__Quality assessment:__

On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3/5 is "mediocre pass", 4/5 is "good", and 5/5 is "excellent"): 4/5. Justification: Kmo had time to develop the necessary points to reasonable length.

All times in these blog "revision histories" are stated in UTC (Universal Coordinated Time/ Temps Universel Coordoné, a precisification of the old GMT, or "Greenwich Mean Time"), in the ISO-prescribed YYYYMMDDThhmmZ timestamping format. UTC currently leads Toronto civil time by 5 hours and currently lags Tallinn civil time by 2 hours.

On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3/5 is "mediocre pass", 4/5 is "good", and 5/5 is "excellent"): 4/5. Justification: Kmo had time to develop the necessary points to reasonable length.

__Revision history:__All times in these blog "revision histories" are stated in UTC (Universal Coordinated Time/ Temps Universel Coordoné, a precisification of the old GMT, or "Greenwich Mean Time"), in the ISO-prescribed YYYYMMDDThhmmZ timestamping format. UTC currently leads Toronto civil time by 5 hours and currently lags Tallinn civil time by 2 hours.

**20171214T1740Z/version 3.3.0: Kmo added remarks in a couple of places on the STOS, now making it clear (he had previously overlooked this) both***(a)*that this week's detailed STOS construction is more elaborate than it has to be, and*(b)*that the elaborate construction does secure an expository advantage, by making the 0s specially and dramatically sparse within the STOS's so-heavy population of 1s. - Kmo reserved the right to make further tiny, nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.3.1, 3.3.2, 3.3.3, ... .**20171213T2039Z/version 3.2.0: Kmo, in further checking, corrected what was in essence an error of substance, in the final words he put into the mouth of the Crown Prosecution during the trial of the Mathematical Logic Three. (It is not that the Mathematical Logic Three could be reasonably be asked for a proof that no IT-orderly sequence Olivia exists. No: the barrister must, rather, aver that they could reasonably be asked for a proof that no**__condition-conformant__IT-orderly sequence exists! One thing we do know by now is that there exist IT-orderly sequences. The STOS, at any rate, is such a sequence.) - Kmo reserved the right to make tiny, nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.2.1, 3.2.2, 3.2.3, ... .**20171213T2023Z/version 3.1.0: Kmo, in checking, found a silly error of substance. He had written, in his proof that there are infinitely many primes, "****So either***G*+ 1 is not itself prime, or" where what was needed was, rather, "So either G + 1 is itself prime, or". - Kmo reserved the right to make tiny, nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.1.1, 3.1.2, 3.1.3, ... .**20171213T1953Z/version 3.0.0: Kmo finished uploading a version in coherent full-sentences prose. He reserved the right to make tiny, nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.0.1, 3.0.2, 3.0.3, ... .****20171212T2042Z/version 2.0.0: Kmo managed to upload a duly polished finegrained outline. (To his alarm, he had found in preparing it that he needed to correct some of the thinking that went even into the coarsegrained outline.) He now proceeded to the first few steps in the approximately 3-hour task which is the conversion of such an outline into coherent full-sentences prose. He imagined that he would under pressure of other duties have to down tools by UTC=20171212T2130Z or so, and that he would manage to complete the job only over the coming 24 hours. Oi veh, oy vey zmir, oi veh gewalt.****20171212T1557Z/version 1.0.0: Kmo, struggling a bit with ideas, and so running rather slowly, managed to upload a coarsegrained outline. He hoped over the coming 3 hours to finish converting this into a finegrained outline, and thereupon to start converting his work into coherent full-sentences prose.**

[CAUTION: A bug in the

[CAUTION: A bug in the

__blogger__server-side software has in some past months shown a propensity to insert inappropriate whitespace at some points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual__blogger__"Posted by Toomas (Tom) Karmo at" appears. - The__blogger__software has also shown a propensity, at any rate when coupled with my erstwhile, out-of-date, Web-authoring uploading browser, to generate HTML that gets formatted in different ways on different downloading browsers. Some downloading browsers have sometimes perhaps not correctly read in the entirety of the "Cascading Style Sheets" (CSS) which on all ordinary Web servers control the browser placement of margins, sidebars, and the like. If you suspect CSS problems in your particular browser, be patient: it is probable that while some content has been shoved into some odd place (for instance, down to the bottom of your browser, where it ought to appear in the right-hand margin), all the server content has been pushed down into your browser in some place or other. - Finally, there may be__blogger__vagaries, outside my control, in font sizing or interlinear spacing or right-margin justification. - Anyone inclined to help with trouble-shooting, or to offer other kinds of technical advice, is welcome to write me via__Toomas.Karmo@gmail.com__.]
I have been slowed down this week not by struggles with emotions (in my particular often-recurring case, this involves depression) but by struggles with ideas. Now, at last, on a Tuesday and a Wednesday rather than on a Monday, I feel able to advance from messy multicoloured pencil jottings at the handwriting desk to the mechanics of a coarsegrained

**blogspot**outline, a finegrained**blogspot**outline (itself departing, alas, nontrivially in some of its ideas from the lately so-solid-seeming coarsegrained**blogspot**outline), and coherent full-sentences**blogspot**prose. - The outlining, I should perhaps remark for some conceivable, hypothetical, reader's hypothetical benefit, is made in the spirit of my 2003 essay "Logic Blueprints Take the Stress out of Writing Nonfiction", on my Web page http://www.metascientia.com/PNNN____lit/PNNN____logic_blueprints.html.
The struggle with ideas included the worried realization, at a couple of serendipitous moments away from handwriting desk and keyboard, both on Monday of this week and on Tuesday of this week, that while I had been duly pessimistic last week regarding Richard von Mises, I may have failed last week to be in a due-diligence way adequately agnostic regarding Martin-Löf with Schnorr and Levin (henceforth "M-L/S/L" - or, as I shall soon also say in a courtroom spirit, the "Mathematical Logic Three").

It will be recalled that the M-L/S/L condition, as I have at any rate managed in an imprecise way to grasp it, is the following: "Infinite bit sequence Roger is such that Roger's successive finite initial segments eventually become, and thereafter forever remain, Kolmogorov-disorderly". This is, to be sure, on my confessedly superficial and imprecise understanding, based only on my rapid glance at encyclopedia materials.

I still think I was right, in discussing the M-L/S/L condition here in "Part T" last week, to reject it as a necessary condition of randomness. Last week, I sketched the construction of a rather sinister infinite sequence Sam that fails the M-L/S/L condition, and yet does sink so low as to be random. My suggested Sam sank to that depth through featuring randomly timed "barbarian incursions" of 0s, disagreeably interrupting his long PAX ROMANA stretches of 1s. Those barbarian incursions proved so long as to render some of Sam's successive finite initial segments Kolmogorov-orderly, with his bouts of finite-initial-segment Kolmogorov-orderliness therefore never ceasing for good.

But now I think that I was wrong in "Part T" to advance the rather confident countervailing suggestion that the M-L/S/L, or "Mathematical Logic Three", condition is at least sufficient for (in other words, is at least a guarantee of) randomness. A more correct decision would have been the reserving of judgement, in an outright agnosticism, in which I would most fulsomely stress my ignorance concerning (perhaps mission-decisive?) technical details. It could be that an infinite bit sequence "Olivia", whose finite initial segments eventually become, and forever thereafter remain, Kolmogorov-disorderly, might nevertheless rise to the modest height of IT-orderlinesss. It is not that the Olivia I am about to cite rises to that magnificent "shimmering height" which is T-orderliness. Nyet/нет, tovarishch Polkovnik. But for mere IT-orderliness, conceivably da/да - at least, "conceivably" enough to make a circumspect agnosticism, rather than a brash recommendation of M-L/S/L as sufficient-for-randomness, the appropriate stance.

And indeed I am going to suggest here that far from having to construct some ingenious new Olivia, we might already possess one - namely, the now-familiar IT-orderly "Specially Troubling Initial Sequence", or STOS, as sketched in "Part R" (from 2017-11-20 or 2017-11-21).

****

Before going too far, it is advisable for me to specify the STOS in more detail than I presented in the "Part R" sketch. To my grief, I now find I did not set up my Tidy Mapping, on which my STOS relies, coherently. I made the mistake of naively specifying "dictionary order" in building the Tidy Mapping. The Tidy Mapping is not coherent unless it results in a straightforward listing of all Turing machines, in set-theory technical terms a listing of "order type omega" rather than of "order type omega plus omga plus omega plus ...". On a dictionary ordering of words in some language whose words are permitted to attain any finite length, we might, alas, already have an infinite number of entries starting with the first letter. (In English-dictionary terms, there might already be an infinite number of words starting with "A" - with

*, for instance, eventually followed by***ABE***, followed by***ABELARD***, and so on forever, as the individually finite-length A-words get longer and longer. Actual natural-language dictionaries are secure against such explosions-into-foreverland only because the total number of words starting with any one given letter is finite.)***ABELARDIAN**
So here is a repair in my hitherto inept "Tidy Mapping". The Turing machines are to be enumerated in a multi-volume dictionary, in which the alphabetically ordered, Turing-machine-encoding, words of just one letter (if there are any) make up the (finite) Volume 1, and the alphabetically ordered, Turing-machine-encoding, words of just two letters (if there are any) make up the (finite) Volume 2, and so on. Then every Turing machine is listed exactly once, one some page or other of some volume or other, with each volume (I reiterate) managing to be finite. (The individual volumes are finite because the underlying alphabet is itself finite.)

With this repair made, we do, at long last, possess a coherently specified "Tidy Mapping", mapping the entire set of Turing machines one-to-one (not just into, but, more strongly, onto) the positive integers. For each positive integer

*i*,*i*is paired with Machine X if and only if X is the*i*th machine to be mentioned in the infinitely many finite-length volumes of the dictionary, or Grand Multi-Volume Listing of All Turing Machines.
We next note the convenient school-mathematics fact that the primes are an infinite subset of the positive integers. (For suppose,

*per absurdum*, that the positive integers harbour a greatest prime, say*P*. Then form the "Grand Product",*G*, of all the primes.*G*is a very large number, namely the product 2 x 3 x 5 x 7 x 11 x 13 x 17 x ... x*P*. Consider, next, the number*G*+ 1. On our supposition,*G*+ 1 cannot be prime, since it is strictly greater than*P*. And yet 2 does not divide evenly into*G*+ 1, since the attempted division leaves a remainder of 1. Likewise, 3 does not divide evenly into*G*+ 1, since the attempted division leaves a remainder of 1, We proceed in this spirit for each prime in the supposedly complete enumeration 2, 3, 5, 7, 11, 13, 17, 19, ... ,*P*, finding in each and every case that the attempted division leaves a remainder of 1. So either*G*+ 1 is itself prime, or else it has some prime factor smaller than itself and nevertheless greater than*P*. In either case,*P*fails to be the last of the primes.)
Consider, then, from the overall Tidy Mapping of Turing Machines 1-to-1 onto the positive integers, just those positive integers

*f*such that*f*denotes a Turing machine that runs forever when started on an all-blanks tape. Let those various special positive integers be*f*-sub-1,*f-*sub-2,*f*-sub-3, ... . (They must themselves be an infinite collection,. If they were a finite collection, then some Turing machine would solve the Halting Problem.) Call the successive primes*p*-sub-1,*p*-sub-2,*p*-sub-3, ... (so, for instance,*p*-sub-1 is 2, and*p*-sub-2 is 3, and*p*-sub-3 is 5, and*p*-sub-6 is 13). Form the STOS, as an infinite sequence overwhelmingly dominated by 1s, with only the exceedingly rare 0 (except that there is no last 0) as follows:- The STOS starts with an unbroken run of exactly (
*p*-sub-1)-raised-to-the-power*-(f*-sub-1) 1s. - The STOS then has exactly one 0.
- Next, the STOS has an unbroken run of exactly (
*p*-sub-2)-raised-to-the-power-*(f*-sub-2) 1s. - The STOS then has exactly one 0.
- Next, the STOS has an unbroken run of exactly (
*p*-sub-3)-raised-to-the-power-*(f-*sub-3) 1s. - (And so on,.)

On a naive and superficial inspection, the STOS looks orderly, since its infinitely many 0s seem so rare within any finite initial STOS-segment. They in fact become so rare that for practical purposes we hardly ever see them. Here we have not Rome under dramatic onslaught from randomly timed barbarian invasions of increasing length, as with last week's Sam, but rather the pleasant-looking world of such well-bred pre-war English detectives as G.K. Chesterton's "Father Brown". Villains are present indeed, and yet they are very much lurking, generally very hard to see, amid the abounding rhododendrons or topiaries of the pre-war English Country House.

And the STOS (to reiterate a point already much stressed) is orderly - just not in the naively Turing sense. In place of being naively T-orderly, it is, more subtly, IT-orderly.

It might fairly be objected there that I have now made the STOS more elaborate than it had to be. (I neglected the objection in all version of this prior to version 3.3.0.) It would have been enough to construct the STOS as follows, avoiding my intricate detour into the sequence

It might fairly be objected there that I have now made the STOS more elaborate than it had to be. (I neglected the objection in all version of this prior to version 3.3.0.) It would have been enough to construct the STOS as follows, avoiding my intricate detour into the sequence

*p*-sub-1,*p*-sub-2,*p*-sub-3, ... of primes:- The STOS starts with an unbroken run of exactly
*f*-sub-1 1s. - The STOS then has exactly one 0.
- Next, the STOS has an unbroken run of exactly
*f*-sub-2 1s. - The STOS then has exactly one 0.
- Next, the STOS has an unbroken run of exactly
*f-*sub-3 1s. - (And so on,.)

****

Before going too far, it is also advisable for me to make a few remarks on variants of the Halting Problem.

A Turing machine functions as a fork with two tines. On the one hand is the overwriteable, so very mutable, infinite tape. This is used as input in cases where input is desired, as output in cases where output is desired, and in essentially all scenarios as scratchpad memory. On the other hand is the immutable finite internal program table. In searching for special, solvable, cases of the Halting Problem, it seems appropriate to direct attention to both tines.

I gather this week from a hasty glance at some Web resource such as Wikipedia that some work has been done in relation to the first-mentioned tine. People have been considering Turing-like machines which run a merely finite tape. The literature somehow describes this in terms of a tape with a pair of "end markers". Perhaps the literature term "Linear Bounded Automaton" is also somehow relevant. I gather that on some definition of a Turingish-machine-with-finite-tape, the Halting Problem has been found in some sense machine-soluble.

What, now, about the second tine? What happens if we confine attention to Turing machines with fewer than some bounding number

*u*("u" for "upper bound") of rows in the internal program table? In terms of the Turing-Davis formalism I introduced in "Part R", what happens if we confine attention to machines defined by a sequence of fewer than*u*5-tuples? Could there be a Turing machine H-sub-*u*which decides, in a finite number of steps, whether any Turing machine of table-of-5-tuples length strictly less than*u*eventually halts when started on an all-blanks tape?
I would be grateful, both in the context of the present discussion and in the interest of a general improved understanding, at my so-inadequate-desk, of mathematical logic, to learn from someone whether this second-tine problem has been studied. As always, e-mails should be sent to

__Toomas.Karmo@gmail.com__. One can of course also reply directly to this blog, under the**blogspot**commenting formalism, provided one respects the rules I set out in my first posting, back in 2016 April. (The rules provide that responses shall be published verbatim, whether or not I agree with them, provided that there is nothing like obscenity or libel, and provided also that posters specify in the body of their posting their true name, their municipality and nation of residence, and one of their duly functioning e-mail addresses.)
****

The best I can now offer, labouring under my various intellectual limitations, is a mock trial, in which a defending barrister argues for L-M/S/L (they stand, quivering in the courtroom dock, as the "Mathematical Logic Three"), and a Crown Prosecutor argues against them. The Defence has to suggest that the conceded orderliness in the STOS is accompanied by, happily, a failure of the condition proffered by the Mathematical Logic Three as guaranteeing randomness. The Prosecution, or "Crown", has to suggest that the proffered condition either holds for the (IT-orderly) STOS or holds for some tricky, and yet (IT-) orderly, Olivia other than the STOS, and that therefore the proffered condition is no guarantee of randomness.

In criminal trials, it is perhaps normal for the prosecution to open. The reason for this is in turn perhaps that it is the prosecution which is considered to carry the burden of proof. Where the burden of proof lies in the trial of the Mathematical Logic Three, I dare not say. For reasons of mere literary convenience, I do choose to have the Defence open proceedings.

****

The Defence begins in a commonsense way, by remarking that 0s are very rare indeed in the STOS (most especially as it has this week been specified, with an excursion into the infinite progression of primes

*p*-sub-1,

*p*-sub-2, ...), and become even progressively rarer as the inspected finite initial STOS-segments get progressively longer. Maybe this banal fact is enough, avers the Defence, eventually to make every finite initial segment compactly computable, in other words Kolmogorov-orderly. Since 1s are overwhelmingly predominant, it suffices (so this opening Defence argument runs) to direct any finite-initial-STOS-segment-generating Turing machine into a counted 1-writing loop, for the overwhelming greater part of its computation. Surely (the suggestion proceeds) the machine's internal program table will thereby be overwhelming shorter than the finite initial STOS-segment the machine is is tasked with writing.

But here much will turn on the precise technical meaning assigned by the Mathematical Logic Three to "overwhelmingly more". And I, alas, as a feeble

*amicus curiae*, am out of my depth. All I can say is that if my argument from last week, regarding the randomness of barbarian-troubled Sam, is a good one, then correspondingly the present Defence suggestion (namely, that the so-sparse-in-0s, and IT-orderly, STOS has the good fortune to fail the proffered M-L/S/L guarantee-of-randomness) looks good. However we proceed (say I, in my modest

*amicus curiae*capacity), we should at least couple an appraisal of this week's dramatically-0-sparse, and yet orderly, prime-numbers-steered STOS construction with an appraisal of last week's dramatically 0-rich, and as I last week thought and this week still think random, Sam construction.)

This is not quite all from the Defence. After the court's luncheon recess, the Defence additionally remarks that some relief might be had from the above-sketched "second-tine" variant on the Halting Problem. The so-rare 0s in the STOS help denote (so runs this additional, post-prandial, argument) a sequence of Turing machines which is in a happy way ordered. First comes the encoding,

*f*-sub-1 (by way of an unbroken string of exactly

*p*-sub-1-to-the-power-

*f*-sub-1 1s), of a short running-forever Turing machine. Next comes the encoding,

*f*-sub-2 (by way of a string of exactly

*p*-sub-2-to-the-power-

*f*-sub-2 1s), of a no-shorter running-forever Turing machine. The pattern continues without end. The successively denoted running-forever Turing machines form an endless parade, says the defending barrister, in which we never witness a forever-running machine with a table of some given length followed by a forever-running machine with a shorter table. (The table lengths of the individual successive forever-running machines in the endless parade, in other words, exhibit monotonic increase, even while perhaps not exhibiting strictly monotonic increase.)

If, the Defence now pleads (luncheon is well digested; tea-time approaches), for every positive integer

*u*the Halting Problem can be Turing-solved for all Turing machines featuring an internal table of length less than

*u*, then further relief is in prospect. Maybe finite initial segments of the STOS can be computed rather compactly. They can be computed (so concludes this post-prandial suggestion) not by the above-mentioned, pre-prandial, commonsense expedient ot sending their generating machines into long 1-writing counted loops, but in a mathematically more sophisticated way, by having their successive generating machines solve successive versions, for a monotonicially increasing positive-integers succession

*u*,

*u*-prime,

*u*-prime-prime, ..., of the "second tine" limited Halting Problem.

****

The Crown (perhaps on Day Two in the trial against the Mathematical Logic Three, after breakfast) might make much of the fact that "overwhelmingly more" is spelled out in perilous technical detail by the Mathematical Logic Three. Perhaps their spelling-out will trigger their downfall, by making it

*false*that the successively initial segments of the STOS eventually are "overwhelmingly" longer than the program tables of the successive machines successively generating them - whether these machines proceed naively, with counted 1-writing loops, or even in a more sophisticated way, on the strength of some solution to the second-tine limited Halting Problem?

And, as I have already remarked, the Crown can raise the spectre that even if the STOS has all its finite initial segments eventually becoming Kolmogorov-orderly, nevertheless some new IT-orderly Olivia, more cunning than the IT-orderly STOS, could succeed in satisfying the perilously proffered guarantee-condition of randomness. Perhaps the Crown will put on a bit of grumpy rhetoric, clutching its lapels as it growls:

*Just as, Ladies and Gentlemen of the Jury, we would appreciate a constructive proof of the allegation that some condition-conformant infinite sequence does exist - this, Ladies and Gentlemen of the Jury, was the wish so reasonably expressed by 'Doctor Karmo' last week, in 'Part T'", when he timidly hoped for a construction or a sketch-of-a-construction - so we really would, Ladies and Gentlemen, now appreciate a rigorous proof that no IT-orderly condition-conformant sequence, no fiendishly cunning condition-conformant "Olivia" of any IT-orderly kind, can exist.*

****

This is as far as I can safely go. At the moment, the verdict has to remain unknown, a circumspect agnosticism being my sole safe stance.

****

With this week's allotment of blogging time running out, I must finish. The homework assigned last week will have to be discussed in the next installment - perhaps next week, perhaps (since I should also perhaps try to blog on something like the theology of Christmas, or on Gregorian chant) only once Christmas is past.

Having already today, however, made some repairs to previous writeups, I should also make something of an improvement in the homework question as set last week. Let the reader draw a big black-ink rectangle, for the (perhaps empty, perhaps nonempty) set of all random infinite bit sequences (call it, this week and henceforth, "Randomia"). As in last week's presentation of this particular homework, let "Greenland" be that subset (perhaps empty, perhaps nonempty) of Randomia which comprises the infinite bit sequences S such that every T-selected infinite subsequence of S is random. As in last week's presentation of this particular homeowork, let "Redland" be that subset (perhaps empty, perhaps nonempty) of Randomia which comprises the infinite bit sequences S such that every IT-selected infinite subsequence of S is random. As in last week's presentation, call the (perhaps empty, perhaps nonempty) intersection of Redland and Greenland "Yule Country".

But now (this is what I in my unfeigned and shocking idiocy overlooked last week) give a name to that part of Randomia which is neither in Redland nor in Greenland. Since it is associated with no rainbow colour at all, call it "Drabland".

Use the terms "Extreme Redland" and "Extreme Greenland" in the same way as last week.

Randomia is thus partitioned into four mutually exclusive, jointly exhaustive subsets, namely Extreme Greenland ("eG"), Extreme Redland ("eR"), Yule Country ("Y"), and Drabland ("D"). Redland is, now as already last week, the union of eR and Y. Greenland is, now as already last week, the union of eG and Y.

Drawing Randomia myself with black ink, and then wielding my trusty erasable green and red pencils for the overlapping Greenland and Redland, and finally for visual vividness putting a dashed erasable-blue-pencil outline around Yule Country and a dashed black-pencil outline around Drabland, I get the following:

(The photo can, as is usual in

**blogger**publishing on

**blogspot**, be enlarged with a mouse-click. In my particular drawing, Redland occupies the upper half of Randomia, and Greenland the left half of Randomia.)

Formally, we now have exactly 16 sets to play with. At one logical extreme is Randomia. At another logical extreme is the empty set. In addition, there are the following fourteen sets:

- eG
- eR
- Y
- D
- union of eG and eR
- union of eG and Y
- union of eG and D
- union of eR and Y
- union of eR and D
- union of Y and D
- union of eR, Y, and D (complement in Randomia of eG)
- union of eG, Y, and D (complement in Randomia of eR)
- union of eG, eR, and D (complement in Randomia of Y)
- union of eG, eR, and Y (complement in Randomia of D)

*Upon conceding, for the same of argument, the (in fact possibly contestable) assumption that Randomia is nonempty, what, if anything, can then be asserted - or perhaps can then somehow more plausibly or less plausibly be conjectured - regarding the emptiness or nonemptiness of these fourteen assorted sets?*How light or grave, in other words, is the agnosticism under which we labour in the fourteenfold taxonomy, under the provisional governing assumption that Randomia is nonempty?

****

Before completely finishing for this week, I may as well direct attention to the availability of Web-based Turing-machine simulators, running displayed sample program tables, with the simulation authors in some or all cases even inviting the public to load and run their own program tables. (One can Google on, for example, the string

**turing machine simulator**.) One or more of these simulators might help make some of the ideas in recent installments vivid for some readers - possibly including those readers for whom Turing machines are somewhat novel, and whose appetites have been whetted by a visit to the "Mike Davey" YouTube clip which I mentioned at the end of "Part S" (on 2017-11-29 or so).

I am averse to spending much of my own scanty time exploring the various available simulators and comparing their respective merits. Some reader, however, may want to take on the task. To help such a hypothetical reader, I list, in an aprioristic spirit, some

*a-priori*-reasonable tests of simulator merit:

- Does the offered simulator in some way visually highlight the particular row of its program table that it is at any particular instant executing?
- Does the offered simulator make it possible to step manually, row by weary row, through a program table, upon repeatedly pressing something like a simulated doorbell-button?
- When manual stepping is turned off, does the offered simulator present a choice of running speeds (allowing the simulation operator, in other words, to run the clock either quickly or less quickly)?
- Does the offered simulator make it possible to start execution at any arbitrarily specified line in the program table?
- Does the offered simulator make it possible, given an already-populated tape, to select any desired arbitrary square as the Boot Square before starting the simulation?
- Is the offered simulator supplied with abundant, and abundantly commented, samples of program tables?
- Are there useful accompanying tutorial materials, or other useful accompanying writeups?

****

I may as well also - this is a small thing, I admit - report the availability on YouTube of vids showing the action of the "Digi-Comp 1" three-flipflop mechanical computer which I mentioned in "Part S":

- In a vid entitled "Digi-Comp 1 Plastic Computer", to a length of 2:16, from YouTube user "perkiert", is a clear view of both sides of the device. The respective views make it plain that the short horizontal white soft-plastic programming tubes in the front have a mechanically different role from the long white soft-plastic programming tubes in the back. (In the front, such a (short) tube arrests the inward movement of a spring-loaded vertical front-side metal rod. In the back, such a (long) tube does not arrest any rod movement, but instead engages with a spring-loaded vertical back-side metal rod in such a way as to cause a red-plastic flipflop to be pushed by the rod in one or the other of the two pertinent horizontal directions.) In my corner of the Web, the vid can be had through the URL https://www.youtube.com/watch?v=Qx5Iawpm5Kg. As of UTC-20171213T1707Z or so, it had garnered 8,155 views.
- In a vid entitled "Digi-Comp 1", to a length of 0:42, likewise from YouTube user "perkiert", is a clear demonstration that Digi-Comp 1 supports infinite loops. (The device is in this vid programmed to count 000, 001, 010, 011, 100, 101, 110, 111, 000, 001, ... .) In my corner of the Web, the vid can be had through the URL https://www.youtube.com/watch?v=mP3eNQ6ftsM. As of UTC=20171213T1707Z or so, it had garnered 9.979 views.

**[This is the end of the current blog posting.]**