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Signed-digit (SD) number representation systems have been defined for any radix r L 3 with digit values ranging over the set {-a,. .. ,-1, 0,1,. * a , a}, where a is an arbitrary integer in the range 112 r < a < r. Such number representation systems possess sufficient redundancy to allow for the annihilation of carry or borrow chains and hence result in fast propagation-free addition and subtraction. In this paper, we refer to the above as "ordinary" SD number systems and define generalized SD number systems which contain them as a special symmetric subclass. It is shown that the generalization not only provides a unified view of all redundant number systems which have proven useful in practice (including stored-carry and stored-borrowed systems), but also leads to new number systems not examined before. Examples of such new number systems are stored-carry-or-borrow systems, stored-double-carry systems, and certain redundant decimal representations.

Sum-& Conclusions-Voting is important in the realization of ultrareliable systems based on the multi-channel computation paradigm. In an earlier paper (1991 Aug) I dealt with voting networks, viz, hardware implementation of certain voting schemes. A voting algorithm specifies how the voting result is obtained from the input data and can be the basis for implementing a hardware voting network or a software voting routine. This paper presents efficient n-way plurality and threshold voting algorithms based on the type of voting (exact, inexact, or approval), rule for output selection (plurality or threshold) and properties of the input object space (size & structure). Exact voting is the most common voting method and is the easiest to implement. Inexact voting algorithms are more complicated due to intransitivity of approximate equality. As an example, when approximate equality a = b for numerical inputs a & b is defined as la-bl 5 E , then a n b and b n c do not imply a P c. In approval voting, each input to the voting process consists of a finite or infinite set of values that have been "approved" by the corresponding computation channel and the value, or set of values, with the highest approval voting must emerge as output. Multiple approved values can result from non-unique answer to a given problem or from uncertainties in the solution process. For exact voting, the complexity of an n-way voting algorithm depends on the structure of the input object space. Threshold voting often requires less time & space, except when the threshold is very small. I extend the techniques for designing efficient exact voting algorithms to inexact and approval voting schemes. My results show that optimal linear-time (i n n) voting algorithms are available when the input object space is small. Next in the time-complexity hierarchy is the case of a totally-ordered object space that supports worstcase order+. log(n)]-time algorithms for both exact and inexact voting as well as for certain approval-voting schemes. These algorithms are intimately related to sorting and have the same time complexity. An unordered input object space leads to worst-case quadratic-time exact and inexact voting algorithms, even when a distance metric can be defined on pairs of input objects. MIT FTMP design [21], Camegie-Mellon University C.vmp system [47], August Systems industrial control computers [56].

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ÐHoneycomb and diamond networks have been proposed as alternatives to mesh and torus architectures for parallel processing. When wraparound links are included in honeycomb and diamond networks, the resulting structures can be viewed as having been derived via a systematic pruning scheme applied to the links of 2D and 3D tori, respectively. The removal of links, which is performed along a diagonal pruning direction, preserves the network's node-symmetry and diameter, while reducing its implementation complexity and VLSI layout area. In this paper, we prove that honeycomb and diamond networks are special subgraphs of complete 2D and 3D tori, respectively, and show this viewpoint to hold important implications for their physical layouts and routing schemes. Because pruning reduces the node degree without increasing the network diameter, the pruned networks have an advantage when the degree-diameter product is used as a figure of merit. Additionally, if the reduced node degree is used as an opportunity to increase the link bandwidths to equalize the costs of pruned and unpruned networks, a gain in communication performance may result.

We introduce the notion of two-valued digit (twit) as a binary variable that can assume one of two different integer values. Posibits, or simply bits, in 0 1 and negabits in 1 0 , commonly used in two's-complement representations and ( ) encoding of binary signed digits, are special cases of twits. A weighted bit-set (WBS) encoding, which generalizes the two's-complement encoding by allowing one or more posibits and/or negabits in each radix-2 position, has been shown to unify many efficient implementations of redundant number systems. A collection of equally weighted twits, including ones with noncontiguous values (e.g., 1 1 or 0 2 ), can lead to wider representation range without the added storage and interconnection costs associated with multivalued digit sets. We present weighted twit-set (WTS) encodings as a generalization of WBS encodings, examine key properties of this new class of encodings, and show that any redundant number system (e.g., generalized signed-digit and hybrid-redundant systems), including those that are based on noncontiguous and/or zero-excluded digit sets, is faithfully representable by WTS encoding. We highlight this broad coverage by a tree chart having WTS representations at its root and various useful redundant representations at its many internal nodes and leaves. We further examine how highly optimized conventional components such as standard full/half-adders and compressors may be used for arithmetic on WTS-encoded operands, thus allowing highly efficient and VLSI-friendly circuit implementations. For example, focusing on the WBS-like subclass of WTS encodings, we describe a twit-based implementation of a particular stored-transfer representation which offers area and speed advantages over other similar designs based on WBS and hybrid-redundant representations.

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A scheme is proposed for representing 2's-complement binary numbers in which there are two least-significant bits (LSBs). Benefits of the extra LSB include making the number representation range symmetric (i.e. from 22 k21 to 2 k21 for k-bit integers), allowing sign change by simple bitwise logical inversion, facilitating multiprecision arithmetic and enabling the truncation of results in lieu of rounding. These advantages justify the added storage and interconnect costs stemming from the extra bit. Operation latencies show little or no change relative to conventional 2's-complement arithmetic, thus making double-LSB representation attractive.