Published: 4th September 2013
DOI: 10.4204/EPTCS.128
ISSN: 2075-2180


Machines, Computations and Universality 2013
Zürich, Switzerland, 9/09/2013 - 11/09/2013

Edited by: Turlough Neary and Matthew Cook

Turlough Neary and Matthew Cook

Invited Talks

Explorations in computer-assisted computer science
Liesbeth De Mol
Universal Pattern Generation by Cellular Automata
Jarkko Kari
Reversible Logic Elements with Memory and Their Universality
Kenichi Morita
Ludicrously fast universal computation and construction with reconfigurable molecular robots
Erik Winfree
Intrinsic universality and the computational power of self-assembly
Damien Woods

Informal Presentations

(An Outline Of) A Fault-Tolerant Turing Machine
Ilir Çapuni
Formal Language Questions for Eulerian Trails
Henning Fernau and Meenakshi Paramasivan
Array Automata on Cayley Grids
Rudolf Freund and Marion Oswald
Fractal Dimension of Space-time Diagrams and the Runtime Complexity of Small Turing Machines
Joost J. Joosten, Fernando Soler-Toscano and Hector Zenil
Causal sets for geometrical Gandy machines and Gandy machines over multisets
Adam Obtułowicz
Machine Spaces: Axioms and Metrics
Jörg Zimmermann and Armin B. Cremers

Contributed Papers

Tiling Problems on Baumslag-Solitar groups.
Nathalie Aubrun and Jarkko Kari
How to Obtain Computational Completeness in P Systems with One Catalyst
Rudolf Freund and Gheorghe Păun
One-dimensional Array Grammars and P Systems with Array Insertion and Deletion Rules
Rudolf Freund, Sergiu Ivanov, Marion Oswald and K.G. Subramanian
Topology and Non-Deterministic Polynomial Time Computation : Avoidance of The Misbehaviour of Hub-Free Diagrams and Consequences
Anthony Gasperin
Satisfiability of cross product terms is complete for real nondeterministic polytime Blum-Shub-Smale machines
Christian Herrmann, Johanna Sokoli and Martin Ziegler
About Strongly Universal Cellular Automata
Maurice Margenstern
Hyperbolic tilings and formal language theory
Maurice Margenstern and K.G. Subramamian
Intrinsic Universality of Causal Graph Dynamics
Simon Martiel and Bruno Martin
AND and/or OR: Uniform Polynomial-Size Circuits
Niall Murphy and Damien Woods
On the Equivalence of Cellular Automata and the Tile Assembly Model
Jacob Hendricks and Matthew J. Patitz
A Small Universal Petri Net
Dmitry A. Zaitsev


This volume contains the papers presented at the 6th conference on Machines, Computations and Universality (MCU 2013). MCU 2013 was held in Zürich, Switzerland, September 9-11, 2013. The MCU series began under the guise Machines et Calculs Universels/Universal Machines and Computations and was organised by Maurice Margenstern in Paris in 1995. Subsequent editions of the conference were held at Metz, France (MCU 1998); Chişinău, Moldova (MCU 2001); Saint-Petersburg, Russia (MCU 2004); and Orléans, France (MCU 2007).

Since its inception MCU has been concerned with gaining a deeper understanding of computation through the study of models of general purpose computation. This volume continues in this tradition and includes new simple universal models of computation, and other results that clarify the relationships between models.

In addition to the contributed papers and invited talks, the program for MCU 2013 included two new features designed to help spread new ideas and foster collaborative efforts: informal presentations and open problem sessions. The informal presentation track allowed researchers to present recent work on topics of relevance for the MCU community. The open problem sessions gave participants the opportunity to share ideas for new research directions and to initiate research collaborations.

We are grateful to the EPTCS staff in particular Rob van Glabbeek for all their help in preparing this volume. We thank all the authors who submitted to MCU 2013, the invited speakers Liesbeth De Mol, Jarkko Kari, Kenichi Morita, Erik Winfree and Damien Woods, the program committee for their sterling work and insightful comments, and the referees who assisted in the evaluation of papers. We would like to thank the MCU steering committee with a special thank you to Maurice Margenstern and Jérôme Durand-Lose for their invaluable advice. Finally we would like to thank our sponsors with a special mention to Rodney Douglas, and also the administrative staff at the Institute of Neuroinformatics for their help in organising this event, in particular Kathrin Aguilar Ruiz-Hofacker, David Lawrence, Simone Schumacher and Frank Hitzemann who were always ready to provide advice and assistance.

Turlough Neary
Matthew Cook

Zürich, September 2013

Program committee

Andrew Adamatzky University of the West of England, UK
Matthew Cook University of Zürich and ETH Zürich (co-chair)
Erzsérbet Csuhaj-Varjú      Eötvös Loránd University, Hungary
Jérôme Durand-Lose University of Orléans, France
Rudolf Freund University of Vienna, Austria
Gabriel Istrate Institute e-Austria, Romania
Jarkko Kari University of Turku, Finland
Lila Kari University of Western Ontario, Canada
Kamala Krithivasan Indian Institute of Technology, India
Maurice Margenstern University of Lorraine, France
Turlough Neary University of Zürich and ETH Zürich (co-chair)
Matthew Patitz University of Arkansas, USA
Igor Potapov University of Liverpool, UK
Klaus Sutner Carnegie Mellon University, USA
Sergey Verlan University of Paris Est, France
Damien Woods California Institute of Technology, USA


Chaim Goodman-Strauss      Jacob Hendricks      Pierre-Etienne Meunier      Amir Simjour

Organising committee

Matthew Cook University of Zürich and ETH Zürich
Turlough Neary        University of Zürich and ETH Zürich

Steering committee

Maurice Margenstern University of Lorraine, France (chair)
Jérôme Durand-Lose University of Orléans, France (vice-chair)
Erzsérbet Csuhaj-Varjú Eötvös Loránd University, Hungary
Natasha Jonoska University of South Florida, USA
Kenichi Morita Hiroshima University, Japan
Gheorghe Păun The Romanian Academy, Romania
Arto Salomaa University of Turku, Finland
K. G. Subramanian University of Science, Malaysia

Sponsoring institutions

Institute of Neuroinformatics (University of Zürich and ETH Zürich)
Swiss National Science Foundation
University of Zürich
ETH Zürich

Explorations in computer-assisted computer science

Liesbeth De Mol
Centre for Logic and Philosophy of Science
Ghent University
elizabeth [dot] demol [at] ugent [dot] be

The aim of this talk is to reflect on the relations, frictions and interactions between theory and (computer) experiments in the context of theoretical computer science. I will draw inspiration from the historiography and philosophy of mathematics where one can observe an increasing interest in what one could call "explorative mathematics". This will be connected to the history of computer science where, as I will argue, one can find a similar school of thought throughout its short history. By way of a case study, I will present some old and new theoretical and experimental results on tag systems to illustrate the difficulties and opportunities involved with "explorative computer science".

Universal Pattern Generation by Cellular Automata

Jarkko Kari
Department of Mathematics and Statistics
University of Turku, Finland
jkari [at] utu [dot] fi

A one-dimensional cellular automaton is a discrete dynamical system where a sequence of symbols evolves synchronously according to a local update rule. We discuss simple update rules that make the automaton perform multiplications of numbers by a constant. If the constant and the number base are selected suitably the automaton becomes a universal pattern generator: all finite strings over its state alphabet appear from a finite seed. In particular we consider the automata that multiply by constants 3 and 3/2 in base 6. We discuss the connections of these automata to some difficult open questions in number theory, and we pose several further questions concerning pattern generation in cellular automata.


  1. Jarkko Kari (2012): Universal pattern generation by cellular automata. Theoretical Computer Science 429, pp. 180 – 184 doi:10.1016/j.tcs.2011.12.037.

Ludicrously fast universal computation and construction with reconfigurable molecular robots

Erik Winfree
California Institute of Technology
Pasadena, California
winfree [at] caltech [dot] edu

I will review recent theoretical results in the 'nubots' model for reconfigurable molecular robots, which combines features of asynchronous cellular automata and L-systems. Like cellular automata, units contain finite state, are arranged on a two-dimensional lattice, and take actions in parallel based on their local neighborhood. Like L-systems, it is possible for units to duplicate (akin to cell division) and therefore the number of units can grow exponentially with time. The key innovation is a movement rule, inspired by molecular motor proteins, that allows one unit to move relative to another -- and potentially pull with it a collection of other units to which it has formed bonds. Thus, the model allows long-range signaling via mechanical motion of large bonded groups of units. This is asymptotically unrealistic since reasonable mechanical signals shouldn't travel faster than the speed of sound, much as the standard boolean circuit model is asymptotically unrealistic since it allows electronic signals to travel faster than light. Nonetheless, the nubots model provides insight into the complexity of 'biological' tasks such as the growth, from a single unit, of a large and complex object. A central result is that a wide class of algorithmically-described shapes can be grown in time polylogarithmic in the number of units in the final shape. This is exponentially faster than would be possible in cellular automata models. Joint work with Damien Woods, Ho-Lin Chen, Scott Goodfriend, Nadine Dabby, and Peng Yin. arXiv:1301.2626 [cs.DS]

(An Outline Of) A Fault-Tolerant Turing Machine

Ilir Çapuni
Department of Computer Engineering
Epoka University
Tirana, Albania
ilir [at] bu [dot] edu

We consider computations of a Turing machine under noise that causes violations of the transition function that we call faults. In this paper we review the construction of a Turing machine given in ([1]) that can simulate any other fault-free machine even when it is subjected to noise that causes bursts of faults of size $ \leq \beta$ that are at least $O(\beta^2)$ apart from each other. Further, we outline how this construction can be used as a building block for a machine that can simulate any other machine when it is subjected to a noise that causes faults that occur independently of each other with some small probability, as presented in ([2]) . This paper is an exposition of a part of a joint work with Peter Gacs.

Isolated bursts of faults

We start by giving a brief overview of a machine $ M_1 $ that simulates machine $ M_2 $, and can withstand isolated bursts of faults.

Each tape cell of the simulated machine $ M_{2} $ will be represented by a block of size $ Q $ of the simulating machine $ M_{1} $ called a colony . Each step of $ M_{2} $ will be simulated by a computation of $ M_{1} $ called a work period. During this time, the head of $ M_{1} $ makes a number of sweeps over the current colony, decodes the represented cell symbol and state, computes the new state, and transfers the necessary information to the neighbor colony that corresponds to the new position of the head of $ M_{2} $.

In order to protect information from the propagation of errors, the tape of $ M_{1} $ is subdivided into tracks: each track corresponds to a field of a cell symbol of $ M_{1} $ viewed as a data record. Each stage of computation will be repeated three times. The results will be stored in separate tracks, and a final cell-by-cell majority vote will recover the result of the work period from them.

All this organization is controlled by a few key fields, and the technically most challenging part of the construction is the protection of this control information from bursts. For example, a burst can reverse the head in the middle of a sweep. Our goal is that such structural disruptions be discovered locally, so we cannot allow the head to go far from the place where it was turned back. Therefore the head's movement will not be straight even during a single sweep: it will make frequent short switchbacks (zigzags). This will trigger alarm, and the start of a recovery procedure if for example a turn-back is detected. It is a significant challenge that the recovery procedure itself can be interrupted (or started) by a burst.

Hierarchical construction

In order to build a machine that can resist faults occuring independently of each other with some small probability, we take the approach suggested in ([4]), and implemented in ([3]) for the case of one-dimensional cellular automata. We will build a hierarchy of simulations: machine $ M_1 $ simulates machine $ M_2 $ which simulates machine $ M_3 $, and so on. For simplicity we assume all these machines have the same program, and all simulations have the same blocksize.

One cell of machine $ M_3 $ is simulated by one colony of machine $ M_2 $. Correspondingly, one cell of $ M_2 $ is simulated by one colony of machine $ M_1 $. So one cell of $ M_3 $ is simulated by $ Q^2 $ cells of $ M_1 $. Further, one step of machine $ M_3 $ is simulated by one work period of $ M_2 $ of, say, $ O(Q^{2}) $ steps. One step of $ M_2 $ is simulated by one work period of $ M_1 $, so one step of $ M_3 $ is simulated by $ O(Q^{4}) $ steps of $ M_1 $.

Per construction, machine $ M_{1} $ can withstand bursts of faults whose size is $ \le \beta $ for some constant parameter $ \beta $, that are separated by some $ O(Q^{2}) $ fault-free steps. Machines $ M_2 $, $ M_3 $, $ \dots $ have the same program, so it would be natural to expect that machine $ M_1 $ can withstand also some additional , larger bursts of size $ \le \beta Q $ if those are separated by, say, at least $ O(Q^{4}) $ steps.

But a new obstacle arises. On the first level, damage caused by a big burst spans several colonies. The repair mechanism of machine $ M_1 $ outlined in the previous section above is too local to recover from such extensive damage. This cannot be allowed, since then the whole hierarchy would stop working. So we add a new mechanism to $ M_{1} $ that, more modestly, will just try to restore a large enough portion of the tape, so it can go on with the simulation of $ M_2 $, even if all original information was lost. For this, $ M_{1} $ may need to rewrite an area as large as a few colonies. This will enable the low-level recovery procedure of machine $ M_{2} $ to restore eventually a higher-level order.

A tricky issue is "forced self-simulation": while we are constructing machine $ M_1 $ we want to give it the feature that it will simulate a machine $ M_{2} $ that works just like $ M_{1} $. The "forced" feature means that this simulation should work without any written program (that could be corrupted). This will be achieved by a construction similar to the proof of the Kleene's fixed-point theorem. We first fix a (simple) programming language to express the transition function of a Turing machine. Then, we write an interpreter for it in this same language. The program of the transition function of $ M_{2} $ in this language, is a string that will be "hard-wired" into the transition function of $ M_{1} $, so that $ M_{1} $, at the start of each work period, can write it on a working track of the base colony. Then the work period will interpret it, applying it to the data found there, resulting in the simulation of $ M_{2} $. In this way, an infinite sequence of simulations arises, in order to withstand larger and larger but sparser and sparser bursts of faults. It can be proven that with the appropriate choice of parameters, with probability 1, eventually nothing is left over from the set of the faulty times, if we first remove bursts of size $\beta_1$ that are $V_1$ apart from each other, and then continue with the removal of bursts of size $\beta_2$ that are $V_2$ apart from each other, and so on.

Since the $ M_1 $ uses the universal interpreter, which in turns simulates the same program, it is natural to ask how machine $ M_1 $ simulates a given Turing machine $ G $ that does the actual useful computation? For this task, we set aside a separate track on each machine $ M_i $, on which some arbitrary other Turing machine can be simulated.


  1. Ilir Capuni & Peter Gacs: A Turing Machine Resisting Isolated Bursts Of Faults. Chicago Journal of Theoretical Computer Science 2013, doi: 10.4086/cjtcs.2013.003
  2. Ilir Capuni (2012): A Fault-tolerant Turing Machine. Ph.D. thesis, Boston University, Commonwealth avenue, Boston MA 02215.
  3. Peter Gacs (2001): Reliable Cellular Automata with Self-Organization. Journal of Statistical Physics 103(1/2), pp. 45-267, doi:10.1023/A:1004823720305 See also arXiv:math/0003117 [math.PR] and the proceedings of STOC '97.
  4. G. L. Kurdyumov (1978): An Example of a Nonergodic Homogenous One-Dimensional Random Medium with Positive Transition Probabilities. Soviet Mathematical Doklady 19(1), pp. 211-214.

Formal Language Questions for Eulerian Trails

Henning Fernau
Abteilung Informatikwissenschaften, Fachbereich IV
Universität Trier, Germany
Meenakshi Paramasivan
Abteilung Informatikwissenschaften, Fachbereich IV
Universität Trier, Germany

We consider several formal languages associated to multigraphs that allow for Eulerian trails. We place these languages into the Chomsky hierarchy and show several normal form properties.

1 Introduction

We consider undirected multigraphs. A walk is an alternating sequence of vertices and edges, beginning and ending with a vertex, where the vertices that precede and follow an edge are the end vertices of that edge. A walk is closed if its first and last vertices are the same, and open if they are different. A trail is a walk in which all the edges are distinct. Closed trails are called tours. A trail is Eulerian if it uses all edges precisely once. Multigraphs admit a Eulerian trail if and only if either every vertex has even degree (in that case, a Eulerian tour is admitted) or all but two vertices have even degree (in that case, the two vertices of odd degree are the endvertices of that trail).

If we draw all vertices of a multigraph G = (V, E) on a horizontal line, then we can associate different integers to vertices by a mapping φ : V → 𝓩 called pseudo-linear drawing such that u is to the left of v in the drawing iff φ (u) < φ (v). If G = (V, E) admits a Eulerian trail, we can describe the edge set of G by a sequence of drawing commands like → |n, meaning that we have to connect the current vertex u to the vertex v satisfying φ (v) = φ (u) + n, with v becoming then the current vertex, i.e., describing a right move by n steps, or ← |n , describing a left move by n steps, Hence, G, φ and a startpoint x ∈ V on the trail uniquely specify a word w(G, φ , x) over the alphabet 𝛴 = {→, ←, |}, called Euler trace. Conversely, every word in ET = 𝛴+ \ ({|}𝛴) describes a multigraph Gw that admits a Eulerian trail. For instance, w =→ | → | ← || describes a triangle, i.e., Gw = C3 . Concepts similar to ET have been studied in [1, 2]. Obviously, ET ⊆ 𝛴 is a regular language. Notice that, as φ (x) is not specified with w ∈ ET, there are infinitely many pseudo-linear drawings φ such that w = w(G, φ , x), but all are obtained from one another by shifting along the horizontal line and hence describe the same multigraph. For the standard pseudo-linear drawing φw of w ∈ ET , we assume that the start vertex x of the Eulerian trail of the specified multigraph Gw obeys φw (x) = 0. We will be interested in certain subsets of ET that describe interesting properties of Euler traces.

2 Normal forms

The word w =→ || describes a (multi)graph on two vertices x, y with one edge xy. Arbitrarily setting φ (x) = 0, φ (y) = 2 is obtained. However, there is no vertex z with φ (z) = 1. We will call any integer n a hole for φ if there exist vertices u, v with φ (u) < n < φ (v) but no vertex z with φ (z) = n. So, in our example, 1 is a hole. It is obvious that every multigraph with a Eulerian trail has a pseudo-linear drawing without any holes. As having holes or not is invariant under shifts, it makes sense to call w ∈ ET hole-free if the standard pseudo-linear drawing φw does not contain any holes. Define EThole-free = {w ∈ ET : w is hole-free}. From our description, it follows that every multigraph G with a Eulerian trail can be described by some w ∈ EThole-free.

Theorem 1. There is a polynomial-time algorithm that, given w ∈ ET, computes some w' ∈ EThole-free with Gw = Gw' .

Theorem 2. ET \ EThole-free is a non-regular language that can be described by some one-counter automaton, while EThole-free is not context-free.

For the first part, notice that a one-counter automaton can guess the hole position and then verify its guess. For the second part, consider L = EThole-free ∩ ({→ |}{|}+ {←}{|}+ {→ |}+ ). A rational transduction can transfer {am bm cm : m ≥ 1} to L = {→ |n ← |m (→ |)l : n > 1 ∧ m ∈ {n - 1, n, n + 1} ∧ l ∈{m - 1, m, m + 1}}, which shows that L is not context-free.

Another observation is that any multigraph G with a Eulerian trail has a pseudo-linear drawing φ that never puts any vertex to the left of the start vertex x. Also this property is invariant under shifting, so that it is also a property of words w ∈ ET (via φw ). This defines the languages ETleft-start and EThole-free,left-start, where the latter collects all hole-free words from ETleft-start.

Theorem 3. There is a polynomial-time algorithm that, given w ∈ ET, computes some w' ∈ EThole-free,left-start with Gw = Gw' .

Theorem 4. Both ETleft-start and ET \ ETleft-start are context-free (in fact, one-counter languages). Neither EThole-free,left-start nor ET \ EThole-free,left-start are context-free.

3 Euler tours

We can likewise define the set ET of descriptions of multigraphs admitting some Euler tour. With the transduction 𝜏 that replaces any sequence of | following → (until the end of word or some ←) by ↑ and the remaining | by ↓, w ∈ ET iff w ∈ ET and 𝜏(w) contains as many ↑ as ↓ symbols.

Theorem 5. ET and ET \ ET are context-free (in fact, one-counter languages).

Although the mentioned normal forms also exist for Euler tours, and they can be produced in polynomial-time, none of the corresponding three languages nor their complements are context-free.

4 Conclusions

We have started looking into the complexity of Eulerian multigraphs from a formal language perspective. This research direction allows for many similar questions, for instance, classifying ETgraph (not context-free, but the complement is), ETloop-free (regular) or also ET2-regular = {w ∈ ET : ∀v ∈ V (Gw ) : d(v) = 2}. Notice that ET \ ET2-regular is a one-counter language, while ET2-regular ∩ {→ |}+ ⋅ {←} ⋅ {|}+ ⋅ {→} ⋅ {|}+ ⋅ {←} ⋅ {|}+ = {(→ |)n ← |m → |l ← |k : n > 0, m > n, l> m, k + m = n + l } is not context-free.


[1] L. Jeganathan, K. Krithivasan & R. Rama (2010): Graph Splicing Systems. In Z. Majkic, M. Sipser, R. Radha & D. Wei, editors: International Conference on Theoretical and Mathematical Foundations of Computer Science, TMFCS 2008, Curran Associates, Inc., pp. 72-79.

[2] M. Paramasivan & N. G. David (2011): Shuffle Operations on Euler Graphs. Mapana Journal of Sciences 10(1), pp. 63-78. Available at

Array Automata on Cayley Grids

Rudolf Freund
Technische Universität Wien
Institut für Computersprachen
Favoritenstr. 9, A-1040 Wien, Austria
rudi [at] emcc [dot] at
Marion Oswald
Technische Universität Wien
Institut für Computersprachen
Favoritenstr. 9, A-1040 Wien, Austria
marion [at] emcc [dot] at

As a natural extension of string languages (see [6]), arrays on the $d$-dimensional grid $\mathbb{Z}^{d}$ have been introduced, and the correspondence between array grammars generating and array automata accepting $k$-connected arrays have been investigated. For any $d\geq 1$, a $d$-dimensional array $\mathcal{A}$ over an alphabet $V$ is a function $\mathcal{A}:\mathbb{Z}^{d}\rightarrow V\cup \left\{ \#\right\}$, where $shape\left( \mathcal{A}\right) =\left\{ v\in \mathbb{Z}^{d}\mid \mathcal{A}\left( v\right) \neq \#\right\}$ is finite and $\#\notin V$ is called the blank symbol. $V^{\ast d}$ denotes the set of all $d$-dimensional arrays over $V$. For any array $\mathcal{A}\in V^{\ast d}$ and any $v\in \mathbb{Z}^{d}$ we define $\tau _{v}\left( \mathcal{A} \right) $ to be the $d$-dimensional array translated by $v$, i.e., $% \left( \tau _{v}\left( \mathcal{A}\right) \right) \left( w\right) =\mathcal{A% }\left( w-v\right) $ for all $w\in \mathbb{Z}^{d}$. Usually (e.g., see [257]) arrays are regarded as equivalence classes of arrays with respect to linear translations, i. e., only the relative positions of the symbols different from $\#$ in $\mathbb{Z}^{d}$ are taken into account: the equivalence class $\left[ \mathcal{A}\right] $ of an~array $\mathcal{A}\in V^{\ast d}$ is defined by $\left[ \mathcal{A} \right] =\{\mathcal{B}\in V^{\ast d}\mid \mathcal{B}=\tau _{v}\left( \mathcal{A}\right) $ for some $v\in \mathbb{Z}^{d}\}$. The set of all equivalence classes of $d$-dimensional arrays over $V$ with respect to linear translations is denoted by $\left[ V^{\ast d}\right]$.

Following the ideas of E. Csuhaj-Varjú and V. Mitrana ([3]), we started to investigate arrays on Cayley grids of finitely generated groups. For a group $G$ generated by the elements $e_{1}$,...,$e_{m}$, we consider the corresponding Cayley graph $C\left( G\right)$ and arrays over an alphabet $V$ on $C\left( G\right)$ with the nodes of $C\left( G\right)$ getting assigned elements of $V\cup \left\{ \#\right\} $. In that sense, with $N\left( C\left( G\right) \right) $ denoting the set of nodesof $C\left( G\right)$, a finite array on $C\left( G\right)$ is a function $\mathcal{A}:N\left( C\left( G\right) \right) \rightarrow V\cup \left\{ \#\right\}$ with $shape\left( \mathcal{A}\right) =\left\{ v\in N\left( C\left( G\right) \right) \mid \mathcal{A} \left( v\right) \neq \#\right\}$ being finite. $V^{\ast }\left\langle C\left(G\right)\right\rangle$ ($\left[ V^{\ast }\left\langle C\left(G\right) \right\rangle \right])$ denotes the set of (equivalence classes) of arrays over $V$ on $C\left( G\right)$. An array $\mathcal{A\in }V^{\ast}\left\langle C\left( G\right) \right\rangle $ is called $k$-connected if for any two elements $v$ and $w$ in $shape\left( \mathcal{A}\right)$ there is a path $v=w_{1}-...-w_{n}=w$ in $C\left( G\right)$ with $\left\{w_{1},...,w_{n}\right\}\subseteq shape\left(\mathcal{A}\right)$ such that for the distance $d\left( w_{i},w_{i-1}\right)$ between $w_{i}$ and $w_{i-1}$ we have $d\left( w_{i},w_{i-1}\right) \leq k$ for all $1<i\leq n$.

$\mathbb{Z}$ is a special case of an Abelian group generated by $\left(1\right)$ and its inverse $\left(-1\right)$; $\mathbb{Z}^{d}$ is an Abelian group generated by the unit vectors $\left( 0,...,1,...0\right)$ and their inverses. It is well known that every finitely generated Abelian group is a direct sum of a torsion group and a free Abelian group where the torsion group may be written as a direct sum of finitely many groups of the form $\mathbb{Z}/p^{k}\mathbb{Z}$ for $p$ being a prime, and the free Abelian group is a direct sum of finitely many copies of $\mathbb{Z}$. As a first example of arrays on a Cayley grid of a non-Abelian group, arrays on the hexagonal grid were considered ([1]).

A derivation step in an array grammar generating a language of arrays on the Cayley grid $C\left( G\right) $ consists of replacing a finite subpattern by another subpattern on $C\left( G\right) $ of the same shape. Let $\mathcal{L}_{gen}\left( C\left( G\right),ARB\right)$ denote the family of array languages of $k$-connected arrays on $C\left( G\right)$ generated by array grammars with arbitrary rules, and let $ENUM\left(C\left(G\right)\right)$ be the family of recursively enumerable array languages of $k$-connected arrays on $C\left( G\right) $ (i.e., the arrays are encoded as strings by using a suitable encoding for the positions in $C\left( G\right)$).

On the other hand, array automata accepting $k$-connected arrays on $C\left(G\right)$ can be defined by generalizing Turing machines working on a one-dimensional tape for accepting strings (e.g., see [4]). Like in the string case, we may consider deterministic and non-deterministic variants of array automata; the families of array languages of arrays on $C\left( G\right)$ accepted by (non-)deterministic array automata are denoted by $\mathcal{L}_{acc}\left(C\left( G\right) ,ATM\right) $ and $\mathcal{L}_{acc}\left( C\left( G\right),detATM\right) $, respectively.

Due to the characterization of finitely generated Abelian groups, we may state the following result:

Theorem 1 For any finitely generated Abelian group, \begin{equation*} \begin{array}[b]{lllll} \mathcal{L}_{gen}\left( C\left( G\right) ,ARB\right) & = & ENUM\left( C\left( G\right) \right) & & \\ & = & \mathcal{L}_{acc}\left( C\left( G\right) ,ATM\right) & = & \mathcal{L} _{acc}\left( C\left( G\right) ,detATM\right). \end{array} \end{equation*}

For non-Abelian groups, at least the equalities \begin{equation*} \begin{array}[b]{lll} \mathcal{L}_{gen}\left( C\left( G\right) ,ARB\right) & = & ENUM\left( C\left( G\right) \right) \\ & = & \mathcal{L}_{acc}\left( C\left( G\right) ,ATM\right) \end{array} \end{equation*} hold as long as the conditions restricting the free group are (Turing) decidable and all paths in the Cayley graph $C(G)$ are (Turing) computable.

There are interesting subclasses of non-Abelian groups $G$ where each element has finite order. As a special variant, there may even exist a special integer $n$ such that $x^{n}=e$ for every $x$ in $G$, with $e$ being the neutral element of $G$ and $n$ being called the exponent of $G$. As we know from algebra, there are infinite groups having this property, e.g., some of the famous Burnside groups $B\left( m,n\right) $: the free Burnside group of rank $m$ and exponent $n$, denoted by $B\left( m,n\right) $, is the maximal group with $m$ distinguished generators $x_{1},...,x_{m}$ and $x^{n}=1$ for each $x\in $ $\left\{ x_{1},...,x_{m}\right\} $. For specific $m$ and $n$, $B\left( m,n\right) $ is infinite. For such an infinite group, finitely generated, with exponent $n$, in contrast to the string case or the usual case of $\mathbb{Z}^{d}$ or even the hexagonal grid, no infinite line within the Cayley grid can be determined, hence, it seems to be impossible to get arbitrary workspace for intermediate computations in a deterministic way. Following this informal argument, we cojecture that the deterministic variant of array automata working on Cayley graphs of such groups has weaker accepting power than the non-deterministic variant:

Conjecture 2 For any infinite free Burnside group $B\left( m,n\right) $, \begin{equation*} \begin{array}[b]{lllll} \mathcal{L}_{gen}\left( C\left( B\left( m,n\right) \right) ,ARB\right) & = & ENUM\left( C\left( B\left( m,n\right) \right) \right) & & \\ & = & \mathcal{L}_{acc}\left( C\left( B\left( m,n\right) \right) ,ATM\right) & \supset & \mathcal{L}_{acc}\left( C\left( B\left( m,n\right) \right) ,detATM\right). \end{array} \end{equation*} It is even not clear if there exists a free Burnside group $B\left( m,n\right) $ such that $\mathcal{L}_{acc}\left( C\left( B\left( m,n\right) \right) ,detATM\right) $ contains non-recursive array languages.


[1]   K. Aizawa & A. Nakamura (1989): Grammars on the hexagonal array. In P. S.-P. Wang, editor: Array Grammars, Patterns and Recognizers, Series in Computer Science 18, World Scientific Publ., pp. 469–477, doi:10.1142/9789814368308_0012.

[2]   C. R. Cook & P. S.-P. Wang (1978): A Chomsky hierarchy of isotonic array grammars and languages. Computer Graphics and Image Processing 8, pp. 144–152, doi:10.1016/S0146-664X(78)80022-7.

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Fractal Dimension of Space-time Diagrams and the Runtime
Complexity of Small Turing Machines

Joost J. Joosten
Universitat de Barcelona
Barcelona, Spain
jjoosten [at] ub [dot] edu
Fernando Soler-Toscano
Universidad de Sevilla
Sevilla, Spain
fsoler [at] us [dot] es
Hector Zenil
Karolinska Institute
Stockholm, Sweden
hectorz [at] labores [dot] eu

Complexity measures are designed to capture complex behaviour and to quantify how complex that particular behaviour is. If a certain phenomenon is genuinely complex this means that it does not all of a sudden becomes simple by just translating the phenomenon to a different setting or framework with a different complexity value. It is in this sense that we expect different complexity measures from possibly entirely different fields to be related to each other.

In this work we look at small one-way infinite Turing machines (TMs) with two and three states and a binary tape alphabet. For any particular such machine τ and any particular input x we consider the space-time diagram of τ(x) as the collection of consecutive tape configurations of the computation τ(x). We look at the spatial representation σ0 of the memory when τ starts on input x. Next we look at σ1: the spatial representation of the memory after one step in the computation and so forth (see Fig. 1).

Space-time diagram of the Busy Beaver in (3,2)
Figure 1: Example of Space-time diagrams for one of the 2 Busy Beavers in (3,2) on the first three inputs (TMs 599 063 and 666 364 in Wolfram's enumeration). Their computation has the largest runtime, space and boxes sequences.

It is on these spatial objects (see Fig. 1) and in particular the limit for x→∞ for which we compute or estimate the fractal dimension d(τ) defined as follows.

Definition 1 (Box dimension of a Turing machine) Let τ be a TM that converges on infinitely many input values x. In case τ(x)↓, let N(τ, x) denote the number of black cells in the space-time diagram of τ on input x and let t(τ, x) denote the number of steps needed for τ to halt on x. We will define the Box dimension of a TM τ and denote it by d(τ). In case t(τ,x) is constant from some x onwards, we define d(τ) : = 2. Otherwise, we define

d(τ) : =
lim inf
x→ ∞ 


It turns out that there is a very strong relation between the fractal dimension of d(τ) and its runtime complexity. In particular we found in small spaces that, a TM runs in at most linear time, if and only if, its fractal dimension is 2, and its dimension is 1, if only if, it runs in super-polynomial time and it uses polynomial space. We prove an upper and a lower bounds for the dimension of Turing machines and we verify experimentally (see Table 1) that if a TM runs in time O(xn), the corresponding dimension is (n+1)/n. We also put forward an Upper Bound Conjecture to the effect that the proven upper bound is actually always attained. For special cases this can also be proved. Moreover, under some additional assumptions this can also be proven in general. In the experiment we test if in our test-space the additional assumptions were also necessary ones and they turn out to be so.

Thus, we have a translation from the computational framework to the geometrical framework. Fractal dimension is clearly related to degrees of freedom and as such related to an amount of information storage. We find the results presented here notable because they relate two completely different complexity measures: the geometrical fractal dimension and the runtime complexity of a computation.

The paper forms part of a research programme where the authors have exhaustively mined and thoroughly investigated the behavior of small Turing machines related to different measures of complexity [1,2,3,4,5,6].

Boxes  Runtime  Space  Machines 
O(n3) O(n2) O(n) 3358
O(n4) O(n3) O(n) 6
o(Ρ) o(Ρ) o(Ρ) 14
Table 1: Distribution of TMs in the space 3 states and 2 symbols of which we had to compute the corresponding dimension over their complexity classes. o(Ρ) denotes the class of super-polynomials.


[1] Jean-Paul Delahaye & Hector Zenil (2012): Numerical evaluation of algorithmic complexity for short strings: A glance into the innermost structure of randomness. Applied Mathematics and Computation 219(1), pp. 63-77, doi:10.1016/j.amc.2011.10.006. Available at

[2] Joost J. Joosten, Fernando Soler-Toscano & Hector Zenil (2011): Program-size versus Time Complexity. Slowdown and Speed-up Phenomena in the Micro-cosmos of Small Turing Machines. International Journal of Unconventional Computing 7(5), pp. 353-387. Available at

[3] Joost J. Joosten, Hector Zenil & Fernando Soler-Toscano (2011): Fractal Dimension as an Indication of the Terminating Runtime of Discrete Dynamical Systems (abstract). In S. Thurner & M. Szell, editors: ECCS'11, Vienna, Austria, p. 214. Available at

[4] Fernando Soler-Toscano, Hector Zenil, Jean-Paul Delahaye & Nicolas Gauvrit (2012): Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines. arXiv:1211.1302 [cs.IT]. Available at

[5] Fernando Soler-Toscano, Hector Zenil, Jean-Paul Delahaye & Nicolas Gauvrit (2013): Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures. Computability 2, doi:10.3233/COM-13019. Available at

[6] Hector Zenil, Fernando Soler-Toscano & Joost J. Joosten (2012): Empirical Encounters with Computational Irreducibility and Unpredictability. Minds and Machines 22(3), pp. 149-165, doi:10.1007/s11023-011-9262-y. Available at

Causal sets for geometrical Gandy machines
and Gandy machines over multisets

Adam Obtułowicz
Institute of Mathematics, Polish Academy of Sciences
Śniadeckich 8, P.O.B. 21, 00-956 Warsaw, Poland

A new approach to the computational complexity beyond the known complexity measures of the consumed time and space of computation is proposed. The approach focuses on the chaotic behaviour and randomness aspects of computational processes and bases on a representation of these processes by causal sets. The abstract systems allowing some synchronized parallelism of computation are investigated within the approach, where the computational processes realized in a discrete time are represented by causal sets like in [2]. The representation of computational processes by causal sets is aimed to provide an abstraction from those features of computational processes which have not a spatial nature such that the abstraction could make visible some new aspects of the processes like an aspect of chaotic behaviour or a fractal shape. The aspects of a chaotic behaviour and a fractal shape inspired by the research area of dynamics of nonlinear systems [11] regarding an unpredictability of the behaviour of these systems¹ could suggest an answer to the following question formulated in [13]: Is the concept of randomness, founded in the concept of absence of computable regularities, the only adequate and consistent one? In which direction, if any, should one look for alternatives? The answers may have an impact on designing pseudorandom number generators applied in statistics, cryptography, and Monte Carlo Method. Thus the proposed approach comprises measuring of complexity of computational processes by a use of graph dimensions [6] and network fractal dimensions [10] in parallel to measuring complexity of random strings in [4] by Hausdorff dimension. The proposed approach concerns investigations of abstract computing devices of two types: The geometrical Gandy machines are some modifications of the known Gandy's mechanisms [3] by assuming that the sets of machine instantaneous descriptions² are skeletal sets, similarly like in [7], with respect to the permutations of urelements, where urelements are n-tuples of rational numbers. Hence the adjective “geometrical” is used. It has been pointed out in [12] that Gandy mechanisms “conform to basic principles of physics”, see also  [1]. The geometrical Gandy machines are aimed to respect a claim (contained in the open problem in [12]) for Gandy mechanisms to be “more consistent with local causation in physics”. In other words, to be more realistic than (possible) imaginary constructs within the theory of hereditarily finite sets and hence less “technically somewhat complicated”. The Gandy machines over multisets are geometrical Gandy machines, where the urelements appearing in the machine instantaneous descriptions are in addition labelled by multiset and the machine rules of local causation respect processing of multisets in the manner of P system evolution rules. These machines are aimed to present some uniform families of P system [5] in terms of a one machine with finite number of schemes of evolution rules. The assumption that the sets of machine instantaneous descriptions are skeletal together with the features of machine local causation rules provide a natural construction of causal sets representing computational processes. The causal sets representing computational processes are here subsets of space-time, i.e. (n+1)-tuples of rational numbers if urelements are n-tuples (forming phase space), and the causality relations are determined by the applications of machine local causation rules, respectively. The examples of Gandy-Păun-Rozenberg machines, including generalized machines, in [7] and [8] give rise to the examples of geometrical Gandy machines after some slight modifications. Some models of computation in [2] may give rise to examples of geometrical Gandy machines whose computational processes are represented by causal sets of fractal shape.


  1. P. Arrighi & G. Dowek (2011): The physical Church--Turing thesis and principles of quantum theory. arXiv: 1102.1612.
  2. T. Bolognesi (2010): Causal sets from simple models of computation. Int. Journal of Unconventional Computing 6, pp. 489–524.
  3. R. Gandy (1980): Church's thesis and principles for mechanisms. In: The Kleene Symposium, North-Holland, Amsterdam, pp. 123–148, doi:10.1016/S0049-237X(08)71257-6
  4. J. H. Lutz (2003): The dimensions of individual strings and sequences. Information and Computation 187, pp. 49–79, doi:10.1016/S0890-5401(03)00187-1
  5. N. Murphy & D. Woods (2011): The computational power of membrane systems under tight uniformity condition. Nat. Computing 10, pp. 613–632, doi:10.1007/s11047-010-9244-7
  6. T. Nowotny & M. Requardt (1998): Dimension theory of graphs and networks. J. Phys. A Math. Gen. 31, pp. 2447–2463, doi:10.1088/0305-4470/31/10/018
  7. A. Obtułowicz (2011): Randomized Gandy–Păun–Rozenberg machines. In: Membrane Computing, Lecture Notes in Computer Science 6501, Springer, Berlin, pp. 305–324, doi:10.1007/978-3-642-18123-8_24
  8. A. Obtułowicz (2012): Generalized Gandy–Păun–Rozenberg machines for tile systems and cellular automata. In: Conference on Membrane Computing 2011, Lecture Notes in Computer Science 7184, Springer, Berlin, pp. 314–322, doi:10.1007/978-3-642-28024-5_21
  9. G. Păun, G. Rozenberg & A. Salomaa (2009): The Oxford Handbook of Membrane Computing. Oxford Univ. Press, Oxford.
  10. H. D. Rozenfeld, L. K. Gallos, Ch. Song & H. A. Makse (2009): Fractal and transfractal scale-free networks. In: Encyclopedia of Complexity and System Science, Springer, Berlin, pp. 3924–3943, doi:10.1007/978-0-387-30440-3_231
  11. S. H. Strogatz (1994): Nonlinear Dynamics and Chaos. Perseus Books Publ., LLC.
  12. K. Sutner (1998): Computation theory of cellular automata. In: MFCS'98 Satellite Workshop on Cellular Automata, Brno, Czech Republic, August 25–27, 1998.
  13. S. B. Volchan (2002): What is a random sequence. Amer. Math. Monthly 109, pp. 46–63, doi:10.2307/2695767

¹ unpredictability due to sensitive dependence on initial conditions–an important feature of deterministic transient chaos [11] often having fractal shape.
² a machine instantaneous description is here a hereditarily finite set which describes potential ‘wireless’ intercommunication between urelements appearing in this set.

Machine Spaces: Axioms and Metrics

Jörg Zimmermann
Institute of Computer Science
University of Bonn, Germany
jz [at] iai [dot] uni-bonn [dot] de
Armin B. Cremers
Institute of Computer Science
University of Bonn, Germany
abc [at] iai [dot] uni-bonn [dot] de

1   Time Axioms

We use the axiomatic method in order to investigate the notion of an abstract machine and its implied set of computable functions and complexity classes. Time is the driver of the computational processes, so we want to isolate the abstract properties of time which are relevant from a computational point of view. Here we model time as a totally ordered monoid, thus allowing discrete, continuous, and transfinite time structures. All time elements can be compared and be combined by an associative operation, denoted by ‘+’, which has a neutral element 0(t). Furthermore, the order structure and the algebraic structure are compatible:

(Compatibility)   ∀ t1, t2, t3: t1 ≤ t2 ⇒ t1+t3 ≤ t2+t3 and t3+t1 ≤ t3+t2.

Compatibility is required in both possible ways because the monoid operation has not to be commutative (which is, for example, the case for the sum of ordinal numbers).

2   Machine Axioms

Machines operate on a state space Σ, about which we want to say nothing specific here. Further we introduce the input space I, output space O, and program space P. An initializer is a mapping init from P × I to Σ and an output operator is a mapping out from Σ to O. A machine wrt. a time structure T and a state space Σ is a mapping M from Σ × T to Σ, denoted by Mt(s). Finally, there is a subset HALT of Σ. States in HALT will be used to signal termination of a computation. Let TERMM(s) denote the set of time points t with Mt(s) ∈ HALT. Now we require the following axioms:

(Start)   ∀ s ∈ Σ: M0(t)(s) = s   (i.e., M0(t) = idΣ),

(Action)   ∀ t1, t2 ∈ T: Mt1+t2 = Mt2 ∘ Mt1.

These two axioms state that the time monoid is operating on the state space via machine M.

(Stop)   ∀ s ∈ Σ, t1, t2 ∈ T: t1 ∈ TERMM(s) and t1 ≤ t2 ⇒ Mt1(s) = Mt2(s).

That is, after reaching a termination state, nothing changes anymore, i.e., termination states are fixpoints of the machine dynamics.

(Well-Termination)   ∀ s ∈ Σ: TERMM(s) ≠ ∅ ⇒ ∃ t1 ∈ TERMM(s) ∀ t2 ∈ TERMM(s): t1 ≤ t2.

Well-termination requires that if a machine terminates on s, i.e., reaches HALT for some point in time, then there is a first point in time when this happens, or, equivalently, the set of time points taking s to HALT has a least element, if it is not empty. If TERMM(s) is non-empty and t* its least element, then define M(s) = out(Mt*(s)), otherwise M(s) is left undefined.

Definition 2.1. A function f: I → O is implemented by p ∈ P on M iff f(x) = M(init(p, x)) for all x ∈ I.

Functions f which are implementable on a machine M are called "M-computable". [p]M denotes the (partial) function implemented by p on M.

3   A Metric on Machine Spaces

We now want to introduce a generalized metric on machine spaces which represents some kind of distance between different machines by relating the resources needed by the machines to solve the same problems. Here we focus on time complexity as our resource measure:

Definition 3.1. timepM(x) = min(TERMM(init(p,x))).

Definition 3.2. τ: T → T is an admissible time transfer function (attf) from M1 to M2 iff τ is monotone and ∀ p1 ∈ P1 ∃ p2 ∈ P2: [p1]M1 = [p2]M2 and ∀ x ∈ I: timep2M2 (x) ≤ τ(timep1M1 (x)).

Two machines M1 and M2 are time-compatible if they operate on the same time structure, input space and output space. Note that the state space and the program space have not to be the same. We now define a generalized metric Δ(t) which takes two time-compatible machines and maps them on an element of ℘(TT), i.e., the distance is represented by a set of functions on the time structure T:

Definition 3.3. Δ(t)(M1, M2) = {τ | τ is an attf from M1 to M2}.

We use the whole set of attfs, because there seems to be no obvious way to single one out. Additionally, one can combine and compare sets of functions much like single functions. However, future research may identify a canonical element in the set of attfs, maybe by introducing additional axioms.

Definition 3.4. Let α, β ⊆ TT. Then define their composition as α ∘ β = {τ1 ∘ τ2 | τ1 ∈ α, τ2 ∈ β}.

Definition 3.5. α ≤ β iff ∀ τ2 ∈ β ∃ τ1 ∈ α: τ1 ≤ τ2.

With these definitions the sets of attfs become a directedly ordered monoid (dom). A directed order is a partial order where two elements always have an upper bound, and directed monoids can be used as ranges for generalized metrics, allowing many standard constructions of topology [2]. Our metric can be classified as a dom-valued directed pseudometric, satisfying the following triangle inequality:

Δ(t)(M1, M3) ≤ Δ(t)(M2, M3) ∘ Δ(t)(M1, M2).

This triangle inequality enables the notion of ε-balls in machine space, which, for example, can be used to explore questions regarding model risk and robustness in statistics. The proposed ideas and concepts are inspired by work started and continued in [1], [4], [5], and [3].


[1] R. Gandy (1980): Church's Thesis and principles for mechanisms. In J. Barwise, H. J. Keisler & K. Kunen, editors: The Kleene Symposium, North-Holland, pp. 123-148, doi:10.1016/S0049-237X(08)71257-6.

[2] R. Kopperman (1988): All topologies come from generalized metrics. The American Mathematical Monthly 95(2), pp. 89-97, doi:10.2307/2323060.

[3] T. Neary & D. Woods (2012): The Complexity of Small Universal Turing Machines: A Survey. In: SOFSEM 2012, LNCS 7147, Springer, pp. 385-405, doi:10.1007/978-3-642-27660-6_32.

[4] M. B. Pour-El & I. Richards (1982): Noncomputability in models of physical phenomena. International Journal of Theoretical Physics 21(6-7), pp. 553-555, doi:10.1007/BF02650184.

[5] W. Sieg (2008): Church Without Dogma: Axioms for Computability. In S. B. Cooper, B. Löwe & A. Sorbi, editors: New Computational Paradigms, Springer, pp. 139-152, doi:10.1007/978-0-387-68546-5_7.