Another line of attack for making the many-one reductions weaker is to only allow isomorphisms.

…

I didn’t read those papers yet, but … by Neil Immerman gives a nice survey of those results, which I read for a start.

Immerman’s survey presentation was from Jan 26, 2017. But the subject is actually much wider than I gathered from that survey presentation and the two papers I linked to. There are good recent survey papers which give a better impression how wide that subject really is:

The Isomorphism Conjecture for NP (23 pages)

https://www.cse.iitk.ac.in/users/manindra/survey/Isomorphism-Conjecture.pdf

by Manindra Agrawal – Dec 19, 2009

Investigations Concerning the Structure of Complete Sets (14 pages)

http://ftp.cs.rutgers.edu/pub/allender/isomorphism.pdf

by Eric Allender – 2014

(It should be clear that I didn’t really try to read those texts, and just browsed through them.)

Well, I worked through the survey “Parallel Algorithms” by G. Blelloch and B. Maggs in the meantime, and finished the first quarter of “Thinking in Parallel: Some Basic Data-Parallel Algorithms and Techniques” by U. Vishkin. The terms EREW, CREW, CRCW (E=exclusive, C=concurent, R=read, W=write) are explained, but their detailed relation to NC^{1}, L, NL, SAC^{1}, and AC^{1} is not discussed. I found such a discussion (including the interesting fact that L is in EREW^{1} by viewing L as directed tree reachability) in section 3 “Models of Parallel Computation” of

A Survey of Parallel Algorithms for Shared-Memory Machines (70 pages)

https://www2.eecs.berkeley.edu/Pubs/TechRpts/1988/5865.html

https://www2.eecs.berkeley.edu/Pubs/TechRpts/1988/CSD-88-408.pdf

by Richard M. Karp and Vijaya Ramachandran – March 1988

What I want to say is that I have read Buss’ paper in principle, but didn’t yet work thoroughly enough through section 7, and I don’t understand yet why Buss says (page 10): “However, we shall use the yet stronger property (c) of deterministic log time reducibility. Although it is not transitive, it is perhaps the strongest possible reasonable notion of reducibility.” Why is DLogTime not transitive?

I worked thoroughly through section 7 in the meantime, and I understand now why DLogTime is not transitive. I also learned to appreciate Emil Jeřábek’s detailed answer on the fine grained relation between LH and AC0 in this context. (Especially the part: “Additionally, we require that the ATM makes only *one* such query, at the very end of the computation, and returns the answer of the oracle as the result of the computation.”) And I found a later paper by S. Buss on the same subject:

Algorithms for Boolean Formula Evaluation and for Tree Contraction (19 pages)

https://www.math.ucsd.edu/~sbuss/ResearchWeb/Boolean3/paper.pdf

by S Buss – Oct 1991

Those papers would have been easy to find, if…

https://www.math.ucsd.edu/~sbuss/ResearchWeb/Boolean

https://www.math.ucsd.edu/~sbuss/ResearchWeb/Boolean2

https://www.math.ucsd.edu/~sbuss/ResearchWeb/Boolean3

I know you have written …, and it has been cited from time to time. I don’t think it is wrong, but I do think that Bell’s theorem is not sloppy. By this I mean that it is not a cheap mathematical trick, but an honest attempt to explain and expose the differences between expectation values arising from quantum mechanics and expectation values arising from typical local hidden parameters.

The “blind spot” for me is rather the laziness not trying to explain why Bohmian mechanics (and Everett’s original formulation of the many-worlds interpretation) are not great physical interpretations. And by laziness I mean that such explanations should stay concrete and point out specific shortcomings of these interpretations, and not just sweeping claims that no ontological interpretations are possible, or that talking about interpretations at all would be a waste of time and physically meaningless.

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