A Turing Machine Constructed with Cubelets Robots

Abstract

We describe the design and implementation of a modular and distributed robot that physically simulates functioning of a 2-symbols Turing machine. The robot is constructed with Cubelets (small autonomous robot-cubes used for teaching basic robotics and programming for kids) and Lego® bricks (thus the robot is called CULET). The Cubelets robotic blocks allow for a high level programming: instead of a traditional code, the robots are programmed by assembling various elementary blocks of robots together. We illustrate how read and write operations are physically implemented by the CULET.

Full text

Journal of Robotics, Networking and Articial Life
Vol . 5(4); March (2019), pp. 265–268
DOI: https://doi.org/10.2991/jrnal.k.190220.013; ISSN 2405-9021 print; 2352-6386 online
https://www.atlantis-press.com/journals/jrnal
Research Article
A Turing Machine Constructed with Cubelets Robots
Ricardo Quezada Figueroa1,*, Daniel Ayala Zamorano1, Genaro Juárez Martínez1,2, Andrew Adamatzky2
1Artificial Life Robotics Lab, Escuela Superior de Cómputo, Instituto Politécnico Nacional, México
2Unconventional Computing Laboratory, University of the West of England, Bristol, UK
1. INTRODUCTION
In the history of Turing machines a large number of physical
mechanical devices have been developed. For example, in 2012,
to celebrate Turing’s 100th birthday, LEGO built an autonomous
mechanical Turing machine adding binary numbers [1]. Being
inspired by ideas of machines emerged from the theory of self-re-
producing cellular automata [2], we decided to make a robot based
in small robotic-cubes Cubelets and pieces Lego®. We called this
robot CULET. A key feature of CULET is the first ever implementa-
tion of the machine constructed with robots and bricks. This robotic
machine is capable for simulating the function of other machine.
Cubelets, enhanced with pieces of Lego bricks, demonstrate how a
robotic machine simulates an abstract computing machine. A dis-
tributed controller of the robot is in the cube which can implement
operation read and other two cubes that move the head and the
arm (which implements write operation). These cubes need to be
re-programed for each table of transitions.
For the implementations we programmed two special Turing
machines. The first one is a universal Turing machine able to sim-
ulate the behaviour of a complex elementary universal cellular
automaton known as Rule 110 [3]. The second Turing machine
programmed in CULET doubles the number of 1’s in the tape.
This is an extended version of a work presented at the International
Conference on Artificial Life and Robotics (ICAROB), 2019
edition [4]. The subjects treated here and not in the original ver-
sion are: (1) the exposition of a similar machine to CULET that is
able to perform logic operations and (2) a clearer explanation of
the operation of the robot.
The paper is structured as follows. In Section 2 we introduce the
Cubelet robots and describe the Turing machines implemented.
Section 3 describes the design. Section 4 describes the implemen-
tation of a robot that computes logic functions. We give some final
remarks in Section 5.
2. PRELIMINARIES
2.1. Cubelets
The Cubelets are small cube-shaped robots designed to teach fun-
damentals of robotics and programming to kids. Cubes can have
different functionality. An idea is to construct robots only through
the concatenation of different cubes. This way Cubelets can be
seen as a formal language, where every valid construction is a
valid expression in the language, with virtually unlimited number
of possible combinations. This can be interpreted as a high level
of programming: a user does not modify the original program of
each cube, but instead, uses this program together with the inter-
action among cubes to produce a new robot. There are three types
of cubes [5]:
•Input sensors: cubes used to obtain data from the environment.
The most common are the proximity and light sensors.
•Actuators: cubes used to execute movements, e.g. drives or
rotors.
•Thinkers: cubes to implement a computation, e.g. inverters,
maximum, minimum, etc.
A single cube communicates with his neighbourhoods through a
shared channel that is established in the concatenation of blocks.
The input sensors-cubes distribute their readings thought the
ARTICLE INFO
Article History
Received 16 October 2018
Accepted 4 December 2018
Keywords
Turing machine
robotics
cubelets
Lego
cellular automata
ABSTRACT
We describe the design and implementation of a modular and distributed robot that physically simulates functioning of a
2-symbols Turing machine. e robot is constructed with Cubelets (small autonomous robot-cubes used for teaching basic
robotics and programming for kids) and Lego® bricks (thus the robot is called CULET). e Cubelets robotic blocks allow for a
high level programming: instead of a traditional code, the robots are programmed by assembling various elementary blocks of
robots together. We illustrate how read and write operations are physically implemented by the CULET.
© 2019 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
*Corresponding author. Email: qf7.ricardo@gmail.com

266 R.Q. Figueroa et al. / Journal of Robotics, Networking and Artificial Life 5(4) 265–268
channels; the thinkers-cubes obtain these readings, process the
information and make a decision, and send the commands to the
actuators.
Also, each cube can be reprogrammed to re-assign its functional-
ity. This reprogramming is done in C or Scratch-like code. CULET,
which is described in the next section, follows this approach.
2.2. Universal Turing Machines
The Turing machines were proposed by Turing in 1936 [6] to solve
problems about completeness and decidability. A Turing machine
is a finite-state machine with an external memory medium, an
infinite tape divided into squares. A head of the machine moves
on the tape and reads the value in each square sequentially [7].
Formally, the Turing machine is a set of quintuples of the form
[Equation (1)]:
qsqsd
ii00 11
,,
,,
++
()
(1)
where q is the set of states, s is the set of symbols and d indicates
the direction of movement of the head in the tape (right or left);
i is the time step. So, the first two elements of the quintuple indicate
the current state and symbol, the next two elements indicate the
following state and symbol, and the last one indicates the direction
of movementa.
One of the greatest achievements of Turing’s work is the conceptu-
alization of the so called universal machines. A universal machine U
can do everything that any machine can do. U receives as input two
things: a description of a machine A, and the input for that machine
M. The machine U writes on the tape the same string of symbols
that the machine A would write in its response to the input M.
2.3. Elementary Cellular Automata
Cellular automata were proposed by von Neumann [2] as means to
study decentralized systems (which later led to designs of distrib-
uted computing, ironically called non-von-Neumann-style com-
puters). The cellular automata consists of an array of cells, each of
which is a finite-state automaton (the same rules for all the cells)
and where the state of each cell in the next generation depends on
its current neighbourhood. Von Neumann developed a very com-
plex automaton with 29 states evolving in two dimensions.
Another important event in cellular automata history is the popu-
larization of Conway’s Game of Life in the 70’s [9]. This is a cellu-
lar automaton with only two states and a neighbourhood of eight
cells (Moore neighbourhood). The automaton is popular because it
shows how a complex and unpredictable behaviour emerges from
simple rules. Berlekamp et al. [10] demonstrated the universality of
this automata by constructing a register machine.
In the 1980’s years, Wolfram [11] described the elementary cellu-
lar automata to understand and formalise the dynamics of cellular
automata. The space is one-dimensional (a row of cells) and a cell’s
neighbourhood is two closest cell-neighbours of the cell. Wolfram
hypothesis was that if we cannot understand such minimal models,
then we could never understand bigger ones. The interesting part
comes when even in those apparently simple automata there are
intriguing patterns of a behaviour.
2.4. Universal ECA: Rule 110
Cook [12] proved that the elementary cellular automaton Rule 110
is universal. The proof is based on showing how, with particle inter-
actions [13] typical of the rule, one can emulate the operation of a
cyclic tag system. This one is the smallest cellular automaton demon-
strated to be computationally universal by simulating a very sophis-
ticated sequential machine via particles collisions in one dimension.
Such a new machine was invented to simulate a computable function
known as a cyclic tag system in this automaton, for details see Cook
[12] and Wolfram [14]. Rule 110 can simulate a cyclic tag system
[12], cyclic tag systems can simulate tag systems [12,14], tag systems
can simulate Turing machines [7], and a Turing machine (developed
by Eppstein and published by Cook [12]) can simulate Rule 110 and
therefore such a Turing machine is universal. The CULET can sim-
ulate the universal Turing machine which simulates the behaviour
of Rule 110, complex behaviour, and collisions of particles [15,16].
The state diagram of Figure 1a shows the operation of the machine.
Every generation in which the head of the machine reaches a right end,
a new generation of the rule 110 is computed. There are some consid-
erations with respect to this machine: the initial condition and the
representation of the cellular automata symbols. The initial condition
of the machine must be, from left to right, an infinite string of zeros,
the encoded version of the initial configuration that the machine is
going to emulate, and an infinite repetition of the string “10”. A “1” of
the cellular automata is represented like “11” in the machine, and a “0”
like “00”; the symbol “10” is used like a blank symbol.
2.5. Duplicator Turing Machine
A duplicator Turing machine was used by Rendell [17] as a part
of the Turing machine constructed in the Game of Life cellular
automaton. We implement this machine in CULET. In Figure 1b,
we show a 3-symbols version: it starts advancing to the right, for
each one scanned, it put another one in the left until there is no
more ones to the right. For the implementation in the CULET we
used a 2-symbols version of this machine.
3. CULET DESIGN
In Figure 2 we show a snapshot of CULET. The t ape is represented by
an array of Lego blocks on rectangles of a black paper; the position
aSome other standard literature, like Davis [8], defines the machine in terms of quadruples
instead of quintuples. Figure 1 | Implemented Turing machines. (a) Rule 110. (b) Duplicator.
(a) (b)

R.Q. Figueroa et al. / Journal of Robotics, Networking and Artificial Life 5(4) 265–268 267
of the block (front or back) indicates the symbol in the cell (0 or 1,
respectively). The Cubelet car moves through the tape reading and
writing in the tape; the read operation is achieved with a distance
sensor that tests the position of the Lego block under scanning; the
writing is done with a lever made of Lego and a rotate cube: depend-
ing on the desired symbol, the lever pushes or pulls the block of the
tape. The movement across the tape is done with a couple of wheels
that move in right or left direction. In the backwards of the car is
attached a rail that helps to maintain the alignment of the car with
respect to the tape.
The rotator cube plays the role of controller: it reads the values of
the distance sensors, move the Lego leaver and instruct the wheels
when to move and in which direction. This controller encodes
the transition rules of the Turing machine being simulated. The
controller program starts reading the current symbol on the tape.
Depending on the value scanned, it determines if a writing oper-
ation is needed, and if is the case, it performs it. After that, the
controller instructs the motor on the new direction to move, left
or right, depending on the automata rules. The car moves until it
reaches a new cell and then the cycle of operations is repeated.
4. LOGIC GATES WITH CUBELETS
Previous to the development of CULET was some others models
with similar capabilities. One of the most successful was a Cubelets-
only car that were able to perform logic operations. CULET:
Cubelets Lego Universal Turing Machine ECA Rule 110 [19] shows
a video with some examples.
Figure 3 shows a photograph of this robot. The fundamental idea
is the same that the one for CULET: a car that moves through a
tape of cell and that is able to read and write in it. Here the tape
is made of cubes: each cell is formed by a sensor distance and a
flashlight; each time that the sensor detects a movement, the state
of the flashlight (on or off) swaps. The car uses a rotator block to
position the brightness sensor on top of the tape; the read operation
is performed by moving the sensor on top of a flashlight; the write
operation is done by moving the sensor on top of a distance sensor
of the tape. The main inconveniences of this model are the follow-
ing. First is the issue of the price: the bigger the program, the bigger
the tape; CULET uses cheaper cell components than the logic gates
machine. Second is the point of the coordination and calibration of
the components: this model depends on delays for make a move-
ment along the tape and to move the brightness sensor.
5. FINAL REMARKS
We developed a robotic Turing machine named CULET, able to sim-
ulate any 2-symbols Turing machine. CULET is constructed with
autonomous cube-robots Cubelets and Lego® bricks. A video showing
the function of the Turing machine simulating Rule 110 with CULET
is available in CULET: Cubelets Lego Universal Turing Machine ECA
Rule 110 [19] and the duplicator Turing machine in CULET: Cubelets
Lego Doubler Turing Machine [20]. The full code to reprogram
Cubelets is available from Code developed for CULET [21]. In the
future work, we will develop machines reading more than two states.
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Figure 2 | Robotic Turing machine. Figure 3 | Logic gates implemented with Cubelets.

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Authors Introduction
Ricardo Q. Figueroa
Ricardo Q. Figueroa is a researcher at
the Artificial Life Robotics Lab in the
Superior School of Computer Sciences,
National Polytechnic Institute, Mexico
City, Mexico. His areas of interest are
robotics, swarm robotics, artificial life,
cryptography.
Daniel A. Zamorano
Daniel A. Zamorano is a researcher at
the Artificial Life Robotics Lab in the
Superior School of Computer Sciences,
National Polytechnic Institute, Mexico
City, Mexico. Currently, his areas or
work are cellular automata, complex
systems, system security and artificial
intelligence.
Genaro J. Martínez
Genaro J. Martínez is a professor at the
School of Computer Sciences, National
Polytechnic Institute, Mexico City, Mexico
and visiting fellow at the Unconventional
Computing Centre, University of the
West of England, Bristol, UK. His does
research in cellular automata, unconven-
tional computing, artificial life, complex
systems, and swarm robotics.
Andrew Adamatzky
Andrew Adamatzky is a professor at
the Department of Computer Science
and Director of the Unconventional
Computing Centre, University of the
West of England, Bristol, UK. He does
research in reaction-diffusion comput-
ing, cellular automata, massive parallel computation, collec-
tive intelligence, bionics, complexity, non-linear science, novel
hardware.