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Provability (other) , also known as provable set
{{disambiguation ...
Provability or provable (and disprovability or disprovable) may refer to: * Provability logic, a modal logic * Provable prime, an integer that has been calculated to be prime * Provable security, computer system security that can be proved * Provably correct, correctness of an algorithm that can be proved * Provably total, function that can be proven to be computable See also * Proof (other) * Proof theory, a branch of mathematical logic * Recursively enumerable set In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provability Logic
Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. Examples There are a number of provability logics, some of which are covered in the literature mentioned in . The basic system is generally referred to as GL (for Gödel– Löb) or L or K4W (W stands for well-foundedness). It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4). Namely, the axioms of GL are all tautologies of classical propositional logic plus all formulas of one of the following forms: * Distribution axiom: * Löb's axiom: And the rules of inference are: * ''Modus ponens'': From ''p'' → ''q'' and ''p'' conclude ''q''; * Necessitation: From \vdash ''p'' conclude \vdash . History The GL model was pioneered by Robert M. Solovay in 1976. Since then, until his death in 1996, the prime inspirer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provable Prime
In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm. Boot-strapping techniques using Pocklington primality test are the most common ways to generate provable primes for cryptography. Contrast with probable prime, which is likely (but not certain) to be prime, based on the output of a probabilistic primality test. In principle, every prime number can be proved to be prime in polynomial time by using the AKS primality test. Other methods which guarantee that their result is prime, but which do not work for all primes, are useful for the random generation of provable primes. Provable primes have also been generated on embedded devices. See also *Primality test *Probable prime In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provable Security
Provable security refers to any type or level of computer security that can be proved. It is used in different ways by different fields. Usually, this refers to mathematical proofs, which are common in cryptography. In such a proof, the capabilities of the attacker are defined by an adversarial model (also referred to as attacker model): the aim of the proof is to show that the attacker must solve the underlying hard problem in order to break the security of the modelled system. Such a proof generally does not consider side-channel attacks or other implementation-specific attacks, because they are usually impossible to model without implementing the system (and thus, the proof only applies to this implementation). Outside of cryptography, the term is often used in conjunction with secure coding and security by design, both of which can rely on proofs to show the security of a particular approach. As with the cryptographic setting, this involves an attacker model and a model of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provably Correct
In theoretical computer science, an algorithm is correct with respect to a specification if it behaves as specified. Best explored is ''functional'' correctness, which refers to the input-output behavior of the algorithm (i.e., for each input it produces an output satisfying the specification). Within the latter notion, ''partial correctness'', requiring that ''if'' an answer is returned it will be correct, is distinguished from ''total correctness'', which additionally requires that an answer ''is'' eventually returned, i.e. the algorithm terminates. Correspondingly, to prove a program's total correctness, it is sufficient to prove its partial correctness, and its termination. The latter kind of proof (termination proof) can never be fully automated, since the halting problem is undecidable. For example, successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon—say an odd perfect number—it is quite easy to write a par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provably Total
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions. Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable''. This term has since come to be identified with the com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof (other)
Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a construct in proof theory * Mathematical proof, a convincing demonstration that some mathematical statement is necessarily true * Proof complexity, computational resources required to prove statements * Proof procedure, method for producing proofs in proof theory * Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects * Statistical proof, demonstration of degree of certainty for a hypothesis Law and philosophy * Evidence, information which tends to determine or demonstrate the truth of a proposition * Evidence (law), tested evidence or a legal proof * Legal burden of proof, duty to establish the truth of facts in a trial * Philosophic burden of proof, obligation on a party in a dispute to pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic that represents Mathematical proof, proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as Recursive data type, inductively-defined data structures such as list (computer science), lists, boxed lists, or Tree (data structure), trees, which are constructed according to the axioms and rule of inference, rules of inference of the logical system. Consequently, proof theory is syntax (logic), syntactic in nature, in contrast to model theory, which is Formal semantics (logic), semantic in nature. Some of the major areas of proof theory include structural proof theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |