NDLI: On circuits and numbers

Content Provider	IEEE Xplore Digital Library
Author	Vuillemin, J.E.
Copyright Year	1968
Abstract	We establish new, yet intimate relationships between the 2-adic integers /sub 2/Z from arithmetics and digital circuits, both finite and infinite, from electronics. 1) Rational numbers with an odd denominator correspond to output only synchronous circuits. 2) Bit-wise 2-adic mappings correspond to combinational circuits. 3) Online functions /spl forall/n/spl isin/N,x/spl isinsub 2/Z:f(x)=f(xmodd2/sup n/)mod2/sup n/), correspond to synchronous circuits. 3) Continuous functions, /sub 2/Z/spl rarrsub 2/Z, correspond to circuits with output enable. The proof is obtained by constructing synchronous decision diagrams SDDs. They generalize to sequential circuits as classical BDD constructs do for combinational circuits. From simple identities over /sub 2/Z, we derive both classical and new bit-serial circuits for computing: {+,-,/spl times/,1/(1-2x), (1+8x)}. The correctness of each circuit directly follows from the 2-adic definition of the corresponding operator. All but the adders (+,-) above are infinite. Yet the use of reset signals reduces all previously infinite operators to finite circuits. The present work lays out the semantic basis of a new language for describing synchronous circuits. Language 2Z incorporates arithmetic synthesis for some of the above bit-serial operators, and for periodic binary constants (logic from chronograms). It also provides for the powerful deeply binding synchronous enable and reset operators, whose meaning is discussed.<>
Sponsorship	IEEE Computer Society Technical Committee on Distributed Process IEEE Computer Society Technical Committee on VLSI IEEE Technical Committee on Computer Architecture IEEE Computer Society
Starting Page	868
Ending Page	879
Page Count	12
File Size	1116185
File Format	PDF
ISSN	00189340
Volume Number	43
Issue Number	8
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	1994-08-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Hardware Combinational circuits Circuit synthesis Programmable logic arrays Silicon Digital arithmetic Digital circuits Sequential circuits Binary decision diagrams Adders
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Computational Theory and Mathematics Software Hardware and Architecture

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