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Computability
About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.
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TL;DR: In this article, the authors define three operators which transform (multi-) representations into admissible ones in such a way that relative computability of functions is preserved, and show that the use of admissible representations rather than of non-admissible ones does not decrease the class of relatively computable functions.
Abstract: The property of admissibility of representations plays an important role in Type–2 Theory of Effectivity (TTE). TTE defines computability on sets with continuum cardinality via representations. Admissibility is known to be indispensable for guaranteeing reasonable effectivity properties of the used representations.
The question arises whether every function that is computable with respect to arbritrary representations is also computable with respect to closely related admissible ones. We define three operators which transform (multi–) representations into admissible ones in such a way that relative computability of functions is preserved. Thus the use of admissible (multi–) representations rather than of non–admissible ones does not decrease the class of relatively computable functions.
15 citations
08 Nov 2004
TL;DR: The Semantic Randomization Theorem is proved, which states that the complexity of an arbitrary self-referential functional is unbounded in the limit of k-limited fine-grained parallel processors.
Abstract: In this paper, we first apply traditional computability theory to prove that the randomization problem, as defined herein, is recursively unsolvable. We then move on to extend traditional computability theory for the case of k-limited fine-grained parallel processors (i. e., temporal relativity). Using this modification, we are able to prove the Semantic Randomization Theorem (SRT). This theorem states that the complexity of an arbitrary self-referential functional (i.e., implying representation and knowledge) is unbounded in the limit. Furthermore, it then follows from the unsolvability of the randomization problem that effective knowledge acquisition in the large must be domain-specific and evolutionary. It is suggested that a generalized operant mechanics will be the fixed-point randomization of a domain-general self-referential randomization. In practice, this provides for the definition of knowledge-based systems that can formally apply analogy in the reasoning process as a consequence of semantic randomization.
15 citations
TL;DR: Two applications of s-computability are discussed: one is to address several open problems regarding classical computable functions; the other is to conduct computable analysis of several important classes of partial differential equations.
15 citations
TL;DR: In this paper, the concept of reducibility in recursive function theory and computational complexity theory is applied to real numbers to investigate the notion of relative computability and relative complexity of real numbers.
15 citations
TL;DR: The result is used to show that the hierarchy of classes of languages accepted by pushdown automata based on the number of alternations collapses at the second level of the hierarchy.
Abstract: Languages accepted by alternating auxiliary pushdown automata using simultaneously a(n) alternations and s(n) space are shown to be members of the class of languages accepted by nondeterministic Turing machines using a(n) 2es(n) space for some c > 0. This result is used to show that the hierarchy of classes of languages accepted by pushdown automata based on the number of alternations collapses at the second level of the hierarchy. The power of alternation bounded pushdown automata without auxiliary storage is also investigated.
15 citations