Output Encoding Research Articles

Self-checking combinational circuit design for single and unidirectional multibit error

This article presents novel input and output encoding techniques such that the resulting circuit is bidirectional error-free. The circuit can be fully optimized and any types of gates can be used. These schemes are used to design the functional part of a self-checking circuit. The input encoding algorithm can be applied to any circuit without significantly increasing the input lines. The output encoding technique involves graph-embedding which is done with heuristic method of polynomial complexity. The heuristic technique produces nearly optimal output encoding. Previously published work restrict the types of gates used in the circuit to non-inversion gates (AND/OR), and use inverters only at the inputs. The proposed techniques have a clear advantage over the currently available techniques because they allow the use of any types of gates. These techniques do not necessarily increase the overhead when applied to different MCNC benchmark circuits as the experimental results indicate. The only restriction is that either the inputs or the outputs have to be symbolic, and the two-level description of a circuit has to be given.

An optimization technique for the design of multiple valued PLA's

An optimization technique for the design of two types of multiple-valued PLAs is described. In a type-I PLA, the multiple-valued function is realized directly, whereas in a type-II PLA, output encoding is used to encode the binary output of the PLA. In both types, multiple function literal circuits are used for the purpose of minimization. It is shown that the proposed technique leads to a considerably reduced size of PLA when compared to the earlier techniques. >

How to solve the N-bit encoder problem with just one hidden unit

We demonstrate that it is possible to construct a three-layer network of the standard feedforward architecture that can solve the N input- N output encoder problem employing just one (linear)hidden unit.

Self-checking sequential circuit design using m -out-of- n codes

A technique for designing sequential circuits which are totally self-checking for single stuck at faults is presented. This technique uses m-out-of-n codes for state assignments and for output encoding. The next stage logic and the output logic are implemented such that any stuck-at-fault will either create a single bit error or unidirectional multibit error at the output. The technique has been applied to MCNC benchmark circuits and the overhead is estimated.

Exact algorithms for output encoding, state assignment, and four-level Boolean minimization

A novel minimization procedure of prime implicant generation and covering that operates on symbolic outputs, rather than binary-valued outputs, is proposed for solving the output encoding problem. An exact solution to this minimization problem is also an exact solution to the encoding problem. While this covering problem is more complex than the classic unate covering problem, a single logic minimization step replaces O(N-factorial) minimizations. The input encoding problem can be exactly solved using multiple-valued Boolean minimization. An exact algorithm is presented for state assignment by generalizing the proposed output encoding approach to the multiple-valued input case. Four-level Boolean minimization entails finding a cascaded pair of two-level logic functions that implement another logic function, such that the sum of the product terms in the two cascaded functions or truth tables is minimum. Four-level Boolean minimization can be formulated as an encoding problem and solved exactly using the proposed algorithms. Preliminary experimental results are presented which indicate that this approach is significantly more efficient than exhaustive search. Computationally efficient heuristic approaches based on the exact algorithms are proposed for output encoding, state assignment, and four-level Boolean minimization. >

Symbolic and neural learning algorithms: an experimental comparison

Despite the fact that many symbolic and neural network (connectionist) learning algorithms address the same problem of learning from classified examples, very little is known regarding their comparative strengths and weaknesses. Experiments comparing the ID3 symbolic learning algorithm with the perception and backpropagation neural learning algorithms have been performed using five large, real-world data sets. Overall, backpropagation performs slightly better than the other two algorithms in terms of classification accuracy on new examples, but takes much longer to train. Experimental results suggest that backpropagation can work significantly better on data sets containing numerical data. Also analyzed empirically are the effects of (1) the amount of training data, (2) imperfect training examples, and (3) the encoding of the desired outputs. Backpropagation occasionally outperforms the other two systems when given relatively small amounts of training data. It is slightly more accurate than ID3 when examples are noisy or incompletely specified. Finally, backpropagation more effectively utilizes a “distributed” output encoding.

How to Solve the N Bit Encoder Problem with Just Two Hidden Units

I demonstrate that it is possible to construct a three-layer network of the standard feedforward architecture that can solve the N input-N output encoder problem with just two hidden units.

Area-Time Optimal Fast Implementation of Several Functions in a VLSI Model

Area and computation time are considered to be important measures with which VLSI circuits are evaluated. In this paper, the area-time complexity for nontrivial n-input m-output Boolean functions, such as a decoder and an encoder, is studied with a model similar to Brent-Kung's model. A lower bound on area-time-product (ATαaα.≥1) for these functions is shown: for example, AT <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> = ω(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α-l</sup> ) for an n-input 2V-output decoder, and AT <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> = ω( n . log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α-1</sup> n) for an n-input ⌈log n⌉-output encoder. The results shown in this paper are complementary to those by Brent-Kung or Thompson, and are useful for a class of functions of rather simple structures, e.g., a priority encoder, a comparator, and symmetric functions.