Boolean differential calculus

From testwiki
Jump to navigation Jump to search

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 each other.[1]

The Boolean differential calculus allows various aspects of dynamical systems theory such as automata theory on finite automata, Petri net theory,[2] and supervisory control theory (SCT), 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 use 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 (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 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 Template:Ill,[15] Christian Posthoff,[15] and Template:Ill[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 differential dx of a Boolean variable x models the relation

dx={0,no change of x1,change of x

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

The differential dx is binary and 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".
Retrieved from "https://en.wiki.beta.math.wmflabs.org/w/index.php?title=Boolean_differential_calculus&oldid=41850"