Boolean function

In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {−1,1}). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory.

A Boolean function takes the form k is a non-negative integer called the arity of the function. In the case where f:{0,1}k→{0,1}m with 1"}}'>m>1 is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography).

There are 2k entries.

Every x1,...,xk, and two propositional formulas are logically equivalent if and only if they express the same Boolean function.

The rudimentary symmetric Boolean functions (logical connectives or logic gates) are:

• NOT, negation or complement - which receives one input and returns true when that input is false ("not")
• AND or conjunction - true when all inputs are true ("both")
• OR or disjunction - true when any input is true ("either")
• XOR or exclusive disjunction - true when one of its inputs is true and the other is false ("not equal")
• NAND or Sheffer stroke - true when it is not the case that all inputs are true ("not both")
• NOR or logical nor - true when none of the inputs are true ("neither")
• XNOR or logical equality - true when both inputs are the same ("equal")

An example of a more complicated function is the majority function (of an odd number of inputs).

A Boolean function may be specified in a variety of ways:

• Truth table: explicitly listing its value for all possible values of the arguments
  • Marquand diagram: truth table values arranged in a two-dimensional grid (used in a Karnaugh map)
  • Binary decision diagram, listing the truth table values at the bottom of a binary tree
  • Venn diagram, depicting the truth table values as a colouring of regions of the plane

Algebraically, as a propositional formula using rudimentary Boolean functions:

• Negation normal form, an arbitrary mix of AND and ORs of the arguments and their complements
• Disjunctive normal form, as an OR of ANDs of the arguments and their complements
• Conjunctive normal form, as an AND of ORs of the arguments and their complements
• Canonical normal form, a standardized formula which uniquely identifies the function:
  • Algebraic normal form or Zhegalkin polynomial, as a XOR of ANDs of the arguments (no complements allowed)
  • Full (canonical) disjunctive normal form, an OR of ANDs each containing every argument or complement (minterms)
  • Full (canonical) conjunctive normal form, an AND of ORs each containing every argument or complement (maxterms)
  • Blake canonical form, the OR of all the prime implicants of the function

Boolean formulas can also be displayed as a graph:

• Propositional directed acyclic graph
  • Digital circuit diagram of logic gates, a Boolean circuit
  • And-inverter graph, using only AND and NOT

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