Karnaugh mapThe Karnaugh map (KM or K-map) is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which was a rediscovery of Allan Marquand's 1881 logical diagram aka Marquand diagram but with a focus now set on its utility for switching circuits. Veitch charts are also known as Marquand–Veitch diagrams or, rarely, as Svoboda charts, and Karnaugh maps as Karnaugh–Veitch maps (KV maps).
Zhegalkin polynomialZhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials (полиномы Жегалкина), also known as algebraic normal form, are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient.
Tautology (logic)In mathematical logic, a tautology (from ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement.
Canonical normal formIn Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form (CDNF) or minterm canonical form, and its dual, the canonical conjunctive normal form (CCNF) or maxterm canonical form. Other canonical forms include the complete sum of prime implicants or Blake canonical form (and its dual), and the algebraic normal form (also called Zhegalkin or Reed–Muller). Minterms are called products because they are the logical AND of a set of variables, and maxterms are called sums because they are the logical OR of a set of variables.
Disjunctive normal formIn boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or (in philosophical logic) a cluster concept. As a normal form, it is useful in automated theorem proving. A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals. A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction.
Boolean algebraIn mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬.
Boolean functionIn 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 , where is known as the Boolean domain and is a non-negative integer called the arity of the function.
