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Fractal dimension versus computational complexity

Complexity measures are designed to capture complex behavior and quantify *how* complex, according to that measure, that particular behavior is. It can be expected that different complexity measures from possibly entirely different fields are related to each other in a non-trivial fashion. Here we study small Turing machines (TMs) with two symbols, and two and three states. For any particular such machine τ and any particular input x we consider what we call the 'space-time' diagram which is the collection of consecutive tape configurations of the computation τ(x). In our setting, we define fractal dimension of a Turing machine as the limiting fractal dimension of the corresponding space-time diagram. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time iff its dimension is 2, and its dimension is 1 iff it runs in super-polynomial time and it uses polynomial space. If a TM runs in time O(x^n) we have empirically verified that the corresponding dimension is (n+1)/n, a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on the one side versus the time complexity of a computation on the other side.

 

Fractal dimension versus computational complexity
Joost J. Joosten, Fernando Soler-Toscano, Hector Zenil

http://arxiv.org/abs/1309.1779


Via Complexity Digest
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"World’s First" Bacteria-Grown Book Combines Science and Culture | The Creators Project

Dr. Simon Park's microorganisms have produced their own copy of 'On the Origin of Species.'
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Libib.com

Libib.com | media archaeology | Scoop.it
ERICKFELINTO: An awesome library collection!
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Benjamin Bratton: 'Geobiopolitics and Planetary-Scale Computation: Sensors, Abstractors, Governors'

Benjamin Bratton: 'Geobiopolitics and Planetary-Scale Computation: Sensors, Abstractors, Governors' | media archaeology | Scoop.it
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At M.I.T., Science Embraces a New Chaos Theory: Art

At M.I.T., Science Embraces a New Chaos Theory: Art | media archaeology | Scoop.it
More than 30 artists have been invited to embed in the labs since the creation of the school’s Center for Art, Science & Technology in 2012.
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Virginia Woolf on Why She Became a Writer and the Shock-Receiving Capacity Necessary for Being an Artist

Virginia Woolf on Why She Became a Writer and the Shock-Receiving Capacity Necessary for Being an Artist | media archaeology | Scoop.it
"Behind the cotton wool is hidden a pattern... the whole world is a work of art... there is no Shakespeare... no Beethoven... no God; we are the words; we are
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The Meaning Of A Sound Place. G. Bohme, The Aesthetics Of The Atmosphere | Digicult | Digital Art, Design and Culture

The Meaning Of A Sound Place. G. Bohme, The Aesthetics Of The Atmosphere | Digicult | Digital Art, Design and Culture | media archaeology | Scoop.it
The theme of space is a cornerstone in the philosophical investigation. It appeals to the world experienced by the subjects in terms of perceptual knowledge, the aim then is never the metrological and scientific analysis of change, but the attempt at clarification of the common sensitive comprehension of those data which the outside world offers to those who meet it. This having been said, space has never ceased to be a current topic, in the two basic forms in which philosophy has investigated it, in the first place the Cartesian spatium understood as the interval between one another and the Aristotelian topos or place. In the latest aesthetic debate, which is developing in Germany as well as in Italy, the topic emerged once again strongly seeking a new path of investigation which could depart from the previous two.
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3 Interesting Tools for Creating Maps with Students

3 Interesting Tools for Creating Maps with Students | media archaeology | Scoop.it

Via Tom D'Amico (@TDOttawa)
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7 CREEPIEST Things On Google Maps

Google Maps street view has taken pictures of some creepy, creepy things. Here are seven pictures guaranteed to make your skin crawl. What creepy ...

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We Talked to the Artist Creating Clouds Inside Cathedrals | The Creators Project

Man-made clouds billow through Neuehouse as Dutch artist Berndnaut Smilde brings his 'Nimbus' project to New York.


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Vanessa Lopes's curator insight, May 24, 2015 3:18 PM
E não é que fizeram a nuvem?
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Do thoughts have a language of their own? The... - Lapidarium notes

Do thoughts have a language of their own? The... - Lapidarium notes | media archaeology | Scoop.it
Do thoughts have a language of their own? The language of thought hypothesis
The language of thought drawing by Robert Horvitz
“ “We dissect nature along lines laid down by our native languages. The...
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Interactions Gallery | British HCI 2016 Conference | 11 – 15th July 2016, Bournemouth

Interactions Gallery | British HCI 2016 Conference | 11 – 15th July 2016, Bournemouth | media archaeology | Scoop.it
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"Mon zombie et moi. La philosophie comme fiction", de Pierre Cassou-Noguès : une peur d'avance

"Mon zombie et moi. La philosophie comme fiction", de Pierre Cassou-Noguès : une peur d'avance | media archaeology | Scoop.it
Cet essai et plusieurs romans de la rentrée illustrent la capacité de la littérature à penser les angoisses du présent en se projetant dans des mondes futurs.
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Grafters: Industrial society in image and word

Grafters: Industrial society in image and word | media archaeology | Scoop.it
The People’s History Museum in Manchester has recently opened a photo exhibition that explores how industrial workers went from objects in photos, to heroic representations of industry and finally …
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DRHA16 - Digital Research in the Humanities and Arts

DRHA16 - Digital Research in the Humanities and Arts | media archaeology | Scoop.it
Facilitating dialogue between academics and practitioners from, Digital Arts and Humanities, Creative Industries and Digital Libraries and Archives
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What is Meditation / Mindfulness Good for? | Information is Beautiful

What is Meditation / Mindfulness Good for? | Information is Beautiful | media archaeology | Scoop.it
Dedicated to distilling the world’s data, information and knowledge into beautiful, interesting and, above all, useful visualizations, infographics and diagrams.
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Rescooped by Lucia Leao from PowerPoint Presentation Tools and Resources
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3D Labyrinth - Hand-drawn Diagrams in PowerPoint

3D Labyrinth - Hand-drawn Diagrams in PowerPoint | media archaeology | Scoop.it
Your search for a Labyrinth PPT template stops here! Capture the attention of your audience by using 3 D slides. Used by consultants, educators and trainers in case studies, decision making modules and in problem solving scenarios.

Via PowerPoint & Keynote Solutions from Chillibreeze
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The Definitive Guide to Free Fonts → Typewolf

The Definitive Guide to Free Fonts → Typewolf | media archaeology | Scoop.it
Discover free alternatives to the most popular (and expensive) fonts used by designers – features not only Google Fonts but free fonts from all over the web.
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ElevatedMaps Launches to Create Artistic 3D Printed Custom Maps of Anyplace in the World

ElevatedMaps Launches to Create Artistic 3D Printed Custom Maps of Anyplace in the World | media archaeology | Scoop.it
I'm in the process of moving into a new home, and boy do I have my hands full. My husband and I have decided to furnish and design the interior of our home comp

Via Brittney
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Brittney's curator insight, June 19, 2015 5:55 AM

ElevatedMaps Launches to Create Artistic 3D Printed Custom Maps of Anyplace in the World http://3dprint.com/74284/3d-printed-maps/