are technically Turing-Complete, but cannot feasibly simulate a Turing Machine within the usual time constraints we hope for. One example of this is Godel's recursive ...
1
vote
3answers
146 views

unbounded complexity

If a language L is decidable, does that imply that the is a computable function f such that L is in O(f(n)) ? For example what would be the complexity class of the language of "provably halting ...
8
votes
2answers
349 views

Where should I learn about Kolmogorov complexity of overlapping substrings?

I would like to know more about the relationship between the Kolmogorov complexity of a string and that of its substrings. The relation that up to an additive constant, $K(x,y) = K(x) + K(y\ |\ x, ...
1
vote
0answers
117 views

Is there an easy decision algorithm for the inhabitation problem for simple types?

Consider the basic system of simple types usually known as $TA_\lambda$. One can prove that (as a consequence of the Subject Reduction Property and the fact that any typable term is strongly ...
2
votes
1answer
296 views

Problem to a solution

Consider an NP hard problem $\frak P$ which takes an input of length n $\frak P$ can be solved partially by a factor $ p_i = p(n,i)\in$ [0,1)... by a polynomial time algorithm $\mathcal A(i)$ ...
0
votes
1answer
146 views

Interaction-based approximation for HP-complete λ-theory?

We are looking for a proof or counter-examples for the following hypothesis. Two combinators $M$ and $N$ are solvable and equivalent in the HP-complete sensible $\lambda$-theory iff either $$ \exists ...
4
votes
3answers
382 views

What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?

Let me begin with an example. Consider the computable function $f(x) = 2x$. A Turing machine can implement this function in $O(|n|)$ steps: simply walk to the end of the input string, write a $0$, ...
3
votes
2answers
192 views

Smallest base to reach partial recursive functions as a closure of unbound search

It is customary to define the class of partial recursive functions by taking the set of primitive recursive functions $PR$ and taking closure over unbound search operation. Do we need the "whole" set ...
1
vote
1answer
146 views

Grzegorczyk-hierarchy, growth-rate and functions with finite image

Grzegorczyk-hierarchy divides primitive recursive functions in distinct classes with respect to their growth-rate. It seems that the higher we go the hierarchy, the more tools we have to define ...
3
votes
1answer
181 views

Various notions of Turing reduction for partial functions

If $f$ and $g$ are partial functions $\mathbb{N} \to \mathbb{N}$, define six preorder relations $f \preceq g$ as follows: $f \mathop{\preceq_{\mathrm{S}}} g$ ("$f$ is strict/Sasso reducible to $g$") ...
9
votes
0answers
268 views

Various definitions of recursion from ordinal machines

Background: I'm trying to get an intuitive understanding of α-recursion and related concepts in higher recursion theory. Once nice book is Peter Hinman's Recursion-Theoretic Hierarchies, available ...
4
votes
1answer
542 views

Deciding equivalence of regular languages

Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows: build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) ...
1
vote
2answers
268 views

Satisfiability problem for FOL[<,R]

Let FOL[<,R] be the fragment of first-order logic enriched with two relational symbols < and R and the first-order axioms that say: < is a strict partial order and R is an irreflexive and ...
4
votes
3answers
289 views

Existential quantification over regular predicates

A regular language over an alphabet $\Sigma$ is a subset of the set of all words over $\Sigma$ that can be accepted by some finite automaton. A regular language identifies a certain property of ...
20
votes
4answers
1k views

Algorithmically unsolvable problems in topology

This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various ...
3
votes
2answers
963 views

Is there an algorithm that can “reverse engineer” a Regular Expression?

Given a Regular language (represented as a black box to which one can apply inputs and get 0/1) Is there an algorithm that can find a finite deterministic automaton that produces that language?
3
votes
2answers
333 views

Representation of μ-recursive functions

Can every μ-recursive function be defined using a single instance of the μ operator applied to a primitive recursive function? According to Wikipedia, any μ-recursive function can be expressed as the ...
6
votes
2answers
875 views

Busy Beaver - Proof for BB(2) = 4

Hi, I need to prove the above claim. I can show that $BB(2)\ge 4$ by building a turing machine, but how can i show that $BB(2) \le 4$? Searched a lot over the web, and saw that Rado proved it in ...
3
votes
2answers
610 views

Kleene's fixed point theorem on recursive subsets of computable functions

I have a question about the possibility to apply/restate the Kleene fixed point theorem on recursive subsets of computable functions. I don't know if this is trivial and/or if related questions have ...
2
votes
1answer
1k views

How many cpus needed to check a 100 million digit prime number efficiently? [closed]

If I had access to potentially unlimited CPUs and wanted to quickly check 100 million digit numbers for primality using a map-reduce architecture, how many CPUs would be necessary? Each of the mapped ...
2
votes
1answer
266 views

Given a PDA M such that L(M) is in DCFL construct a DPDA N such that L(N) = L(M)

Is it possible to construct an algorithm which takes as input a pushdown automaton $M$ along with the information that the language accepted by this automaton $L(M)$ is a deterministic context-free ...
26
votes
3answers
2k views

“Simpler” statements equivalent to Con(PA) or Con(ZFC)?

Given any reasonable formal system F (e.g., Peano Arithmetic or ZFC), we all know that one can construct a Turing machine that runs forever iff F is consistent, by enumerating the theorems of F and ...
1
vote
1answer
542 views

final step(s) for a proof that a function is not primitive recursive

My function is $f:\mathbb{N} \rightarrow \mathbb{N},\ f(n)=2\uparrow ^n 3$ , the Ackermann(-Péter) function, with the second argument fixed to 3 (and "$\uparrow$" the Knuth up-arrow), which I believe ...
5
votes
2answers
461 views

A Query regarding the Halting Problem (Omega): Halting Probability for Given Input Size

I was studying the Halting Problem in context of the Probability and had a few doubts regarding it. Hope someone could help me out. I am aware of the probability of a Random program halting on a ...
15
votes
7answers
1k views

Between mu- and primitive recursion

It is well known that primitive recursion is not powerful enough to express all functions, Ackermann function being probably the best known example. Now, in the logic courses (that I have had look ...
6
votes
1answer
423 views

post correspondence problem variant

Is there an algorithm which takes as input two lists of words $v_1,...,v_n$ and $w_1,...,w_n$ over an alphabet $X$ and decides if there is an infinite sequence $(k_i)$ where $1 \leq k_i \leq n$ for ...
2
votes
0answers
183 views

Constructing hard inputs for the complement of bounded halting

If there is always a hard input for the complement of bounded halting, can that input be constructed? More precisely, suppose that for any deterministic TM $M$ accepting $$ ...
1
vote
2answers
952 views

post correspondence problem

I have read a couple of proofs for the undecidability of the post correspondence problem, but neither reference gave a concrete example of two lists of words over a fixed alphabet such that the ...
6
votes
5answers
2k views

Aren't “oracle machines” unsound concepts?

From Wikipedia (bold emphasis at the end is mine): In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as ...
14
votes
2answers
1k views

Structure theorems for Turing-decidable languages?

Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages. Such ...
4
votes
2answers
683 views

What are the limits of non-halting?

It's easy enough to build Turing Machines that don't halt. But how complex can we make these? For example, suppose a machine has access to its state transition table and can write to it like a C ...
12
votes
2answers
1k views

Why is Kleene's notion of computability better than Banach-Mazur's?

In this post about the difference between the recursive and effective topos, Andrej Bauer said: If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos ...
2
votes
1answer
2k views

What is a universal function?

This question stems from Dick Lipton's recent blog post on the Axiom of Choice. I asked there but got no takers. I promise I'm not an inept Googler, but I couldn't find a satisfactory answer. I ...
4
votes
5answers
5k views

Prove a function is primitive recursive

Hey, I'm taking a course in computability theory, but I'm struggling with primitive recursion. More specifically we are often asked to prove that some arbitrary function is primitive recursive, but I ...
11
votes
3answers
743 views

Alive dynamical system

Intuitively, one can say that a dynamical system is alive if one can build a universal Turing machine inside. So, Conway's Game of Life is alive and shift space should be dead. I fail to make this ...
2
votes
2answers
748 views

Why is every finite set Diophantine?

I understand that every finite set is recursively enumerable, as I see that one could just encode each element of some finite set on a Turing Machines tape, and then have the machine check each member ...