Boolean function

In mathematics and logic, a Boolean function is a function whose arguments, as well as the function itself, assume values from a two-element set (usually {0,1}).[1] As a result, it is sometimes referred to as a "switching function".

Not to be confused with Binary function.

A Boolean function takes the form ${\displaystyle f:\{0,1\}^{k}\to \{0,1\}}$, where ${\displaystyle \{0,1\}}$ is called a Boolean domain and ${\displaystyle k}$ is a non-negative integer called the arity of the function. In the case where ${\displaystyle k=0}$, the "function" is essentially a constant element of ${\displaystyle \{0,1\}}$.

Every ${\displaystyle k}$-ary Boolean function can be expressed as a propositional formula in ${\displaystyle k}$ variables ${\displaystyle x_{1},...,x_{k}}$, and two propositional formulas are logically equivalent if and only if they express the same Boolean function. There are ${\displaystyle 2^{2^{k}}}$ ${\displaystyle k}$-ary functions for every ${\displaystyle k}$.

Boolean functions in applications

A function that can be utilized to evaluate any Boolean output in relation to its Boolean input by logical type of calculations. Such functions play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers. The properties of Boolean functions play a critical role in cryptography, particularly in the design of symmetric key algorithms (see substitution box).

Boolean functions are often represented by sentences in propositional logic, and sometimes as multivariate polynomials over GF(2), but more efficient representations are binary decision diagrams (BDD), negation normal forms, and propositional directed acyclic graphs (PDAG).

In cooperative game theory, monotone Boolean functions are called simple games (voting games); this notion is applied to solve problems in social choice theory.

In order to optimize electronic circuits, Boolean functions can be minimized using the Quine–McCluskey algorithm or Karnaugh map.

See also

• Algebra of sets
• Balanced boolean function
• Boolean algebra
• Boolean algebra topics
• Boolean differential calculus
• Boolean-valued function
• Decision tree model
• Evasive Boolean function
• Indicator function
• Logical connective
• Symmetric Boolean function
• Pseudo-Boolean function
• Read-once function
• Signed set
• Truth function
• Truth table

References

1. Boolean function, Encyclopedia of Math

Further reading

• Crama, Y; Hammer, P. L. (2011), Boolean Functions : Theory, Algorithms, and Applications, Cambridge University Press, doi:10.1017/CBO9780511852008, ISBN 9780511852008.
• "Boolean function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Janković, Dragan; Stanković, Radomir S.; Moraga, Claudio (November 2003). "Arithmetic expressions optimisation using dual polarity property" (PDF). Serbian Journal of Electrical Engineering. 1 (71–80, number 1). Archived from the original (PDF) on 2016-03-05. Retrieved 2015-06-07.
• Bradford Henry Arnold (1 January 2011). Logic and Boolean Algebra. Courier Corporation. ISBN 978-0-486-48385-6.
• Mano, M. M.; Ciletti, M. D. (2013), Digital Design, Pearson.