I. 3.1 (Partial) Recursive Functions and Sets
VII.4 Open questions about the filter of pseudo-deep degrees
Theorem VII.2.4 and Corollary VII.2.5 give important structural information about the filter of pseudo-deep degrees. However, there remain open questions about that structure, especiallyFpseudo∖Fdeep.
Question VII.4.1. What is the cardinality ofFpseudo∖Fdeep? I.e., how many pseudo-deep degrees are there which aren’t deep degrees?
Corollary VII.2.5 puts constraints on∣Fpseudo∖Fdeep∣.
Proposition VII.4.2. ∣Fpseudo∖Fdeep∣ ∈ {1,ℵ0}.
Proof. Because degw(L) ∈Fpseudo∖Fdeep, we know 1≤ ∣Fpseudo∖Fdeep∣. Ew is countable, so ∣Fpseudo∖ Fdeep∣ ≤ ℵ0.
If 1 < ∣Fpseudo∖Fdeep∣ < ℵ0, then Fpseudo∖ (Fdeep∪ {degw(L)}) has a minimal element, and such a minimal element is a minimal element of Fpseudo∖ {degw(L)}since Fdeep is upward-closed, contradicting Corollary VII.2.5.
Currently, the only two pseudo-deep degrees known to not be deep are degw(L) and degw(LUAslow), though they are not known to be distinct.
Question VII.4.3. AreLand LUAslow weakly equivalent?
Something slightly stronger than asking whetherL≡wLUAslow is the following.
Question VII.4.4. Given a deep Π01 class P, does there exist a slow-growing order function p∶N→ (1,∞) such that LUA(p) ≤wP?
Proposition VII.4.5.
(a) An affirmative answer to Question VII.4.4 gives an affirmative answer to Question VII.4.3.
(b) An affirmative answer to Question VII.4.4 gives an affirmative answer to Question VI.6.3, i.e., for all slow-growing order functionsp∶N→ (1,∞)andq∶N→ (1,∞)there exists a slow-growing order function r∶N→ (1,∞)such that LUA(r) ≤wLUA(p) ∪LUA(q).
(c) An answer to Question VII.4.1 of ‘1’ gives an affirmative answer to Question VII.4.3.
An answer to Question VII.4.1 of ‘ℵ0’ suggests further structural questions about antichains inFpseudo∖ Fdeep and related properties.
Question VII.4.6.
(a) Do there exist weakly incomparable elements ofF ∖F ?
(b) Do there exist infinitely many pairwise weakly incomparable elements ofFpseudo∖Fdeep?
(c) Is degw(L)meet-irreducible? I.e., are there no pseudo-deep degreesp,qsuch that inf{p,q} =degw(L)?
In contrast, if ∣Fpseudo∖Fdeep∣ = ℵ0, thenFpseudo∖Fdeepcontains infinite chains.
One possible approach to answering Question VII.4.1 would be by showing that SC is not of deep degree, as it is of pseudo-deep degree and we know that SC≰wLUAslow by Theorem VI.4.3.
We observe that the known lattice theoretic properties available are not enough to determine the structure ofFpseudo∖Fdeep.
Remark VII.4.7. Some lattices ⟨P,≤⟩ and filters F1 ⊂−−F2 ⊆P which give some idea of how the boundary betweenFpseudo andFdeep might look include the following:
• Consider the lattice ⟨P,≤⟩ ∶= ⟨{S⊆ [0,1] ∣ ∣S∣ ≤ ℵ0} ∪ {[0,1]},⊇⟩and the filters F1∶= {S⊆ [0,1] ∣ ∣S∣ <
ℵ0} andF2=P. In this case,∣F2∖F1∣ = ℵ0and the minimum ofF2,[0,1], is meet-irreducible.
• Consider the lattice⟨P,≤⟩ ∶= ⟨{S⊆N∣ ∣S∣ < ℵ0∨ ∣N∖S∣ < ℵ0},⊇⟩and the filtersF1∶= {S⊆N∣ ∣S∣ < ℵ0} andF2=P. In this case,∣F2∖F1∣ = ℵ0 and the minimum ofF2,N, is not meet-irreducible.
REFERENCES
[1] Laurent Bienvenu and Ludovic Patey, Diagonally non-computable functions and fireworks, 2016.
[2] Laurent Bienvenu and Christopher P. Porter,DeepΠ01classes, The Bulletin of Symbolic Logic22(2016), no. 2, 249–286.
[3] Stephen Binns, A splitting theorem for the Medvedev and Muchnik lattices, Mathematical Logic Quar- terly49(2003), no. 4, 327–335.
[4] Stephen Binns, Richard A. Shore, and Stephen G. Simpson, Mass problems and density, Journal of Mathematical Logic16(2016), no. 02, 1650006.1 – 1650006.10, 10 pages.
[5] Cristian S. Calude, Ludwig Staiger, and Sebastiaan A. Terwijn,On partial randomness, Annals of Pure and Applied Logic138(2006), no. 1, 20–30.
[6] Rodney G. Downey and Denis R. Hirschfeldt, Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer-Verlag, New York, 2010, xxviii + 855 pages.
[7] Bruno Durand, Leonid Levin, and Alexander Shen, Complex tilings, Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (New York, NY, USA), STOC ’01, Association for Computing Machinery, 2001, pp. 732–739.
[8] Bruno Durand, Leonid A. Levin, and Alexander Shen, Complex tilings, J. Symbolic Logic73 (2008), no. 2, 593–613.
[9] Noam Greenberg and Joseph S. Miller, Diagonally non-recursive functions and effective Hausdorff di- mension, Bulletin of the London Mathematical Society43(2011), no. 4, 636–654.
[10] Kojiro Higuchi, WM. Phillip Hudelson, Stephen G. Simpson, and Keita Yokoyama, Propagation of partial randomness, Annals of Pure and Applied Logic165(2014), no. 2, 742–758.
[11] Denis R. Hirschfeldt, Andr´e Nies, and Frank Stephan, Using random sets as oracles, Journal of the London Mathematical Society75(2007), no. 3, 610–622.
[12] W. M. Phillip Hudelson, Mass problems and initial segment complexity, Journal of Symbolic Logic79 (2014), no. 1, 20–44.
[13] WM. Phillip Hudelson, Partial randomness and Kolmogorov complexity, Ph.D. thesis, 2013, URL:
https://etda.libraries.psu.edu/files/final_submissions/8355.
[14] Carl G. Jockusch and Robert I. Soare, Π01 classes and degrees of theories, Transactions of the American Mathematical Society173(1972), 35–56.
[15] Mushfeq Khan,Shift-complex sequences, Bull. Symbolic Logic19(2013), no. 2, 199–215.
[16] Mushfeq Khan and Joseph S. Miller,Forcing with bushy trees, The Bulletin of Symbolic Logic23(2017), no. 2, 160–180.
[17] Bjørn Kjos-Hanssen, Wolfgang Merkle, and Frank Stephan, Kolmogorov complexity and the recursion theorem, STACS 2006 (Berlin, Heidelberg) (Bruno Durand and Wolfgang Thomas, eds.), Springer Berlin Heidelberg, 2006, pp. 149–161.
[18] Joseph S. Miller,Two notes on subshifts, Proceedings of the American Mathematical Society140(2012), no. 5, 1617–1622.
[19] ,Assorted results in and about effective randomness, unpublished manuscript, 27 pages, 2020.
[20] Hartley Rogers, Godel numberings of partial recursive functions, J. Symbolic Logic 23 (1958), no. 3, 331–341.
[21] Hartley Rogers, Jr,Theory of Recursive Functions and Effective Computability, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, 1967, xix + 482 pages.
[22] A. Yu. Rumyantsev and M. A. Ushakov,Forbidden substrings, kolmogorov complexity and almost peri- odic sequences, STACS 2006 (Berlin, Heidelberg) (Bruno Durand and Wolfgang Thomas, eds.), Springer Berlin Heidelberg, 2006, pp. 396–407.
[23] Andrey Yu. Rumyantsev,Everywhere complex sequences and the probabilistic method, 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011, March 10-12, 2011, Dortmund, Germany (Thomas Schwentick and Christoph D¨urr, eds.), LIPIcs, vol. 9, Schloss Dagstuhl - Leibniz- Zentrum f¨ur Informatik, 2011, pp. 464–471.
[24] Stephen G. Simpson,Mass problems and randomness, Bulletin of Symbolic Logic11(2005), no. 1, 1–27.
[25] ,An extension of the recursively enumerable Turing degrees, Journal of the London Mathematical Society75(2007), no. 2, 287–297.
[26] Stephen G. Simpson, Computability, Unsolvability, Randomness, 2009, URL:http://www.personal.
psu.edu/t20/notes/cur.pdf. Last visited on 2021/02/23.