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-4
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32 views

Simultaneous Recursion Problem [on hold]

i think: We can apply simultaneous recursion with more than two functions in a similar manner, defining each of then at n + 1 in terms of the values at n. We can also define simultaneous recursion ...
6
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0answers
24 views

Different definitions of optimal decompressors

Let $B^{<\omega}$ be the set of finite binary strings. I will only consider functions from $B^{<\omega}$ to $B^{<\omega}$. I recall the definition of the algorithmic complexity of a string ...
3
votes
2answers
253 views

Are two-qubit unitaries necessary for universal quantum computation?

I was going through Fast Universal Quantum Computation with Railroad-switch Local Hamiltonians by Daniel Nagaj. In the first sentence of the fifth paragraph on the fourth page, he said, Two-qubit ...
0
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0answers
39 views

Can there exist a single Turing machine complete for PTIME, or for $\#P_1$?

In "The Complexity of Enumeration and Reliability Problems", Valiant mentions the existence of a single Turing machine that is complete for the class $\#P_1$ (i.e., $\#P$ with unary input). On page ...
6
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1answer
190 views

Complete problems and universal simulator machines

I'm trying to get straight in my mind the relation between complete problems and universal simulator machines. Some notions of computability have universal machines (Turing-computability) and some ...
11
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3answers
3k views

What is a practical non-von Neumann architecture?

Are there any practical applications for non-von Neumann programming models? What are the most widely adopted non-von Neumann programming languages?
2
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1answer
108 views

Speed-up of universal computation by caching

A universal computer is a program that can execute any other program. It is interesting to ask whether there are "booster" computers that execute programs faster than they execute "on their own". In ...
3
votes
1answer
219 views

Are randomly generated infinite patterns computable?

Fix a prefix-free universal Turing machine $U$. Consider the following random process*. The state of the process is a bit-string $s$, initialized with the empty string (say). Suppose the value of the ...
6
votes
1answer
260 views

Computing power sums

I know some schemes to compute power sums (I mean $1^k + 2^k + ... + n^k$) (here I assume that every integer multiplication can be done in $O(1)$ time for simplicity): one using just fast algorithm ...
2
votes
2answers
648 views

Was Babbage's Analytical Engine really turing-complete?

According to literature, Babbage's Analytical Engine is turing-complete because it supports conditional branching: it can perform different operations depending on the sign of the result last ...
7
votes
1answer
248 views

Proofs techniques related to Curry–Howard correspondence

I am looking for sources about formalized notion of programs. This seems to be closely related to Curry-Howard correspondence, but one could also track this back to Universal Turing Machines and its ...
0
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0answers
68 views

What is the best known hardness result for Factoring? [duplicate]

Possible Duplicate: Are the problems PRIMES, FACTORING known to be P-hard? In particular, do we know that $\mathrm{Factoring}$ is hard for $\mathsf{AC^0[p]}$, $\mathsf{ACC}$, or ...
1
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3answers
261 views

Complexity of universal computation on random input

This post is a refinement of a previous question which turned out to be trivial It is also related to another previous question Motivation: A property commonly ascribed to artificial intelligence is ...
1
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1answer
131 views

Average performance of universal computation

This post is related to my previous question but takes a different angle There are several related questions in here Consider an input-less algorithm $A$ generated randomly using a prefix-free ...
2
votes
1answer
479 views

Initial conditions for universal Rule 110

In A New Kind of Science, Wolfram proves that the Rule 110 cellular automaton can emulate a cyclic tag system, and is therefore a universal computer. I was wondering what specific initial conditions ...
2
votes
0answers
153 views

Particle collisions for universal computation

Proof of universality of Game of Life is straightforward (CAFAQ): (two annihilating glider streams with gaps (ie. 0s) are colliding, one is "data" and the second is all glider filled, ie.: ...
6
votes
2answers
305 views

Which model of computation to simulate to prove universality?

I am starting out in theoretical computer science. I have a model of computation based on observations of auto-associative memory in the brain. I believe (with little evidence) that I can do ...
3
votes
1answer
228 views

Abstract definition of universal computation

There are many universal computation systems. Turing machines, tag systems, rewrite systems, cellular automata to name just a few. The universality of a system is proved via reduction from a known ...
4
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0answers
218 views

Are there connections between Turing machines and symbolic dynamic systems?

On a course, when shift systems were being introduced, the lector said that "if the shift of symbols sequence reminds you Turing machine, then it is a very correct association": $\sigma(\ldots, ...
4
votes
2answers
565 views

Transition Diagram of a Universal Turing Machine

I have searched the web for the transition diagram of a universal Turing machine without luck. Is anyone aware of such a diagram? I need this as a reference, so preferably a book or a published ...
17
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5answers
1k views

What's the simplest noncontroversial 2-state universal Turing machine?

I'm wanting to encode a simple Turing machine in the rules of a card game. I'd like to make it a universal Turing machine in order to prove Turing completeness. So far I've created a game state ...
2
votes
2answers
334 views

Can reversible computations alone be used to create a computer?

We are able to perform universal computations with the reversible model. Basically, during the computations, no information should be erased, so that no involved entropy increase would occur. Are ...
12
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2answers
583 views

Dark Integers: General Purpose Computations on Internet Routers

Greg Egan in his fiction "Dark Integers" (story about two universes with two different mathematics communicating by means of proving theorems around of inconsistence in arithmetic) claims that it is ...
7
votes
2answers
244 views

Restricting entries of unitary operators to real numbers and universal gate sets

In Bernstein and Vazirani's seminal paper "Quantum Complexity Theory", they show that a $d$-dimensional unitary transformation can be efficiently approximated by a product of what they call ...
13
votes
5answers
791 views

Which models of computation can be expressed through grammars?

This is a reformulation of Are grammars programs? previous asked by Vag and with many suggestions from the commenters. In what way can a grammar be seen as specifying a model of computation? If, for ...
2
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1answer
187 views

Terminology for types of universal computation

Some models of computation are universal in the sense they can compute any arbitrary computable function $f:\mathbb{N} \rightarrow \mathbb{N}$. Other models are universal only as far as the input and ...
23
votes
2answers
595 views

Justification of log f in DTIME hierarchy theorem

If we look at DTIME hierarchy theorem, we've got a log due to the overhead in simulation of a deterministic Turing Machine by a universal machine : $DTIME(\frac{f}{\log f}) \subsetneq DTIME(f)$ We ...
2
votes
1answer
366 views

Assistance in showing computational equivalence between software and hardware computing systems

I'm in the middle of a discussion regarding the equivalence of software and hardware computing systems, and I would like some comment. Any arguments and/or documentation (supporting or dissenting) ...
20
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3answers
823 views

Is there a non Turing-complete model of computation whose halting problem is undecidable?

I cannot think of any such model, maybe some form of typed lambda calculus? some elementary cellular automaton? This would almost disprove Wolfram's "Principle of Computational Equivalence": ...
5
votes
2answers
689 views

Universal Turing Machines in “Computational Complexity” by Papadimitriou

The first part of this question has been solved (see comments). In the book "computational complexity" by Papadimitriou, a Universal Turing Machine is given. But this machine is not concrete, in the ...
9
votes
2answers
240 views

Is there an official name for a notion of “reusably universal”?

There are several different (probably inequivalent) notions of computational universality (see for example the last couple pages of ...