[jnqlnxts] classification of extensive-form games by computational requirements

game length:

1. infinite-length games possible.  (if finite game state, then game must be loopy.)
2. games are guaranteed to be finite length but finite games of arbitrary length possible.  (e.g., players take turns naming smaller positive integers, Kirby-Paris hydra.)
3. maximum game length known at start.

this classification is practically fuzzy: maximum game length could be finite but so large as to effectively be infinite (future post hxfdvhup).  (e.g., loopy games with special rules prohibiting repetition.)  or, infinite or extremely long games are theoretically possible but require both players to be silly.

game state:

bounded or unbounded amount of information needed to describe game state.

common case of unbounded: loopy game has special rules regarding repeating a previous position (e.g., chess 3-fold repetition, go 囲碁 super-ko) so, as the game progresses, game state includes a growing list of positions already seen (in practice, a hash table).

this classification is practically fuzzy: theoretically bounded but so huge as to effectively be infinite.  chess and go 囲碁 have finite numbers of possible positions so keeping track of all of them classifying each as already seen or not requires only a finite bounded (but huge) amount of information.

the above classifications matter for computer implementations of games: how many bits wide should the move counter be to avoid overflow?

other famous classifications: perfect vs. imperfect information (players hide information).  complete vs. incomplete information (randomness or chance).

[tgtdtucz] Fourier series for ellipse in polar coordinates

below is the formula for an ellipse in polar coordinates, with eccentricity e and one focus at the origin (multiply by scaling factor to embiggen):

r = 1 / (1 - e*cos(t))

the formula has cosine in denominator.  we compute its Fourier transform to get a formula with the sines and cosines only in the numerators.  (only cos is needed.)

Mathematica:

FourierCosSeries[1/(1-e*Cos[t]), t, 7, Assumptions-> e < 1 && 0 <= e] //Timing

{245.35853, 1/Sqrt[1 - e^2] + (2*(-1 + 1/Sqrt[1 - e^2])*Cos[t])/e - (2*(-2 + e^2 + 2*Sqrt[1 - e^2])*Cos[2*t])/(e^2*Sqrt[1 - e^2]) + (2*(-4 + 4/Sqrt[1 - e^2] + e^2*(1 - 3/Sqrt[1 - e^2]))*Cos[3*t])/e^3 + (2*(8 + e^4 - 8*Sqrt[1 - e^2] + 4*e^2*(-2 + Sqrt[1 - e^2]))*Cos[4*t])/(e^4*Sqrt[1 - e^2]) - (2*(16 - 16/Sqrt[1 - e^2] + e^4*(1 - 5/Sqrt[1 - e^2]) + e^2*(-12 + 20/Sqrt[1 - e^2]))*Cos[5*t])/e^5 - (2*(e^6 + e^2*(48 - 32*Sqrt[1 - e^2]) + 6*e^4*(-3 + Sqrt[1 - e^2]) + 32*(-1 + Sqrt[1 - e^2]))*Cos[6*t])/(e^6*Sqrt[1 - e^2]) + (2*(e^6*(-7 + Sqrt[1 - e^2]) - 64*(-1 + Sqrt[1 - e^2]) - 8*e^4*(-7 + 3*Sqrt[1 - e^2]) + 16*e^2*(-7 + 5*Sqrt[1 - e^2]))*Cos[7*t])/(e^7*Sqrt[1 - e^2])}

what is a general formula for the coefficients (to get an infinite series)?  this has almost certainly already been solved, but I could not find such a formula published anywhere.  it seems like the kind of thing Joseph Fourier might have done himself.

investigate Fourier transforms of generalizations of the original formula with cosine in the denominator, for example, r(theta) = 1/(sum(n, A[n] * cos(n*theta + B[n]))).  what is the Fourier transform of the reciprocal of a Fourier transform?  previously, polar plots of regular (not reciprocal) Fourier series.  under what conditions is the shape convex?

[hgulalrr] subtracting times of day with units

time elapsed between:
4th day of the month 11:59:59 PM
6th day of the month 6:06:06 AM

$ units
You have: (6 day + 6 hr + 6 min + 6 s) - (4 day + 23 hr + 59 min + 59 s)
You want: day;hr;min;s
1 day + 6 hr + 6 min + 7 s

the semicolon syntax is documented in the units man page in the section entitled "UNIT LISTS: CONVERSION TO SUMS OF UNITS".

computing elapsed time requires more effort if endpoints are in different months, e.g., January 31, February 1 = January 32, February n = January (31 + n),... March 1 = February ( {28 or 29} + 1) = January (31 + {28 or 29} + 1).

incorrect if Daylight Saving Time changes within the interval; incorrect if endpoints are in different time zones; incorrect if a leap second occurs; incorrect if crossing the boundary between Gregorian and Julian calendars.

for such complicated cases, first convert each endpoint into a format which increases continuously and uniformly, then subtract, then convert with units to "day;hour;minute;second".

note well that Unix time does NOT increase uniformly: it stops at a leap second (or more precisely, it discontinuously decreases by one second).  subtracting two Unix times yields an elapsed time value that is incorrect by the number of leap seconds which occurred during the interval.

use the "right" counter instead of POSIX Unix time.  note that, as of Debian Trixie, you need to install the tzdata-legacy package to get the "right" counter value of a given date and time.

[gjjimaii] estimating pi with primes

we compute an approximation of Riemann zeta(2) = pi^2/6 using the Euler product formula over primes, then solve for pi:

? for(n=0, 10, z=1; forprime(p=1, 10^n, z*=(1-(1.0/p^2))); print("10^"n" ",sqrt(6/z)))

pi estimated from the product of all primes less than 10^n:

10^0 2.4494897427831780981972840747058913920
10^1 3.0935921676911454192536940842087145469
10^2 3.1387371950717209555231899287777680519
10^3 3.1413931847879282123372947308786365411
10^4 3.1415772351603451399115522305263190950
10^5 3.1415913934202453316750296375573376397
10^6 3.1415925471279879791648160810012226847
10^7 3.1415926443848380990491814377585237930
10^8 3.1415926527790129584805149679019884804
10^9 3.1415926535173431626391708456483210697
10^10 3.1415926535832448165410657025850891080

cpu time = 7min, 50,161 ms, real time = 7min, 50,326 ms.

each power of 10 yields slightly more than one more correct digit.  asymptotically, probably just one digit, so this is not an efficient method of calculating pi, though it is nice to have gotten the digits above without too much effort.

previously, estimating e with primes.

[pglmhtip] how to exit visual mode in ex

everyone wants to know the opposite direction, how to exit line-editing "ex mode" and return to visual mode in vi, so it is tough to find information for this direction.

the correct answer is "Q", but vim 8.2 (package vim-runtime) in Ubuntu 22.04 has remapped Q to something completely unrelated ("formatting") so Q is inaccessible.  (because everyone who hates line-editing also wants to configure vi never to accidentally enter "ex mode").  unlike emacs using M-x, there is no way to call a function by its long name if you just want to use it once.  to make Q exit visual mode for the rest of the session: ":unmap Q".  but of course that changes state, so you are not returning exactly to the same "ex" state that you started with: what if you then want to enter a visual mode with Q again having its remapped "formatting" behavior?  I do not know of a way to restore Q to whatever it was before (without knowing what it was before) (unlike emacs, there is no local-set-key, local-unset-key).  this is a mess.

to start "ex" (or "vi") with Q being available again: "ex -u NONE".  or maybe "ex -u NONE -N".  but "-u NONE" disables all initialization files, not just the Q remapping.  (-u NONE also makes vim behave more like original vi, a large set of differences in behavior.  -N modifies -u NONE to keep vim behaving like vim, which is good if you are used to vim not vi.)  I could not find any way to disable just the Q remapping from the command line.  this is a mess.

(if you want Q permanently available, you can create a user initialization file to override the system initialization file.)

the almost correct answer, even with "Q" remapped and inaccessible, is "gQ".  but "Q" and "gQ" are subtly different (in a way I do not understand).  when vim is invoked as "ex", it starts in "Q" mode, not "gQ" (according :h Q), and, as described above, there is no easy way to go back to Q after entering visual the first time.  this is a mess.

Q mode (but not gQ mode) has been removed from neovim (nvim).  if invoked as "ex", neovim presumably starts in gQ mode, but this is not documented anywhere I could find.

[fixtyxpa] walk may repeat the batter

if a baseball batter is awarded a base on balls, let the manager have the option of instead having the next batter ("batter on deck") take first base and the walked batter repeat the at bat.

not sure if it should be a temporary or permanent change in the batting order.

or, any pinch runner instead of batter on deck.

[ujzkawfu] 45 minutes from now, the human race could be extinct

two clocks, one labeled "Time now" and the other set and labeled "45 minutes into the future".  above them, a sign reading "45 minutes from now, the human race could be be extinct."  in between the clocks, a button with a cover over it labeled "LAUNCH".

more fancy: the T+45min clock stops once the launch button is pressed.  the launch button remains depressed -- it does not retract -- once pushed, indicating it has been pressed.  the T-0 clock runs for 45 minutes until its time agrees with the stopped T+45min clock, then it too stops.  the clocks resume and the button resets after 12 hours (for art).

too fancy: graphical display of missiles in flight.

(low earth orbit = 90 minutes, so max 45 minutes for an ICBM to reach anywhere on earth.  extinction could be quicker than that if we strategically position nukes all over the earth, which we do (sigh).  although it will be difficult to directly kill submarine crews, they are doomed to die from starvation or radiation poisoning after salted nuclear weapons.)

[jxtzkavi] magic hexagon base 6

it seems appropriate to present the unique nontrivial magic hexagon with its numbers expressed in base 6.  final number = 31 (base 6) = 19 (base 10).  magic sum = 102 (base 6) = 38 (base 10).

     3  25  30
  31  11   1  15
24   2   5  10  13
  20   4  12  22
    14  21  23

[typdfiqs] Amtrak Southwest Chief schedule change starting late May 2026

according to "Schedules" on https://www.amtrak.com/ , 2026-05-07:

Thursday, May 28, 2026: Train 3 departs CHI 2:25p arrives LAX 7:57a (43h 32m).  this is the last train on the old schedule.

Friday, May 29, 2026: Train 1003 (a special number) departs CHI 1:30p arrives LAX 8:12a (44h 42m)

Saturday, May 30, 2026: Train 3 departs CHI 1:30p arrives LAX 8:12a (44h 42m)

there is no information about this on "Service Alerts and Notices" https://www.amtrak.com/alert

[ynwwppvz] estimating e by counting primes

let n = 2^22 = 4194304.  there are p = primepi(n) = 295947 primes less than n.

using the prime number theorem n/log(n) ~= primepi(n), then e ~= n^(p/n) ~= 2.93 .

using n/(log(n) - 1) ~= primepi(n), then e ~= n^(p/(n+p)) ~= 2.732 .

(the same n was used previously computing e with primorial.  using the values of primes as we did then yielded slightly more precision (2.717) than using just their count as we do now.)

the typical way of computing fractional powers is with log and exp, but if you have such functions, then e = exp(1).

we chose n to be a power of 2 to permit easily computing the fractional power ^(p/n) in the first approximation without needing log and exp: the 1/n power in the can be computed by repeated square roots.  n^(p/n) = (n^p)^(1/n).  compute n^p first by augmenting machine precision floating point with an exponent that can become much larger than provided by the machine type.  the only operations one needs are multiplication and square root.  example implementation in Haskell.  there are some bits (pun!) of symmetry between exponentiation by squaring for the numerator of p/n followed by repeated square root for the denominator.

for the second approximation n^(p/(n+p)), one could have attempted to choose a different n such that n+p is a power of two.  but 3916608 + primepi(3916608) = 2^22-1 and 3916609 + primepi(3916609) = 2^22+1 (because 3916609 is prime), so it would have required some fudging.

better is to approximate the fractional exponent p/(n+p) as round(2^64 * p/(n+p))/2^64 .  (64 bits should be enough for everyone.)  for n = 2^22, p = 295947, this works out to to 1215802539408625636 / 2^64.

or, use Newton's method to compute the (n+p)th root of x^p.

comparing with methods that don't involve primes, but inputs of a similar scale:

e ~= (1+1/n)^n : for n=2^22, e ~= 2.718281504 .

sum(n=0, 10, 1.0/n!) ~= 2.71828180 , choosing 10 because 10! is the largest factorial less than 2^22 .

[esowrowp] speaking leading zeroes

zeroty ten twenty thirty forty fifty sixty seventy eighty ninety

0 zero
1 one
00 zeroty
01 zeroty one
000 zero hundred
001 zero hundred one
0000 zero thousand
0001 zero thousand one
00000 zeroty thousand
00001 zeroty thousand one
01000 zeroty one thousand
000000 zero hundred thousand

or "ohty" instead of zeroty to avoid similarity between zero and zeroty?  but ohty sounds similar to eighty.

or "aughty"?  also sounds similar to eighty.

or "noughty"?  context probably distinguishes it from naughty.

"nohty" ("notey")?

typically digit strings that are not numbers should be read as digit strings, but place value indicators inserted into the string serve as useful signposts.  in this case, also consider speaking internal zeroes: 

000000 zero hundred zeroty zero thousand, zero hundred zeroty zero
20 twenty zero

zip codes

10 "ten" remains irregular.  though could we do something like 10 one-ty zero, 20 two-ty zero (future post knlwtvtb).

[pfbjvubj] environment variables modifying Makefile variables

consider the following GNU Makefile and invocations in the bash shell:

C = c
Q ?= q
.PHONY:default
default:
        @echo C=$(C)
        @echo Q=$(Q)

$ make
C=c
Q=q

$ make C=notc
C=notc
Q=q

$ make Q=notq
C=c
Q=notq

$ C=notc make
C=c
Q=q

$ Q=notq make
C=c
Q=notq

either type of assignment can be overridden by specifying a new value after "make" on the command line, but only variables assigned with ?= can also be overridden by the environment.  I've deliberately done both: makefiles in which make communicates with its caller via environment variables, and makefiles in which make should ignore any identically named environment variables.

Next, experimenting with += :

P += p
.PHONY:default
default:
        @echo P=$(P)

$ make
P=p

$ make P=prefix
P=prefix

$ P=prefix make
P=prefix p

like with ?= , only user-specified values in the environment can be appended to with += .  this is especially useful is the following idiom: if a Makefile specifies "CFLAGS += necessary flags", then the user can add additional flags to CFLAGS just for one build without editing the Makefile.  for += to work keeping the necessary CFLAGS specified in the Makefile, the user must specify CFLAGS in the environment as demonstrated in the last example above, not in a command-line argument to make.

an attacker could surreptitiously set CFLAGS="something evil" in your environment, thereby affecting all builds that use exclusively CFLAGS += , but if you are worried about an attacker who can set and export arbitrary variables in your environment, you probably have bigger problems.

future work: the "override" makefile directive.

[obaxedbg] which half point

rate chess players by how often they turn a drawn position into a lost one, and by how often they turn a winning position into a drawn one.  (turning a winning position into a lost one counts as one of each.)  the best at avoiding bad moves of the first type might not be the best at avoiding the other type.

devilish detail of determining ground truth of a position.  maybe play out by an engine.

devilish detail of normalization: some positions are easier to make mistakes in, and some players probably (as a result of play style) encounter such positions more frequently.

skill difference might matter: a subtle (but still half-point) mistake against a beginner might not matter, because the beginner might not be able to exploit it, or might be expected to blunder it back later in the game.

having a high rate at either or both of these categories might not equate to bad results -- perhaps you have a playing style that, even though you make a lot of mistakes, you induce your opponents to make even more.

having a low rate of making mistakes might not equate to good results: only one mistake per game is enough to lose every game.

opposite measure: how often a player induces an opponent to make each of the half-point mistakes.  similar devilish details, including normalization.

[rgvdheir] random convex polytope

first, create a random smooth shape by picking random weights for the first few spherical harmonics and summing them.  reject if the object is not convex: determining this seems difficult.  can the coefficients be chosen to favor convexity?  previously 2D.

then, uniformly randomly sample some points on the surface and construct tangent planes, forming a random convex polyhedron (might not be closed).  uniform sampling of surface points on an arbitrary surface seems difficult.  maybe a random walk of small steps each of approximately constant geodesic length.  how many steps?

or, permit starting with a smooth shape not necessarily convex.  reject a tangent plane if its inward half space does not contain the origin.

extend this construction to higher dimensions.  what are the hyper-spherical harmonics?  curse of dimensionality probably somehow happens.

a 4D polytope (polychoron) is interesting because its cells are random polyhedra whose faces fit together pairwise but not further in 3D.  how can a collection of specified irregular polyhedra be manufactured?  3D printing is of course one way.  or, mill some faces, but include nubs on one or more faces.  then, reorient the piece upside down, holding it in place from below by the nubs.  mill the remaining faces.  then, somehow remove the nubs.

mill two negative halves then cast then join somehow?

create a net and fold (origami)?

the edges of a random convex 3D polyhedron define a random planar graph.  do analogous graphs for convex polytopes in higher dimensions have a property analogous to planarity?

Delaunay triangulation is another way to get a random planar graph, but faces are generally always triangles.  but are most of the faces produced by the tangent method described above also triangles?

3D convex hull of random points results in a convex polyhedron, and this generalizes to arbitrary dimensions.  I think most faces will again be triangles (simplices).  can the points be sampled in a manner to get interesting shapes?  for example, after the first few points, avoid new points that are way outside the convex hull so far, drastically changing the polyhedron.

aside: the minimal surface in 3D that contains all given points is not necessarily the convex hull as it is in 2D.  the minimal surface might have hyperbolic paraboloid faces.

a random convex polytope generates a random linear or quadratic programming problem.

applying a linear transformation to a convex polytope maintains its convexity.  proof: assume false, meaning that after the linear transformation there is a line segment that starts inside, goes outside, then comes back inside (definition of non-convex).  apply the inverse linear transformation, which is linear.  the line segment remains a line segment.  it still goes in out in.  the existence of such a line segment in the initial polytope means it was not convex.

consider relaxing the requirement of convexity and only require non-self-intersecting.  one can have holes and other weird features, though perhaps some so weird that it might be good to forbid them.  what weird features, especially in higher dimensions?  polygonal faces can be non-convex with holes.  that is probably OK, but faces with intersecting edges (e.g., pentagram) are not OK.

previous thoughts on random polyhedra..

[ibrwpoia] full coverage by uniform sampling

B different balls in an urn.  uniformly draw with replacement N times.  what is the probability that all B balls are seen during the N draws?

given B, for what value of N is the probability (about) 50%?

simulation in Pari/GP:

u(trials, B, N)= my(success=0); for(c=1, trials, my(a=vector(B)); for(i=1, N, r=1+random(B); a[r]=1); good=1; for(i=1, B, if(0==a[i], good=0; break)); success+=good); 1.0*success/trials;

B N
2 2
4 7
8 19-20
10 26-27
16 50
20 67
30 113
32 122
40 162
50 213-214
60 267-268
64 289-290

B=64 balls, N=290 draws:
u(10^5, 64, 290)
0.50303

cpu time = 21,920 ms, real time = 21,924 ms.

exact answer: 1 - probability of failure by the inclusion-exclusion principle, simplifies to:

v(B, N)= sum(i=0, B, (-1)^i * (B-i)^N * binomial(B, i)) /B^N

1.*v(64,290)
0.504257282

exact bounds:

10 [26, 27]
20 [66, 67]
30 [112, 113]
40 [161, 162]
50 [213, 214]
60 [267, 268]
70 [322, 323]
80 [379, 380]
90 [437, 438]
100 [496, 497]
110 [556, 557]
120 [618, 619]
130 [679, 680]
140 [742, 743]
150 [806, 807]
160 [870, 871]
170 [934, 935]
180 [1000, 1001]
190 [1066, 1067]
200 [1132, 1133]
210 [1199, 1200]
220 [1266, 1267]
230 [1334, 1335]
240 [1402, 1403]
250 [1471, 1472]
260 [1540, 1541]

2 [2, 2]
4 [6, 7]
8 [19, 20]
16 [50, 51]
32 [122, 123]
64 [289, 290]
128 [667, 668]
256 [1512, 1513]
512 [3381, 3382]
1024 [7472, 7473]
2048 [16364, 16365]
4096 [35569, 35570]

the growth rate of N seems to be O(B*log B).  let's use N = B*log B exactly and calculate the probability of full coverage:

v(B, round(B*log(B)))

B N v
4 6 0.380859375
8 17 0.365562211
16 44 0.347014424
32 111 0.363466498
64 266 0.364065310
128 621 0.366295109
256 1420 0.367804277
512 3194 0.367507854
1024 7098 0.367763404
2048 15615 0.367750482
4096 34070 0.367873303

the probability seems to be converging to exp(-1) = 0.367879441 .

how many draws N are necessary to achieve at least a given probability of full coverage?

previously similar: Euler's Game of Coincidence

[fieutgtv] sum of the reciprocals of known primes

at the time of writing, the largest known prime number is the Mersenne prime 2^136279841 - 1, a 41-million-digit number.

the sum of the reciprocals of all primes up to the largest known prime can be estimated to be approximately

log(log(2^136279841 - 1)) + 0.261497 ~= 18.6252

, where the offset 0.261497 is the Meissel-Mertens constant (known to many more digits).  (how many more?)

(previously, on log(2^n +1).)

however, the sum of the reciprocals of all primes up to infinity is infinity, i.e., the infinite sum diverges (proven by Euler).  despite our finite sum being up to a huge prime, 18.6 is still a long way from infinity.

the reciprocal of the next prime after the largest known prime ("the first unknown prime") will add approximately 10^-41000000 to our partial sum 18.6252 , a tiny step.  and each additional step will be yet smaller, because reciprocals keep getting smaller.  yet infinity will still eventually be "reached": the sum will exceed any finite number you can think of: Graham's number, Rayo's number, Loader's number, etc., (and all finite numbers you can't think of).

previously, computing the sum one button press at a time.

[wlxarken] hope nand comfort

hypothesis: a society may grant its lower classes hope for upward social mobility, or a comfortable existence, but not both.

this is a depressing hypothesis similar in spirit to those that originally got economics labeled the dismal science.

the general ideas justifying the hypothesis are the following.  if you grant the lower classes the possibility of upward social mobility, only some of them will be permitted to achieve it, so they will need to compete against each other in a zero-sum game for who achieves it.  zero-sum games are cutthroat, not comfortable.  if you grant (enough) comfort, there won't be incentive to seek upward social mobility (this might require more thought).  if you grant comfortable upward social mobility, then the lower classes will become depopulated, so society won't be able to function because it needs lower classes to do lower-class stuff.

also seemingly in support of this hypothesis are highly unequal, highly stratified societies which have remained stable for long periods of time.  they presumably deny upward mobility but grant enough relative comfort to squelch social upheaval.

unfortunately, or fortunately, this hypothesis is only a vague idea which could be incorrect in many ways.

Solow famously disproved Malthus's original "dismal science" with the economic concept of technology.  however, our dismal hypothesis survives overall positive sum: technology may lift the entire pyramid, but it still remains a pyramid.

society can grant its lower classes neither hope nor comfort, which is why the conjunction is nand not xor.  but a situation with neither hope nor comfort might be unstable, threatening civil war.

[utglyvmo] dark matter cloud

what would we observe if the earth, or the solar system, passed though a dense cloud of dark matter?

assume WIMP.  we would see not the dark matter directly but only indirect effects.  hypothesize that orbits will change.  will the sun gravitationally capture dark matter and increase in mass, increasing its fusion rate?

do dense clouds of dark matter exist?  how dense might they be?

update: a low-surface-brightness galaxy could be considered a galaxy-sized dense cloud of dark matter.  but I don't think their average dark matter density is that much higher than that of the Milky Way.

previously on dark matter in the solar system.

[sebqohhz] Star Wars by ancient Greek playwrights

pretend that the spirits of ancient Greek playwrights had possessed George Lucas and teamed up within his mind to write the actual screenplay to Star Wars.  who wrote which part?

Aeschylus:

[opening shot: stars.  small spaceship.  laser fire.  huge Imperial Star Destroyer.]

also:

Obi-Wan Kenobi: The Force is what gives a Jedi his power.  It's an energy field created by all living things.  It surrounds us and penetrates us.  It binds the galaxy together.

Sophocles:

Ben Kenobi: Yes, I was once a Jedi knight, the same as your father.
Luke Skywalker: I wish I'd known him.
Ben Kenobi: He was the best star pilot in the galaxy, and a cunning warrior.  I understand that you've become quite a good pilot yourself.
[sorrowfully]
Ben Kenobi: And he was a good friend.

Euripides:

Luke: But I was going into Tosche Station to pick up some power converters!

Aristophanes: Robot Chicken

[pmycapmx] horse meat

cow = beef
pig = pork
sheep = mutton
deer = venison
horse = ?

we have alternate names for the meat of animals that we maybe do not want to think of as animals when we eat them.  Americans and British generally do not like to eat horse.  would we be more willing if there were an alternate English name for horse meat?  in cultures which do eat horse, do they have an alternate name for its meat (for any meat)?  other animals (e.g., dog)?

in English, chicken = chicken, fish = fish, but these we consider "lower" animals.  however, lamb = lamb despite sheep = mutton.

maybe horse will become demoted to lower animal soon because of the ascendancy of the automobile.  will horse-riding as a hobby of the upper class affect its transition to acceptability as food?  how is it currently affecting?  on one hand, horses are (expensive) friends not food to those who ride them.  on the other hand, horses are literally below them when ridden.