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Symmetric Boolean Function
In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones (or zeros) in the input.Ingo Wegener, "The Complexity of Symmetric Boolean Functions", in: ''Computation Theory and Logic'', ''Lecture Notes in Computer Science'', vol. 270, 1987, pp. 433–442 For this reason they are also known as Boolean counting functions. There are 2''n''+1 symmetric ''n''-ary Boolean functions. Instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an ''n''-variable symmetric Boolean function: the (''n'' + 1)-vector, whose ''i''-th entry (''i'' = 0, ..., ''n'') is the value of the function on an input vector with ''i'' ones. Mathematically, the symmetric Boolean functions correspond one-to-one with the functions that map ''n+1'' elements to two elements, f: \ \rightarrow \. Symmetric Boolean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NOR Gate
The NOR gate is a digital logic gate that implements logical NOR - it behaves according to the truth table to the right. A HIGH output (1) results if both the inputs to the gate are LOW (0); if one or both input is HIGH (1), a LOW output (0) results. NOR is the result of the negation of the OR operator. It can also in some senses be seen as the inverse of an AND gate. NOR is a functionally complete operation—NOR gates can be combined to generate any other logical function. It shares this property with the NAND gate. By contrast, the OR operator is ''monotonic'' as it can only change LOW to HIGH but not vice versa. In most, but not all, circuit implementations, the negation comes for free—including CMOS and TTL. In such logic families, OR is the more complicated operation; it may use a NOR followed by a NOT. A significant exception is some forms of the domino logic family. The original Apollo Guidance Computer used 4,100 integrated circuits (IC), each one containing o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Function
In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\left(x_2,x_1\right) for all x_1 and x_2 such that \left(x_1,x_2\right) and \left(x_2,x_1\right) are in the domain of f. The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k-tensors on a vector space V is isomorphic to the space of homogeneous polynomials of degree k on V. Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry. Symmetrization Given any function f in n variables wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horn Clause
In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951. Definition A Horn clause is a clause (a disjunction of literals) with at most one positive, i.e. unnegated, literal. Conversely, a disjunction of literals with at most one negated literal is called a dual-Horn clause. A Horn clause with exactly one positive literal is a definite clause or a strict Horn clause; a definite clause with no negative literals is a unit clause, and a unit clause without variables is a fact;. A Horn clause without a positive literal is a goal clause. Note that the empty clause, consisting of no literals (which is equivalent to ''false'') is a goal clause. These three kinds of Horn clauses are illustrated in the following propositional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majority Function
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. Representing true values as 1 and false values as 0, we may use the (real-valued) formula: :\langle p_1,\dots,p_n \rangle = \operatorname \left ( p_1,\dots,p_n \right ) = \left \lfloor \frac + \frac \right \rfloor. The "−1/2" in the formula serves to break ties in favor of zeros when the number of arguments ''n'' is even. If the term "−1/2" is omitted, the formula can be used for a function that breaks ties in favor of ones. 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 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas's Theorem
In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient \tbinom by a prime number ''p'' in terms of the base ''p'' expansions of the integers ''m'' and ''n''. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas. Statement For non-negative integers ''m'' and ''n'' and a prime ''p'', the following congruence relation holds: :\binom\equiv\prod_^k\binom\pmod p, where :m=m_kp^k+m_p^+\cdots +m_1p+m_0, and :n=n_kp^k+n_p^+\cdots +n_1p+n_0 are the base ''p'' expansions of ''m'' and ''n'' respectively. This uses the convention that \tbinom = 0 if ''m'' < ''n''. Proofs There are several ways to prove Lucas's theorem.Consequences * A binomial coefficient is divisible by a prime ''p'' if and only if at least one of the base ''p'' digits of ''n'' is greater than the corresponding digit of ''m''. * In particular, is |
Zhegalkin Polynomial
Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials (russian: 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. Exponents are redundant because in arithmetic mod 2, ''x''2 = ''x''. Hence a polynomial such as 3''x''2''y''5''z'' is congruent to, and can therefore be rewritten as, ''xyz''. __TOC__ Boolean equivalent Prior to 1927, Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, and so on. Zhegalkin showed that all Boolean operations could be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Normal Form
In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), '' Zhegalkin normal form'', or '' Reed–Muller expansion'' is a way of writing logical formulas in one of three subforms: * The entire formula is purely true or false: *: 1 *: 0 * One or more variables are combined into a term by AND (\and), then one or more terms are combined by XOR (\oplus) together into ANF. Negations are not permitted: : a \oplus b \oplus \left(a \and b\right) \oplus \left(a \and b \and c\right) * The previous subform with a purely true term: : 1 \oplus a \oplus b \oplus \left(a \and b\right) \oplus \left(a \and b \and c\right) Formulas written in ANF are also known as Zhegalkin polynomials and Positive Polarity (or Parity) Reed–Muller expressions (PPRM). Common uses ANF is a normal form, which means that two equivalent formulas will convert to the same ANF, easily showing whether two formulas are equivalent for automated theorem proving. Unlike other normal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Weight
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string, or the digit sum of the binary representation of a given number and the ''ℓ''₁ norm of a bit vector. In this binary case, it is also called the population count, popcount, sideways sum, or bit summation. History and usage The Hamming weight is named after Richard Hamming although he did not originate the notion. The Hamming weight of binary numbers was already used in 1899 by James W. L. Glaisher to give a formula for the number of odd binomial coefficients in a single row of Pascal's triangle. Irving S. Reed introduced a concept, equivalent to Hamming weight in the binary case, in 1954. Hamming weight is used in several disciplines including information theory, coding theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
XNOR Gate
The XNOR gate (sometimes XORN'T, ENOR, EXNOR or NXOR and pronounced as Exclusive NOR. Alternatively XAND, pronounced Exclusive AND) is a digital logic gate whose function is the logical complement of the Exclusive OR (XOR gate, XOR) gate. It is equivalent to the logical connective (\leftrightarrow) from mathematical logic, also known as the material biconditional. The two-input version implements logical equality, behaving according to the truth table to the right, and hence the gate is sometimes called an "equivalence gate". A high output (1) results if both of the inputs to the gate are the same. If one but not both inputs are high (1), a low output (0) results. The Boolean algebra, algebraic notation used to represent the XNOR operation is S = A \odot B. The algebraic expressions (A + \overline) \cdot (\overline + B) and A \cdot B + \overline A \cdot \overline B both represent the XNOR gate with inputs ''A'' and ''B''. Symbols There are logic gate#Symbols, two symbols for XN ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
