Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 236806, 8 pages

http://dx.doi.org/10.1155/2015/236806

## Simulation of Turing Machine with uEAC-Computable Functions

^{1}School of Automation, Beijing Institute of Technology, Beijing 100081, China^{2}Department of Electrical and Computer Engineering, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202, USA

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.

#### Abstract

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 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 [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 continuous-valued 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 -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 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 boundary-valued LLA functions determine the computation of the uEAC forms. In the present study, a fully connected single-input, single-output uEACs array is proposed and its computational capabilities are investigated by showing that any Turing machine can be simulated with uEAC-computable functions, even in the case that some noise is added to the initial configuration of 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 with uEAC-computable 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 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 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 () the current state of the machine, () the symbol in the cell being scanned by the head, and () 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: means moving one cell to the right, means moving one cell to the left if there are cells to the left, and means not to move. If the machine reaches a situation in which no transition rule will be carried out, then the machine halts.

*Definition 1. *A single tape Turing machine is -tuple , where(i)is a nonempty finite set of states;(ii) is the tape alphabet that describes the contents of cells of the tape;(iii) is the transfer function;(iv) is the initial state.

*Example 2. *Consider an example of single tape Turing machine with three states and is the initial state. As discussed above, let be a blank symbol and . The transfer function is given by Table 1.

Given the input , the computation performed by the machine iswhere the symbol “” marks the position of the read-write head.