Lower Bounds for the Computational Power of Networks of Spiking Neurons

1996 ◽  
Vol 8 (1) ◽  
pp. 1-40 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Formal Model ◽  
Random Access ◽  
Spiking Neurons ◽  
Turing Machines ◽  
Computational Power ◽  
Computational Tool ◽  
Threshold Functions ◽  
Basic Assumptions ◽  
Threshold Circuits ◽  
Phase Differences

We investigate the computational power of a formal model for networks of spiking neurons. It is shown that simple operations on phase differences between spike-trains provide a very powerful computational tool that can in principle be used to carry out highly complex computations on a small network of spiking neurons. We construct networks of spiking neurons that simulate arbitrary threshold circuits, Turing machines, and a certain type of random access machines with real valued inputs. We also show that relatively weak basic assumptions about the response and threshold functions of the spiking neurons are sufficient to employ them for such computations.

Computational Power of Threshold Circuits of Energy at most Two

2018 ◽  
Vol E101.A (9) ◽  
pp. 1431-1439
Author(s):  
Hiroki MANIWA ◽  
Takayuki OKI ◽  
Akira SUZUKI ◽  
Kei UCHIZAWA ◽  
Xiao ZHOU
Keyword(s):  
Computational Power ◽  
Threshold Circuits

QUANTUM COMPUTATION WITH RESTRICTED AMPLITUDES

2003 ◽  
Vol 14 (05) ◽  
pp. 853-870 ◽  
Author(s):  
HARUMICHI NISHIMURA
Keyword(s):  
Quantum Computation ◽  
Polynomial Time ◽  
Exact Algorithms ◽  
Quantum Computers ◽  
Turing Machines ◽  
Computational Power ◽  
Polynomial Time Algorithms ◽  
Exact Quantum ◽  
Bounded Error ◽  
Transition Amplitudes

In this paper, we explore the power of quantum computers with restricted transition amplitudes. In 1997 Adleman, DeMarrais, and Huang showed that quantum Turing machines (QTMs) with the amplitudes from [Formula: see text] are computationally equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial-time algorithms. We show that QTMs with the amplitudes from [Formula: see text] is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms, i.e., algorithms that output correct answers with certainty. By extending this result, it is shown that exact quantum computers with rational biased coins are equivalent to classical computers. Moreover, we discuss the computational power of exact quantum computers with multiple types of coins. We also show that, from the viewpoint of zero-error polynomial-time algorithms, [Formula: see text] is not more powerful than [Formula: see text] as the set of amplitudes taken by QTMs; however, it is sufficient to solve the factoring problem.

ON THE NECESSITY OF FORMAL MODELS FOR REAL-TIME PARALLEL COMPUTATIONS

2001 ◽  
Vol 11 (02n03) ◽  
pp. 353-361 ◽  
Author(s):  
STEFAN D. BRUDA ◽  
SELIM G. AKL
Keyword(s):  
Positive Integer ◽  
Real Time ◽  
Complexity Theory ◽  
Turing Machine ◽  
Formal Model ◽  
Parallel Computations ◽  
General Nature ◽  
Turing Machines ◽  
Formal Models ◽  
Language L

We assume the multitape real-time Turing machine as a formal model for parallel real-time computation. Then, we show that, for any positive integer k, there is at least one language Lk which is accepted by a k-tape real-Turing machine, but cannot be accepted by a (k - 1)-tape real-time Turing machine. It follows therefore that the languages accepted by real-time Turing machines form an infinite hierarchy with respect to the number of tapes used. Although this result was previously obtained elsewhere, our proof is considerably shorter, and explicitly builds the languages Lk. The ability of the real-time Turing machine to model practical real-time and/or parallel computations is open to debate. Nevertheless, our result shows how a complexity theory based on a formal model can draw interesting results that are of more general nature than those derived from examples. Thus, we hope to offer a motivation for looking into realistic parallel real-time models of computation.

ON THE POWER OF ONE-WAY GLOBALLY DETERMINISTIC SYNCHRONIZED ALTERNATING TURING MACHINES AND MULTIHEAD AUTOMATA

1995 ◽  
Vol 06 (04) ◽  
pp. 431-446 ◽  
Author(s):  
ANNA SLOBODOVÁ
Keyword(s):  
Turing Machine ◽  
Simple Form ◽  
Space Complexity ◽  
Complexity Classes ◽  
Turing Machines ◽  
Computational Power ◽  
Parallel Processes ◽  
Open Questions ◽  
The One ◽  
Alternating Turing Machines

The alternating model augmented by a special simple form of communication among parallel processes—the so-called synchronized alternating (SA) model, provides (besides others) nice characterizations of the space complexity classes defined by nondeterministic Turing machines. The model investigated in this paper — globally deterministic synchronized alternating (GDSA) model—is obtained by a feasible restriction of nondeterminism in SA. It is known that it characterizes the deterministic counterparts of the nondeterministic space classes characterized by the SA model. In the paper we resume in the investigation of GDSA solving the open questions about the computational power of the one-way GDSA models. It is known that in the case of space-bounded Turing machine and multihead automata, the one-way SA models are equivalent to their two-way counterparts. We show that the same holds for GDSA models. The results contribute to the knowledge about the model and imply new characterizations of the deterministic space complexity classes.

scholarly journals The computing power of Turing machine based on quantum logic

10.29007/k8cb ◽  
2018 ◽  
Author(s):  
Yun Shang ◽  
Xian Lu ◽  
Ruqian Lu
Keyword(s):  
Quantum Logic ◽  
Turing Machine ◽  
Turing Machines ◽  
Computational Power ◽  
Orthomodular Lattices ◽  
Locally Finite ◽  
Computing Power ◽  
Truth Value ◽  
Quantum Language

Turing machines based on quantum logic can solve undecidableproblems. In this paper we will give recursion-theoreticalcharacterization of the computational power of this kind of quantumTuring machines. In detail, for the unsharp case, it is proved that&#931<sup>0</sup><sub>1</sub>&#8746&#928<sup>0</sup><sub>1</sub>&#8838L<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))&#8838&#928<sup>0</sup><sub>2</sub>when the truth value lattice is locally finite and the operation &#8743is computable, whereL<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))denotes theclass of quantum language accepted by these Turing machine indepth-first model (respectively, width-first model);for the sharp case, we can obtain similar results for usual orthomodular lattices.

Universality of SNQ P Systems Using One Type of Spikes and Restrictive Rule Application

2020 ◽  
Vol 31 (01) ◽  
pp. 117-132
Author(s):  
Andrei Păun ◽  
Florin-Daniel Bîlbîe
Keyword(s):  
P Systems ◽  
Current Work ◽  
Turing Machines ◽  
Computational Power ◽  
Process Information ◽  
Spiking Neural P Systems ◽  
Rule Application ◽  
The Way

We investigate the spiking neural P systems with communication on request (SNQ P systems) that are devices in the area of neural like P systems abstracting the way in which neurons work and process information. Here we discuss the SNQ P systems using the rule application strategy as defined by Linqiang Pan and collaborators and we are able to improve their result of universality of such systems using two types of spikes. In the current work, we prove that only one type of spikes is sufficient for reaching the computational power of Turing Machines for these devices, bringing closer to implementation such a device. The result holds both in maximum parallel manner application of the rules as well as the maximum-sequentiality application of rules.

Concurrent Program Schemes

1987 ◽  
Vol 10 (4) ◽  
pp. 337-361
Author(s):  
A.J. Kfoury ◽  
P. Urzyczyn
Keyword(s):  
Concurrent Programs ◽  
Turing Machines ◽  
Computational Power ◽  
Sequential Programs ◽  
Concurrent Program ◽  
Nondeterministic Choice ◽  
Flow Diagrams ◽  
Alternating Turing Machines

We study the programming formalism FD of “flow-diagrams” to which we gradually add various features of concurrency. The weakest form of concurrency is introduced by the construct “and”, which is dual to the nondeterministic choice “or” and plays a role similar to universal states in alternating Turing machines. Stronger (and more realistic) forms of concurrency are obtained when processes are allowed to communicate. We consider communication by channels and communication by messages. We calibrate the computational power of classes of concurrent programs FD+α against that of sequential programs, where α is the addition of one of the following features: {and}, {and, or}, {and, or, channels}, or {and, or, messages}.

On the Computational Power of Threshold Circuits with Sparse Activity

2006 ◽  
Vol 18 (12) ◽  
pp. 2994-3008 ◽  
Author(s):  
Kei Uchizawa ◽  
Rodney Douglas ◽  
Wolfgang Maass
Keyword(s):  
Neural Circuits ◽  
Circuit Complexity ◽  
Complexity Measure ◽  
Computational Power ◽  
Complexity Measures ◽  
Threshold Circuit ◽  
Polynomial Size ◽  
Threshold Circuits ◽  
Size Threshold ◽  
The Brain

Circuits composed of threshold gates (McCulloch-Pitts neurons, or perceptrons) are simplified models of neural circuits with the advantage that they are theoretically more tractable than their biological counterparts. However, when such threshold circuits are designed to perform a specific computational task, they usually differ in one important respect from computations in the brain: they require very high activity. On average every second threshold gate fires (sets a 1 as output) during a computation. By contrast, the activity of neurons in the brain is much sparser, with only about 1% of neurons firing. This mismatch between threshold and neuronal circuits is due to the particular complexity measures (circuit size and circuit depth) that have been minimized in previous threshold circuit constructions. In this letter, we investigate a new complexity measure for threshold circuits, energy complexity, whose minimization yields computations with sparse activity. We prove that all computations by threshold circuits of polynomial size with entropy O(log n) can be restructured so that their energy complexity is reduced to a level near the entropy of circuit states. This entropy of circuit states is a novel circuit complexity measure, which is of interest not only in the context of threshold circuits but for circuit complexity in general. As an example of how this measure can be applied, we show that any polynomial size threshold circuit with entropy O(log n) can be simulated by a polynomial size threshold circuit of depth 3. Our results demonstrate that the structure of circuits that result from a minimization of their energy complexity is quite different from the structure that results from a minimization of previously considered complexity measures, and potentially closer to the structure of neural circuits in the nervous system. In particular, different pathways are activated in these circuits for different classes of inputs. This letter shows that such circuits with sparse activity have a surprisingly large computational power.

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