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Euler diagram for P, NP, NP-complete, and NP-hard set of problems (excluding the empty language and its complement, which belong to P but are not NP-complete) Main article: NP-completeness To attack the P = NP question, the concept of NP-completeness is very useful.
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science.
Product = 2 × 2 × 2 × 2 × 3 × 2 × 2 × 3 × 3 × 5 = 8640. This also works for the greatest common divisor (gcd), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the gcd of 48 and 180 is 2 × 2 × 3 = 12.
However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise problems, etc. The model of computation: The most common model of computation is the deterministic Turing machine, but many complexity classes are based on non-deterministic Turing machines, Boolean circuits , quantum Turing ...
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
Euler diagram for P, NP, NP-complete, and NP-hard set of problems.The left side is valid under the assumption that P≠NP, while the right side is valid under the assumption that P=NP (except that the empty language and its complement are never NP-complete)
Folds can be regarded as consistently replacing the structural components of a data structure with functions and values. Lists, for example, are built up in many functional languages from two primitives: any list is either an empty list, commonly called nil ([]), or is constructed by prefixing an element in front of another list, creating what is called a cons node ( Cons(X1,Cons(X2,Cons ...
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable.It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings.