BBOX is a language for black box algorithms. We present a BBOX interpreter for GAP.
Suppose we have the two BBOX programs M22.find (a finder for M22) and M22.check (a checker for M22). We can load them into GAP as follows:
gap> find := prepareblackbox("bbox/spor/M22.find");
[ [ "set", "V", 0 ], [ "rand", 1 ], [ "ord", 1, "A" ], [ "incr", "V" ],
[ "_if", "V", "gt", 1000, "then", "timeout" ],
[ "_if", "A", "notin", 1, 2, 3, 4, 5, 6, 7, 8, 11, "then", "fail" ],
[ "_if", "A", "noteq", 8, "then", "jmp", 2 ], [ "mu", 1, 1, 3 ],
[ "mu", 3, 3, 2 ], [ "set", "X", 0 ], [ "incr", "X" ],
[ "_if", "X", "gt", 1000, "then", "timeout" ], [ "rand", 4 ],
[ "cjr", 3, 4 ], [ "mu", 2, 3, 5 ], [ "ord", 5, "D" ],
[ "_if", "D", "notin", 2, 3, 4, 5, 6, 7, 8, 11, "then", "fail" ],
[ "_if", "D", "noteq", 11, "then", "jmp", 11 ], [ "mu", 5, 3, 6 ],
[ "mu", 5, 6, 7 ], [ "ord", 7, "E" ],
[ "_if", "E", "noteq", 11, "then", "jmp", 11 ], [ "oup", 2, 2, 3 ] ]
gap> check := prepareblackbox("bbox/spor/M22.check");
[ [ "chor", 1, 2 ], [ "chor", 2, 4 ], [ "mu", 1, 2, 3 ], [ "chor", 3, 11 ],
[ "mu", 3, 2, 4 ], [ "mu", 3, 4, 5 ], [ "chor", 5, 11 ], [ "mu", 5, 2, 6 ],
[ "chor", 6, 6 ] ]
Now let us use the finder to find standard generators for M22:
gap> G := MathieuGroup(22);
Group([ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22),
(1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),
(1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ])
gap> g := blackbox(G, find, [], rec());;
gap> a := g.gens[1];
(1,5)(2,8)(7,12)(9,22)(10,14)(11,21)(13,18)(15,16)
gap> b := g.gens[2];
(2,4,5,10)(3,18)(6,12)(7,17,22,8)(9,11,20,15)(13,16,14,19)
We can also use the checker to test them:
gap> ans := blackbox(G, check, [a,b], rec(verbose := true));
1. chor 1 2
2. chor 2 4
3. mu 1 2 3
4. chor 3 11
5. mu 3 2 4
6. mu 3 4 5
7. chor 5 11
8. mu 5 2 6
9. chor 6 6
rec( multiply := 4, invert := 0, power := 0, order := 5, class := 0,
random := 0, timetaken := 0, conjugate := 0, conjugateinplace := 0,
commutator := 0, vars := [ ], callstack := [ ], result := true )
Because ans.result is true, the checker succeeded.
Last updated 23rd September 2005