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Parameter Definability in the Recursively Enumerable Degrees
"... The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly defin ..."
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The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...
1998], The Π3theory of the computably enumerable Turing degrees is undecidable
 Trans. Amer. Math. Soc
, 1998
"... Abstract. We show the undecidability of the Π3theory of the partial order of computably enumerable Turing degrees. Recursively enumerable (henceforth called computably enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets o ..."
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Abstract. We show the undecidability of the Π3theory of the partial order of computably enumerable Turing degrees. Recursively enumerable (henceforth called computably enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups. Putting the computably enumerable sets
Cellular Automata, Decidability and Phasespace
 FUNDAMENTA INFORMATICAE
"... Cellular automata have rich computational properties and, at the same time, provide plausible models of physicslike computation. We study decidability issues in the phasespace of these automata, construed as automatic structures over infinite words. In dimension one, slightly more than the first or ..."
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Cited by 3 (2 self)
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Cellular automata have rich computational properties and, at the same time, provide plausible models of physicslike computation. We study decidability issues in the phasespace of these automata, construed as automatic structures over infinite words. In dimension one, slightly more than the first order theory is decidable but the addition of an orbit predicate results in undecidability. We comment on connections between this “what you see is what you get” model and the lack of natural intermediate degrees.
Universality and Cellular Automata
"... Abstract. The classification of discrete dynamical systems that are computationally complete has recently drawn attention in light of Wolfram’s “Principle of Computational Equivalence”. We discuss a classification for cellular automata that is based on computably enumerable degrees. In this setting ..."
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Abstract. The classification of discrete dynamical systems that are computationally complete has recently drawn attention in light of Wolfram’s “Principle of Computational Equivalence”. We discuss a classification for cellular automata that is based on computably enumerable degrees. In this setting the full structure of the semilattice of the c.e. degrees is inherited by the cellular automata. 1 Intermediate Degrees and Computational Equivalence One of the celebrated results of recursion theory in the 20th century is the positive solution to Post’s problem: there are computably enumerable sets whose Turing degree lies strictly between ∅, the degree of any recursive set, and ∅ ′, the degree of the Halting set or any other complete computably enumerable set. The result was obtained independently and almost simultaneously by R. M. Friedberg and A. A. Muchnik, see [8, 14]. The method used in their construction of an intermediate degree is remarkable since it departs significantly from earlier attempts by Post and others to obtain such degrees by imposing structural
The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
Computational Processes, Observers and Turing Incompleteness
"... We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physicslike computation. These processes admit a natural classification into decidable, intermediate and comp ..."
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We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physicslike computation. These processes admit a natural classification into decidable, intermediate and complete, where intermediate processes correspond to recursively enumerable sets of intermediate degree in the classical setting. It is shown that a standard finite injury priority argument will not suffice to establish the existence of an intermediate computational process.
Extension of embeddings in the recursively enumerable degrees
"... The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an a ..."
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The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an algebraic insight into R assert either extension or nonextension of embeddings. We extend, strengthen, and unify these results and their proofs to produce complete and complementary criteria and techniques to analyze instances of extension and nonextension. We conclude that the full extension of embeddings problem is decidable.
Computational Processes and Incompleteness
, 906
"... We introduce a formal definition of Wolfram’s notion of computational process based on cellular automata, a physicslike model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury pri ..."
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We introduce a formal definition of Wolfram’s notion of computational process based on cellular automata, a physicslike model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process. 1 Computational Processes Degrees of unsolvability were introduced in two important papers by Post [21] and Kleene and Post [12]. The object of these papers was the study of the complexity of decision problems and in particular their relative complexity: how does a solution to one problem contribute to the solution of another, a notion that can be formalized in terms of Turing reducibility and Turing degrees. Post was particularly interested in the degrees of recursively enumerable (r.e.) degrees. The Turing degrees of r.e. sets together with Turing reducibility form a partial order and in fact an upper semilattice R. It is easy to see that R has least element /0, the degree of decidable sets, and a largest element /0 ′ , the degree of the halting set. Post asked whether there are any other r.e. degrees and embarked on a program to establish the existence of such an intermediate degree by constructing a suitable r.e. set. Post’s efforts produced a number of interesting ideas such as simple, hypersimple and hyperhypersimple sets but failed to produce
Universality, Turing Incompleteness and Observers
"... The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics a ..."
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The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics and construct a finite set of axioms that are strong enough to prove all proper theorems, but no more. Thus a proof of consistency and a proof of completeness were required. These proofs should be carried only by strictly finitary means so as to be beyond any reasonable criticism. As Hilbert pointed out [19], to carry out this project one needs to develop a better understanding of proofs as objects of mathematical discourse: To reach our goal, we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves. Furthermore, Hilbert hoped to find a single, mechanical procedure that would, at least in principle, provide correct answers to all welldefined questions
1 Introduction Degrees of Unsolvability
, 2006
"... Modern computability theory began with Turing [Turing, 1936], where he introduced ..."
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Modern computability theory began with Turing [Turing, 1936], where he introduced