Majority function

Last updated

In Boolean logic, the majority function (also called the median operator) is the Boolean function that evaluates to false when half or more arguments are false and true otherwise, i.e. the value of the function equals the value of the majority of the inputs.

Contents

Boolean circuits

Three-bit majority circuit Majority Logic.png
Three-bit majority circuit
Four-bit majority circuit Four-Bit Majority Circuit.png
Four-bit majority circuit

A majority gate is a logical gate used in circuit complexity and other applications of Boolean circuits. A majority gate returns true if and only if more than 50% of its inputs are true.

For instance, in a full adder, the carry output is found by applying a majority function to the three inputs, although frequently this part of the adder is broken down into several simpler logical gates.

Many systems have triple modular redundancy; they use the majority function for majority logic decoding to implement error correction.

A major result in circuit complexity asserts that the majority function cannot be computed by AC0 circuits of subexponential size.

Properties

For any x, y, and z, the ternary median operator ⟨x, y, z⟩ satisfies the following equations.

An abstract system satisfying these as axioms is a median algebra.

Ties

Most applications deliberately force an odd number of inputs so they don't have to deal with the question of what happens when exactly half the inputs are 0 and exactly half the inputs are 1. The few systems that calculate the majority function on an even number of inputs are often biased towards "0" – they produce "0" when exactly half the inputs are 0 – for example, a 4-input majority gate has a 0 output only when two or more 0's appear at its inputs. [1] In a few systems, the tie can be broken randomly. [2]

Monotone formulae for majority

For n = 1 the median operator is just the unary identity operation x. For n = 3 the ternary median operator can be expressed using conjunction and disjunction as xy + yz + zx. Remarkably this expression denotes the same operation independently of whether the symbol + is interpreted as inclusive or or exclusive or.

For an arbitrary n there exists a monotone formula for majority of size O(n5.3). This is proved using probabilistic method. Thus, this formula is non-constructive. [3]

Approaches exist for an explicit formula for majority of polynomial size:

See also

Notes

  1. ↑ Peterson, William Wesley; Weldon, E.J. (1972). Error-correcting Codes . MIT Press. ISBN   9780262160391.
  2. ↑ Chaouiya, Claudine; Ourrad, Ouerdia; Lima, Ricardo (July 2013). "Majority Rules with Random Tie-Breaking in Boolean Gene Regulatory Networks". PLOS ONE. 8 (7). Public Library of Science: e69626. doi: 10.1371/journal.pone.0069626 . PMC   3724945 .
  3. ↑ Valiant, Leslie (1984). "Short monotone formulae for the majority function". Journal of Algorithms. 5 (3): 363–366. doi:10.1016/0196-6774(84)90016-6.
  4. ↑ Amano, Kazuyuki (2018). "Depth Two Majority Circuits for Majority and List Expanders". 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). 117 (81). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik: 1–13. doi:10.4230/LIPIcs.MFCS.2018.81.
  5. ↑ Hoory, Shlomo; Magen, Avner; Pitassi, Toniann (2006). "Monotone Circuits for the Majority Function". Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science. Vol. 4110. Springer. pp. 410–425. doi:10.1007/11830924_38. ISBN   978-3-540-38044-3.

Related Research Articles

<span class="mw-page-title-main">Boolean algebra (structure)</span> Algebraic structure modeling logical operations

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.

In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.

In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size n is in NC if there exist constants c and k such that it can be solved in time O((log n)c) using O(nk) parallel processors. Stephen Cook coined the name "Nick's class" after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.

<span class="mw-page-title-main">Multiplexer</span> A device that selects between several analog or digital input signals

In electronics, a multiplexer, also known as a data selector, is a device that selects between several analog or digital input signals and forwards the selected input to a single output line. The selection is directed by a separate set of digital inputs known as select lines. A multiplexer of inputs has select lines, which are used to select which input line to send to the output.

<span class="mw-page-title-main">Combinational logic</span> Type of digital logic implemented by boolean circuits

In automata theory, combinational logic is a type of digital logic that is implemented by Boolean circuits, where the output is a pure function of the present input only. This is in contrast to sequential logic, in which the output depends not only on the present input but also on the history of the input. In other words, sequential logic has memory while combinational logic does not.

<span class="mw-page-title-main">Exclusive or</span> True when either but not both inputs are true

Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ. With multiple inputs, XOR is true if and only if the number of true inputs is odd.

In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A.

In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional.

<span class="mw-page-title-main">Boolean function</span> Function returning one of only two values

In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set. Alternative names are switching function, used especially in older computer science literature, and truth function, used in logic. Boolean functions are the subject of Boolean algebra and switching theory.

In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form (CDNF), minterm canonical form, or Sum of Products as a disjunction (OR) of minterms. The De Morgan dual is the canonical conjunctive normal form (CCNF), maxterm canonical form, or Product of Sums which is a conjunction (AND) of maxterms. These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.

Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them.

XOR gate is a digital logic gate that gives a true output when the number of true inputs is odd. An XOR gate implements an exclusive or from mathematical logic; that is, a true output results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results. XOR represents the inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is "must have one or the other but not both".

Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1. Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under some operations satisfying certain equations.

<span class="mw-page-title-main">Boolean circuit</span> Model of computation

In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length.

Logic optimization is a process of finding an equivalent representation of the specified logic circuit under one or more specified constraints. This process is a part of a logic synthesis applied in digital electronics and integrated circuit design.

<span class="mw-page-title-main">Circuit complexity</span> Model of computational complexity

In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that compute them. A related notion is the circuit complexity of a recursive language that is decided by a uniform family of circuits .

In mathematics, a median algebra is a set with a ternary operation satisfying a set of axioms which generalise the notions of medians of triples of real numbers and of the Boolean majority function.

In computational complexity theory, arithmetic circuits are the standard model for computing polynomials. Informally, an arithmetic circuit takes as inputs either variables or numbers, and is allowed to either add or multiply two expressions it has already computed. Arithmetic circuits provide a formal way to understand the complexity of computing polynomials. The basic type of question in this line of research is "what is the most efficient way to compute a given polynomial ?"

In theoretical computer science, the circuit satisfiability problem is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true. In other words, it asks whether the inputs to a given Boolean circuit can be consistently set to 1 or 0 such that the circuit outputs 1. If that is the case, the circuit is called satisfiable. Otherwise, the circuit is called unsatisfiable. In the figure to the right, the left circuit can be satisfied by setting both inputs to be 1, but the right circuit is unsatisfiable.

In 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 ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations.

References

Commons-logo.svg Media related to Majority functions at Wikimedia Commons

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.