P ≠ NP ∩ co-NP for Infinite Time Turing Machines

Deolalikar, Vinay; Hamkins, Joel David; Schindler, Ralf

doi:10.1093/logcom/exi022

Cited by 23 publications

(18 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Schindler defined P + = P f 0 where f 0 (x) = ω x 1 ck + 1. [14] (Thm 4) shows directly P f 0 = P ω ck 1 . Underlying this is perhaps the following Bounding Lemma:…”

Section: Complexity Of Ittm-computationmentioning

confidence: 98%

See 1 more Smart Citation

Transfinite machine models

Welch¹

2014

Turing's Legacy

View full text Add to dashboard Cite

“…Schindler defined P + = P f 0 where f 0 (x) = ω x 1 ck + 1. [14] (Thm 4) shows directly P f 0 = P ω ck 1 . Underlying this is perhaps the following Bounding Lemma:…”

Section: Complexity Of Ittm-computationmentioning

confidence: 98%

“…We refer the reader to this paper for the discussion and question of the P SP ACE f classes that arise. [28] showed that for almost all f , N P f = P f and later [14] showed that for many α P α = N P α ∩ co-N P α including those α that begin a gap in the clockable ordinals. (Such are admissible by [83] Thm.…”

Section: Complexity Of Ittm-computationmentioning

confidence: 99%

Transfinite machine models

Welch¹

2014

Turing's Legacy

View full text Add to dashboard Cite

“…Schindler proved P = NP for infinite time Turing machines in [Sch03], using methods from descriptive set theory to analyze the complexity of the classes P and NP. This has now been generalized in joint work [DHS05] to the following, where the class co-NP consists of the complements of sets in NP. This proof appears in [DHS05].…”

Section: Some Applications and Extensionsmentioning

confidence: 99%

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

Coskey¹,

Hamkins²

2013

Effective Mathematics of the Uncountable

Self Cite

View full text Add to dashboard Cite

ABSTRACT. We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class ∆ ∼ 1 2 in a satisfying way.Infinite time Turing machines fruitfully extend the operation of ordinary Turing machines into transfinite ordinal time and by doing so provide a robust theory of computability on the reals. In a mixture of methods and ideas from set theory, descriptive set theory and computability theory, the approach provides infinitary concepts of computability and decidability on the reals, which climb nontrivially into the descriptive set-theoretic hierarchy (at the level of ∆ 1 2 ) while retaining a strongly computational nature. With infinite time Turing machines, we have infinitary analogues of numerous classical concepts, including the infinite time Turing degrees, infinite time complexity theory, infinite time computable model theory, and now also the infinite time analogue of the theory of Borel equivalence relations under Borel reducibility.In this article, we shall give a brief review of the machines and their basic theory, and then explain in a bit more detail our recent application of infinite time computability to an analogue of Borel equivalence relation theory, a full account of which is given in [CH11]. The basic idea of this application is to replace the concept of Borel reducibility commonly used in that theory with forms of infinite time computable reducibility, and study the accompanying hierarchy of equivalence relations. This approach retains much of the Borel analysis and results, while also illuminating a part of the hierarchy of equivalence relations that seems beyond the reach of the Borel theory, including many highly canonical

show abstract

“…As the Web 2.0 age is dawning for mathematics, more and more mathematical development is moving online; not just publications. An example of this is the PolyMath site, where upon the recent announcement of a proof of P = N P , the mathematics community has organized itself in a WiKi and found a significant gap in the proof within two weeks; see [4]. The PlanetMath community which has collaborated on 8500 graduate-level encyclopedia articles over 10 years [20] is another, and also the Mizar community, who have formalized more than 60000 definitions, assertions, and proofs and have machine-checked them over the last 40 years.…”

Section: Introductionmentioning

confidence: 99%

Workflows for the Management of Change in Science, Technologies, Engineering and Mathematics

Autexier

David

Dietrich

et al. 2011

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Mathematical knowledge is a central component in science, engineering, and technology (documentation). Most of it is represented informally, and -in contrast to published research mathematics -subject to continual change. Unfortunately, machine support for change management has either been very coarse grained and thus barely useful, or restricted to formal languages, where automation is possible. In this paper, we report on an effort to extend change management to collections of semi-formal documents which flexibly intermix mathematical formulas and natural language and to integrate it into a semantic publishing system for mathematical knowledge. We validate the long-standing assumption that the semantic annotations in these flexiformal documents that drive the machine-supported interaction with documents can support semantic impact analyses at the same time. But in contrast to the fully formal setting, where adaptations of impacted documents can be automated to some degree, the flexiformal setting requires much more user interaction and thus a much tighter integration into document management workflows.

show abstract

P ≠ NP ∩ co-NP for Infinite Time Turing Machines

Cited by 23 publications

References 7 publications

Transfinite machine models

Transfinite machine models

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

Workflows for the Management of Change in Science, Technologies, Engineering and Mathematics

Contact Info

Product

Resources

About