## Turing Computability With Neural Nets (1991)

Venue: | Applied Mathematics Letters |

Citations: | 63 - 13 self |

### BibTeX

@ARTICLE{Siegelmann91turingcomputability,

author = {Hava T. Siegelmann and Eduardo D. Sontag},

title = {Turing Computability With Neural Nets},

journal = {Applied Mathematics Letters},

year = {1991},

volume = {4},

pages = {77--80}

}

### Years of Citing Articles

### OpenURL

### Abstract

. This paper shows the existence of a finite neural network, made up of sigmoidal neurons, which simulates a universal Turing machine. It is composed of less than 10 5 synchronously evolving processors, interconnected linearly. High-order connections are not required. 1. Introduction This paper addresses the question: What ultimate limitations, if any, are imposed by the use of neural nets as computing devices? In particular, and ignoring issues of training and practicality of implementation, one would like to know if every problem that can be solved by a digital computer is also solvable --in principle-- using a net. This question has been asked before in the literature. Indeed, Jordan Pollack ([7]) showed that a certain recurrent net model --which he called a "neuring machine," for "neural Turing"-- is universal. In his model, all neurons synchronously update their states according to a quadratic combination of past activation values. In general, one calls high-order nets those in...

### Citations

387 | On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines - Blum, Shub, et al. - 1989 |

352 | Mathematical Control Theory. Deterministic Finite Dimensional Systems
- Sontag
- 1990
(Show Context)
Citation Context ...u + c), where �� is a hard threshold (Heaviside) function, can only simulate a finite automaton, as all states are essentially binary. Many other types of "machines" may be used for univ=-=ersality (see [9], especi-=-ally Chapter 2, for general definitions of continuous machines). For instance, with a similar proof we can show that systems evolving according to equations x + = x+��(Ax+bu+c), where �� takes... |

62 | On the computational power of sigmoid versus Boolean threshold circuits - Maass, Schnitger, et al. - 1991 |

52 |
On Connectionist Models of Natural Language Processing
- Pollack
- 1987
(Show Context)
Citation Context ...ld like to know if every problem that can be solved by a digital computer is also solvable --in principle-- using a net. This question has been asked before in the literature. Indeed, Jordan Pollack (=-=[7]) showed that a cert-=-ain recurrent net model --which he called a "neuring machine," for "neural Turing"-- is universal. In his model, all neurons synchronously update their states according to a quadra... |

37 |
Efficient simulation of finite automata by neural nets
- Alon, Dewdney, et al.
- 1991
(Show Context)
Citation Context ... of McCulloch and Pitts in the 1940s, it has been clear how to simulate logic gates by networks of threshold (binary-valued) neurons, and hence how to obtain finite automata using such nets (see e.g. =-=[1]-=- for more recent work on that problem). One can simulate Turing machines if one allows a potentially unbounded number of neurons; see e.g. [4] for variations on this theme and relations to cellular au... |

27 |
Neural computability
- Garzon, Franklin
- 1989
(Show Context)
Citation Context ...e how to obtain finite automata using such nets (see e.g. [1] for more recent work on that problem). One can simulate Turing machines if one allows a potentially unbounded number of neurons; see e.g. =-=[4]-=- for variations on this theme and relations to cellular automata. Since we insist on a fixed number of neurons, which does not increase during the computation, our problem is different. Supported in p... |

16 |
Dynamics of iterated-map neural networks
- Marcus, Westervelt
- 1989
(Show Context)
Citation Context ... oe(x) := 0 if x ! 0, oe(x) := x if 0sxs1, and oe(x) := 1 if x ? 1. Processor nets as above appear frequently in neural network studies, and their dynamic properties are of interest (see for instance =-=[6]-=-); the continuous-time analogue is also studied in the literature (see some comments in the concluding section below). Given any infinite sequence of rational numbers ! = u(0); u(1); u(2); : : : (thou... |

11 |
Turing equivalence of neural networks with second order connection weights
- Sun, Chen, et al.
- 1991
(Show Context)
Citation Context ...chronously update their states according to a quadratic combination of past activation values. In general, one calls high-order nets those in which activations are combined using multiplications; see =-=[11]-=- for related work and many other references to such nets. Pollack left open the question of establishing if high-order connections are really necessary in order to achive universality; the feeling amo... |

7 | Remarks on interpolation and recognition using neural nets - Sontag - 1991 |

6 |
Dynamics of iterated-map neural networks,” Phys
- Marcus, Westervelt
- 1989
(Show Context)
Citation Context ...near function σ(x):=0ifx<0, σ(x):=x if 0 ≤ x ≤ 1, and σ(x):=1ifx>1. Processor nets as above appear frequently in neural network studies, and their dynamic properties are of interest (see for instance =-=[6]-=-); the continuous-time analogue is also studied in the literature (see some comments in the concluding section below). Given any infinite sequence of rational numbers ω = u(0),u(1),u(2),... (thought o... |

5 |
Neural Nets Are Universal Computing Devices
- Siegelmann, Sontag
- 1991
(Show Context)
Citation Context ...--zsm 00 : : : where m may be zero or positive. Iff OE(n) is undefined, m = 0; otherwise OE(n) = m \Gamma 1. We next describe the main ideas of the proof; details can be found in the technical report =-=[8]. As in th-=-e textbook approach of "counter machines," one can encode an infinite tape into real-valued activations; the only problem is how to do this while preserving a purely linear-interconnection a... |