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3-satisfiability
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) asks whether there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the formula's variables can be consistently replaced by the values TRUE or FALSE to make the formula evaluate to TRUE. If this is the case, the formula is called ''satisfiable'', else ''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. SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT. There is no know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schaefer's Dichotomy Theorem
In computational complexity theory, a branch of computer science, Schaefer's dichotomy theorem, proved by Thomas Jerome Schaefer, states necessary and sufficient conditions under which a finite set ''S'' of relations over the Boolean domain yields polynomial-time or NP-complete problems when the relations of ''S'' are used to constrain some of the propositional variables. It is called a dichotomy theorem because the complexity of the problem defined by ''S'' is either in P or is NP-complete, as opposed to one of the classes of intermediate complexity that is known to exist (assuming P ≠ NP) by Ladner's theorem. Special cases of Schaefer's dichotomy theorem include the NP-completeness of SAT (the Boolean satisfiability problem) and its two popular variants 1-in-3 SAT and not-all-equal 3SAT (often denoted by NAE-3SAT). In fact, for these two variants of SAT, Schaefer's dichotomy theorem shows that their monotone versions (where negations of variables are not allowed) are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P Versus NP Problem
The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved. Here, "quickly" means an algorithm exists that solves the task and runs in polynomial time (as opposed to, say, exponential time), meaning the task completion time is bounded above by a polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is " P" or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be ''verified'' in polynomial time is "NP", standing for "nondeterministic polynomial time".A nondeterministic Turing machine can move to a state that is not determined by the previous state. Such a machine could solve an NP problem in polynomial time by falling into t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negation
In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. For example, if P is "Spot runs", then "not P" is "Spot does not run". An operand of a negation is called a ''negand'' or ''negatum''. Negation is a unary operation, unary logical connective. It may furthermore be applied not only to propositions, but also to notion (philosophy), notions, truth values, or interpretation (logic), semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). 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 f:\^k \to \, where \ is known as the Boolean domain and k is a non-negative integer called the arity of the function. In the case where k=0, the function is a constant element of \. A Boolean function with multiple outputs, f:\^k \to \^m with m>1 is a vectorial or ''vector-valued'' Boolean function (an S-box in symmetric cryptography). There are 2^ different Boolean functions with k arguments; equal to the number of different truth tables with 2^k entries. Every k-ary Boolean function can be expressed as a propositional formula in k variables x_1,...,x_k, and two propositional formulas a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Normal Form
In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. In automated theorem proving, the notion "''clausal normal form''" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals. Definition A logical formula is considered to be in CNF if it is a conjunction of one or more disjunctions of one or more literals. As in disjunctive normal form (DNF), the only propositional operators in CNF are or (\vee), and (\and), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for CNF: : ''CNF'' \, \to \, ''Disjunct'' \, \mid \, ''Disjunct'' \, \land \, ''CNF'' : ''Disjunct'' \, \to \, ''Literal'' \, \mid\, ''Literal'' ... [...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 that gives it useful properties for use in logic programming, formal specification, universal algebra 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 disjunctive 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. 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 follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithmics
In mathematics and computer science, an algorithm () is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an effective method, an algorithm can be expressed within a finite amount of space and time"Any classical mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
