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An Algebra of Scans
 In Mathematics of Program Construction
, 2004
"... A parallel prefix circuit takes n inputs x1 , x2 , . . . , xn and produces the n outputs x1 , x1 x2 , . . . , x1 x2 xn , where `#' is an arbitrary associative binary operation. Parallel prefix circuits and their counterparts in software, parallel prefix computations or scans, have numerou ..."
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A parallel prefix circuit takes n inputs x1 , x2 , . . . , xn and produces the n outputs x1 , x1 x2 , . . . , x1 x2 xn , where `#' is an arbitrary associative binary operation. Parallel prefix circuits and their counterparts in software, parallel prefix computations or scans, have numerous applications ranging from fast integer addition over parallel sorting to convex hull problems. A parallel prefix circuit can be implemented in a variety of ways taking into account constraints on size, depth, or fanout. Traditionally, implementations are either defined graphically or by enumerating the underlying graph. Both approaches have their pros and cons. A figure if well drawn conveys the possibly recursive structure of the scan but it is not amenable to formal manipulation. A description in form of a graph while rigorous obscures the structure of a scan and is equally hard to manipulate. In this paper we show that parallel prefix circuits enjoy a very pleasant algebra. Using only two basic building blocks and four combinators all standard designs can be described succinctly and rigorously. The rules of the algebra allow us to prove the circuits correct and to derive circuit designs in a systematic manner. lord darlington. . . . [Sees a fan lying on the table.] And what a wonderful fan! May I look at it? lady windermere. Do. Pretty, isn't it! It's got my name on it, and everything. I have only just seen it myself. It's my husband's birthday present to me. You know today is my birthday?  Oscar Wilde, Lady Windermere's Fan 1
Synchronous Digital Circuits as Functional Programs
"... Functional programming techniques have been used to describe synchronous digital circuits since the early 1980s and have proven successful at describing certain types of designs. Here we survey the systems and formal underpinnings that constitute this tradition. We situate these techniques with resp ..."
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Functional programming techniques have been used to describe synchronous digital circuits since the early 1980s and have proven successful at describing certain types of designs. Here we survey the systems and formal underpinnings that constitute this tradition. We situate these techniques with respect to other formal methods for hardware design and discuss the work yet to be done.
Technical Report Number 682 Translating HOL
, 2007
"... Translating HOL functions to hardware ..."
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