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Finite State Machines and Recurrent Neural Networks  Automata and Dynamical Systems Approaches
 Neural Networks and Pattern Recognition
, 1998
"... We present two approaches to the analysis of the relationship between a recurrent neural network (RNN) and the finite state machine M the network is able to exactly mimic. First, the network is treated as a state machine and the relationship between the RNN and M is established in the context of alg ..."
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Cited by 29 (11 self)
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We present two approaches to the analysis of the relationship between a recurrent neural network (RNN) and the finite state machine M the network is able to exactly mimic. First, the network is treated as a state machine and the relationship between the RNN and M is established in the context of algebraic theory of automata. In the second approach, the RNN is viewed as a set of discretetime dynamical systems associated with input symbols of M. In particular, issues concerning network representation of loops and cycles in the state transition diagram of M are shown to provide a basis for the interpretation of learning process from the point of view of bifurcation analysis. The circumstances under which a loop corresponding to an input symbol x is represented by an attractive fixed point of the underlying dynamical system associated with x are investigated. For the case of two recurrent neurons, under some assumptions on weight values, bifurcations can be understood in the geometrical c...
21 Addition and Related Arithmetic Operations with Threshold Logic
, 1996
"... In this paper we investigate the reduction of the size for small depth feedforward linear threshold networks performing binary addition, comparison, and related functions. For n bit operands we propose a depth3 O( n 2 log n ) asymptotic size network for the binary addition with polynomially bou ..."
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Cited by 12 (6 self)
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In this paper we investigate the reduction of the size for small depth feedforward linear threshold networks performing binary addition, comparison, and related functions. For n bit operands we propose a depth3 O( n 2 log n ) asymptotic size network for the binary addition with polynomially bounded weights. We propose also a depth3 addition of optimal O(n) asymptotic size network and a depth2 comparison of O( p n) asymptotic size network, both with O(2 p n ) asymptotic size of weight values. For existing architectural formats we show that our schemes, with equal or smaller depth networks, substantially outperform existing schemes in terms of size and fanin requirements and in occasions in weight requirements. Keywords Computer Arithmetic, Binary Adders, Binary Comparison, Majority Circuits, Threshold Logic, Neural Networks. I. Introduction and Main Results A linear threshold gate with a Boolean output F (X) is defined by: F (X) = sgn(F(X)) = 8 ? ! ? : 1 if F(X) 0 0 i...
Periodic Symmetric Functions, Serial Addition and Multiplication with Neural Networks
, 1998
"... This paper investigates threshold based neural networks for periodic symmetric Boolean functions and some related operations. It is shown that any ninput variable periodic symmetric Boolean function can be implemented with a feedforward linear threshold based neural network with size of O(log n) a ..."
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Cited by 8 (4 self)
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This paper investigates threshold based neural networks for periodic symmetric Boolean functions and some related operations. It is shown that any ninput variable periodic symmetric Boolean function can be implemented with a feedforward linear threshold based neural network with size of O(log n) and depth also of O(log n), both measured in terms of neurons. The maximum weight and fanin values are in the order of O(n). Under the same assumptions on weight and fanin values, an asymptotic bound of O(log n) for both size and depth of the network is also derived for symmetric Boolean functions that can be decomposed into a constant number of periodic symmetric Boolean subfunctions. Based on this results neural networks for serial binary addition and multiplication of nbit operands are also proposed. It is shown that the serial addition can be computed with polynomially bounded weights and a maximum fanin in the order of O(log n) in O( n log n ) serial cycles, where a serial cycle c...
Block Save Addition with Threshold Logic
, 1995
"... In this paper we investigate small depth linear threshold element networks for multioperand addition. We consider depth2 linear threshold element networks and block save addition. We improve the overall cost of the block save addition, both in terms of gates and wires (actually the number of gates ..."
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Cited by 8 (6 self)
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In this paper we investigate small depth linear threshold element networks for multioperand addition. We consider depth2 linear threshold element networks and block save addition. We improve the overall cost of the block save addition, both in terms of gates and wires (actually the number of gates each multiplied by its number of inputs), with the inclusion of the telescopic sums proposed by Minnick together with a minimization technique based on gates sharing. We show that previously proposed schemes require about twice the number of linear threshold gates for common operand lengths. Furthermore, we show that the number of wires required by an implementation for previously proposed schemes is also about two times higher than the number of wires required for the scheme we describe for commonly architected operand sizes. 1 Introduction A linear threshold gate computing a Boolean function F (X) is defined by the following: F (X) = sgn(F(X)) = ae 1 if F(X) 0 0 if F(X) ! 0 with F(...
Low Weight and FanIn Neural Networks for Basic Arithmetic Operations
 In 15 th IMACS World Congress 1997 on Scientific Computation, Modelling and Applied Mathematics, volume 4 Artificial Intelligence and Computer Science
, 1997
"... In this paper we investigate low weight and fanin neural networks for the precise computation of some basic arithmetic operations. First we assume one bit per serial cycle LSB first operand reception and introduce a pipeline network performing serial binary addition in O(n) time constructed with 11 ..."
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Cited by 8 (5 self)
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In this paper we investigate low weight and fanin neural networks for the precise computation of some basic arithmetic operations. First we assume one bit per serial cycle LSB first operand reception and introduce a pipeline network performing serial binary addition in O(n) time constructed with 11 threshold gates, a maximum weight of 2 and a maximum fanin of 4. Further we prove that serial multiplication can be implemented with a threshold network constructed with 11(n \Gamma 1) threshold gates and the same maximal values for fanin and weights. The achieved delay performance is in the order of 2n \Gamma 1 + 2dlog ne. Consequently we propose schemes for the addition of 32bit operands based on a "carry look ahead" approach. In particular we show that the 32bit 2 \Gamma 1 addition can be implemented in depth8=7=5, with a maximum fanin of 4=4=6 and a maximum weight of 2=4=5, respectively. We finally show that the 2 \Gamma 1 binary addition using redundant represented operands can...
Block Save Addition with Telescopic Sums
, 1995
"... In this paper we investigate small depth linear threshold element networks for multioperand addition. In particular, we consider depth2 linear threshold element networks and block save addition. We improve the overall cost in terms of gates and wires of the block save addition with the inclusion o ..."
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Cited by 6 (3 self)
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In this paper we investigate small depth linear threshold element networks for multioperand addition. In particular, we consider depth2 linear threshold element networks and block save addition. We improve the overall cost in terms of gates and wires of the block save addition with the inclusion of the telescopic sums proposed by Minnick. We show that previously proposed schemes require about twice the number of linear threshold gates for common operand lengths. Furthermore, we show that the number of wires required by an implementation for previously proposed schemes is also about two times higher than the number of wires required for the scheme we describe for commonly architected operand sizes. 1 Introduction A linear threshold gate computing a Boolean function F (X) is defined by the following: F (X) = sgn(F(X)) = ae 1 if F(X) 0 0 if F(X) ! 0 with F(X) = n X i=1 ! i x i + / The threshold element comprises a set of input variables, X = (x 1 ; x 2 ; : : : ; xn\Gamma1 ; ...
Highspeed Hybrid ThresholdBoolean Logic
 In: Proceedings of the 45th IEEE International Midwest Symposium on Circuits and Systems
, 2002
"... In this paper we propose a highspeed hybrid ThresholdBoolean logic style suitable for Boolean symmetric functions implementation. First, we present a depth2 hybrid implementation scheme for arbitrary symmetric Boolean functions, based on differential Threshold logic gates as circuit style. Finally ..."
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Cited by 4 (2 self)
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In this paper we propose a highspeed hybrid ThresholdBoolean logic style suitable for Boolean symmetric functions implementation. First, we present a depth2 hybrid implementation scheme for arbitrary symmetric Boolean functions, based on differential Threshold logic gates as circuit style. Finally, we present the hybrid logic design of a counter. The simulation results, suggest that the hybrid designed in higher speed when compared with traditional Threshold logic and Boolean logic counterparts, at expense of between transistors.
Implementation of Threshold Logic
, 1998
"... Traditionally, logic circuits have been, and are still being implemented using Boolean logic. Although there has been a tremendous increase in the performance of the technology used to implement Boolean logic primitives, the underlying paradigm has remained unchanged over the years. Since the ear ..."
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Cited by 3 (1 self)
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Traditionally, logic circuits have been, and are still being implemented using Boolean logic. Although there has been a tremendous increase in the performance of the technology used to implement Boolean logic primitives, the underlying paradigm has remained unchanged over the years. Since the early 1960's there is a fundamentally more powerful alternative for Boolean logic available, called Threshold Logic (TL). Although implementations of TL gates have grown with the advances in technology, none of these have ever proven widely applicable. Because of this TL was never a practical success. Recently a new technology called Capacitive Threshold Logic (CTL) was disclosed which holds the promise of being the first practically applicable TL technology. This thesis investigates the issues related to the application of CTL, and particularly to the implementation of arithmetic operations. It starts with a very general introduction into TL. After this it deals with a number of probl...
On the Power of Democratic Networks
 Swiss Federal Institute of Technology, Department of Mathematics
, 1996
"... . Linear Threshold Boolean units (LTUs) are the basic processing components of artificial neural networks of Boolean activations. Quantization of their parameters is a central question in hardware implementation, when numerical technologies are used to store the configuration of the circuit. In the ..."
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Cited by 2 (1 self)
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. Linear Threshold Boolean units (LTUs) are the basic processing components of artificial neural networks of Boolean activations. Quantization of their parameters is a central question in hardware implementation, when numerical technologies are used to store the configuration of the circuit. In the previous studies on the circuit complexity of feedforward neural networks, no differences had been made between a network with "small" integer weights and one composed of majority units (LTUs with weights in f\Gamma1; 0; +1g), since any connection of weight w (w integer) can be simulated by jwj connections of value sgn(w). This paper will focus on the circuit complexity of democratic networks, i.e. circuits of majority units with at most one connection between each pair of units. The main results presented are the following: any Boolean function can be computed by a depth3 nondegenerate democratic network and can be expressed as a linear threshold function of majorities; ATLEASTk and AT...
Block Save Addition with Threshold Gates
, 1995
"... We investigate the realization of small depth feedforward linear threshold networks capable of performing multioperand addition. We focus on parallel designs for high performance implementations. More in particular we discuss depth2 networks capable of reducing the multioperand addition into a t ..."
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Cited by 2 (2 self)
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We investigate the realization of small depth feedforward linear threshold networks capable of performing multioperand addition. We focus on parallel designs for high performance implementations. More in particular we discuss depth2 networks capable of reducing the multioperand addition into a two operand addition which is performed by a high speed two operand adder for the final reduction to the sum. First we provide some background information related to the design of high performance multioperand addition using ordinary Boolean logic. Consequently we focus on design techniques adaptable to linear threshold gates. We review existing schemes and we present some new results regarding the design of multioperand addition with feedforward linear threshold networks. We propose a scheme that improves the overall cost substantially for the block save addition and for entire multioperand addition network. We estimate, for a number of block save adders of practical interest, the overal...