Editing Novikov self-consistency principle (section)

==Time-loop logic==
{{Main|Time loop}}

Time-loop logic, coined by [[roboticist]] and futurist [[Hans Moravec]],<ref>{{cite web|url=http://www.frc.ri.cmu.edu/users/hpm/project.archive/general.articles/1991/TempComp.html|title=Time Travel and Computing|first=Hans|last=Moravec|year=1991|access-date=2008-07-28|author-link=Hans Moravec|archive-url=https://web.archive.org/web/20090129114503/http://www.frc.ri.cmu.edu/users/hpm/project.archive/general.articles/1991/TempComp.html|archive-date=2009-01-29|url-status=dead}}</ref> is a hypothetical system of computation that exploits the Novikov self-consistency principle to compute answers much faster than possible with the standard model of [[computational complexity theory|computational complexity]] using [[Turing machine]]s. In this system, a computer sends a result of a computation [[time travel|backwards through time]] and relies upon the self-consistency principle to force the sent result to be correct, provided the machine can reliably receive information from the future and provided the algorithm and the underlying mechanism are [[Formal verification|formally correct]]. An incorrect result or no result can still be produced if the time travel mechanism or algorithm are not guaranteed to be accurate.

A simple example is an [[iterative method]] algorithm. Moravec states:

{{quote| Make a computing box that accepts an input, which represents an approximate solution to some problem, and produces an output that is an improved approximation. Conventionally you would apply such a computation repeatedly a finite number of times, and then settle for the better, but still approximate, result. Given an appropriate negative delay something else is possible: [...] the result of each iteration of the function is brought back in time to serve as the "first" approximation. As soon as the machine is activated, a so-called "fixed-point" of F, an input which produces an identical output, usually signaling a perfect answer, appears (by an extraordinary coincidence!) immediately and steadily. [...] If the iteration does not converge, that is, if F has no fixed point, the computer outputs and inputs will shut down or hover in an unlikely intermediate state.}}

===Quantum computation with a negative delay===

Physicist [[David Deutsch]] showed in 1991 that this model<!-- details required --> of computation could solve NP problems in [[Time complexity#Polynomial time|polynomial time]],<ref name="Deutsch1991">{{cite journal | first=David | last=Deutsch | url= http://journals.aps.org/prd/abstract/10.1103/PhysRevD.44.3197 | title= Quantum mechanics near closed timelike lines | journal = Physical Review D | volume = 44 | issue = 10 | year=1991 | doi= 10.1103/PhysRevD.44.3197 | pages=3197–3217 | bibcode=1991PhRvD..44.3197D | pmid= 10013776| url-access= subscription }}</ref> and [[Scott Aaronson]] later extended this result to show that the model could also be used to solve [[PSPACE]] problems in polynomial time.<ref>{{cite journal|journal=Scientific American|date=March 2008 | first= Scott | last= Aaronson| title= The Limits of Quantum Computers |volume=298 |issue=3 |pages=68–69 |doi=10.1038/scientificamerican0308-62 |pmid=18357822 |bibcode=2008SciAm.298c..62A |url= http://www.scottaaronson.com/writings/limitsqc-draft.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.scottaaronson.com/writings/limitsqc-draft.pdf |archive-date=2022-10-09 |url-status=live | via= scottaaronson.com }}</ref><ref>{{cite journal | first1= Scott | last1= Aaronson | first2= John |last2= Watrous | url=http://www.scottaaronson.com/papers/ctc.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.scottaaronson.com/papers/ctc.pdf |archive-date=2022-10-09 |url-status=live | title=Closed Timelike Curves Make Quantum and Classical Computing Equivalent | journal = Proceedings of the Royal Society A | volume = 465 | year=2009 | issue = 2102 | doi= 10.1098/rspa.2008.0350 | pages= 631–647 | bibcode=2009RSPSA.465..631A|arxiv = 0808.2669 | s2cid= 745646 | via= scottaaronson.com}}</ref> Deutsch shows that quantum computation with a negative delay&mdash;backwards time travel&mdash;produces only self-consistent solutions, and the chronology-violating region imposes constraints that are not apparent through classical reasoning.<ref name="Deutsch1991" /> Researchers published in 2014 a simulation in which they claim to have validated Deutsch's model with photons.<ref name=RingbauerEtAl2014>{{cite journal| first1= Martin | last1= Ringbauer | first2= Matthew A. | last2= Broome | first3= Casey R. | last3= Myers | first4= Andrew G. | last4= White | first5= Timothy C. | last5= Ralph|title=Experimental simulation of closed timelike curves|journal=Nature Communications| date= 19 June 2014| volume= 5| doi= 10.1038/ncomms5145|arxiv = 1501.05014 |bibcode = 2014NatCo...5.4145R| pmid= 24942489| page= 4145| s2cid= 12779043 }}</ref> However, it was shown in an article by Tolksdorf and Verch that Deutsch's self-consistency condition can be fulfilled to arbitrary precision in any quantum system described according to relativistic [[quantum field theory]] even on spacetimes which do not admit closed timelike curves, casting doubts on whether Deutsch's model is really characteristic of quantum processes simulating closed timelike curves in the sense of [[general relativity]].<ref>{{cite journal
| last1        = Tolksdorf
| first1       = Juergen
| last2        = Verch
| first2       = Rainer
|date=2018
| title        = Quantum physics, fields and closed timelike curves: The D-CTC condition in quantum field theory
| journal      = Communications in Mathematical Physics
| volume       = 357
| issue        = 1
| pages        = 319–351
| arxiv        = 1609.01496
| bibcode      =2018CMaPh.357..319T 
| doi          =  10.1007/s00220-017-2943-5
| s2cid = 253751446
}}</ref>
In a later article,<ref>{{cite journal
| last1        = Tolksdorf
| first1       = Juergen
| authorlink1  = 
| last2        = Verch
| first2       = Rainer
| authorlink2  = 
|date=2021
| title        = The D-CTC condition is generically fulfilled in classical (non-quantum) statistical systems
| journal      = Foundations of Physics
| volume       = 51
| issue        = 93
| series       = 
| page = 93
| arxiv        = 1912.02301
| bibcode      = 2021FoPh...51...93T
| doi          =  10.1007/s10701-021-00496-z
| s2cid = 208637445
}}</ref>
the same authors show that Deutsch's CTC fixed point condition can also be fulfilled in any system
subject to the laws of classical [[statistical mechanics]], even if it is not built up by quantum systems. The authors conclude that hence, 
Deutsch's condition is not specific to quantum physics, nor does it depend on the quantum nature of a physical system so that it can be fulfilled.
In consequence, Tolksdorf and Verch argue that Deutsch's condition is not sufficiently specific to allow statements about time travel scenarios or their hypothetical realization by quantum physics.

=== Lloyd's prescription ===
An alternative proposal was later presented by [[Seth Lloyd]]<ref>{{cite journal
| last1        = Lloyd
| first1       = Seth
| authorlink1  = Seth Lloyd
| last2        = Maccone
| first2       = Lorenzo
| last3        = Garcia-Patron
| first3       = Raul
| last4        = Giovannetti
| first4       = Vittorio
| last5        = Shikano
| first5       = Yutaka
| last6        = Pirandola
| first6       = Stefano
| last7        = Rozema
| first7       = Lee A.
| last8        = Darabi
| first8       = Ardavan
| last9        = Soudagar
| first9       = Yasaman
| last10        = Shalm
| first10       = Lynden K.
| last11        = Steinberg
| first11       = Aephraim M.
| authorlink11  = Aephraim M. Steinberg
| date         = 27 January 2011
| title        = Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency
| journal      = Physical Review Letters
| volume       = 106
| issue        = 4
| pages        = 040403
| doi          = 10.1103/PhysRevLett.106.040403
|bibcode = 2011PhRvL.106d0403L
 |arxiv = 1005.2219
 | pmid=21405310
| s2cid = 18442086
}}</ref><ref>{{cite journal
| last1        = Lloyd
| first1       = Seth
| authorlink1  = Seth Lloyd
| last2        = Maccone
| first2       = Lorenzo
| last3        = Garcia-Patron
| first3       = Raul
| last4        = Giovannetti
| first4       = Vittorio
| last5        = Shikano
| first5       = Yutaka
| year         = 2011
| title        = The quantum mechanics of time travel through post-selected teleportation
| arxiv        = 1007.2615
| doi=10.1103/PhysRevD.84.025007
| volume=84
| issue=2
| pages = 025007
| journal=Physical Review D
| bibcode= 2011PhRvD..84b5007L
| s2cid = 15972766
}}</ref> based upon [[post-selection]] and path integrals. In particular, the path integral is over single-valued fields, leading to self-consistent histories.