mirror of
https://github.com/zenorogue/hyperrogue.git
synced 2024-12-20 23:50:27 +00:00
537 lines
16 KiB
C++
537 lines
16 KiB
C++
// RogueViz - Grigorchuk group
|
|
// Copyright (C) 2011-2019 Zeno and Tehora Rogue, see 'hyper.cpp' for details
|
|
|
|
/** \file rogueviz/grigorchuk.cpp
|
|
* \brief Grigorchuk group
|
|
*
|
|
* This is a visualization of the Grigorchuk group. It is the first known group with
|
|
* intermediate growth (i.e., superpolynomial and subexponential).
|
|
*
|
|
* The implementation is based on:
|
|
*
|
|
* Rostislav Grigorchuk, Igor Pak,
|
|
* Groups of Intermediate Growth: an Introduction for Beginners
|
|
* https://arxiv.org/pdf/math/0607384.pdf
|
|
*
|
|
* which presents the material in a simple way.
|
|
*
|
|
* This creates a map whose tiles correspond to the elements of the Grigorchuk group.
|
|
* More precisely, the tiles correspond to the subgroup of index 2 generated by ac, ca, and b
|
|
* (this is "more playable"). The three tiles adjacent to g are gb, gac, and gca.
|
|
*
|
|
* The 'lines' drawn split each tile into two halves, which correspond to the elements of the
|
|
* actual Grigorchuk group (g and ga; ga is the one close to gac).
|
|
*
|
|
* Every element of the Grigorchuk group has finite order. Therefore, if you choose a specific
|
|
* way of travelling (e.g. turn left, go, turn right, go) you will always eventually reach the
|
|
* starting point.
|
|
*
|
|
* Command line options:
|
|
*
|
|
* -grigorchuk -- play on the Grigorchuk group
|
|
* -grig-limit 100000 -canvas G -- color the tiles according to the distance from the starting point
|
|
* (i.e., the neutral element), the number is the number of tiles colored
|
|
* -grig-nolines -- show no splitting lines (also can be switched in the experiments menu)
|
|
* -grig-nolabels -- show no labels (also can be switched in the experiments menu)
|
|
*
|
|
*/
|
|
|
|
#include "rogueviz.h"
|
|
|
|
namespace grigorchuk {
|
|
|
|
using namespace hr;
|
|
|
|
typedef tuple<bool, string, string> splitter;
|
|
|
|
void add(string& s, char c) {
|
|
if(s.size() == 0) s.push_back(c);
|
|
else if(c == s.back()) s.pop_back();
|
|
else if(c != 'a' && s.back() != 'a')
|
|
s.back() = s.back() ^ c ^ 'd' ^ 'b' ^ 'c';
|
|
else s.push_back(c);
|
|
}
|
|
|
|
splitter split(string s) {
|
|
bool swapped = false;
|
|
string s0, s1;
|
|
for(char c: s) {
|
|
if(c == 'b') add(s0, swapped?'a':'c'), add(s1, swapped?'c':'a');
|
|
if(c == 'c') add(s0, swapped?'a':'d'), add(s1, swapped?'d':'a');
|
|
if(c == 'd') add(swapped ? s1 : s0, 'b');
|
|
if(c == 'a') swapped = !swapped;
|
|
}
|
|
return splitter{swapped, s0, s1};
|
|
}
|
|
|
|
splitter split_slow(string s) {
|
|
bool swapped = false;
|
|
string s0, s1;
|
|
for(char c: s) {
|
|
if(c == 'b') (s0 += swapped?'a':'c'), (s1 += swapped?'c':'a');
|
|
if(c == 'c') (s0 += swapped?'a':'d'), (s1 += swapped?'d':'a');
|
|
if(c == 'd') ((swapped ? s1 : s0) += 'b'), ((swapped ? s0 : s1) += '-');
|
|
if(c == 'a') swapped = !swapped, s0 += '-', s1 += '-';
|
|
}
|
|
return splitter{swapped, s0, s1};
|
|
}
|
|
|
|
string reduce(const string& x) {
|
|
string res;
|
|
for(char c: x) add(res, c);
|
|
return res;
|
|
}
|
|
|
|
#define Split(x) auto sw = split(x); auto swapped = get<0>(sw); auto s0 = get<1>(sw); auto s1 = get<2>(sw)
|
|
|
|
bool empt(const string& x) {
|
|
Split(x); // auto [swapped, s0, s1] = split(x);
|
|
if(x == "") return true;
|
|
if(x == "d") return false;
|
|
if(swapped) return false;
|
|
return empt(s0) && empt(s1);
|
|
}
|
|
|
|
bool empt_slow(const string& x) {
|
|
Split(x); // auto [swapped, s0, s1] = split_slow(x);
|
|
printf("%s -> %d %s %s\n", x.c_str(), swapped, s0.c_str(), s1.c_str());
|
|
if(x == "") return true;
|
|
if(x == "d") return false;
|
|
if(swapped) return false;
|
|
return empt(s0) && empt(s1);
|
|
}
|
|
|
|
typedef const struct rep* prep;
|
|
|
|
struct rep {
|
|
bool swapped;
|
|
prep a0;
|
|
prep a1;
|
|
mutable char last;
|
|
mutable bool visited = false;
|
|
mutable int len;
|
|
rep(bool s, prep a0, prep a1, char l, bool vis = false) : swapped(s), a0(a0), a1(a1), last(l), visited(vis), len(-1) {}
|
|
};
|
|
|
|
bool operator < (const rep a, const rep b) {
|
|
return tie(a.swapped, a.a0, a.a1) < tie(b.swapped, b.a0, b.a1);
|
|
}
|
|
|
|
bool operator == (const rep a, const rep b) {
|
|
return tie(a.swapped, a.a0, a.a1) == tie(b.swapped, b.a0, b.a1);
|
|
}
|
|
|
|
rep grig_I = rep{false, &grig_I, &grig_I, 0, false};
|
|
extern rep grig_a, grig_b, grig_c, grig_d;
|
|
|
|
rep grig_a = rep{true, &grig_I, &grig_I, 'a', false};
|
|
rep grig_b = rep{false, &grig_a, &grig_c, 'b', false};
|
|
rep grig_c = rep{false, &grig_a, &grig_d, 'c', false};
|
|
rep grig_d = rep{false, &grig_I, &grig_b, 'd', false};
|
|
|
|
// (ab) c = a (a,c) (a,d) = a(a,c) (a,d) =
|
|
map<rep, char> all_reps; // = {grigid, &grigid};
|
|
|
|
prep lookup(rep x) {
|
|
if(x == grig_I) return &grig_I;
|
|
else if(x == grig_a) return &grig_a;
|
|
else if(x == grig_b) return &grig_b;
|
|
else if(x == grig_c) return &grig_c;
|
|
else if(x == grig_d) return &grig_d;
|
|
else if(all_reps.count(x)) return &(all_reps.find(x)->first);
|
|
else return &(all_reps.emplace(x, 0).first->first);
|
|
}
|
|
|
|
/*prep add_a(prep x) {
|
|
return lookup({!x->swapped, x->a0, x->a1, 'a'});
|
|
}
|
|
prep add_d(prep x) {
|
|
if(x == &grig_I) return &grig_d;
|
|
if(x == &grig_d) return &grig_I;
|
|
return lookup({x->swapped, x->swapped?add_d(x->a0):x->a0, x->swapped?x->a1:add_d(x->a1), 'd'});
|
|
}
|
|
prep add_c(prep x) { return lookup({x->swapped, (x->swapped?add_a:add_d)(x->a0), (x->swapped?add_d:add_a)(x->a1), 'c'}); }
|
|
prep add_b(prep x) { return lookup({x->swapped, (x->swapped?add_a:add_c)(x->a0), (x->swapped?add_c:add_a)(x->a1), 'b'}); }
|
|
*/
|
|
|
|
/* ostream& operator << (ostream& os, prep x) {
|
|
if(x == &grig_I) return os << "I";
|
|
// else if(x == &grig_a) return os << "a";
|
|
else if(x == &grig_b) return os << "b";
|
|
else if(x == &grig_c) return os << "c";
|
|
else if(x == &grig_d) return os << "d";
|
|
else {
|
|
if(x->swapped) os << "a";
|
|
os << "(" << x->a0 << "," << x->a1 << ")";
|
|
return os;
|
|
}
|
|
} */
|
|
|
|
prep mul (prep x, prep y) {
|
|
if(x == &grig_I) return y;
|
|
if(y == &grig_I) return x;
|
|
if(x == &grig_a && y == &grig_a) return &grig_I;
|
|
if(x == &grig_b && y == &grig_b) return &grig_I;
|
|
if(x == &grig_c && y == &grig_c) return &grig_I;
|
|
if(x == &grig_d && y == &grig_d) return &grig_I;
|
|
if(x == &grig_b && y == &grig_c) return &grig_d;
|
|
if(x == &grig_c && y == &grig_b) return &grig_d;
|
|
if(x == &grig_b && y == &grig_d) return &grig_c;
|
|
if(x == &grig_d && y == &grig_b) return &grig_c;
|
|
if(x == &grig_c && y == &grig_d) return &grig_b;
|
|
if(x == &grig_d && y == &grig_c) return &grig_b;
|
|
if(!y->swapped) return lookup(rep{x->swapped, mul(x->a0, y->a0), mul(x->a1, y->a1), y->last});
|
|
else return lookup(rep{!x->swapped, mul(x->a1, y->a0), mul(x->a0, y->a1), y->last});
|
|
}
|
|
|
|
string encode(string s) {
|
|
if(s == "") return "I";
|
|
else if( s == "d") return "d";
|
|
else {
|
|
Split(s); // auto [swapped, s0, s1] = split(s);
|
|
return (swapped ? "a(" : "(") + encode(s0) + "," + encode(s1) + ")";
|
|
}
|
|
}
|
|
|
|
set<string> seen;
|
|
|
|
void addmore(const string& s, int more) {
|
|
if(more == 0) {
|
|
string sr = s;
|
|
reverse(sr.begin(), sr.end());
|
|
for(string q: seen) {
|
|
string qo = q;
|
|
for(char cr: sr) add(q, cr);
|
|
if(empt(q)) {
|
|
// printf("%s = %s /%s\n", s.c_str(), qo.c_str(), sr.c_str());
|
|
return;
|
|
}
|
|
}
|
|
seen.insert(s);
|
|
// printf("%s\n", s.c_str());
|
|
return;
|
|
}
|
|
for(char c: {'a', 'b', 'c', 'd'}) {
|
|
string s1 = s;
|
|
add(s1, c);
|
|
if(isize(s1) != isize(s)+1) continue;
|
|
addmore(s1, more-1);
|
|
}
|
|
}
|
|
|
|
string deform(prep x2) {
|
|
string t = "";
|
|
while(x2 != &grig_I) {
|
|
if(x2->last == 'a') t += 'a', x2 = mul(x2, &grig_a);
|
|
else if(x2->last == 'b') t += 'b', x2 = mul(x2, &grig_b);
|
|
else if(x2->last == 'c') t += 'c', x2 = mul(x2, &grig_c);
|
|
else if(x2->last == 'd') t += 'd', x2 = mul(x2, &grig_d);
|
|
else if(x2->last == 'A') t += "ca", x2 = mul(mul(x2, &grig_c), &grig_a);
|
|
else if(x2->last == 'C') t += "ac", x2 = mul(mul(x2, &grig_a), &grig_c);
|
|
else return "?" + t + "?";
|
|
}
|
|
reverse(t.begin(), t.end());
|
|
return t;
|
|
}
|
|
|
|
bool prepared = false;
|
|
|
|
prep ac, ca;
|
|
|
|
int grig_limit = 10000;
|
|
|
|
int prepared_dists = 0;
|
|
int next;
|
|
int length = 0;
|
|
|
|
vector<prep> all;
|
|
|
|
void visit(prep x, char l, int d) {
|
|
if(!x->visited) x->visited = true, x->last = l, all.push_back(x), x->len = d;
|
|
}
|
|
|
|
void prepare_to_next(bool verbose) {
|
|
while(true) {
|
|
int i = prepared_dists++;
|
|
prep x = all[i];
|
|
if(!x->visited) println(hlog, "visited or not");
|
|
if(1) {
|
|
// printf("%s\n", deform(x).c_str());
|
|
}
|
|
visit(mul(x, &grig_b), 'b', x->len + 1);
|
|
visit(mul(x, ac), 'A', x->len + 1);
|
|
visit(mul(x, ca), 'C', x->len + 1);
|
|
|
|
if(i == next) {
|
|
if(verbose)
|
|
addMessage("there are "+its(i)+" elements in distance up to "+its(length));
|
|
println(hlog, "Grigorchuk: ", tie(length, i));
|
|
next = all.size(), length++;
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
|
|
void prepare() {
|
|
prepared = true;
|
|
|
|
// rep* grigid = lookup(rep { false, NULL, NULL });
|
|
|
|
ac = mul(&grig_a, &grig_c);
|
|
ca = mul(&grig_c, &grig_a);
|
|
// prep where = &grig_I;
|
|
|
|
string s = "";
|
|
all.clear();
|
|
|
|
|
|
/*
|
|
for(int a=0; a<=32; a++) {
|
|
// printf("%p -> %d %p %p\n", where, where->swapped, where->a0, where->a1);
|
|
cout << where << " | " << encode(s) << "\n";
|
|
where = ((a&1) ? add_a : add_b) (where);
|
|
s += (a&1) ? 'a' : 'b';
|
|
}
|
|
|
|
string test = "ba";
|
|
string pw = "";
|
|
for(int i=0; i<=16; i++) {
|
|
printf("%d: %d\n", i, empt(pw));
|
|
pw += test;
|
|
}
|
|
*/
|
|
|
|
// printf("TEST %s\n", encode("bcd").c_str());
|
|
|
|
visit(&grig_I, 0, 0);
|
|
length = 0;
|
|
next = all.size();
|
|
|
|
prepared_dists = 0;
|
|
|
|
while(prepared_dists < grig_limit) prepare_to_next(false);
|
|
|
|
prep test = &grig_b;
|
|
test = mul(test, &grig_a);
|
|
test = mul(test, &grig_d);
|
|
test = mul(test, &grig_a);
|
|
test = mul(test, &grig_d);
|
|
printf("badad = %s\n", deform(test).c_str());
|
|
}
|
|
|
|
bool view_labels = true, view_lines = true;
|
|
|
|
}
|
|
|
|
namespace hr {
|
|
|
|
struct hrmap_grigorchuk : hrmap_standard {
|
|
|
|
heptagon *origin;
|
|
heptagon *getOrigin() override { return origin; }
|
|
|
|
map<heptagon*, grigorchuk::prep> dec;
|
|
map<grigorchuk::prep, heptagon*> enc;
|
|
|
|
void gtie(heptagon* h, grigorchuk::prep p) {
|
|
dec[h] = p;
|
|
enc[p] = h;
|
|
}
|
|
|
|
hrmap_grigorchuk() {
|
|
if(!grigorchuk::prepared) grigorchuk::prepare();
|
|
origin = tailored_alloc<heptagon> (S7);
|
|
origin->s = hsOrigin;
|
|
origin->emeraldval = 0;
|
|
origin->zebraval = 0;
|
|
origin->fiftyval = 0;
|
|
origin->fieldval = 0;
|
|
origin->rval0 = origin->rval1 = 0;
|
|
origin->cdata = NULL;
|
|
origin->alt = NULL;
|
|
origin->c7 = NULL;
|
|
origin->distance = 0;
|
|
origin->c7 = newCell(3, origin);
|
|
gtie(origin, &grigorchuk::grig_I);
|
|
}
|
|
|
|
heptagon *create_step(heptagon *p, int d) override {
|
|
auto pr = dec[p];
|
|
// auto pr1 = pr;
|
|
|
|
switch(d) {
|
|
using namespace grigorchuk;
|
|
case 0: pr = mul(mul(pr, &grig_a), &grig_c); break;
|
|
case 1: pr = mul(mul(pr, &grig_c), &grig_a); break;
|
|
case 2: pr = mul(pr, &grig_b); break;
|
|
}
|
|
|
|
heptagon *h;
|
|
|
|
if(enc.count(pr)) {
|
|
h = enc[pr];
|
|
// println(hlog, deform(pr), "*", "acd"[d], " = ", deform(pr1));
|
|
}
|
|
else {
|
|
if(!pr->visited) pr->last = "ACb" [d];
|
|
|
|
h = tailored_alloc<heptagon> (S7);
|
|
h->s = hsOrigin;
|
|
h->emeraldval = 0;
|
|
h->zebraval = 0;
|
|
h->fiftyval = 0;
|
|
h->fieldval = 0;
|
|
h->rval0 = h->rval1 = 0;
|
|
h->cdata = NULL;
|
|
h->alt = NULL;
|
|
h->c7 = newCell(3, h);
|
|
h->distance = p->distance + 1;
|
|
gtie(h, pr);
|
|
}
|
|
|
|
h->c.connect(d == 2 ? 2 : 1-d, p, d, false);;
|
|
return h;
|
|
}
|
|
|
|
void draw_at(cell *at, const shiftmatrix& where) override {
|
|
|
|
dq::clear_all();
|
|
dq::enqueue_by_matrix(at->master, where * currentmap->master_relative(centerover, true));
|
|
|
|
while(!dq::drawqueue.empty()) {
|
|
auto& p = dq::drawqueue.front();
|
|
heptagon *h = get<0>(p);
|
|
shiftmatrix V = get<1>(p);
|
|
dq::drawqueue.pop();
|
|
|
|
cell *c = h->c7;
|
|
if(!do_draw(c, V)) continue;
|
|
|
|
if(grigorchuk::view_lines) queueline(V * ddspin(c, 2) * xpush0(cgi.tessf/2), V * ddspin(c, 2) * xpush0(-cgi.tessf), 0xFF00FFFF, 2);
|
|
|
|
if(grigorchuk::view_labels) queuestr(V, 0.3, grigorchuk::deform(dec[c->master]), 0xFFFFFF);
|
|
|
|
if(patterns::whichCanvas == 'G' && c->landparam == 0)
|
|
c->landparam = 0x102008 * (1 + ((hrmap_grigorchuk*)currentmap)->dec[c->master]->len);
|
|
|
|
drawcell(c, V * currentmap->master_relative(c, false));
|
|
|
|
for(int i=0; i<3; i++) if(c->move(i))
|
|
dq::enqueue_by_matrix(h->cmove(i), optimized_shift(V * adj(h, i)));
|
|
}
|
|
}
|
|
|
|
transmatrix relative_matrixh(heptagon *h2, heptagon *h1, const hyperpoint& hint) override {
|
|
if(gmatrix0.count(h2->c7) && gmatrix0.count(h1->c7))
|
|
return inverse_shift(gmatrix0[h1->c7], gmatrix0[h2->c7]);
|
|
return Id;
|
|
}
|
|
|
|
transmatrix relative_matrixc(cell *c2, cell *c1, const struct hyperpoint& hint) override {
|
|
if(gmatrix0.count(c2) && gmatrix0.count(c1))
|
|
return inverse_shift(gmatrix0[c1], gmatrix0[c2]);
|
|
return Id;
|
|
}
|
|
};
|
|
|
|
eGeometry gGrigorchuk(eGeometry(-1));
|
|
|
|
void create_grigorchuk_geometry() {
|
|
if(gGrigorchuk != eGeometry(-1)) return;
|
|
ginf.push_back(ginf[gNormal]);
|
|
gGrigorchuk = eGeometry(isize(ginf) - 1);
|
|
auto& gi = ginf[gGrigorchuk];
|
|
gi.sides = 3;
|
|
gi.vertex = 8;
|
|
gi.flags = qANYQ | qEXPERIMENTAL;
|
|
gi.tiling_name = "{3,8}";
|
|
gi.quotient_name = "Grigorchuk";
|
|
gi.menu_displayed_name = "Grigorchuk group";
|
|
gi.shortname = "Grig";
|
|
gi.default_variation = eVariation::pure;
|
|
}
|
|
|
|
int readArgsG() {
|
|
using namespace arg;
|
|
|
|
if(0) ;
|
|
else if(argis("-grig-limit")) {
|
|
shift(); grigorchuk::grig_limit = argi();
|
|
}
|
|
|
|
else if(argis("-grigorchuk")) {
|
|
|
|
PHASEFROM(3);
|
|
|
|
stop_game();
|
|
create_grigorchuk_geometry();
|
|
set_geometry(gGrigorchuk);
|
|
set_variation(eVariation::pure);
|
|
}
|
|
|
|
else if(argis("-grig-nolines")) {
|
|
grigorchuk::view_lines = false;
|
|
}
|
|
|
|
else if(argis("-grig-nolabels")) {
|
|
grigorchuk::view_labels = false;
|
|
}
|
|
|
|
else return 1;
|
|
return 0;
|
|
}
|
|
|
|
auto hook = addHook(hooks_args, 100, readArgsG)
|
|
+ addHook(hooks_newmap, 100, [] { return geometry == gGrigorchuk ? new hrmap_grigorchuk : nullptr; })
|
|
+ addHook(patterns::hooks_generate_canvas, 100, [] (cell* c) {
|
|
if(patterns::whichCanvas == 'G' && geometry == gGrigorchuk)
|
|
return 0x102008 * (1 + ((hrmap_grigorchuk*)currentmap)->dec[c->master]->len);
|
|
return -1;
|
|
})
|
|
+ addHook(dialog::hooks_display_dialog, 100, [] () {
|
|
if(current_screen_cfunction() == showEuclideanMenu && geometry == gGrigorchuk) {
|
|
dialog::addBoolItem_action(XLAT("Grigorchuk lines"), grigorchuk::view_lines, 'L');
|
|
dialog::addBoolItem_action(XLAT("Grigorchuk labels"), grigorchuk::view_labels, 'M');
|
|
}
|
|
})
|
|
+ addHook(hooks_initialize, 100, create_grigorchuk_geometry)
|
|
+ addHook_rvslides(140, [] (string s, vector<tour::slide>& v) {
|
|
if(s != "mixed") return;
|
|
using namespace rogueviz::pres;
|
|
v.push_back(tour::slide{
|
|
"Grigorchuk group", 10, tour::LEGAL::NONE,
|
|
|
|
"This is a visualization of the Grigorchuk group. It is the first known group with "
|
|
"intermediate growth (i.e., superpolynomial and subexponential).\n\n"
|
|
"Each tile corresponds to two elements of the Grigorchuk group.\n\n"
|
|
"Every element of the Grigorchuk group has finite order. Therefore, if you choose a specific "
|
|
"way of travelling (e.g. turn left, go, turn right, go) you will always eventually reach the "
|
|
"starting point.\n\n"
|
|
"Cells are color-coded by the distance to the origin. Distance is only known for a given number of cells "
|
|
"(initially 10000); if you want to compute more distances, press '5'. Press 'o' to enable/disable lines.\n\n"
|
|
"See grigorchuk.cpp for more comments.",
|
|
|
|
[] (tour::presmode mode) {
|
|
slide_url(mode, 'p', "a paper about Grigorchuk group", "https://arxiv.org/pdf/math/0607384.pdf");
|
|
if(mode == pmStart) {
|
|
grigorchuk::grig_limit = 10000;
|
|
gamestack::push();
|
|
slide_backup(patterns::whichCanvas, 'G');
|
|
slide_backup(firstland, laCanvas);
|
|
slide_backup(specialland, laCanvas);
|
|
set_geometry(gGrigorchuk);
|
|
start_game();
|
|
resetview();
|
|
}
|
|
if(mode == pmKey) {
|
|
grigorchuk::prepare_to_next(true);
|
|
}
|
|
if(mode == pmStop) {
|
|
gamestack::pop();
|
|
slide_restore_all();
|
|
}
|
|
}}
|
|
);});
|
|
}
|
|
|