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13
On Folk Theorems
, 1980
"... this paper is to refine this definition somewhat, adapting it to the purposes of the research community in computer science. Accordingly, we shall attempt to provide a reasonable definition of or, rather, criteria for folk theorems, followed by a detailed example illustrating the ideas. The latter e ..."
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this paper is to refine this definition somewhat, adapting it to the purposes of the research community in computer science. Accordingly, we shall attempt to provide a reasonable definition of or, rather, criteria for folk theorems, followed by a detailed example illustrating the ideas. The latter endeavor might take one of two possible forms. We could take a piece of folklore and show that it is a theorem, or take a theorem and show that it is folklore. As an example of the first form we could have shown that the statement P NP, which is folklore, is also a theorem. However, since we have resolved to introduce no new technical material in this paper, and moreover, since researchers in our community seem to be less familiar with folklore than with theorems, Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission
Systemic computation: A model of interacting systems with natural characteristics,” International journal of parallel, emergent and distributed systems
, 2007
"... Abstract. Natural systems provide unique examples of computation in a form very different from contemporary computer architectures. Biology also demonstrates capabilities such as adaptation, selfrepair and selforganisation that are becoming increasingly desirable for our technology. To address the ..."
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Abstract. Natural systems provide unique examples of computation in a form very different from contemporary computer architectures. Biology also demonstrates capabilities such as adaptation, selfrepair and selforganisation that are becoming increasingly desirable for our technology. To address these issues a new computer model and architecture with natural characteristics is presented. Systemic computation is Turing Complete; it is designed to support biological algorithms such as neural networks, evolutionary algorithms and models of development, and shares the desirable capabilities of biology not found in conventional architectures. Systemic computation may also be implemented using natural systems, enabling the potential for future computational analysis and control of biology.
On groups whose word problem is solved by a nested stack automaton. arXiv:math.GR/9812028
, 1998
"... Abstract. Accessible groups whose word problems are accepted by a deterministic nested stack automaton with limited erasing are virtually free. 1. Introduction. During the past several years combinatorial group theory has received an infusion of ideas both from topology and from the theory of formal ..."
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Abstract. Accessible groups whose word problems are accepted by a deterministic nested stack automaton with limited erasing are virtually free. 1. Introduction. During the past several years combinatorial group theory has received an infusion of ideas both from topology and from the theory of formal languages. The resulting interplay between groups, the geometry of their Cayley diagrams, and associated formal languages has led to several developments including
Every Computably Enumerable Random Real Is Provably Computably Enumerable Random
, 2009
"... We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefixfree Turing machine ” can be prov ..."
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We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefixfree Turing machine ” can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA. We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine U such that, based on U, PA cannot prove the randomness of its halting probability. The paper also includes a sharper form of the KraftChaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle.
Effectivizing Inseparability
, 1991
"... Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and show that ..."
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Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and show that the pairs of index sets effectively inseparable in Smullyan's sense are the same as those effectively inseparable in ours. In fact we characterize the pairs of index sets effectively inseparable in either sense thereby generalizing Rice's Theorem. For subrecursive index sets we have sufficient conditions for various inseparabilities to hold. For inseparability by sets in the same subrecursive class we have a characterization. The latter essentially generalizes Kozen's (and Royer's later) Subrecursive Rice Theorem, and the proof of each result about subrecursive index sets is presented "Rogers style" with care to observe subrecursive restrictions. There are pairs of sets effectively inseparab...
Linear Recursive Functions
"... With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turingcomplete computation model. However, erasing and copying do not need to be explicitly included ..."
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With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turingcomplete computation model. However, erasing and copying do not need to be explicitly included in Turingcomplete computation models: in this paper we show that the class of partial recursive functions that are syntactically linear (that is, partial recursive functions where no argument is erased or copied) is Turingcomplete.
Formal Languages for Linguists: Classical and Nonclassical Models
, 2001
"... The basics of classical formal language theory are introduced, as well as a wide coverage is given of some new nonstandard devices motivated in molecular biology, which are challenging traditional conceptions, are making the theory revived and could have some linguistic relevance. Only definitions a ..."
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The basics of classical formal language theory are introduced, as well as a wide coverage is given of some new nonstandard devices motivated in molecular biology, which are challenging traditional conceptions, are making the theory revived and could have some linguistic relevance. Only definitions and a few results are presented, without including any proof. The chapter can be profitably read without any special previous mathematical background. A long list of references completes the chapter, which intends to give a flavour of the field and to encourage young researchers to go deeper into it.
Not Multi, but ManyCore: Designing Integral Parallel Architectures for Embedded Computation
"... ..."
FUNDAMENTALS OF SIMULATION MODELING
"... We start with basic terminology and concepts of modeling, and decompose the art of modeling as a process. This overview of the process helps clarify when we should or should not use simulation models. We discuss some common missteps made by many inexperienced modelers, and propose a concrete approac ..."
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We start with basic terminology and concepts of modeling, and decompose the art of modeling as a process. This overview of the process helps clarify when we should or should not use simulation models. We discuss some common missteps made by many inexperienced modelers, and propose a concrete approach for avoiding those mistakes. After a quick review random number and random variate generation, we view the simulation model as a blackbox which transforms inputs to outputs. This helps frame the need for designed experiments to help us gain better understanding of the system being modeled. 1 BACKGROUND AND TERMINOLOGY We use models in an attempt to gain understanding and insights about some aspect of the real world. There are