Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2015, Article ID 236806, 8 pages
http://dx.doi.org/10.1155/2015/236806

Research Article
Simulation of Turing Machine with
uEAC-Computable Functions
Yilin Zhu,1 Feng Pan,1 Lingxi Li,2 Xuemei Ren,1 and Qi Gao1
1

School of Automation, Beijing Institute of Technology, Beijing 100081, China
Department of Electrical and Computer Engineering, Indiana University-Purdue University Indianapolis,
Indianapolis, IN 46202, USA

2

Correspondence should be addressed to Qi Gao; gaoqi@bit.edu.cn
Received 10 June 2015; Accepted 3 November 2015
Academic Editor: Jean J. Loiseau
Copyright © 2015 Yilin Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The micro-Extended Analog Computer (uEAC) is an electronic implementation inspired by Rubel’s EAC model. In this study, a
fully connected uEACs array is proposed to overcome the limitations of a single uEAC, within which each uEAC unit is connected
to all the other units by some weights. Then its computational capabilities are investigated by proving that a Turing machine M can
be simulated with uEAC-computable functions, even in the presence of bounded noise.

1. Introduction
Analog computer almost disappeared since the blossom of
digital computer in the second half of the last century.
Actually, the first “computer” in the world, the Antikythera
mechanism [1], which was used to predict astronomical
positions and eclipses, was an analog computer. Recently
analog computer is again regaining interest, and this stems
partly from the development of various unconventional
computational techniques, such as quantum computation,
DNA computation, and cellular automaton.
The first significant paradigm of analog computer is the
General Purpose Analog Computer (GPAC) [2] introduced
by Shannon as mathematical model of the Differential Analyzer [3]. Shannon proved that GPAC was able to generate
differentially algebraic functions, such as polynomials, the
exponential functions, the trigonometric functions and sums,
products, and compositions of them. More generally, he
claimed that a function could be generated by a GPAC
if it satisfied some algebraic differential equations. Rubel
showed that the Dirichlet problem on the disc cannot be
solved by a GPAC and he defined the Extended Analog
Computer (EAC) [4], which was able to directly compute
partial differential equations, solve the inverse of functions,
and implement spatial continuity. Mycka pointed out that

the set of GPAC-computable functions was a proper subset
of EAC-computable functions [5]. Graca et al. proved that
Turing machine could be robustly simulated by flows defined
by polynomial ordinary differential equations (ODEs) [6] and
pointed out that the solution of the initial value problems
defined by some ODEs was computable by GPAC; hence,
it followed that GPACs could simulate Turing machines.
Piekarz compared the computational capabilities of the EAC
and partial recursive functions and proved that EAC could
generate any partial recursive function defined over N [7].
In his paper that proposed the EAC model, Rubel stressed
that the EAC was a conceptual computer and whether it could
be realized by actual physical, chemical, or biological devices
was not known. However, researches into the continuousvalued Lukasiewicz logic as a computational paradigm led
to an electronic implementation of the EAC [8]. Mills and
colleagues designed and built an electronic implementation
inspired by Rubel’s EAC model, called the micro-Extended
Analog Computer (uEAC) [9, 10], after a decade’s research
[11–13]. Moreover, Mills introduced the Δ-digraph [10], a
diagrammatic tool, to demonstrate the relationship of the
nature, Rubel’s EAC model, and uEAC, and, particularly,
he related the EAC model to uEAC by dividing the “black
boxes” of the EAC model into explicit functions and implicit
functions. The current version of uEAC was designed in 2005

2
at Indiana University [10, 14] and had been applied to letter
recognition [15, 16], exclusive-OR (XOR) problem [10, 17],
Cyclotron Beam Control [18], and biologically derived circuit
pattern construction [19], and so forth. It mainly consists of a
conductive sheet in which currents can be injected and read
at different locations, analog-to-digital and digital-to-analog
converters that are used to interface the conductive sheet
to the onboard controller, a microprocessor that controls
the input/output array and emulates Lukasiewicz logic array
(LLA) functions. The topology of the conductive sheet, the
material from which it is constructed, and the boundaryvalued LLA functions determine the computation of the
uEAC forms. In the present study, a fully connected singleinput, single-output uEACs array is proposed and its computational capabilities are investigated by showing that any
Turing machine M can be simulated with uEAC-computable
functions, even in the case that some noise is added to the
initial configuration of M or during the iteration of the
system. Turing machine [20] is the standard paradigm for
digital computation since the work of Turing in the 1930s;
we will prove the main result of this paper by constructing
a robust simulation of Turing machine M with uEACcomputable functions.
The paper can be outlined as follows. Section 2 provides
some basic notations about Turing machine, Rubel’s EAC
model, and the uEAC. The fully connected uEACs array is
presented and discussed in detail in Section 3. Section 4 states
the main result of this paper: a Turing machine M can be
robustly simulated by uEAC-computable functions, even in
the presence of bounded noise. We prove the theorem in
Section 5 by constructing a robust Turing machine M simulation with uEAC-computable functions. Some conclusions
and suggestions are given in Section 6.

2. Preliminaries
2.1. Turing Machine. A Turing machine can be seen as a state
machine; at each moment the machine is in one of a finite
number of states. It has an infinite one-dimensional tape
which is divided into cells and accessed by a read-write head.
By infinite one-dimensional tape, we mean that the cells are
arranged in a left-right orientation, and the tape has a leftmost
cell and stretches infinitely far to the right. Each cell contains
one symbol; the read-write head can move left and right along
the tape to scan successive cells.
The action of a Turing machine is determined completely
by (1) the current state of the machine, (2) the symbol in the
cell being scanned by the head, and (3) a table of transition
rules. At each step, the machine reads the symbol under the
head and then checks the transition rule and executes two
operations: writing a new symbol into the current cell under
the head of the tape and moving the head one position to
the left or to the right or making no move. The tape head
moves in the following manner: 𝑅 means moving one cell to
the right, 𝐿 means moving one cell to the left if there are cells
to the left, and 𝑁 means not to move. If the machine reaches a
situation in which no transition rule will be carried out, then
the machine halts.

Mathematical Problems in Engineering
Table 1: Transfer function 𝛿 of the Turing machine M.
𝛿
𝑞0
𝑞1
𝑞2

𝐵
𝑞1 , 𝐵, 𝑅
𝑞2 , 1, 𝑅
𝑞2 , 𝐵, 𝑅

1
𝑞0 , 1, 𝑅
𝑞1 , 1, 𝑅
𝑞2 , 𝐵, 𝑁

(1, B, N)

(1, 1, R)

q0

(B, B, R)

(B, 1, R)

q1

q2

Figure 1: The computation performed by the Turing machine M.

Definition 1. A single tape Turing machine is 4-tuple M =
⟨Q, Σ, 𝛿, 𝑞0 ⟩, where
(i) Q is a nonempty finite set of states;
(ii) Σ is the tape alphabet that describes the contents of
cells of the tape;
(iii) 𝛿 : Q × Σ → Q × Σ × {𝐿, 𝑁, 𝑅} is the transfer function;
(iv) 𝑞0 ∈ Q is the initial state.
Example 2. Consider an example of single tape Turing
machine M = ⟨{𝑞0 , 𝑞1 , 𝑞2 }, Σ, 𝛿, 𝑞0 ⟩ with three states
{𝑞0 , 𝑞1 , 𝑞2 } and 𝑞0 is the initial state. As discussed above, let
𝐵 be a blank symbol and Σ = {1, 𝐵}. The transfer function 𝛿
is given by Table 1.
Given the input 𝜔 = 1𝑚 𝐵𝐵1𝑛 , the computation performed
by the machine is
(𝑞0 , /1𝑚 𝐵𝐵1𝑛 ) ,
(𝑞0 , 1/1𝑚−1 𝐵𝐵1𝑛 ) ,
(𝑞0 , 12 /1𝑚−2 𝐵𝐵1𝑛 ) ,
..
.
𝑚

(1)
𝑛

(𝑞0 , 1 /𝐵𝐵1 ) ,
(𝑞1 , 1𝑚 𝐵/𝐵1𝑛 ) ,
(𝑞2 , 1𝑚 𝐵1/1𝑛 ) ,
(𝑞2 , 1𝑚 𝐵1𝐵/1𝑛−1 ) ,
where the symbol “/” marks the position of the read-write
head.
In Figure 1, states of the Turing machine {𝑞0 , 𝑞1 , 𝑞2 } are
represented by circles, with the concentric circle being the
initial state 𝑞0 . A transition is represented as a labeled arrow
with 3 tuples; the first term is the content of the cell under
the read-write head, the second one is the content of the cell

Mathematical Problems in Engineering

3

after this transition, and the third one is the movement of the
read-write head.
2.2. Rubel’s EAC Model and the uEAC. In Rubel’s definition,
the EAC works on a hierarchy of levels, getting more versatile
as one goes to higher levels in the hierarchy. At the lowest
level 0, it produces and manipulates real polynomials of any
finite number of real variables (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ), while, at level
1 and higher, it produces differentially algebraic real-analytic
functions (C𝜔 ). The outputs of the machine at level 𝑁 − 1 can
be used as inputs at level 𝑁 or higher. An important feature of
the EAC is that it is “extremely well-posed,” which means that
when the inputs at some level are modified by small errors,
then the outputs differ from the original outputs only by a
small amount on each compact set.
Definition 3. The function 𝑦 ∈ C𝜔 is generated by EAC at
level 𝑁 ≥ 1 (𝑦 ∈ EAC𝑁), if 𝑦 is a function such that
(i) 𝑦(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) = 𝑢1 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) + 𝑢2 (𝑥1 , 𝑥2 ,
. . . , 𝑥𝑘 ), where 𝑢1 , 𝑢2 ∈ EAC𝑁−1 ;
(ii) 𝑦(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) = 𝑢1 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) × 𝑢2 (𝑥1 , 𝑥2 , . . .,
𝑥𝑘 ), where 𝑢1 , 𝑢2 ∈ EAC𝑁−1 ;
(iii) 𝑦(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) = V(𝑢1 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ), 𝑢2 (𝑥1 , 𝑥2 , . . .,
𝑥𝑘 ), . . . , 𝑢𝑚 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 )), where V, 𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ∈
EAC𝑁−1 ;
(iv) 𝑦(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) = {𝑥𝑘+1 , 𝑥𝑘+2 , . . . , 𝑥𝑘+𝑙 } = {𝑓1 (𝑥1 , 𝑥2 ,
. . . , 𝑥𝑘 ), 𝑓2 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ), . . . , 𝑓𝑙 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 )}, and
{𝑥𝑘+1 , 𝑥𝑘+2 , . . . , 𝑥𝑘+𝑙 } is the solution of
𝑦1 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 , 𝑥𝑘+1 , 𝑥𝑘+2 , . . . , 𝑥𝑘+𝑙 ) = 0,
𝑦2 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 , 𝑥𝑘+1 , 𝑥𝑘+2 , . . . , 𝑥𝑘+𝑙 ) = 0,
(2)

..
.
𝑦𝑙 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 , 𝑥𝑘+1 , 𝑥𝑘+2 , . . . , 𝑥𝑘+𝑙 ) = 0,

where 𝑓1 , 𝑓2 , . . . , 𝑓𝑙 are C𝜔 functions and 𝑦1 , 𝑦2 ,
. . . , 𝑦𝑙 ∈ EAC𝑁−1 ;
(v) 𝑦(𝑥1 ,𝑥2 , . . . , 𝑥𝑘 ) = 𝐷𝑓(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) = 𝜕𝛼1+𝛼2 +⋅⋅⋅+𝛼𝑛 𝑓/
𝛼
𝛼
𝛼
𝜕𝑥1 1 𝜕𝑥2 2 ⋅ ⋅ ⋅ 𝜕𝑥𝑘 𝑘 , where 𝑓 ∈ EAC𝑁−1 ;
(vi) 𝑦(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) = {Ω, Ω󸀠 } ∈ EAC𝑁+1/2 , for any 𝑦󸀠 ∈
EAC𝑁 defined on set Λ, where
Ω = {(𝑥1 , 𝑥1 , . . . , 𝑥𝑘 ) ∈ Λ : 𝑦󸀠 (𝑥1 , 𝑥1 , . . . , 𝑥𝑘 ) > 0} ,
󸀠

󸀠

Ω = {(𝑥1 , 𝑥1 , . . . , 𝑥𝑘 ) ∈ Λ : 𝑦 (𝑥1 , 𝑥1 , . . . , 𝑥𝑘 ) ≥ 0} .

(3)

Moreover, for any function (𝑦, Ω) produced at level 𝑁
and for any subset Ω∗ of Ω produced at level 𝑁 − 1/2,
the function (𝑦|Ω∗ , Ω∗ ) can be produced at level 𝑁;
(vii) 𝑦(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) = 𝑦∗ , 𝑦∗ is a unique analytic
continuation of 𝑦 from Ω ∩ Ω∗ to all Ω∗ , where
(𝑦, Ω) ∈ EAC𝑁 and Ω ∩ Ω∗ ≠ 0;

(viii) 𝑦(𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) is a solution of a set of differential
functions of the form
𝐹 (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 : 𝑦󸀠 , 𝑦1󸀠 , 𝑦2󸀠 , . . . , 𝑦𝑙󸀠 ) = 0

(4)

on set Ω subject to certain boundary requirements,
where Ω ∈ EAC𝑁−1/2 , 𝑦󸀠 ∈ EAC𝑁−1 , and
𝑦󸀠 , 𝑦1󸀠 , 𝑦2󸀠 , . . . , 𝑦𝑙󸀠 are partial derivatives of 𝑦;
(ix) for (𝑥1 , 𝑥2 , . . . , 𝑥𝑘 ) ∈ Ω and (𝑥10 , 𝑥20 , . . . , 𝑥𝑘0 ) ∈ 𝛾,
𝑦 (𝑥10 , 𝑥20 , . . . , 𝑥𝑘0 ) =

lim

𝑦󸀠 (𝑥) ,

(𝑥1 ,𝑥2 ,...,𝑥𝑘 ) → (𝑥10 ,𝑥20 ,...,𝑥𝑘0 )

(5)

where 𝑦󸀠 ∈ EAC𝑁−1 , Ω ∈ EAC𝑁−1/2 , and 𝛾 ∈
EAC𝑁−1/2 are subsets of 𝜕Ω.
The EAC is an extension of Shannon’s GPAC. Rubel
proved that every C𝜔 -function that could be computed by a
GPAC could also be computed by an EAC [4], and, moreover,
Euler’s gamma function Γ(𝑥) and Riemann zeta function
𝜁(𝑠) can also be computed by an EAC [21], while the GPAC
cannot solve these problems. Rubel stressed that the EAC
was a conceptual computer and whether it could be realized
by actual physical, chemical, or biological devices was not
known, and most computer scientists also regarded the EAC
as a machine that was theoretically impossible to be built.
However, Mills and his colleagues designed and built an
electronic implementation inspired by the EAC, the microExtended Analog Computer (uEAC), and the current version
was designed in 2005. Readers who are interested in the
hardware of the uEAC are referred to [8, 10, 14] for more
details.
Suppose that current 𝐼 is injected to the conductive sheet
at location 𝑂. The distances from 𝑂 to two arbitrary points 𝑖, 𝑗
on the same radius are 𝑟𝑖 , 𝑟𝑗 , respectively, and 𝑉𝑖𝑗 is the voltage
between 𝑖 and 𝑗. Without loss of generality, we suppose that
𝑗 is located outside of 𝑖 and there are 𝑚 resistances between 𝑖
and 𝑗, the length of every resistance is Δ𝑟 (i.e., 𝑟𝑗 −𝑟𝑖 = Δ𝑟⋅𝑚),
and we have
𝑉𝑖𝑗 =

𝐼
(𝑅 + 𝑅2 + ⋅ ⋅ ⋅ + 𝑅𝑚 )
𝑛 1

𝜌 ⋅ Δ𝑟
𝐼 𝑚
= ∑
.
𝑛 𝑘=1 [𝑟𝑖 + (2𝑘 − 1) ⋅ (Δ𝑟/2)] ⋅ (2𝜋/𝑛)

(6)

Let Δ𝑟 → 0; we have
𝑉𝑖𝑗 =

𝐼𝜌 𝑟𝑗 𝑑𝑟 𝐼𝜌 𝑟𝑗
=
ln ,
∫
2𝜋 𝑟𝑖 𝑟
2𝜋 𝑟𝑖

(7)

where 𝜌 is the electrical resistivity of the conductive sheet.
Let 𝑘𝑖𝑗 = (𝜌/2𝜋) ln(𝑟𝑗 /𝑟𝑖 ), then 𝑉𝑖𝑗 = 𝐼 ⋅ 𝑘𝑖𝑗 , and we note
that 𝑘𝑖𝑗 is dependent on 𝜌, 𝑟𝑖 , and 𝑟𝑗 ; in other words, once the
current input locations and voltage output locations on the
conductive material are fixed, 𝑘𝑖𝑗 is a positive constant and
can be seen as a coefficient of the input current 𝐼 and output
voltage 𝑉𝑖𝑗 . The output of uEAC is L(𝑉𝑖𝑗 ), where L is the LLA
basic function [22].

4

Mathematical Problems in Engineering
y(t)

u1 (t + 1)

u2 (t + 1)

uEAC_1

u1 (t)

x(t)

u3 (t + 1)

uEAC_2

u2 (t)

un (t + 1)

uEAC_3 · · ·

u3 (t)

uEAC_n

un (t)

Figure 2: Structure of the fully connected uEACs array.

The uEAC is based on the resistance property of the
conductive sheet, which makes its input-output relation
linear; thus a single uEAC unit is very limited when applied
to nonlinear problems. In the next section, we present a
fully connected uEACs array to expand the computation
capabilities of uEAC.

3. The Fully Connected uEACs Array
We present a fully connected single-input, single-output
uEACs array (see Figure 2) in this section, within which
each uEAC unit is connected to all the other units by some
weights, and 𝑥 corresponds to the external input variable, 𝑢
to a state variable, and 𝑦 to the output variable. In the uEACs
array shown as in Figure 2, the output of the array 𝑦 is not
necessarily restricted to uEAC 1; it can be the output of any
uEAC unit or a combination of the outputs of several units.
Each uEAC unit weights and sums its inputs and updates its
state as the following function:
𝑛

𝑢𝑖 (𝑡 + 1) = L ( ∑ 𝜔𝑖𝑗 𝑢𝑗 (𝑡) + 𝑎𝑖 𝑥 (𝑡)) ,

(8)

𝑗=1

where 𝑛 is the number of uEAC units, 𝜔𝑖𝑗 , 𝑎𝑖 are fixed real
valued weights, and L is the LLA function. It is important to
distinguish this uEACs array model from the Analog Recurrent Neural Network (ARNN). In ARNN, the activation
function of neurons is a saturated-linear function [23, 24],
while, in the uEACs array, L is a piecewise linear function
implemented by LLA basic functions. Another significant
feature that distinguishes uEACs array from the ARNN is that
uEAC is an actual programmable physical implementation
inspired by Rubel’s EAC model, and, in this paper, the
computational capability of such a fully connected uEACs
array is studied. We can rewrite the mathematical model in
vector form as
𝑢̃ = L (𝑊𝑢 + 𝐴𝑥) ,

(9)

where now 𝑢 and 𝐴 are vectors of size 𝑛 and 𝑊 is a real matrix
of size 𝑛 × 𝑛.

The state of this dynamic system at each instant is a real
vector; that is, 𝑢(𝑡) = (𝑢1 (𝑡), 𝑢2 (𝑡), . . . , 𝑢𝑛 (𝑡)) ∈ 𝑅𝑛 . The
𝑖th coordinate of the vector represents the value of the 𝑖th
uEAC units at time 𝑡. In particular, in the physical structure
of uEAC, the term “real” corresponds to values of resistances,
capacitances, and electrical field, which may not be directly
measured, but they affect the dynamic behavior dramatically.
For instance, everything on the earth obeys the exact value
of gravitational acceleration 𝐺 even though we are not able
to measure it. We may replace 𝐺 with a rational number and
observe similar qualitative behavior in finite time simulation;
nonetheless, the infinite-time characteristics depend on the
true value. Another example is 𝜋. When modeling this uEACs
array, some real values are involved, and, for finite time
interval, one may replace these real values by some rational
values, and the same qualitative behavior is observed, while
the long-term characteristics depend on the true values.
The array is said to be fully connected because there is
a weight between every two uEAC units. The status of the
weights can be seen either as unknown parameters to be
estimated, or as constant after being optimized. This prompts
two different views of this uEACs array. When the weights
are considered as unknown parameters, the uEACs array is
an adaptive topology that approximates some input-output
mapping by means of parameter optimization. When the
weights are considered constant, the uEACs array performs
exact computation. We should note that 𝑤𝑖𝑗 may equal 0,
which means that there is no connection between units 𝑖 and
𝑗. Thus this fully connected array can be seen as a general
model of a variety of uEACs arrays, including those in which
only a subset of the uEAC units are used. Moreover, all the
units in this fully connected structure are in the same layer
and compute in parallel. The number of uEAC units in the
network is countable. We assume that the structure of the
array, including the interconnection relationship of the uEAC
units and the values of the interconnection weights, remains
constant. What changes in time are the state values, that
is, outputs of every uEAC unit, which are used in the next
iteration.

Mathematical Problems in Engineering

5

4. Main Results
Before stating the main results of this paper, we introduce
several useful notations. For 𝑥 ∈ R, ⌈𝑥⌉ = min{𝑠 ∈ Z :
𝑠 ≥ 𝑥}. Let ‖(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 )‖∞ = max1<𝑖<𝑛 |𝑥𝑖 |, ‖𝑓‖∞ =
sup𝑥∈R |𝑓(𝑥)|, and 𝑓[𝑘] denotes its 𝑘th iteration; that is,

𝑓

[3]

(10)

= 𝑓 (𝑓 (𝑓 (𝑥))) , . . .

Definition 4. A function 𝑓 : R → R is uEAC-computable if
letting 𝑥 be an arbitrary element in the domain of 𝑓 and given
an output precision 2−𝑛 , there is a uEACs array that is able to
compute a rational approximation of 𝑓 with precision 2−𝑛 .
We should note that the exponential function, the
trigonometric functions, and their compositions are uEACcomputable as basic analytic functions. We may now present
the main result of this paper which states the computation
capabilities of the uEACs array.
Theorem 5. For a Turing machine M, there is a uEACs array
that can robustly simulate it.
Actually, we will prove the following theorem.
Theorem 6. Let 𝜓 : N3 → N3 be the transfer function of
a Turing machine M; there is a uEACs array that is able to
simulate M robustly in the following sense: let 0 < 𝜀 < 1/2 be
some bounded noise added to the initial configuration of M;
there is a uEAC-computable function 𝑓 : R3 → R3 that, for
all 𝑥0 ∈ R3 satisfying ‖𝑥0 − 𝑥0 ‖∞ ≤ 𝜀, we have
󵄩󵄩
󵄩
󵄩󵄩𝑓 (𝑥0 ) − 𝜓 (𝑥0 )󵄩󵄩󵄩∞ ≤ 𝜀,

(11)

where 𝑥0 ∈ N3 represents an initial configuration of M.
If 𝑥 is a halting configuration of M, we have 𝜓(𝑥) = 𝑥. We
will prove this theorem by showing that 𝑓 : R3 → R3 can be
obtained by composing some uEAC-computable functions,
such as exponential function and trigonometric functions.

5. Robust Simulation of Turing Machine with
uEAC-Computable Functions
For a Turing machine M with 10 symbols and 𝑚 states, its
tape contents can be represented as
⋅ ⋅ ⋅ 𝐵𝛼−𝑚 𝛼−𝑚+1 ⋅ ⋅ ⋅ 𝛼−1 𝛼0 𝛼1 ⋅ ⋅ ⋅ 𝛼𝑛 𝐵 ⋅ ⋅ ⋅ ,

(12)

where 𝛼𝑖 are symbols on the tape and 𝐵 is the blank symbol.
Define the encoding functions as
𝑦1 = 𝛼0 + 𝛼1 10 + ⋅ ⋅ ⋅ + 𝛼𝑛 10𝑛 ,
𝑦2 = 𝛼−1 + 𝛼−2 10 + ⋅ ⋅ ⋅ + 𝛼−𝑚 10𝑚−1 .

4

𝜔 (𝑡) = 𝑝0 + ∑ (𝑝ℎ cos (
ℎ=1

𝑓[1] = 𝑓 (𝑥) ,
𝑓[2] = 𝑓 (𝑓 (𝑥)) ,

Let 𝑞 be the current state of M and then the triple (𝑦1 , 𝑦2 , 𝑞) ∈
N3 is the current configuration of M. We use a periodic
function 𝜔 to read the symbols written on the tape; by
trigonometric interpolation we may take

(13)

ℎ𝜋
ℎ𝜋
𝑡) + 𝑞ℎ sin ( 𝑡))
5
5

(14)

+ 𝑝5 cos (𝜋𝑡) ,
where 𝑝0 , 𝑝1 , . . . , 𝑝5 , 𝑞1 , 𝑞2 , . . . , 𝑞4 are coefficients that can
be obtained by solving a system of linear equations. From
the form of 𝜔 we can get that it is uEAC-computable as
a composition of trigonometric functions. Note that 𝜔 is
continuous in R, and, for every given 𝜀, there is some 𝜉𝜀 > 0
that ∀𝑛 ∈ N, 𝑥 ∈ [𝑛 − 𝜉𝜀 , 𝑛 + 𝜉𝜀 ] and we have |𝜔(𝑥) −
𝑛 mod 10| ≤ 𝜀.
Actually, when simulating M, we are dealing with a series
of approximations of (𝑦1 , 𝑦2 , 𝑞), that is, (𝑦1 , 𝑦2 , 𝑞), in the
following sense for given 𝜀:
󵄩󵄩
󵄩
(15)
󵄩󵄩(𝑦1 , 𝑦2 , 𝑞) − (𝑦1 , 𝑦2 , 𝑞)󵄩󵄩󵄩∞ ≤ 𝜀,
and thus we need to keep the error under control during the
iterations. An error control function can be defined as 𝜎(𝑥) =
𝑥−0.2 sin(2𝜋𝑥) to keep the error under control when reading
symbols and states of the Turing machine, and it is a uniform
contraction in a neighborhood of integers.
Proposition 7 (see [6]). Let 𝑛 ∈ Z and 𝜀 ∈ [0, 1/2); there is
some contracting factor 𝜆 𝜀 ∈ (0, 1) that ∀𝛿 ∈ [−𝜀, 𝜀], |𝜎(𝑛 +
𝛿) − 𝑛| < 𝜆 𝜀 𝛿.
Another two uEAC-computable error control functions 𝑢
and 𝑢󸀠 are defined as follows.
Lemma 8 (see [6]). Let 𝑎 ∈ {0, 1}, for any 𝑎, 𝑦 ∈ R satisfying
|𝑎 − 𝑎| ≤ 1/4 and 𝑦 > 0; there is a function 𝑢 : R2 → R
given by 𝑢(𝑥, 𝑦) = (1/𝜋) arctan(4𝑦(𝑥 − 1/2)) + 1/2 such that
|𝑎 − 𝑢(𝑎, 𝑦)| < 1/𝑦.
Lemma 9 (see [6]). Let 𝑎 ∈ {0, 1, 2}, for any 𝑎, 𝑦 ∈ R
satisfying |𝑎 − 𝑎| ≤ 𝜀 and 𝑦 ≥ 2; there is a function 𝑢󸀠 : R2 →
R given by
2

𝑢󸀠 (𝑥, 𝑦) = 𝑢 ((𝜎[𝑠+1] (𝑥) − 1) , 3𝑦)
𝜎[𝑠] (𝑥)
⋅ (2𝑢 (
, 3𝑦) − 1) + 1
2

(16)

such that |𝑎 − 𝑢󸀠 (𝑎, 𝑦)| < 1/𝑦, where
0
𝜀 ≤ 1/4
{
{
𝑠 = { log (4𝜀)
{⌈−
⌉ 𝜀 > 1/4.
log 𝜆 𝜀
{

(17)

Without loss of generality, we suppose that the symbol
being read by M is 𝛼0 . By |𝑦1 − 𝑦1 | ≤ 𝜀, we have
󵄨󵄨
󵄨
󵄨󵄨𝛼0 − 𝜔 (𝜎[𝑡] (𝑦1 ))󵄨󵄨󵄨 ≤ 𝜀,
(18)
󵄨
󵄨

6

Mathematical Problems in Engineering

where 𝑡 = ⌈| log(𝜉𝜀 /𝜀)/ log(𝜆 𝜀 )|⌉. Then 𝜔(𝜎[𝑡] (𝑦1 )) can be seen
as an approximation of the symbol being currently read with
error bounded by 𝜀, and it is uEAC-computable. With the
approximation of the current symbol, we can determine the
next state by polynomial interpolation. Recall that M has 𝑚
states and 10 symbols, let 𝑦 be the symbol being currently
read and 𝑞 the current state, and the next state of M can be
represented as

where
∗

𝐴󸀠1 = 𝜎[𝑠+2] (𝜔 (𝜎[𝑡] (𝑦2 ))) + 10 (𝜎[𝑠+4] (𝑙 )
+ 𝜎[𝑠+4] (𝑦1 ) − 𝜎[𝑠+4] (𝜔 (𝜎[𝑡] (𝑦1 )))) ,
∗

𝐴󸀠2 = 𝜎[𝑠+2] (𝑙 ) + 𝜎[𝑠+2] (𝑦1 )

(22)

− 𝜎[𝑠+2] ((𝜔 (𝜎[𝑡] (𝑦1 )))) ,
9

𝐴󸀠3 =

𝑚
𝑦−𝑟
𝑞−𝑥
)( ∏
) 𝑞𝑖𝑗 ,
𝑞 = ∑∑( ∏
𝑖=0 𝑗=1 𝑟=0,𝑟=𝑖̸ 𝑖 − 𝑟
𝑥=1,𝑥=𝑗̸ 𝑗 − 𝑥
∗

9

9

𝑚

𝜎[V] (𝑦) − 𝑟
)
𝑖−𝑟
𝑟=0,𝑟=𝑖̸

∗

9

𝑚

𝑞∗ = ∑ ∑ ( ∏
𝑖=0 𝑗=1

(19)

𝜎[V] (𝑞) − 𝑥
) 𝑞𝑖𝑗 ,
𝑖−𝑥
𝑥=1,𝑥=𝑗̸
𝑚

where 𝑞∗ is an approximation of 𝑞∗ satisfying |𝑞∗ − 𝑞∗ | ≤ 𝜀
and 𝑞𝑖𝑗 is the state that follows symbols 𝑖 and state 𝑗. This map
returns the next state of M and is also uEAC-computable.
Using the similar construction, the symbol to be written on
the tape, 𝑙∗ , and the direction of the move of the head, 𝑑∗ ,
can also be approximated with precision 𝜀, respectively; that
∗
∗
is, |𝑙 − 𝑙∗ | ≤ 𝜀 and |𝑑 − 𝑑∗ | ≤ 𝜀.
Let 𝑑 = 0 denote a move of the head to the left, let 𝑑 = 1
denote stay, and let 𝑑 = 2 denote a move to the right. In the
absence of error, the next value of 𝑦1 , that is, 𝑦1∗ , is a function
of 𝑦1 , 𝑦2 , 𝑙∗ , and 𝑑∗ ,

(1 − 𝑑∗ ) (2 − 𝑑∗ )
+ 𝐴 2 𝑑∗ (2 − 𝑑∗ )
2

𝑑∗ (1 − 𝑑∗ )
+ 𝐴3
,
−2

(20)

where 𝐴 1 = 𝜔(𝑦2 ) + 10(𝑙∗ + 𝑦1 − 𝜔(𝑦1 )), 𝐴 2 = 𝑙∗ + 𝑦1 −
𝜔(𝑦1 ), and 𝐴 3 = (1/10)(𝑦1 − 𝜔(𝑦1 )). Consider the error, let 𝐷
be an additional approximation of 𝑑∗ to be determined later,
we define three functions 𝐴󸀠1 , 𝐴󸀠2 , and 𝐴󸀠3 to approximate the
tape contents after the head moves left, stays, and moves right,
respectively, and then 𝑦1∗ can be approximated as

𝑦∗1 = 𝐴󸀠1

(1 − 𝐷) (2 − 𝐷)
+ 𝐴󸀠2 𝐷 (2 − 𝐷)
2

+ 𝐴󸀠3

𝐷 (1 − 𝐷)
,
−2

If we take 𝐷 = 𝑑 directly, the error of the term ((1 − 𝐷)(2 −
𝐷))/2 will be amplified when multiplied by 𝐴󸀠1 . To obtain a
sufficiently good approximation of 𝑦1∗ , we have to guarantee
that the error |𝐷 − 𝑑∗ | is bounded; this can be achieved with
the following definition:
∗
1
𝐷 = 𝑢󸀠 (𝑑 , 12000 (𝑦1 + ) + 2) .
2

⋅( ∏

𝑦1∗ = 𝐴 1

1
(𝜎[𝑠+1] (𝑦1 ) − 𝜎[𝑠+1] (𝜔 (𝜎[𝑡] (𝑦1 )))) .
10

(21)

(23)

By the definition of 𝑢󸀠 , we get that |𝐷−𝑑∗ | ≤ 𝜙, where 𝜙 =
1/(12000(𝑦1 +1/2)+2). Then we can obtain sufficient good 𝑦∗1
such that |𝑦∗1 −𝑦1∗ | < 𝜀. We should note that 𝐷, 𝐴󸀠1 , 𝐴󸀠2 , and 𝐴󸀠3
are defined as a composition of uEAC-computable functions
so they are also uEAC-computable. Similarly we can get some
𝑦∗2 satisfying |𝑦∗2 − 𝑦2∗ | < 𝜀. Putting together the maps defined
above, we can define a uEAC-computable function 𝑔 : R3 →
R3 as 𝑔(𝑦1 , 𝑦2 , 𝑞) = (𝑦∗1 , 𝑦∗2 , 𝑞∗ ), such that
󵄩󵄩
󵄩
󵄩󵄩𝜓 (𝑦1 , 𝑦2 , 𝑞) − 𝑔 (𝑦1 , 𝑦2 , 𝑞)󵄩󵄩󵄩∞
(24)
󵄩
󵄩
= 󵄩󵄩󵄩(𝑦1∗ , 𝑦2∗ , 𝑞∗ ) − (𝑦∗1 , 𝑦∗2 , 𝑞∗ )󵄩󵄩󵄩∞ ≤ 𝜀.
Let 0 ≤ 𝛿 ≤ 𝜀 and 𝑖 ∈ N satisfying 𝜎[𝑖] (𝜀) ≤ 𝜀 − 𝛿. We can
define a map 𝑓󸀠 (𝑥) = 𝜎[𝑖] (𝑔(𝑥))(for a 3-dimensional input
𝑥 = (𝑥1 , 𝑥2 , 𝑥3 ), 𝜎(𝑥) = (𝜎(𝑥1 ), 𝜎(𝑥2 ), 𝜎(𝑥3 ))) that if 𝑥0 ∈ N3
is an initial configuration of M and ‖𝑥0 − 𝑥0 ‖∞ ≤ 𝜀, we have
󵄩󵄩󵄩𝑓󸀠 (𝑥 ) − 𝜓 (𝑥 )󵄩󵄩󵄩 = 󵄩󵄩󵄩𝜎[𝑖] (𝑔 (𝑥 )) − 𝜓 (𝑥 )󵄩󵄩󵄩
󵄩󵄩
0
0 󵄩
0
0 󵄩
󵄩∞ 󵄩󵄩
󵄩∞
(25)
≤ 𝜀 − 𝛿.
By the triangle inequality, if ‖𝑥0 −𝑥0 ‖∞ ≤ 𝜀, then, for a uEACcomputable function 𝑓 satisfying ‖𝑓 − 𝑓󸀠 ‖∞ ≤ 𝛿, we have
󵄩
󵄩
󵄩󵄩
󵄩
󸀠
󵄩󵄩𝑓 (𝑥0 ) − 𝜓 (𝑥0 )󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩󵄩𝑓 (𝑥0 ) − 𝑓 (𝑥0 )󵄩󵄩󵄩󵄩∞
󵄩
󵄩
(26)
+ 󵄩󵄩󵄩󵄩𝑓󸀠 (𝑥0 ) − 𝜓 (𝑥0 )󵄩󵄩󵄩󵄩∞
≤ 𝛿 + (𝜀 − 𝛿) = 𝜀.
From the discussion above we get that one can construct
a robust Turing machine simulation with uEAC-computable
functions.

6. Conclusion
uEAC is a novel electronic implementation inspired by
Rubel’s EAC model. Based on the mathematical model of

Mathematical Problems in Engineering
uEAC, we propose a fully connected uEACs array and
investigate its computational capabilities. By proving that any
Turing machine can be simulated with uEAC-computable
functions, we conclude that the proposed uEACs array is at
least as powerful as Turing machine.
This work can be extended in several directions. The
structure of the uEACs array, including the interconnection
relationship of the uEAC units and the values of the interconnection weights, is not necessary to remain constant during
the iteration. Ainslie et al. use genetic algorithm (GA) to
evolve a uEAC to solve XOR problem [25], but their research
is restricted to a single uEAC unit and it is hard to say that
GA is the most suitable evolutionary algorithm for uEAC.
The proposed uEACs array shows great advantages, while its
optimization is much more difficult since we have to optimize
the topology of the array and the particular structure of the
single uEAC unit simultaneously. From the mathematical
point of view, all the heuristic algorithms, such as particle
swarm optimizer (PSO), ant colony algorithm, and simulated
annealing algorithm (SA), can be used to optimize the uEACs
array, but we must consider the computation complexity
and efficiency of these algorithms. Zhu et al. proposed a
comprehensive uEACs array optimization strategy based on
particle swarm optimizer (PSO) [22], and the simulation
results are promising.
Moreover, in the definition of the fully connected uEACs
array, we use uEAC 1, uEAC 2, . . . , uEAC 𝑛 to denote the
uEAC units in the array, while it is still an open question:
how many uEAC units are needed in the array to guarantee
its computational capabilities? In other words, we employ an
indeterminate item “𝑛” to represent the number of uEAC
units, but it is far from satisfied. A comprehensive analysis
to determine the minimum “𝑛” that guarantees the robust
simulation of Turing machine is of great significance, and this
is an emphasis of the future research. The conclusion of this
paper can be rewritten as “a robust simulation of the transfer
function of a Turing machine can be constructed with
uEAC-computable functions.” Actually, when studying the
computational capability of the uEACs array, two questions
are considered. (1) Does a Turing machine can calculate any
uEAC-computable function? (2) Is the function generated by
a Turing machine also uEAC-computable? If the answers of
these two questions are both yes, we can get a more concrete
conclusion that the uEACs array and Turing machine are
equivalent! This paper focuses on the second question while
the first one can be seen as another valuable future research
direction.

Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.

Acknowledgments
This work is supported by National Natural Science Foundation of China (no. 61433003 and no. 61273150), Beijing Higher

7
Education Young Elite Teacher Project, and National Basic
Research Program of China (973 Program, 2012CB821206).

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