# Detection of Total or Partial Symmetry of a Switching Function with the Use of Decomposition Charts

About: This article is published in IEEE Transactions on Electronic Computers.The article was published on 1963-10-01. It has received 37 citations till now. The article focuses on the topics: Function (mathematics) & Decomposition (computer science).

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24 Jul 2006

TL;DR: A distributed reconfigurable fabric inserted at RTL provides a debug platform that can be configured and operated post-silicon via the JTAG port and can be repeatedly reused to configure many debug structures such as assertions checkers, transaction identifiers, triggers, and event counters.

Abstract: In this paper we present a design-for-debug (DFD) reconfigurable infrastructure for SoCs to support at-speed in-system functional debug. A distributed reconfigurable fabric inserted at RTL provides a debug platform that can be configured and operated post-silicon via the JTAG port. The platform can be repeatedly reused to configure many debug structures such as assertions checkers, transaction identifiers, triggers, and event counters.

351 citations

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TL;DR: A new method for detecting groups of symmetric variables of completely specified Boolean functions using the canonical Generalized Reed-Muller forms and a set of signatures that allow for detecting symmetries of any number of inputs simultaneously.

Abstract: In this paper, we present a new method for detecting groups of symmetric variables of completely specified Boolean functions. The canonical Generalized Reed-Muller (GRM) forms are used as a powerful analysis tool. To reduce the search space we have developed a set of signatures that allow us to identify quickly sets of potentially symmetric variables. Our approach allows for detecting symmetries of any number of inputs simultaneously. Totally symmetric functions can be detected very quickly. The traditional definitions of symmetry have also been extended to include more types. This extension has the advantage of grouping input variables into more classes. Experiments have been performed on MCNC benchmark cases and the results verify the efficiency of our method.

73 citations

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TL;DR: This decomposition procedure optimally uses gates with a fan in of two, but post-synthesis reconfiguration for amalgamation into larger fan-in gates is available.

Abstract: A new mathematically based method of synthesis for combinational digital networks is presented, based upon the recognition of symmetry patterns in the functions being synthesized. The design procedure is structured such that at each stage of the synthesis a relevant gate assembly is indicated, which leaves a simplified "remainder" function to be synthesized at the next stage of the realization. This decomposition procedure optimally uses gates with a fan in of two, but post-synthesis reconfiguration for amalgamation into larger fan-in gates is available.

71 citations

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06 Jun 1994TL;DR: A new method for Boolean matching of completely specified Boolean functions of the canonical Generalized Reed-Muller forms is presented that can detect symmetries of any number of inputs simultaneously.

Abstract: In this paper we present a new method for Boolean matching of completely specified Boolean functions. The canonical Generalized Reed-Muller forms are used as a powerful analysis tool. Input permutation, as well as input and output negation for matching are handled simultaneously. To reduce the search space for input correspondence, we have developed a method that can detect symmetries of any number of inputs simultaneously. Experiments on MCNC benchmark circuits are very encouraging.

51 citations

### Cites background from "Detection of Total or Partial Symme..."

...Functions are p-equivalent under P1 and np-equivalent under P1 and P2. npn-equivalent functions allow all three transformations....

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TL;DR: A new algorithm to detect four basic types of two-variable symmetries is presented, which is faster than other known methods, and in some cases achieves a speedup of several orders of magnitude.

Abstract: Symmetry detection in completely specified Boolean functions is important for several applications in logic synthesis, technology mapping, binary decision diagram (BDD) minimization, and testing. This paper presents a new algorithm to detect four basic types of two-variable symmetries. The algorithm detects all pairs of symmetric variables in one pass over the shared BDD of the multioutput function. The worst case complexity of this method is cubic in the number of BDD nodes, but on typical logic synthesis benchmarks the complexity appears to be linear. The computation is particularly efficient when the functions have multiple symmetries or no symmetries. Experiments show that the algorithm is faster than other known methods, and in some cases achieves a speedup of several orders of magnitude.

50 citations

### Cites methods from "Detection of Total or Partial Symme..."

...If decision diagrams [1][14] are used to represent the input parameters of a recursive procedure, the partial results are cached to prevent multiple calls with the same input parameters....

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##### References

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TL;DR: It will be shown that several of the well-known theorems on impedance networks have roughly analogous theorem in relay circuits, including the delta-wye and star-mesh transformations, and the duality theorem.

Abstract: In the control and protective circuits of complex electrical systems it is frequently necessary to make intricate interconnections of relay contacts and switches Examples of these circuits occur in automatic telephone exchanges, industrial motor-control equipment, and in almost any circuits designed to perform complex operations automatically In this article a mathematical analysis of certain of the properties of such networks will be made Particular attention will be given to the problem of network synthesis Given certain characteristics, it is required to find a circuit incorporating these characteristics The solution of this type of problem is not unique and methods of finding those particular circuits requiring the least number of relay contacts and switch blades will be studied Methods will also be described for finding any number of circuits equivalent to a given circuit in all operating characteristics It will be shown that several of the well-known theorems on impedance networks have roughly analogous theorems in relay circuits Notable among these are the delta-wye (δ-Y) and star-mesh transformations, and the duality theorem

922 citations

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TL;DR: A basic part of the general synthesis problem is the design of a two-terminal network with given operating characteristics, and this work shall consider some aspects of this problem.

Abstract: THE theory of switching circuits may be divided into two major divisions, analysis and synthesis. The problem of analysis, determining the manner of operation of a given switching circuit, is comparatively simple. The inverse problem of finding a circuit satisfying certain given operating conditions, and in particular the best circuit is, in general, more difficult and more important from the practical standpoint. A basic part of the general synthesis problem is the design of a two-terminal network with given operating characteristics, and we shall consider some aspects of this problem.

774 citations

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Bell Labs

^{1}TL;DR: The problem in this area which has been attacked most energetically is that of the synthesis of efficient combinational that is, nonsequential, logic circuits.

Abstract: THE SEARCH for simple abstract techniques to be applied to the design of switching systems is still, despite some recent advances, in its early stages The problem in this area which has been attacked most energetically is that of the synthesis of efficient combinational that is, nonsequential, logic circuits

610 citations

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02 May 1952

TL;DR: T h e a p p l i c a t i o n o f t h e s e t e c h n i q u e s h a s b e e n u n d e r i n c r e a s e d s t u d y .

Abstract: S i n c e C l a u d e S h a n n o n I d i s c o v e r e d o r r e d i s c o v e r e d t h a t t r u t h f u n c t i o n l o g i c ( p r o p o s i t i o n a l c a l c u l u s ) c a n b e u s e d i n t h e a n a l y s i s a n d d e s i g n o f c i r c u i t s comp o s e d o f t w o v a l u e d e l e m e n t s , t h e a p p l i c a t i o n o f t h e s e t e c h n i q u e s h a s b e e n u n d e r i n c r e a s e d s t u d y . I n u t i l i z i n g t h i s m e t h o d , o n c e t h e s y s t e m h a s b e e n f o r m u l a t e d i n t e r m s o f some l o g i c a l e q u a t i o n o r e q u a t i o n s , t h e p r o b l e m a r i s e s o f how t o m a n i p u l a t e t h i s e q u a t i o n i n t o t h e f o r m w h i c h c o r r e s p o n d s t o t h e b e s t p h y s i c a l e q u i v a l e n t . T h i s g e n e r a l p r o b l e m c a n b e v e r y comp l i c a t e d a n d d i f f i c u l t . No t o n l y d o e s t h e c o m p l e x i t y i n c r e a s e g r e a t l y w i t h t h e n u m b e r o f i n p u t s a n d o u t p u t s , b u t t h e c r i t e r i a o f t h e b e s t c i r c u i t w i l l v a r y w i t h t h e e q u i p m e n t i n v o l v e d . T he v a r i o u s u s e o f r e l a y s , c r y s t a l d i o d e s , d i f f e r e n t t y p e s o f v a c u u m t u b e c i r c u i t s , o r o t h e r & l a m e n t s , a l l o f w h i c h h a v e d i f f e r e n t a d v a n t a g e s a n d l i m i t a t i o n s o f t i m i n g , l o a d i n g , e t c . , w i l l h a v e a n i m p o r t a n t e f f e c t u p o n t h e c h o i c e f i n a l l y made . E x a m p l e s a r e t h e d e s i r a b i l i t y o f e q u a l i z i n g t h e c o n t a c t l o a d i n g i n a r e l a y c l r ~ u l t , a n d t h e n e c e s s i t y f o r k e e p i n g c h a i n s o f c r y s t a l g a t e s a n d m i x e r s t o a m i n i m u m l e n g t h . A n o t h e r

119 citations

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TL;DR: In this paper, a method for determining whether a Boolean function possesses any group invariance is presented, that is, whether there are any permutations or primings of the independent variables which leave the function unchanged.

Abstract: A method is presented for determining whether a Boolean function possesses any group invariance; that is, whether there are any permutations or primings of the independent variables which leave the function unchanged This method is then extended to the detection of functions which are totally symmetric

60 citations