Boolean differential calculus

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Boolean differential calculus (BDC) (German: Boolescher Differentialkalkül (BDK)) is a subject field of Boolean algebra discussing changes of Boolean variables and Boolean functions.

Boolean differential calculus concepts are analogous to those of classical differential calculus, notably studying the changes in functions and variables with respect to another/others.[1]

The Boolean differential calculus allows various aspects of dynamical systems theory such as

to be discussed in a united and closed form, with their individual advantages combined.

History and applications

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Originally inspired by the design and testing of switching circuits and the utilization of error-correcting codes in electrical engineering, the roots for the development of what later would evolve into the Boolean differential calculus were initiated by works of Irving S. Reed,[3] David E. Muller,[4] David A. Huffman,[5] Sheldon B. Akers Jr.[6] and A. D. Talantsev (A. D. Talancev, А. Д. Таланцев)[7] between 1954 and 1959, and of Frederick F. Sellers Jr.,[8][9] Mu-Yue Hsiao[8][9] and Leroy W. Bearnson[8][9] in 1968.

Since then, significant advances were accomplished in both, the theory and in the application of the BDC in switching circuit design and logic synthesis.

Works of André Thayse,[10][11][12][13][14] Marc Davio[11][12][13] and Jean-Pierre Deschamps[13] in the 1970s formed the basics of BDC on which Dieter Bochmann (de),[15] Christian Posthoff[15] and Bernd Steinbach (de)[16] further developed BDC into a self-contained mathematical theory later on.

A complementary theory of Boolean integral calculus (German: Boolescher Integralkalkül) has been developed as well.[15][17]

BDC has also found uses in discrete event dynamic systems (DEDS)[18] in digital network communication protocols.

Meanwhile, BDC has seen extensions to multi-valued variables and functions[15][19][20] as well as to lattices of Boolean functions.[21][22]

Overview

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Boolean differential operators play a significant role in BDC. They allow the application of differentials as known from classical analysis to be extended to logical functions.

The differentials dxi of a Boolean variable xi models the relation:

dxi={0,no change of xi1,change of xi

There are no constraints in regard to the nature, the causes and consequences of a change.

The differentials dxi are binary. They can be used just like common binary variables.

See also

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References

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  1. ^ H. Wehlan, Boolean Algebra in Encyclopedia of Mathematics
  2. ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (8 pages)
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  7. ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (10 pages)
  8. ^ a b c Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (8 pages)
  9. ^ a b c Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (21 of xviii+295 pages)
  10. ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (76 pages)
  11. ^ a b Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (18 pages)
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  14. ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (144 pages)
  15. ^ a b c d Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (397 pages) (NB. Per DNB-IDN 368893146 a Russian translation of this work was released in 1986.)
  16. ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (303 pages + 5.25-inch floppy disk)
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  20. ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (421 pages) Translation of: Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (452 pages)
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  22. ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (216 pages)

Further reading

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  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (14 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (462 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (9 pages) Translation of: Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (9 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (18 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (NB. Also: Chemnitz, Technische Universität, Dissertation.) (147 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (15 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (392 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (xxii+232 pages) [1] (NB. Per DNB-IDN 1010457748 this hardcover edition has been rereleased as softcover edition in 2010.)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (49 pages)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (24 of 153 pages)
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  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). with Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

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