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Undecidable Problems
Undecidable may refer to: * Undecidable problem in computer science and mathematical logic, a decision problem that no algorithm can decide, formalized as an undecidable language or undecidable set * "Undecidable", sometimes also used as a synonym of independent, something that can neither be proved nor disproved within a mathematical theory * Undecidable figure An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-dime ..., a two-dimensional drawing of something that cannot exist in 3d, such as appeared in some of the works of M. C. Escher See also * Decidable (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undecidable Problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run. Background A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns ''yes''. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence (mathematical Logic)
In mathematical logic, independence is the unprovability of a sentence from other sentences. A sentence σ is independent of a given first-order theory ''T'' if ''T'' neither proves nor refutes σ; that is, it is impossible to prove σ from ''T'', and it is also impossible to prove from ''T'' that σ is false. Sometimes, σ is said (synonymously) to be undecidable from ''T''; this is not the same meaning of " decidability" as in a decision problem. A theory ''T'' is independent if each axiom in ''T'' is not provable from the remaining axioms in ''T''. A theory for which there is an independent set of axioms is independently axiomatizable. Usage note Some authors say that σ is independent of ''T'' when ''T'' simply cannot prove σ, and do not necessarily assert by this that ''T'' cannot refute σ. These authors will sometimes say "σ is independent of and consistent with ''T''" to indicate that ''T'' can neither prove nor refute σ. Independence results in set theory Many inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undecidable Figure
An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-dimensional object. Impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline. Notable examples Notable impossible objects include: * Borromean rings — although conventionally drawn as three linked circles in three-dimensional space, any realization must be non-circular * Impossible cube — invented by M.C. Escher for ''Belvedere'', a lithograph in which a boy seated at the foot of the building holds an impossible cube. * Penrose stairs – created by Oscar Reutersvärd and later independently devised and popularised by Lionel Penrose and his mathematician son Roger Penrose. A variation on the Penrose triangle, it is a two-dimensional depiction of a staircase in which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
