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# Boolean Function

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Boolean Function

In mathematics, a finitary boolean function is a function of the form f : Bk ? B, where BÂ =Â {0,Â 1} is a boolean domain and where k is a nonnegative integer. In the case where kÂ =Â 0, the “function” is simply a constant element of B.

More generally, a function of the form f : X ? B, where X is an arbitrary set, is a boolean-valued function] (see below). If X = M = {1,Â 2,Â 3,Â â€¦}, then f is a binary sequence, that is, an infinite sequence of 0’s and 1’s. If X = [k] = {1,Â 2,Â 3,Â â€¦,Â k}, then f is a binary sequence of length k.

There are

2arg&isin;-→(:-4(x;font-size:12(x;“>2arg&isin;-→(:-4(x;font-size:12(x;“>k
such functions. These 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 S-box).

A boolean mask operation on boolean-valued functions combines values point-wise (for example, by XOR, or other boolean operators).

## Algebraic Normal Form

A boolean function can be written uniquely as a sum (XOR) of products (AND). This is known as the Algebraic Normal Form (ANF).

 f(xarg∈-→(:4(x;font-size:12(x;“>1 xarg∈-→(:4(x;font-size:12(x;“>2 lderiv(⋅)s xarg∈-→(:4(x;font-size:12(x;“>n) = aarg∈-→(:4(x;font-size:12(x;“>0 + aarg∈-→(:4(x;font-size:12(x;“>1xarg∈-→(:4(x;font-size:12(x;“>1 + aarg∈-→(:4(x;font-size:12(x;“>2xarg∈-→(:4(x;font-size:12(x;“>2 + lderiv(⋅)s + aarg∈-→(:4(x;font-size:12(x;“>nxarg∈-→(:4(x;font-size:12(x;“>n + aarg∈-→(:4(x;font-size:12(x;“>12xarg∈-→(:4(x;font-size:12(x;“>1xarg∈-→(:4(x;font-size:12(x;“>2 + aarg∈-→(:4(x;font-size:12(x;“>13xarg∈-→(:4(x;font-size:12(x;“>1xarg∈-→(:4(x;font-size:12(x;“>3 + lderiv(⋅)s + aarg∈-→(:4(x;font-size:12(x;“>n-1nxarg∈-→(:4(x;font-size:12(x;“>n-1xarg∈-→(:4(x;font-size:12(x;“>n + lderiv(⋅)s + aarg∈-→(:4(x;font-size:12(x;“>12lderiv(⋅)snxarg∈-→(:4(x;font-size:12(x;“>1xarg∈-→(:4(x;font-size:12(x;“>2lderiv(⋅)s xarg∈-→(:4(x;font-size:12(x;“>n

where
aarg&isin;-→(:4(x;font-size:12(x;“>0 aarg&isin;-→(:4(x;font-size:12(x;“>1 lderiv(⋅)s aarg&isin;-→(:4(x;font-size:12(x;“>12lderiv(⋅)sn &isin; 01arg&isin;-→(:-4(x;font-size:12(x;“>*
.

The values of the sequence
aarg&isin;-→(:4(x;font-size:12(x;“>0aarg&isin;-→(:4(x;font-size:12(x;“>1lderiv(⋅)saarg&isin;-→(:4(x;font-size:12(x;“>12lderiv(⋅)sn
can therefore also uniquely represent a boolean function. The algebraic degree of a boolean function is defined as the highest number of
xarg&isin;-→(:4(x;font-size:12(x;“>i
that appear in a product term. Thus
f(xarg&isin;-→(:4(x;font-size:12(x;“>1xarg&isin;-→(:4(x;font-size:12(x;“>2xarg&isin;-→(:4(x;font-size:12(x;“>3) = xarg&isin;-→(:4(x;font-size:12(x;“>1 + xarg&isin;-→(:4(x;font-size:12(x;“>3
has degree 1 (linear), whereas
f(xarg&isin;-→(:4(x;font-size:12(x;“>1xarg&isin;-→(:4(x;font-size:12(x;“>2xarg&isin;-→(:4(x;font-size:12(x;“>3) = xarg&isin;-→(:4(x;font-size:12(x;“>1 + xarg&isin;-→(:4(x;font-size:12(x;“>1xarg&isin;-→(:4(x;font-size:12(x;“>2xarg&isin;-→(:4(x;font-size:12(x;“>3
has degree 3 (cubic).

## Efficient representations

Boolean functions are often represented by sentences in propositional logic, but more efficient representations are binary decision diagrams (BDD), negation normal forms, or more generally by propositional directed acyclic graphs (PDAG).

## Boolean-Valued Function

A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain.

A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. A boolean variable x is a variable that takes its value from a boolean domain, as x ∈ B. A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, for example, 0 = false and 1 = true.

In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression.

In formal_semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value.

## References

• Brown,_Frank_Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

• Kohavi,_Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGrawâ€“Hill, 1970. 2nd edition, McGrawâ€“Hill, 1978.

• Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi ItÃ´ (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM.

### Equivalent Concepts

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Some content adapted from the Pseudopedia article “Boolean_function” under the GNU Free Documentation License.
[ last updated: 8:05pm EDT - Sat, Apr 07 2007 ]
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