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. Analysis Properties A Boolean function can have a variety of properties: Constant: Is always true or always false regardless of its arguments. Monotone: for every combination of argument values, changing an argument from false to true can only cause the output to switch from false to true and not from true to false. A function is said to be unate in a certain variable if it is monotone with respect to changes in that variable. Linear: for each variable, flipping the value of the variable either always makes a difference in the truth value or never makes a difference (a parity function). Symmetric: the value does not depend on the order of its arguments. Read-once: Can be expressed with conjunction, disjunction, and negation with a single instance of each variable. Balanced: if its truth table contains an equal number of zeros and ones. The Hamming weight of the function is the number of ones in the truth table. Bent: its derivatives are all balanced (the autocorrelation spectrum is zero) Correlation immune to mth order: if the output is uncorrelated with all (linear) combinations of at most m arguments Evasive: if evaluation of the function always requires the value of all arguments A Boolean function is a Sheffer function if it can be used to create (by composition) any arbitrary Boolean function (see functional completeness) The algebraic degree of a function is the order of the highest order monomial in its algebraic normal form Circuit complexity attempts to classify Boolean functions with respect to the size or depth of circuits that can