Degrees of Unsolvability

Ambos-Spies, Klaus; Fejer, Peter A.

doi:10.1016/b978-0-444-51624-4.50010-1

Cited by 8 publications

(7 citation statements)

References 196 publications

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“…These sets are said to be intermediate or of intermediate degree, a degree being a collection of all sets of the same level of complexity. The question of whether such sets exist is commonly referred to as Post's problem and its solution required the invention of a new proof technique, the so-called priority method, which has since become one of the hallmarks of computability theory, see [40] for a slightly dated but excellent overview or [1] for a more recent account. Somewhat surprisingly, the method was discovered independently and almost simultaneously on different continents by Friedberg and Muchnik, see [12,29].…”

Section: Computation and Physicsmentioning

confidence: 99%

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Universality, Turing Incompleteness and Observers

Sutner¹

2012

A Computable Universe

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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 well-defined questions in mathematics [20]:The Entscheidungsproblem is solved when one knows a procedure by which one can decide in a finite number of operations whether a given logical expression is generally valid or is satisfiable. The solution of the Entscheidungsproblem is of fundamental importance for the theory of all fields, the theorems of which are at all capable of logical development from finitely many axioms.

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Section: Computation and Physicsmentioning

confidence: 99%

“…More recently, Ambos-Spies and Fejer in [1] are no less blunt in their complaint about the lack of natural intermediate problems:…”

Section: Computation and Physicsmentioning

confidence: 99%

Universality, Turing Incompleteness and Observers

Sutner¹

2012

A Computable Universe

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“…2 As illustrated in Fig. 1, ABM agents can represent a broad spectrum of entities ranging from passive physical materials governed by relatively simple dynamical methods, such as physical decay, to individual or group decision-making agents (DMAgents) with social capabilities.…”

Section: Abm Agentsmentioning

confidence: 99%

“…Upward-pointing (black) arrows denote "is a" relationships and downward-pointing (red) arrows denote "has a" relationships. 2 Object-oriented programming (OOP) represents "objects" as encapsulated bundles of data and methods. ABMs are now commonly implemented either directly in OOP languages or by means of toolkits based on OOP languages.…”

Section: Abm Agentsmentioning

confidence: 99%

Agent-based Modeling: The Right Mathematics for the Social Sciences?

Borrill¹,

Tesfatsion²

2011

The Elgar Companion to Recent Economic Methodology

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Abstract:This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research. The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely the duality of ABM: Real-world systems can not only be simulated with verisimilitude using ABM; they can also be efficiently and robustly designed and constructed on the basis of ABM principles.

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“…Indeed, embeddings of nondistributive lattices is closely tied to the structure of R. For instance, following the work in Downey and Lempp , it is shown in Ambos- Spies and Fejer [1996] that the degrees which are tops of N 5 (see Figure 3) are precisely the "noncontiguous" degrees which are not "locally distributive" in R.…”

Section: Introductionmentioning

confidence: 99%

Intervals without Critical Triples

Cholak¹,

Downey²,

Shore³

2017

Logic Colloquium '95

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Abstract. This paper is concerned with the construction of intervals of computably enumerable degrees in which the lattice M 5 (see Figure 1) cannot be embedded. Actually, we construct intervals I of computably enumerable degrees without any weak critical triples (this implies that M 5 cannot be embedded in I, see Section 2). Our strongest result is that there is a low 2 computably enumerable degree e such that there are no weak critical triples in either of the intervals [0, e] or [e, 0 ].

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Degrees of Unsolvability

Cited by 8 publications

References 196 publications

Universality, Turing Incompleteness and Observers

Universality, Turing Incompleteness and Observers

Agent-based Modeling: The Right Mathematics for the Social Sciences?

Intervals without Critical Triples

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