Boolean Function

In mathematics, a finitary boolean function is a function of the form f : B^k ? B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. In the case where k = 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, 2, 3, …}, then f is a binary sequence, that is, an infinite sequence of 0’s and 1’s. If X = [k] = {1, 2, 3, …, k}, then f is a binary sequence of length k.

There are

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 .

The values of the sequence can therefore also uniquely represent a boolean function. The algebraic degree of a boolean function is defined as the highest number of that appear in a product term. Thus has degree 1 (linear), whereas 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, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.

• Korfhage,_Robert_R. (1974), Discrete Computational Structures, Academic Press, New York, NY.

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

• Minsky,_Marvin_L., and Papert,_Seymour,_A. (1988), Perceptrons, An Introduction to Computational Geometry, MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.

See also

Boolean algebra Boolean domain Boolean logic

Propositional calculus Truth table Zeroth order logic

Equivalent Concepts

• Characteristic function
• Indicator function
• Predicate, in some senses.
• Proposition, in some senses.

See Also

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• Algebra of Sets
• Boolean Algebra
• Boolean_algebra_topics

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• Boolean Logic
• Boolean Value

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• Logical Connective
• Truth Function
• Zeroth-Order Logic

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External links

• Boolean Planet — boolean functions in cryptography.

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Some content adapted from the Pseudopedia article “Boolean_function” under the GNU Free Documentation License.