Boolean differential calculus

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Template:Short description Template:Use dmy dates Boolean differential calculus (BDC) (German: Template:Lang (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

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 Template:Lang (Template:Lang, Template:Lang)[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 Template:Lang,[10][11][12][13][14] Marc Davio[11][12][13] and Template:Lang[13] in the 1970s formed the basics of BDC on which Template:Lang,[15] Template:Lang[15] and Template:Lang[16] further developed BDC into a self-contained mathematical theory later on.

Template:AnchorA complementary theory of Boolean integral calculus (German: Template:Lang) has been developed as well.[15][17]

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

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

Overview

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

References

Template:Reflist

Further reading

Template:Authority control

  1. ↑ H. Wehlan, Boolean Algebra in Encyclopedia of Mathematics
  2. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Scheuring_Wehlan_1991_Petri
  3. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Reed_1954
  4. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Muller_1954
  5. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Huffman_1958
  6. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Akers_1959
  7. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Talantsev_1959
  8. ↑ 8.0 8.1 8.2 Cite error: Invalid <ref> tag; no text was provided for refs named Sellers_Hsiao_Bearnson_1968_1
  9. ↑ 9.0 9.1 9.2 Cite error: Invalid <ref> tag; no text was provided for refs named Sellers_Hsiao_Bearnson_1968_2
  10. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Thayse_1970
  11. ↑ 11.0 11.1 Cite error: Invalid <ref> tag; no text was provided for refs named Thayse_1971
  12. ↑ 12.0 12.1 Cite error: Invalid <ref> tag; no text was provided for refs named Thayse_Davio_1973
  13. ↑ 13.0 13.1 13.2 Cite error: Invalid <ref> tag; no text was provided for refs named Davio_Deschamps_Thayse_1978
  14. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Thayse_1981
  15. ↑ 15.0 15.1 15.2 15.3 Cite error: Invalid <ref> tag; no text was provided for refs named Bochmann_Posthoff_1981
  16. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Bochmann_Steinbach_1991
  17. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Steinbach_Posthoff_2013_2
  18. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Scheuring_Wehlan_1991_DEDS
  19. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Yanushkevich_1998
  20. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Bochmann_2006_2008
  21. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Steinbach_Posthoff_2013_1
  22. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Steinbach_Posthoff_2017
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