Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Proof theoryProof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
Proof (truth)A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.
Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
Proof calculusIn mathematical logic, a proof calculus or a proof system is built to prove statements. A proof system includes the components: Language: The set L of formulas admitted by the system, for example, propositional logic or first-order logic. Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics.
Constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Proof assistantIn computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.
Chernoff boundIn probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than exponential (e.g. sub-Gaussian). It is especially useful for sums of independent random variables, such as sums of Bernoulli random variables. The bound is commonly named after Herman Chernoff who described the method in a 1952 paper, though Chernoff himself attributed it to Herman Rubin.
Random number generationRandom number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular outcome sequence will contain some patterns detectable in hindsight but unpredictable to foresight. True random number generators can be hardware random-number generators (HRNGs), wherein each generation is a function of the current value of a physical environment's attribute that is constantly changing in a manner that is practically impossible to model.
Concentration inequalityIn probability theory, concentration inequalities provide bounds on how a random variable deviates from some value (typically, its expected value). The law of large numbers of classical probability theory states that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results show that such behavior is shared by other functions of independent random variables.
Full dress uniformFull dress uniform, also known as a ceremonial dress uniform or parade dress uniform, is the most formal type of uniforms used by military, police, fire and other public uniformed services for official parades, ceremonies, and receptions, including private ones such as marriages and funerals. Full dress uniforms typically include full-size orders and medals insignia. Styles tend to trace back to uniforms used during the 19th century, although the 20th century saw the adoption of mess dress-styled full-dress uniforms.
Hoeffding's inequalityIn probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963. Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small.
