Boolean domain

Concept in mathematical logic

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1},[1][2][3][4][5] or B . {\displaystyle \mathbb {B} .} [6][7]

The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean domain.

In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.

Generalizations

The Boolean domain {0, 1} can be replaced by the unit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x , {\displaystyle 1-x,} conjunction (AND) is replaced with multiplication ( x y {\displaystyle xy} ), and disjunction (OR) is defined via De Morgan's law to be 1 − ( 1 − x ) ( 1 − y ) = x + y − x y {\displaystyle 1-(1-x)(1-y)=x+y-xy} .

Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.

See also

  • Boolean-valued function
  • GF(2)

References

  1. ^ Dirk van Dalen, Logic and Structure. Springer (2004), page 15.
  2. ^ David Makinson, Sets, Logic and Maths for Computing. Springer (2008), page 13.
  3. ^ George S. Boolos and Richard C. Jeffrey, Computability and Logic. Cambridge University Press (1980), page 99.
  4. ^ Elliott Mendelson, Introduction to Mathematical Logic (4th. ed.). Chapman & Hall/CRC (1997), page 11.
  5. ^ Eric C. R. Hehner, A Practical Theory of Programming. Springer (1993, 2010), page 3.
  6. ^ Parberry, Ian (1994). Circuit Complexity and Neural Networks. MIT Press. pp. 65. ISBN 978-0-262-16148-0.
  7. ^ Cortadella, Jordi; et al. (2002). Logic Synthesis for Asynchronous Controllers and Interfaces. Springer Science & Business Media. p. 73. ISBN 978-3-540-43152-7.

Further reading

  • Steinbach, Bernd [in German], ed. (2014-04-01) [2013-09-25]. Recent Progress in the Boolean Domain (1 ed.). Newcastle upon Tyne, UK: Cambridge Scholars Publishing. ISBN 978-1-4438-5638-6. Retrieved 2019-08-04. [1] (455 pages) [2] (NB. Contains extended versions of the best manuscripts from the 10th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2012-09-19/21.)
  • Steinbach, Bernd [in German], ed. (2016-05-01). Problems and New Solutions in the Boolean Domain (1 ed.). Newcastle upon Tyne, UK: Cambridge Scholars Publishing. ISBN 978-1-4438-8947-6. Retrieved 2019-08-04. (480 pages) [3] (NB. Contains extended versions of the best manuscripts from the 11th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2014-09-17/19.)
  • Steinbach, Bernd [in German], ed. (2018-01-01). Further Improvements in the Boolean Domain (1 ed.). Newcastle upon Tyne, UK: Cambridge Scholars Publishing. ISBN 978-1-5275-0371-7. Retrieved 2019-08-04. [4] Archived 2019-08-04 at the Wayback Machine (536 pages) [5] (NB. Contains extended versions of the best manuscripts from the 12th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2016-09-22/23.)
  • Drechsler, Rolf; Soeken, Mathias, eds. (2020) [March 2019]. Written at Bremen, Germany. Advanced Boolean Techniques - Selected Papers from the 13th International Workshop on Boolean Problems (1 ed.). Cham, Switzerland: Springer Nature Switzerland AG. doi:10.1007/978-3-030-20323-8. ISBN 978-3-030-20322-1. S2CID 240782759. (vii+265+7 pages) [6] (NB. Contains extended versions of the best manuscripts from the 13th International Workshop on Boolean Problems (IWSBP 2018) held in Bremen, Germany on 2018-09-19/21.)
  • Drechsler, Rolf; Große, Daniel, eds. (2021-04-30). Recent Findings in Boolean Techniques - Selected Papers from the 14th International Workshop on Boolean Problems (1 ed.). Springer Nature Switzerland AG. ISBN 978-3-030-68070-1. (204 pages) [7] (NB. Contains extended versions of the best manuscripts from the 14th International Workshop on Boolean Problems (IWSBP 2020) held virtually on 2020-09-24/25.)