In fact, it becomes an NP-hard problem, which is basically means that it's practically uncomputable.
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I'm also super interested in the idea that the initial data of the universe could contain irrational or uncomputable numbers.
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Alan Turing discovered that the halting problem was uncomputable in 1936, when he devised his breakthrough mathematical model of a computer, now known as a Turing machine.
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Technically, the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses instead on the computation of deterministic, rather than random, uncomputable functions.
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Unfortunately, Solomonoff also proved that Solomonoff's induction is uncomputable. In fact, he showed that computability and completeness are mutually exclusive: any complete theory must be uncomputable. The proof of this is derived from a game between the induction and the environment. Essentially, any computable induction can be tricked by a computable environment, by choosing the computable environment that negates the computable induction's prediction.
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For instance, the level \Sigma^0_0=\Pi^0_0=\Delta^0_0 of the arithmetical hierarchy classifies computable, partial functions. Moreover, this hierarchy is strict such that at any other class in the arithmetic hierarchy classifies strictly uncomputable functions.
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If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula., chapter V. A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.
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Yuri Ivanovich Manin (; born February 16, 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable.
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Avoiding the huge data transfer requires a suitable (as stated in Overview) computational problem, whose description is short. Dziembowski et al. achieve this by constructing what they call an (m − δ, ε)-uncomputable hash function, which can be computed in quadratic time using memory of size m, but with memory of size m − δ it can be computed with at most a negligible probability ε.
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While a general proof can be given that almost all real numbers are normal (meaning that the set of non-normal numbers has Lebesgue measure zero), this proof is not constructive, and only a few specific numbers have been shown to be normal. For example, Chaitin's constant is normal (and uncomputable). It is widely believed that the (computable) numbers , , and e are normal, but a proof remains elusive.
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K. Vela Velupillai wrote of The unreasonable ineffectiveness of mathematics in economics. To him "the headlong rush with which economists have equipped themselves with a half-baked knowledge of mathematical traditions has led to an un-natural mathematical economics and a non-numerical economic theory." His argument is built on the claim that :mathematical economics is unreasonably ineffective. Unreasonable, because the mathematical assumptions are economically unwarranted; ineffective because the mathematical formalisations imply non-constructive and uncomputable structures.
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Recursion theory, also called computability theory, studies the properties of computable functions and the Turing degrees, which divide the uncomputable functions into sets that have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Recursion theory grew from the work of Rózsa Péter, Alonzo Church and Alan Turing in the 1930s, which was greatly extended by Kleene and Post in the 1940s. Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers.
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There are close relationships between the Turing degree of a set of natural numbers and the difficulty (in terms of the arithmetical hierarchy) of defining that set using a first-order formula. One such relationship is made precise by Post's theorem. A weaker relationship was demonstrated by Kurt Gödel in the proofs of his completeness theorem and incompleteness theorems. Gödel's proofs show that the set of logical consequences of an effective first-order theory is a recursively enumerable set, and that if the theory is strong enough this set will be uncomputable.
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It is known to be undecidable when 9 pairs are used (however, Stephen Wolfram (2002) suggested that it is also undecidable with just 3 pairs). The undecidability of his Post correspondence problem turned out to be exactly what was needed to obtain undecidability results in the theory of formal languages. In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's problem, stimulated much research.
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In Language and World (2020), Gaskin develops the theory of linguistic idealism and defends it against several objections. He addresses the problem that some mathematical entities, in particular uncomputable sets of real numbers, cannot be distinguished by language; he does this by developing a ‘split-level' version of linguistic idealism. In his approach all the basic entities of the world can be named in language, and all further entities, even if they cannot be named, can be derived from these basic entities by describable constructive operations. In Tragedy and Redress in Western Literature: A Philosophical Perspective (2018), Gaskin argues that not even the tragic aspects of life (such as pain and suffering) are beyond language, an objection commonly raised against the idea that language is omnicompetent to talk about and describe reality.
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Description numbers play a key role in many undecidability proofs, such as the proof that the halting problem is undecidable. In the first place, the existence of this direct correspondence between natural numbers and Turing machines shows that the set of all Turing machines is denumerable, and since the set of all partial functions is uncountably infinite, there must certainly be many functions that cannot be computed by Turing machines. By making use of a technique similar to Cantor's diagonal argument, it is possible exhibit such an uncomputable function, for example, that the halting problem in particular is undecidable. First, let us denote by U(e, x) the action of the universal Turing machine given a description number e and input x, returning 0 if e is not the description number of a valid Turing machine.
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Thus an oracle machine with a noncomputable oracle will be able to compute sets that a Turing machine without an oracle cannot. Informally, a set of natural numbers A is Turing reducible to a set B if there is an oracle machine that correctly tells whether numbers are in A when run with B as the oracle set (in this case, the set A is also said to be (relatively) computable from B and recursive in B). If a set A is Turing reducible to a set B and B is Turing reducible to A then the sets are said to have the same Turing degree (also called degree of unsolvability). The Turing degree of a set gives a precise measure of how uncomputable the set is. The natural examples of sets that are not computable, including many different sets that encode variants of the halting problem, have two properties in common: #They are recursively enumerable, and #Each can be translated into any other via a many-one reduction.
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Some agents can be assigned an explicit "goal function"; an agent is considered more intelligent if it consistently takes actions that successfully maximize its programmed goal function. The "goal function" encapsulates all of the goals the agent is driven to act on; in the case of rational agents, the function also encapsulates the acceptable trade-offs between accomplishing conflicting goals. (Terminology varies; for example, some agents seek to maximize or minimize a "utility function", "objective function", or "loss function".) The theoretical and uncomputable AIXI design is a maximally intelligent agent in this paradigm; however, in the real world, AI is constrained by finite time and hardware resources, and scientists compete to produce algorithms that can achieve progressively higher scores on benchmark tests with real-world hardware. Systems that are not traditionally considered agents, such as knowledge-representation systems, are sometimes subsumed into the paradigm by framing them as agents that have a goal of (for example) answering questions as accurately as possible; the concept of an "action" is here extended to encompass the "act" of giving an answer to a question.
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Radó's 1962 paper proved that if f: ℕ → ℕ is any computable function, then Σ(n) > f(n) for all sufficiently large n, and hence that Σ is not a computable function. Moreover, this implies that it is undecidable by a general algorithm whether an arbitrary Turing machine is a busy beaver. (Such an algorithm cannot exist, because its existence would allow Σ to be computed, which is a proven impossibility. In particular, such an algorithm could be used to construct another algorithm that would compute Σ as follows: for any given n, each of the finitely many n-state 2-symbol Turing machines would be tested until an n-state busy beaver is found; this busy beaver machine would then be simulated to determine its score, which is by definition Σ(n).) Even though Σ(n) is an uncomputable function, there are some small n for which it is possible to obtain its values and prove that they are correct. It is not hard to show that Σ(0) = 0, Σ(1) = 1, Σ(2) = 4, and with progressively more difficulty it can be shown that Σ(3) = 6 and Σ(4) = 13 .
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