%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: ntfl.dvi %%CreationDate: Sun Jan 18 07:45:45 1998 %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: _dvips ntfl.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.01.18:0745 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 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y(monoid)g(o)o(v)o(er)i(\006,)g(that)g(is,)g(the)h(set)g(of)f(all)f (\014nite)h(strings)h(of)f(sym)o(b)q(ols)e(c)o(hosen)k(from)300 1812 y(\006,)d(with)h(concatenation)g(as)g(the)h(monoid)c(op)q (eration.)18 b(Th)o(us,)c(if)f(\006)e(=)h Fp(f)p Fu(0)p Fs(;)7 b Fu(1)p Fp(g)p Fu(,)12 b(then)652 1904 y(\006)682 1887 y Fm(\003)713 1904 y Fu(=)g Fp(f)p Fs(\017;)7 b Fu(0)p Fs(;)g Fu(1)p Fs(;)g Fu(00)p Fs(;)f Fu(01)p Fs(;)g Fu(10)p Fs(;)g Fu(11)p Fs(;)g Fu(000)o Fs(;)h(:)f(:)h(:)l Fp(g)p Fs(;)300 1996 y Fu(where)18 b Fs(\017)e Fu(is)h(the)g(notation)f (for)g(the)h(empt)o(y)f(string.)26 b(A)17 b Ft(formal)f(language)p Fu(,)i(or)f(just)300 2046 y Ft(language)p Fu(,)d(is)g(de\014ned)h(to)f (b)q(e)g(an)o(y)g(subset)h(of)e(\006)1056 2031 y Fm(\003)1075 2046 y Fu(.)375 2096 y(Let)i Fs(L;)7 b(L)525 2102 y Fn(1)544 2096 y Fs(;)g(L)591 2102 y Fn(2)624 2096 y Fu(b)q(e)15 b(languages.)21 b(W)m(e)14 b(de\014ne)i(the)f(concatenation)g(of)f (languages)300 2146 y(as)g(follo)o(ws:)642 2238 y Fs(L)670 2244 y 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(example,)e([35)o(].)p eop %%Page: 3 3 3 2 bop 611 187 a Fy(NUMBER)12 b(THEOR)m(Y)i(AND)f(F)o(ORMAL)g(LANGUA)o (GES)230 b Fu(3)337 291 y(De\014ne)16 b Fs(L)495 276 y Fn(0)527 291 y Fu(=)d Fp(f)p Fs(\017)p Fp(g)p Fu(,)h(and)g Fs(L)766 276 y Fq(i)793 291 y Fu(=)f Fs(LL)894 276 y Fq(i)p Fm(\000)p Fn(1)966 291 y Fu(for)h Fs(i)f Fp(\025)h Fu(1.)20 b(W)m(e)14 b(de\014ne)i(the)f Ft(Kle)n(ene)h(closur)n(e)337 341 y Fu(of)e(a)f(language)g(b)o(y)901 442 y Fs(L)929 424 y Fm(\003)960 442 y Fu(=)1008 402 y Fr([)1004 491 y Fq(i)p Fm(\025)p Fn(0)1065 442 y Fs(L)1093 424 y Fq(i)1107 442 y Fs(:)337 579 y Fu(A)18 b Ft(r)n(e)n(gular)f(expr)n(ession)k Fu(o)o(v)o(er)c(an)g(alphab)q(et)h(\006)f(is)g(a)g(w)o(a)o(y)g(to)g (denote)h(certain)g(lan-)337 628 y(guages)f(|)e(a)h(\014nite)g (expression)h(using)e(the)i(sym)o(b)q(ols)d(in)i(\006)g(together)h (with)e(+)h(\(to)337 678 y(denote)d(union\),)d Fp(\003)h Fu(\(to)h(denote)g(Kleene)g(closure\),)g Fs(\017)f Fu(\(to)h(denote)g (the)f(empt)o(y)g(string\),)337 728 y Fp(;)16 b Fu(\(to)g(denote)h(the) g(empt)o(y)d(set\),)k(and)d(paren)o(theses)k(for)c(grouping.)24 b(F)m(or)15 b(example,)337 778 y(the)j(regular)e(expression)i(\()p Fs(\017)11 b Fu(+)g(1\)\(0)g(+)g(01\))1031 763 y Fm(\003)1066 778 y Fu(denotes)18 b(the)f(set)g(of)f(all)f(strings)i(o)o(v)o(er)337 828 y Fp(f)p Fu(0)p Fs(;)7 b Fu(1)p Fp(g)13 b Fu(con)o(taining)g(no)h (t)o(w)o(o)g(consecutiv)o(e)i(1's.)j(If)14 b(a)g(language)f(can)i(b)q (e)g(represen)o(ted)337 877 y(b)o(y)f(a)g(regular)g(expression,)g(it)g (is)f(said)h(to)g(b)q(e)g Ft(r)n(e)n(gular)p Fu(.)412 956 y Fz(3.)24 b(Num)o(b)q(er)f(represen)o(tati)o(on)o(s.)f Fu(In)f(order)g(to)g(talk)f(ab)q(out)h(n)o(um)o(b)q(ers)f(in)337 1005 y(formal)e(language)i(theory)h(terms,)g(w)o(e)f(need)i(a)e(w)o(a)o (y)f(to)i(represen)o(t)h(n)o(um)o(b)q(ers)e(as)337 1055 y(strings)d(of)e(sym)o(b)q(ols)g(o)o(v)o(er)h(a)g(\014nite)g(alphab)q (et.)25 b(Let)16 b(us)h(b)q(egin)f(with)f(the)i(in)o(tegers.)337 1105 y(A)d(classical)g(w)o(a)o(y)f(to)h(do)g(this)g(is)f(the)i (canonical)e(represen)o(tation)i(in)f(base)g Fs(k)q Fu(:)412 1157 y Fk(Theorem)i(3.1.)21 b Ft(L)n(et)13 b Fs(k)i Ft(b)n(e)g(an)f (inte)n(ger)g Fp(\025)e Fu(2)p Ft(.)18 b(Then)c(every)g(p)n(ositive)g (inte)n(ger)g Fs(n)337 1206 y Ft(c)n(an)h(b)n(e)g(r)n(epr)n(esente)n(d) f(uniquely)h(in)f(the)g(form)g Fs(n)e Fu(=)1143 1175 y Fr(P)1186 1219 y Fn(0)p Fm(\024)p Fq(i)p Fm(\024)p Fq(r)1292 1206 y Fs(a)1314 1212 y Fq(i)1328 1206 y Fs(k)1351 1191 y Fq(i)1364 1206 y Ft(,)i(wher)n(e)g(the)h Fs(a)1599 1212 y Fq(i)1627 1206 y Ft(ar)n(e)337 1256 y(inte)n(gers)g(with)f Fu(0)d Fp(\024)h Fs(a)680 1262 y Fq(i)705 1256 y Fs(<)g(k)q Ft(,)j(and)g Fs(a)902 1262 y Fq(r)932 1256 y Fp(6)p Fu(=)d(0)p Ft(.)412 1308 y Fu(By)k(asso)q(ciating)f Fs(n)g Fu(with)g(the)h(string) f Fs(a)1042 1314 y Fq(r)1060 1308 y Fs(a)1082 1314 y Fq(r)q Fm(\000)p Fn(1)1150 1308 y Fp(\001)7 b(\001)g(\001)e Fs(a)1227 1314 y Fn(1)1246 1308 y Fs(a)1268 1314 y Fn(0)1286 1308 y Fu(,)15 b(this)h(theorem)f(giv)o(es)g(a)337 1357 y(bijection)g(b)q(et)o(w)o(een)i(the)e(p)q(ositiv)o(e)g(in)o(tegers)h (and)f(the)g(set)h(of)f(strings)g(giv)o(en)g(b)o(y)g(the)337 1407 y(regular)d(expression)h(\(\006)720 1413 y Fq(k)746 1407 y Fp(\000)5 b(f)p Fu(0)p Fp(g)p Fu(\)\006)892 1392 y Fm(\003)892 1419 y Fq(k)912 1407 y Fu(,)11 b(where)i(\006)1083 1413 y Fq(k)1115 1407 y Fu(=)f Fp(f)p Fu(0)p Fs(;)7 b Fu(1)p Fs(;)g Fu(2)p Fs(;)g(:)g(:)f(:)t(;)h(k)f Fp(\000)f Fu(1)p Fp(g)p Fu(.)16 b(W)m(e)11 b(de\014ne)337 1457 y(\()p Fs(n)p Fu(\))394 1463 y Fq(k)425 1457 y Fu(to)e(b)q(e)i(the)f (string)g Fs(a)727 1463 y Fq(r)745 1457 y Fs(a)767 1463 y Fq(r)q Fm(\000)p Fn(1)835 1457 y Fp(\001)d(\001)g(\001)e Fs(a)912 1463 y Fn(1)931 1457 y Fs(a)953 1463 y Fn(0)981 1457 y Fu(represen)o(ting)11 b Fs(n)f Fu(in)f(base)h Fs(k)q Fu(.)17 b(W)m(e)9 b(also)g(de\014ne)337 1507 y(the)15 b(in)o(v)o(erse)f(map)e([)p Fs(w)q Fu(])693 1513 y Fq(k)725 1507 y Fu(to)i(b)q(e)g(the)g(v)n(alue)f(of)g(the)h(string)g Fs(w)g Fu(when)g(in)o(terpreted)h(as)f(a)337 1557 y(base-)p Fs(k)i Fu(n)o(um)o(b)q(er.)h(W)m(e)d(de\014ne)g(\(0\))878 1563 y Fq(k)910 1557 y Fu(=)e Fs(\017)i Fu(and)g([)p Fs(\017)p Fu(])1107 1563 y Fq(k)1138 1557 y Fu(=)d(0.)412 1608 y(There)18 b(are)f(man)o(y)e(relationships)h(b)q(et)o(w)o(een)i (base-)p Fs(k)g Fu(represen)o(tation)g(and)e(ele-)337 1658 y(men)o(tary)h(n)o(um)o(b)q(er)g(theory)m(.)30 b(Here)19 b(is)f(just)g(one)g(example.)28 b(Giv)o(en)17 b(an)h(in)o(teger)g Fs(n)p Fu(,)337 1708 y(w)o(e)e(ma)o(y)d(form)g Fs(s)609 1714 y Fq(k)630 1708 y Fu(\()p Fs(n)p Fu(\),)i(the)h(sum)e(of)g(its)h (base-)p Fs(k)i Fu(digits.)j(F)m(or)15 b(a)g(prime)f Fs(p)p Fu(,)g(let)i Fs(\027)1607 1714 y Fq(p)1625 1708 y Fu(\()p Fs(n)p Fu(\))337 1758 y(denote)h(the)g(exp)q(onen)o(t)f(of)g (the)g(highest)g(p)q(o)o(w)o(er)g(of)g Fs(p)f Fu(dividing)f Fs(n)p Fu(.)24 b(Then)17 b(w)o(e)f(ha)o(v)o(e)337 1807 y(the)f(follo)o(wing)c(classical)j(theorem)f(of)h(Legendre)h([61)o(,)e (V)m(ol.)g(I,)g(p.)g(10]:)412 1859 y Fk(Theorem)k(3.2.)k Ft(L)n(et)15 b Fs(p)f Ft(b)n(e)h(a)g(prime)g(numb)n(er.)k(Then)c(for)f (al)r(l)g Fs(n)e Fp(\025)g Fu(0)i Ft(we)h(have)832 1975 y Fs(\027)853 1981 y Fq(p)871 1975 y Fu(\()p Fs(n)p Fu(!\))d(=)1000 1947 y Fs(n)d Fp(\000)h Fs(s)1095 1953 y Fq(p)1115 1947 y Fu(\()p Fs(n)p Fu(\))p 1000 1966 172 2 v 1040 2004 a Fs(p)f Fp(\000)h Fu(1)1177 1975 y Fs(:)412 2091 y Fu(One)18 b(anno)o(y)o(ance)f(is)g(that)g(the)h(canonical)e(represen)o(tation)j (in)d(base)i Fs(k)g Fu(su\013ers)337 2141 y(from)g(the)h(\\leading)f (zeros")i(problem)d(|)i(that)f(is,)i(the)f(map)f Fs(w)i Fp(!)g Fu([)p Fs(w)q Fu(])1538 2147 y Fq(k)1576 2141 y Fu(is)e(not)337 2191 y(one-one)c(if)e Fs(w)g Fp(2)g Fu(\006)638 2176 y Fm(\003)638 2202 y Fq(k)658 2191 y Fu(.)18 b(F)m(or)12 b(example,)g([101])1022 2197 y Fn(2)1050 2191 y Fu(=)g([0101])1202 2197 y Fn(2)1230 2191 y Fu(=)g([00101])1403 2197 y Fn(2)1431 2191 y Fu(=)g(5.)17 b(One)d(w)o(a)o(y)337 2241 y(around)f(this)h(di\016cult)o(y)e(is)h(the)g(follo)o(wing)e (simple)g(\\folk)h(theorem",)g(whose)i(precise)337 2290 y(origins)g(are)h(unkno)o(wn)f(to)g(me)f(\(but)i(see)g([87)o(,)f(Note)h (9.1,)d(pp.)i(90{91],)e([101)o(,)i(p.)g(24],)337 2340 y(and)g([40)o(]\):)412 2392 y Fk(Theorem)22 b(3.3.)37 b Ft(L)n(et)19 b Fs(k)h Ft(b)n(e)g(an)g(inte)n(ger)f Fp(\025)h Fu(2)p Ft(.)32 b(Then)20 b(every)f(non-ne)n(gative)337 2441 y(inte)n(ger)g(c)n(an)g(b)n(e)f(r)n(epr)n(esente)n(d)g(uniquely)h (in)g(the)g(form)f Fs(n)g Fu(=)1326 2410 y Fr(P)1370 2454 y Fn(0)p Fm(\024)p Fq(i)p Fm(\024)p Fq(r)1475 2441 y Fs(a)1497 2447 y Fq(i)1511 2441 y Fs(k)1534 2426 y Fq(i)1548 2441 y Ft(,)h(wher)n(e)337 2491 y(the)c Fs(a)428 2497 y Fq(i)457 2491 y Ft(ar)n(e)g(inte)n(gers)f(with)g Fu(1)d Fp(\024)h Fs(a)870 2497 y Fq(i)896 2491 y Fp(\024)f Fs(k)q Ft(.)p eop %%Page: 4 4 4 3 bop 300 187 a Fu(4)475 b Fy(JEFFREY)12 b(SHALLIT)375 291 y Fu(F)m(or)j(example,)f(13)g(=)g(2)c Fp(\001)g Fu(4)g(+)h(2)f Fp(\001)f Fu(2)i(+)f(1)g Fp(\001)g Fu(1.)22 b(This)16 b(theorem)f(giv)o(es)g(a)g(bijection)300 341 y(b)q(et)o(w)o(een)j Fj(N)p Fu(,)f(the)g(non-negativ)o(e)f(in)o(tegers,)h(and)g(the)g (regular)f(language)g(\(1)11 b(+)g(2)g(+)300 391 y Fp(\001)c(\001)g (\001)h Fu(+)h Fs(k)q Fu(\))438 376 y Fm(\003)457 391 y Fu(.)375 441 y(There)k(are)g(man)o(y)d(other)i(w)o(a)o(ys)g(to)g (represen)o(t)i(the)f(non-negativ)o(e)e(in)o(tegers.)18 b(F)m(or)300 490 y(example,)f(let)g(the)i(Fib)q(onacci)e(n)o(um)o(b)q (ers)g(b)q(e)h(de\014ned)h(b)o(y)f Fs(F)1278 496 y Fn(0)1314 490 y Fu(=)g(0,)f Fs(F)1441 496 y Fn(1)1477 490 y Fu(=)h(1,)g(and)300 540 y Fs(F)327 546 y Fq(n)363 540 y Fu(=)c Fs(F)436 546 y Fq(n)p Fm(\000)p Fn(1)510 540 y Fu(+)d Fs(F)580 546 y Fq(n)p Fm(\000)p Fn(2)645 540 y Fu(.)21 b(The)16 b(follo)o(wing)c (theorem)j(giv)o(es)g(the)g(so-called)g Ft(Ze)n(ckendorf)300 590 y Fu(or)f Ft(Fib)n(onac)n(ci)g Fu(represen)o(tation)h([65,107)n(]:) 375 640 y Fk(Theorem)f(3.4.)20 b Ft(Every)12 b(non-ne)n(gative)i(inte)n (ger)e(c)n(an)h(b)n(e)f(r)n(epr)n(esente)n(d)g(uniquely)300 690 y(in)j(the)g(form)520 659 y Fr(P)564 702 y Fn(2)p Fm(\024)p Fq(i)p Fm(\024)p Fq(r)669 690 y Fs(a)691 696 y Fq(i)705 690 y Fs(F)732 696 y Fq(i)746 690 y Ft(,)f(wher)n(e)g Fs(a)912 696 y Fq(i)938 690 y Fp(2)d(f)p Fu(0)p Fs(;)c Fu(1)p Fp(g)p Ft(,)13 b(and)j Fs(a)1209 696 y Fq(i)1222 690 y Fs(a)1244 696 y Fq(i)p Fn(+1)1312 690 y Fp(6)p Fu(=)c(1)p Ft(.)375 739 y Fu(This)19 b(theorem)f(giv)o(es)h(a)f (bijection)h(b)q(et)o(w)o(een)h Fj(N)f Fu(and)g(the)g(regular)g (language)300 789 y Fs(\017)9 b Fu(+)h(1\(0)f(+)g(01\))534 774 y Fm(\003)553 789 y Fu(.)18 b(Notice)d(that)f(in)f(all)g(three)i (cases)h(w)o(e)e(ha)o(v)o(e)g(examined,)e(the)j(set)g(of)300 839 y(\\v)n(alid")f(represen)o(tations)k(is)d(a)h(regular)g(language.) 23 b(This)16 b(observ)n(ation)g(raises)g(the)300 889 y(question,)h(for)f(what)h(n)o(umeration)e(systems)h(is)h(the)g(set)g (of)f(v)n(alid)f(represen)o(tations)300 939 y(regular?)j(See,)c(for)g (example,)e([91)o(,48)o(,67)o(].)375 989 y(As)18 b(ab)q(o)o(v)o(e,)f (if)f Fs(m)i Fu(and)f Fs(n)g Fu(are)h(in)o(tegers,)g(then)g(w)o(e)f (can)h(uniquely)f(write)g Fs(m)h Fu(=)300 1038 y(2)321 1023 y Fq(a)339 1027 y Fi(1)359 1038 y Fu(+)r Fp(\001)7 b(\001)g(\001)r Fu(+)r(2)500 1023 y Fq(a)518 1027 y Fh(c)546 1038 y Fu(and)k Fs(n)g Fu(=)h(2)725 1023 y Fq(b)740 1027 y Fi(1)760 1038 y Fu(+)r Fp(\001)7 b(\001)g(\001)r Fu(+)r(2)901 1023 y Fq(b)916 1027 y Fh(d)935 1038 y Fu(,)k(where)g Fs(a)1096 1044 y Fn(1)1127 1038 y Fs(<)g Fp(\001)c(\001)g(\001)j Fs(<)i(a)1296 1044 y Fq(c)1323 1038 y Fu(and)f Fs(b)1419 1044 y Fn(1)1449 1038 y Fs(<)g Fp(\001)c(\001)g(\001)j Fs(<)i(b)1614 1044 y Fq(d)1633 1038 y Fu(.)300 1088 y(W)m(e)h(clearly)h (ha)o(v)o(e)749 1183 y Fs(mn)d Fu(=)883 1144 y Fr(X)865 1233 y Fn(1)p Fm(\024)p Fq(i)p Fm(\024)p Fq(c)988 1144 y Fr(X)967 1233 y Fn(1)p Fm(\024)p Fq(j)r Fm(\024)p Fq(d)1075 1183 y Fu(2)1096 1166 y Fq(a)1114 1170 y Fh(i)1127 1166 y Fn(+)p Fq(b)1167 1170 y Fh(j)1185 1183 y Fs(:)300 1317 y Fu(Kn)o(uth)16 b([57)o(])g(found)f(a)h(surprising)g(generalization)f (of)g(this)h(iden)o(tit)o(y:)22 b(if)15 b(the)h(Zec)o(k-)300 1367 y(endorf)i(represen)o(tation)h(of)d Fs(m)i Fu(is)g Fs(F)884 1373 y Fq(a)902 1377 y Fi(1)931 1367 y Fu(+)12 b Fs(F)1002 1373 y Fq(a)1020 1377 y Fi(2)1050 1367 y Fu(+)f Fp(\001)c(\001)g(\001)j Fu(+)i Fs(F)1224 1373 y Fq(a)1242 1377 y Fh(c)1260 1367 y Fu(,)18 b(and)f(the)h(Zec)o(k)o (endorf)300 1417 y(represen)o(tation)d(of)f Fs(n)f Fu(is)h Fs(F)726 1423 y Fq(b)741 1427 y Fi(1)768 1417 y Fu(+)c Fs(F)837 1423 y Fq(b)852 1427 y Fi(2)878 1417 y Fu(+)g Fp(\001)d(\001)g(\001)g Fu(+)j Fs(F)1046 1423 y Fq(b)1061 1427 y Fh(d)1080 1417 y Fu(,)j(de\014ne)726 1514 y Fs(m)d Fp(\016)f Fs(n)i Fu(=)899 1475 y Fr(X)882 1564 y Fn(1)p Fm(\024)p Fq(i)p Fm(\024)p Fq(c)1004 1475 y Fr(X)984 1564 y Fn(1)p Fm(\024)p Fq(j)r Fm(\024)p Fq(d)1092 1514 y Fs(F)1119 1520 y Fq(a)1137 1524 y Fh(i)1150 1520 y Fn(+)p Fq(b)1190 1524 y Fh(j)1207 1514 y Fs(:)300 1649 y Fu(Then)j(the)h Fp(\016)e Fu(m)o(ultiplication)e(is)j(asso)q(ciativ)o (e!)k(Also)13 b(see)i([7,43)o(].)375 1698 y(W)m(e)10 b(no)o(w)g(turn)g(to)g(the)h(represen)o(tation)h(of)d(rational)g(n)o (um)o(b)q(ers.)17 b(Let)11 b([)p Fs(a)1479 1704 y Fn(0)1497 1698 y Fs(;)c(:)g(:)g(:)t(;)g(a)1611 1704 y Fq(n)1633 1698 y Fu(])300 1748 y(b)q(e)15 b(an)e(abbreviation)g(for)h(the)g Ft(c)n(ontinue)n(d)i(fr)n(action)784 1855 y Fs(a)806 1861 y Fn(0)834 1855 y Fu(+)1043 1826 y(1)p 881 1845 347 2 v 881 1905 a Fs(a)903 1911 y Fn(1)931 1905 y Fu(+)1089 1877 y(1)p 977 1895 245 2 v 977 1955 a Fs(a)999 1961 y Fn(2)1027 1955 y Fu(+)9 b Fp(\001)e(\001)g(\001)h Fu(+)1184 1927 y(1)p 1172 1945 45 2 v 1172 1984 a Fs(a)1194 1990 y Fq(n)1232 1855 y Fs(:)-944 b Fu(\(3.1\))375 2064 y Fk(Theorem)12 b(3.5.)19 b Ft(Every)11 b(r)n(ational)f(numb)n(er)g(in)h Fu(\(0)p Fs(;)c Fu(1\))j Ft(c)n(an)h(b)n(e)g(expr)n(esse)n(d)g (uniquely)300 2114 y(in)k(the)g(form)823 2203 y Fu([0)p Fs(;)7 b(a)897 2209 y Fn(1)914 2203 y Fs(;)g(a)955 2209 y Fn(2)973 2203 y Fs(;)g(:)g(:)g(:)e(;)i(a)1088 2209 y Fq(n)1110 2203 y Fu(])300 2292 y Ft(wher)n(e)14 b(the)h Fs(a)508 2298 y Fq(i)537 2292 y Ft(ar)n(e)f(p)n(ositive)h(inte)n(gers)f (and)i Fs(a)1014 2298 y Fq(n)1048 2292 y Fp(\025)c Fu(2)p Ft(.)375 2342 y Fu(As)20 b(an)f(application)f(of)h(this)g(theorem,)h(w) o(e)g(pro)o(v)o(e)f(the)h(follo)o(wing)d(theorem,)300 2392 y(inspired)d(b)o(y)g([77)o(]:)375 2441 y Fk(Theorem)19 b(3.6.)28 b Ft(Ther)n(e)16 b(is)g(a)i(bije)n(ction)e Fs(r)g Fu(:)f Fj(N)g Fp(!)g Fj(Q)i Ft(such)h(that)e(b)n(oth)h Fs(r)h Ft(and)300 2491 y Fs(r)320 2476 y Fm(\000)p Fn(1)379 2491 y Ft(ar)n(e)d(c)n(omputable)g(in)g(p)n(olynomial)g(time.)p eop %%Page: 5 5 5 4 bop 611 187 a Fy(NUMBER)12 b(THEOR)m(Y)i(AND)f(F)o(ORMAL)g(LANGUA)o (GES)230 b Fu(5)412 291 y Ft(Pr)n(o)n(of)p Fu(.)18 b(It)c(su\016ces)h (to)f(giv)o(e)f(suc)o(h)i(a)e(bijection)h(b)q(et)o(w)o(een)h Fj(N)f Fu(and)g Fj(Q)c Fp(\\)e Fu(\(0)p Fs(;)f Fu(1\).)412 341 y(Let)15 b Fs(f)507 347 y Fq(k)541 341 y Fu(:)d Fj(N)h Fp(!)f Fu(\(1)d(+)h(2)g(+)g Fp(\001)d(\001)g(\001)h Fu(+)h Fs(k)q Fu(\))963 326 y Fm(\003)997 341 y Fu(b)q(e)15 b(the)g(map)e(that)i(tak)o(es)g(a)f(non-negativ)o(e)337 391 y(in)o(teger)f(to)f(its)g(represen)o(tation)h(in)f(base)h Fs(k)g Fu(using)e(digits)h Fp(f)p Fu(1)p Fs(;)7 b Fu(2)g Fs(:)g(:)g(:)t(;)g(k)q Fp(g)p Fu(,)k(as)h(discussed)337 441 y(in)j(Theorem)f(3.3,)f(and)h(let)h Fs(f)808 423 y Fm(\000)p Fn(1)804 453 y Fq(k)867 441 y Fu(b)q(e)g(the)g(in)o(v)o (erse)h(map.)i(Let)d Fs(g)h Fu(b)q(e)f(the)g(map)e(whic)o(h)337 490 y(tak)o(es)g(a)f(string)h(o)o(v)o(er)f(\(1)6 b(+)g(2)g(+)g(3\))862 475 y Fm(\003)893 490 y Fu(as)13 b(an)f(argumen)o(t)f(and)h(returns)i (a)e(list)g(of)f(strings,)337 540 y(where)22 b(the)f(3's)g(are)g (treated)h(as)e(delimiters.)37 b(F)m(or)21 b(example,)f Fs(g)q Fu(\(121313322\))h(=)337 590 y(\(121)p Fs(;)7 b Fu(1)p Fs(;)g(\017;)g Fu(22\).)15 b(Let)g Fs(h)f Fu(b)q(e)g(the)h (map)d(suc)o(h)i(that)549 675 y Fs(h)p Fu(\()p Fs(a)611 681 y Fn(1)630 675 y Fs(;)7 b(a)671 681 y Fn(2)689 675 y Fs(;)g(:)g(:)g(:)e(;)i(a)804 681 y Fq(k)823 675 y Fu(\))12 b(=)g(\(0)p Fs(;)7 b(a)973 681 y Fn(1)1000 675 y Fu(+)j(1)p Fs(;)d(:)g(:)g(:)t(;)g(a)1177 681 y Fq(k)q Fm(\000)p Fn(1)1249 675 y Fu(+)i(1)p Fs(;)e(a)1352 681 y Fq(k)1381 675 y Fu(+)i(2\))p Fs(:)337 761 y Fu(Then)15 b(w)o(e)f(de\014ne)h(the)f (bijection)g Fs(r)h Fu(as)f(follo)o(ws:)773 846 y Fs(r)q Fu(\()p Fs(n)p Fu(\))d(=)h([)p Fs(h)p Fu(\()p Fs(f)981 829 y Fm(\000)p Fn(1)977 857 y(2)1026 846 y Fu(\()p Fs(g)q Fu(\()p Fs(f)1099 852 y Fn(3)1119 846 y Fu(\()p Fs(n)p Fu(\)\)\)\)])p Fs(;)337 932 y Fu(where)k(the)f(function)f Fs(f)717 914 y Fm(\000)p Fn(1)713 943 y(2)777 932 y Fu(is)g(extended)i (in)e(the)h(ob)o(vious)e(w)o(a)o(y)h(to)g(op)q(erate)h(on)g(lists)337 982 y(of)f(strings.)412 1031 y(F)m(or)h(example,)e(consider)j(the)f (case)h Fs(n)d Fu(=)h(12590.)19 b(Then)d(its)e(represen)o(tation)j(in) 337 1081 y(base)c(3)f(using)g(digits)g Fp(f)p Fu(1)p Fs(;)7 b Fu(2)p Fs(;)g Fu(3)p Fp(g)j Fu(is)i(121313322.)j(This)d(is)g (transformed)g(b)o(y)g Fs(g)h Fu(in)o(to)f(the)337 1131 y(list)17 b(\(121)p Fs(;)7 b Fu(1)p Fs(;)g(\017;)g Fu(22\),)13 b(whic)o(h)k(is)f(mapp)q(ed)g(b)o(y)g Fs(f)1078 1113 y Fm(\000)p Fn(1)1074 1142 y(2)1140 1131 y Fu(in)o(to)g(\(9)p Fs(;)7 b Fu(1)p Fs(;)g Fu(0)p Fs(;)g Fu(6\).)24 b(Then)17 b Fs(h)g Fu(maps)337 1181 y(this)d(to)g(\(0)p Fs(;)7 b Fu(10)p Fs(;)g Fu(2)p Fs(;)g Fu(1)p Fs(;)g Fu(8\).)15 b(Hence)g Fs(r)q Fu(\(12590\))c(=)g([0)p Fs(;)c Fu(10)p Fs(;)g Fu(2)p Fs(;)g Fu(1)p Fs(;)g Fu(8)o(])h(=)k(26)p Fs(=)p Fu(269.)412 1231 y(It)i(remains)e(to)h(see)h(that)f Fs(r)h Fu(and)f Fs(r)949 1216 y Fm(\000)p Fn(1)1007 1231 y Fu(can)g(b)q(e)h(computed)f(in)f(p)q(olynomial)e(time.)337 1280 y(That)i Fs(f)459 1286 y Fn(3)490 1280 y Fu(and)g Fs(f)593 1263 y Fm(\000)p Fn(1)589 1292 y(2)650 1280 y Fu(can)g(b)q(e)g(computed)g(in)f(p)q(olynomial)e(time)h(is)i(easy)m (,)g(and)f(is)h(left)g(to)337 1330 y(the)k(reader.)k(F)m(or)14 b(the)i(p)q(olynomial)11 b(time)i(computabilit)o(y)e(of)j(con)o(tin)o (ued)h(fractions,)337 1380 y(see,)g(for)f(example,)e([8)o(,)h(Chapter)i (4].)p 1665 1355 18 2 v 1665 1378 2 24 v 1681 1378 V 1665 1380 18 2 v 412 1430 a(There)h(are)f(man)o(y)e(other)j(formal)c (language)i(asp)q(ects)i(of)e(con)o(tin)o(ued)h(fractions.)337 1480 y(Some)f(of)g(these)i(deal)e(with)g(the)h(so-called)g(\\LR")e(or)i (\\Stern-Bro)q(cot")h(represen)o(ta-)337 1530 y(tion)e(of)f(rational)g (n)o(um)o(b)q(ers)g([44)o(].)18 b(If)842 1615 y Fs(\022)13 b Fu(=)f([)p Fs(a)952 1621 y Fn(0)970 1615 y Fs(;)7 b(a)1011 1621 y Fn(1)1029 1615 y Fs(;)g(a)1070 1621 y Fn(2)1088 1615 y Fs(;)g(:)g(:)g(:)n Fu(])p Fs(;)337 1700 y Fu(then)15 b(the)g(LR-represen)o(tation)f(of)g Fs(\022)h Fu(is)f(the)g(string)840 1786 y Fs(R)872 1769 y Fq(a)890 1773 y Fi(0)909 1786 y Fs(L)937 1769 y Fq(a)955 1773 y Fi(1)973 1786 y Fs(R)1005 1769 y Fq(a)1023 1773 y Fi(2)1041 1786 y Fs(L)1069 1769 y Fq(a)1087 1773 y Fi(3)1113 1786 y Fp(\001)7 b(\001)g(\001)e Fs(:)337 1871 y Fu(Let)18 b Fs(a;)7 b(b;)g(c;)g(d)15 b Fu(b)q(e)i(in)o(tegers)h(with)f Fs(ad)11 b Fp(\000)g Fs(bc)17 b Fp(6)p Fu(=)g(0.)27 b(Raney)17 b([83)o(])f(ga)o(v)o(e)h(a)g (\014nite-state)337 1921 y(transducer)j(to)e(compute)g(the)h (LR-expansion)e(of)g Fs(\034)24 b Fu(=)19 b(\()p Fs(a\022)13 b Fu(+)g Fs(b)p Fu(\))p Fs(=)p Fu(\()p Fs(c\022)g Fu(+)f Fs(d)p Fu(\))18 b(from)337 1971 y(that)d(of)f Fs(\022)q Fu(.)20 b(Using)14 b(Raney's)g(theorem,)g(one)h(can)f(giv)o(e)g(a)g (purely)h(formal-langua)o(ge-)337 2021 y(theoretic)i(pro)q(of)d(of)h (the)g(fact)g(that)g Fs(\022)i Fu(has)e(b)q(ounded)h(partial)e(quotien) o(ts)h(i\013)g Fs(\034)20 b Fu(do)q(es)337 2070 y([90].)412 2143 y Fz(4.)k(The)f(Th)o(ue-Morse)e(sequence.)i Fu(Recall)c(from)f (the)j(previous)f(section)337 2192 y(that)14 b Fs(s)446 2198 y Fn(2)466 2192 y Fu(\()p Fs(n)p Fu(\))g(denotes)h(the)f(sum)f(of) g(the)i(bits)f(in)f(the)i(base-2)f(represen)o(tation)h(of)e Fs(n)p Fu(.)412 2242 y(No)o(w)23 b(de\014ne)g(an)g(in\014nite)f(w)o (ord)h Fz(t)i Fu(=)i Fs(t)1090 2248 y Fn(0)1108 2242 y Fs(t)1123 2248 y Fn(1)1142 2242 y Fs(t)1157 2248 y Fn(2)1182 2242 y Fp(\001)7 b(\001)g(\001)21 b Fu(o)o(v)o(er)i Fp(f)p Fu(0)p Fs(;)7 b Fu(1)p Fp(g)p Fu(,)23 b(as)f(follo)o(ws:)337 2292 y Fs(t)352 2298 y Fq(n)389 2292 y Fu(=)13 b Fs(s)453 2298 y Fn(2)473 2292 y Fu(\()p Fs(n)p Fu(\))e(mo)q(d)f(2.)22 b(This)15 b(in\014nite)g(w)o(ord)g(is)g(sometimes)e(called)i(the)g(Th)o (ue-Morse)337 2342 y(sequence,)j(b)q(ecause)f(b)q(oth)f(Th)o(ue)g([99)o (])f(and)g(Morse)h([75)o(])f(examined)f(its)i(prop)q(erties)337 2392 y(near)e(the)g(b)q(eginning)f(of)f(this)h(cen)o(tury)m(.)19 b(But)14 b(Prouhet)g(implicitly)c(used)k(the)f(de\014ni-)337 2441 y(tion)h(of)g Fz(t)f Fu(in)h(an)g(1851)f(pap)q(er)i(\([82)o(];)f (also)f(see)j([104)o(]\))d(that)i(ga)o(v)o(e)e(a)h(solution)g(to)g(the) 337 2491 y(m)o(ultigrade)e(problem.)p eop %%Page: 6 6 6 5 bop 300 187 a Fu(6)475 b Fy(JEFFREY)12 b(SHALLIT)375 291 y Fu(The)j Fg(m)o(ultigrade)d(problem)h Fu(\(or)i(T)m(arry-Escott)g (problem;)e(see)i([62)o(]\))f(is)h(to)f(\014nd)300 341 y(disjoin)o(t)e(sets)j Fs(U;)7 b(V)22 b Fu(that)711 310 y Fr(P)754 353 y Fq(u)p Fm(2)p Fq(U)831 341 y Fs(u)855 326 y Fq(i)880 341 y Fu(=)924 310 y Fr(P)968 353 y Fq(v)q Fm(2)p Fq(V)1044 341 y Fs(v)1065 326 y Fq(i)1092 341 y Fu(for)13 b Fs(i)f Fu(=)g(0)p Fs(;)7 b Fu(1)p Fs(;)g(:)g(:)g(:)t(;)g (k)h Fp(\000)g Fu(1.)18 b(Prouhet)300 395 y(observ)o(ed)f(that)f(one)h (could)f(tak)o(e)g Fs(U)k Fu(=)15 b Fp(f)p Fu(0)g Fp(\024)g Fs(n)h(<)f Fu(2)1157 380 y Fq(k)1192 395 y Fu(:)g Fs(t)1234 401 y Fq(n)1272 395 y Fu(=)g(0)p Fp(g)h Fu(and)g Fs(V)24 b Fu(=)16 b Fp(f)p Fu(0)f Fp(\024)300 445 y Fs(n)c(<)h Fu(2)401 430 y Fq(k)433 445 y Fu(:)f Fs(t)471 451 y Fq(n)505 445 y Fu(=)h(1)p Fp(g)p Fu(.)17 b(F)m(or)d(example,)e(w)o(e)i(ha)o(v)o (e)654 528 y(0)675 511 y Fq(i)698 528 y Fu(+)c(3)761 511 y Fq(i)783 528 y Fu(+)g(5)846 511 y Fq(i)869 528 y Fu(+)f(6)931 511 y Fq(i)956 528 y Fu(=)j(1)1021 511 y Fq(i)1044 528 y 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3075 m gs 1 -1 sc (B) col-1 sh gr /Times-Roman ff 210.00 scf sf 1575 3525 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 210.00 scf sf 1950 3375 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 900 3450 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 525 4500 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 1350 4125 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 210.00 scf sf 825 3975 m gs 1 -1 sc (B) col-1 sh gr /Times-Roman ff 210.00 scf sf 1950 3975 m gs 1 -1 sc (B-2) col-1 sh gr /Times-Roman ff 210.00 scf sf 2775 3300 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 210.00 scf sf 3225 3975 m gs 1 -1 sc (B) col-1 sh gr /Times-Roman ff 210.00 scf sf 3900 3825 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 4350 3975 m gs 1 -1 sc (B+2) col-1 sh gr /Times-Roman ff 210.00 scf sf 4800 4425 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 210.00 scf sf 2550 4875 m gs 1 -1 sc (B-2) col-1 sh gr /Times-Roman ff 210.00 scf sf 3825 4875 m gs 1 -1 sc (B) col-1 sh gr /Times-Roman ff 210.00 scf sf 2325 5400 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 3675 5400 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 4350 4950 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 210.00 scf sf 2700 4350 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 3375 4425 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 210.00 scf sf 3375 3450 m gs 1 -1 sc (0) col-1 sh gr /Times-Roman ff 210.00 scf sf 4275 3375 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 210.00 scf sf 3150 5025 m gs 1 -1 sc (1) col-1 sh gr $F2psEnd rs %%EndDocument @endspecial 439 1461 a Ff(Fig.)12 b(5.2)p Fy(.)k Fe(A)o(utomaton)d (gener)n(ating)i(the)e(c)n(ontinue)n(d)i(fr)n(action)e(exp)n(ansion)h (of)f Fv(f)t Fy(\()p Fv(B)r Fy(\))762 1598 y Fu(=)838 1558 y Fr(X)836 1647 y Fq(n)p Fm(\025)p Fn(0)906 1598 y Fs(t)921 1604 y Fq(n)944 1598 y Fs(X)981 1581 y Fn(2)p Fq(n)1030 1598 y Fu(+)c Fs(X)1117 1558 y Fr(X)1116 1647 y Fq(n)p Fm(\025)p Fn(0)1179 1598 y Fu(\()p Fs(t)1210 1604 y Fq(n)1242 1598 y Fu(+)g(1\))p Fs(X)1357 1581 y Fn(2)p Fq(n)762 1728 y Fu(=)42 b Fs(T)6 b Fu(\()p Fs(X)919 1711 y Fn(2)938 1728 y Fu(\))k(+)f Fs(X)s(T)d Fu(\()p Fs(X)1125 1711 y Fn(2)1145 1728 y Fu(\))k(+)f Fs(X)1308 1700 y Fu(1)p 1255 1719 128 2 v 1255 1757 a(1)g Fp(\000)g Fs(X)1363 1745 y Fn(2)1388 1728 y Fs(:)337 1838 y Fu(Hence)16 b(w)o(e)e(ha)o(v)o(e,)f(o)o(v)o(er)h Fs(GF)6 b Fu(\(2\),)627 1931 y(\(1)j(+)h Fs(X)s Fu(\))768 1914 y Fn(3)787 1931 y Fs(T)c Fu(\()p Fs(X)s Fu(\))886 1914 y Fn(2)915 1931 y Fu(+)k(\(1)f(+)g Fs(X)s Fu(\))1097 1914 y Fn(2)1117 1931 y Fs(T)d Fu(\()p Fs(X)s Fu(\))k(+)f Fs(X)16 b Fu(=)11 b(0)p Fs(:)412 2025 y Fu(The)23 b(theorem)f(of)g(Christol)g(is)h (remark)n(able)e(b)q(ecause)j(it)e(relates)h(a)f(purely)337 2074 y(n)o(um)o(b)q(er-theoretic)10 b(fact)f(\(algebraicit)o(y)g(in)g (\014nite)g(c)o(haracteristic\))i(to)e(a)g(purely)g(mac)o(hine-)337 2124 y(theoretic)18 b(fact)e(\(generation)g(b)o(y)g(a)g(\014nite)g (automaton\).)22 b(As)17 b(a)f(consequence,)i(one)337 2174 y(ma)o(y)10 b(obtain)h(transcendence)j(results)e(in)f(\014nite)h (c)o(haracteristic)h(b)o(y)e(pro)o(ving)f(that)i(no)337 2224 y(\014nite)g(automaton)d(can)i(generate)h(the)g(sequence)h(of)d (co)q(e\016cien)o(ts)i(of)f(an)f(appropriate)337 2274 y(formal)i(p)q(o)o(w)o(er)j(series.)20 b(F)m(or)14 b(example,)e(Allouc) o(he)i([2])f(used)j(this)e(tec)o(hnique)h(to)f(giv)o(e)337 2323 y(a)j(new)g(pro)q(of)f(of)g(the)h(transcendence)j(of)c Fs(\031)1043 2329 y Fq(q)1061 2323 y Fu(,)h(the)g(analogue)e(of)h Fs(\031)i Fu(in)e(the)h(\014eld)g(of)337 2373 y(formal)12 b(Lauren)o(t)i(series)i(o)o(v)o(er)d Fs(GF)6 b Fu(\()p Fs(q)q Fu(\).)412 2424 y(Other)12 b(results)f(along)f(this)g(line)g (include)g(those)h(of)f(Berth)o(\023)-20 b(e)12 b([11)o(,12)o(],)e(who) g(pro)o(v)o(ed)337 2481 y(that)434 2460 y Fq(\020)449 2464 y Fh(q)466 2460 y Fn(\()p Fq(n)p Fn(\))p 434 2472 79 2 v 453 2495 a Fq(\031)473 2487 y Fh(n)472 2504 y(q)532 2481 y Fu(is)16 b(transcenden)o(tal)g(for)f(1)f Fp(\024)h Fs(n)f Fp(\024)g Fs(q)d Fp(\000)g Fu(2,)k(a)g(result)h(previously)f (pro)o(v)o(ed)p eop %%Page: 10 10 10 9 bop 300 187 a Fu(10)454 b Fy(JEFFREY)12 b(SHALLIT)300 291 y Fu(b)o(y)k(Y)m(u)g([105)o(])g(for)g(ev)o(ery)i Fs(n)e Fu(suc)o(h)h(that)g(\()p Fs(q)12 b Fp(\000)f Fu(1\))p Fp(j)-16 b Fs(=)o(n)p Fu(.)26 b(Here)18 b Fs(\020)1263 297 y Fq(q)1298 291 y Fu(is)e(the)h(Carlitz)f(zeta-)300 341 y(function,)h(the)h(formal)c(p)q(o)o(w)o(er)k(series)g(analogue)e (of)h(the)g(ordinary)g(zeta-function.)300 391 y(Rec)o(her)11 b([84)o(])e(obtained)g(transcendence)k(results)e(for)e(p)q(erio)q(ds)i (of)e(generalized)h(Carlitz)300 441 y(exp)q(onen)o(tials,)g(i.e.,)f(of) g(generalizations)g(of)g Fs(\031)1009 447 y Fq(q)1027 441 y Fu(.)16 b(Berth)o(\023)-20 b(e)11 b([13)o(])e(pro)o(v)o(ed)h (transcendence)300 490 y(results)17 b(for)e(the)h(Carlitz)g(logarithm)d (and)i(ga)o(v)o(e)h(results)g(on)g(linear)f(expressions)i(in)305 524 y Fq(\020)320 528 y Fh(q)337 524 y Fn(\()p Fq(n)p Fn(\))p 305 536 79 2 v 324 560 a Fq(\031)344 551 y Fh(n)343 568 y(q)403 545 y Fu(for)d(1)f Fp(\024)g Fs(n)f Fp(\024)i Fs(q)c Fp(\000)g Fu(2)15 b([14)o(].)20 b(Allouc)o(he)14 b([3)o(])h(pro)o(v)o(ed)f(the)i(transcendence)h(of)d(the)300 605 y(v)n(alues)d(of)f(the)h(Carlitz-Goss)g(gamm)o(a)d(function)j(for)f (all)g Fs(p)p Fu(-adic)g(rational)g(argumen)o(ts)300 655 y(that)15 b(are)h(not)f(natural)g(n)o(um)o(b)q(ers,)g(and)g(Mend)o (\022)-20 b(es)17 b(F)m(rance)f(and)f(Y)m(ao)g([73)o(])f(extended)300 705 y(the)j(result)f(to)g Ft(al)r(l)k Fu(the)c(v)n(alues)g(of)g(the)g (Carlitz-Goss)g(gamm)o(a)d(function)j(at)f Fs(p)p Fu(-adic)300 755 y(argumen)o(ts)k(that)h(are)g(not)g(natural)f(n)o(um)o(b)q(ers.)35 b(Thakur)20 b(pro)o(v)o(ed)g([98)o(])f(that)h(the)300 804 y(p)q(erio)q(d)14 b(of)g(the)g(T)m(ate)g(elliptic)f(curv)o(e)i(is)e (transcenden)o(tal.)375 883 y Fz(6.)24 b(Automatic)11 b(real)h(n)o(um)o(b)q(ers.)19 b Fu(Giv)o(en)10 b(a)h Fs(k)q Fu(-automatic)f(sequence)j(\()p 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Fu(0110100110010)o(110)7 b Fp(\001)f(\001)g(\001)t Fs(:)300 1374 y Fu(It)22 b(follo)o(ws)e(from)g (a)i(general)g(result)g(of)f(Mahler)h([71)o(])g(that)f Fp(T)32 b Fu(is)22 b(transcenden-)300 1424 y(tal.)h(Mahler's)15 b(pro)q(of)h(tec)o(hnique)g(w)o(as)g(later)f(redisco)o(v)o(ered)j(b)o (y)d(Cobham)f([26)o(])h(and)300 1474 y(Dekking)e([30)o(].)540 1459 y Fn(3)375 1525 y Fu(It)h(ma)o(y)e(b)q(e)j(am)o(using)d(to)i(note) g(that)g(the)h(n)o(um)o(b)q(er)e Fp(T)24 b Fu(app)q(ears)15 b(\\naturally")e(as)300 1575 y(a)i(certain)h(probabilit)o(y)d(in)i (formal)e(language)h(theory)m(.)23 b(Let)15 b Fp(P)k Fu(b)q(e)d(the)g(probabilit)o(y)300 1625 y(that)e(a)g(randomly-c)o (hosen)f(language)g(o)o(v)o(er)h Fp(f)p Fu(0)p Fs(;)7 b Fu(1)p Fp(g)12 b Fu(con)o(tains)i(at)g(least)g(one)h(w)o(ord)f(of)300 1675 y(ev)o(ery)19 b(p)q(ossible)g(length.)32 b(\(Our)19 b(mo)q(del)e(is)i(to)f(decide)h(the)h(mem)o(b)q(ership)c(of)i(eac)o(h) 300 1725 y(w)o(ord)c(in)f Fs(L)h Fu(uniformly)d(at)j(random,)e(with)i (probabilit)o(y)1209 1708 y Fn(1)p 1209 1715 17 2 v 1209 1739 a(2)1230 1725 y Fu(.\))19 b(Then)425 1849 y Fp(P)45 b Fu(=)573 1810 y Fr(Y)573 1899 y Fq(i)p Fm(\025)p Fn(0)641 1849 y Fu(\(1)9 b Fp(\000)g Fu(2)749 1832 y Fm(\000)p Fn(2)792 1820 y Fh(i)807 1849 y Fu(\))j(=)878 1810 y Fr(X)879 1899 y Fq(j)r Fm(\025)p Fn(0)950 1821 y Fu(\()p Fp(\000)p Fu(1\))1035 1806 y Fq(t)1048 1810 y Fh(j)p 950 1840 116 2 v 989 1878 a Fu(2)1010 1866 y Fq(j)1082 1849 y Fu(=)1126 1810 y Fr(X)1127 1899 y Fq(j)r Fm(\025)p Fn(0)1198 1821 y Fu(1)d Fp(\000)g Fu(2)p Fs(t)1305 1827 y Fq(j)p 1198 1840 125 2 v 1241 1878 a Fu(2)1262 1866 y Fq(j)1339 1849 y Fu(=)j(2)d Fp(\000)g Fu(4)p Fp(T)h Fs(:)375 1989 y Fu(This)k(result)g(suggests)i(the)e(follo)o(wing)375 2040 y Fk(Conjecture)e(2.)19 b Ft(L)n(et)11 b Fs(k)q(;)c(b)k Ft(b)n(e)g(inte)n(gers)g Fp(\025)h Fu(2)p Ft(.)17 b(If)11 b Fu(\()p Fs(s)1197 2046 y Fq(i)1212 2040 y Fu(\))1228 2046 y Fq(i)p Fm(\025)p Fn(0)1296 2040 y Ft(is)g(a)g(non-ultimately-) 300 2090 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b(Kaneps)g(&)h(F)m(reiv)n(alds)e([50)o(];)g (Shallit)g(&)h(Breitbart)h([93)o(,94)o(],)e(P)o(omerance,)h(Rob-)300 2441 y(son,)h(&)h(Shallit)f([80)o(],)f(Glaister)i(&)g(Shallit)e([42)o (],)h(and)g(Shallit)g([92)o(].)k(Kosk)n(as)d(and)f(de)300 2491 y(Mathan)e(\(w)o(ork)g(in)g(progress,)i(1996\))d(sho)o(w)h(ho)o(w) g(to)g(apply)g(automaticit)o(y)e(to)i(obtain)p eop %%Page: 15 15 15 14 bop 611 187 a Fy(NUMBER)12 b(THEOR)m(Y)i(AND)f(F)o(ORMAL)g (LANGUA)o(GES)209 b Fu(15)337 291 y(irrationalit)o(y)12 b(measures)i(in)g(\014nite)g(c)o(haracteristic.)412 341 y(One)h(of)e(the)i(nicest)g(results)g(in)e(the)i(area)f(is)f(Karp's)i (theorem)e([51)o(]:)412 391 y Fk(Theorem)k(8.3.)k Ft(L)n(et)15 b Fs(L)c Fp(\022)h Fu(\006)893 376 y Fm(\003)927 391 y Ft(b)n(e)j(a)g(nonr)n(e)n(gular)g(language.)20 b(Then)841 465 y Fs(A)872 471 y Fq(L)897 465 y Fu(\()p Fs(n)p Fu(\))11 b Fp(\025)h Fu(\()p Fs(n)d Fu(+)h(3\))p Fs(=)p Fu(2)337 539 y Ft(for)15 b(in\014nitely)g(many)g Fs(n)p Ft(.)412 589 y 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b(accepting)h(an)e Fs(n)p Fu(-th)g(order)i(appro)o(ximation)10 b(to)k Fs(L)f Fu(can)h(no)o(w)f(b) q(e)h(con-)337 2342 y(structed)i(as)e(follo)o(ws:)446 2392 y Fp(\017)20 b Fu(guess)15 b(the)g(correct)g(o)q(dd)f(prime)f(p)q (o)o(w)o(er)h Fs(p)1148 2377 y Fq(k)1180 2392 y Fp(\024)e Fu(5)7 b(log)f Fs(n)p Fu(;)446 2441 y Fp(\017)20 b Fu(v)o(erify)14 b(that,)f(on)h(input)g(0)894 2426 y Fq(r)912 2441 y Fu(,)f(w)o(e)h(ha)o (v)o(e)615 2491 y Fp(\003)20 b Fs(r)13 b Fp(6\021)f Fu(0)h(\(mo)q(d)g Fs(p)897 2476 y Fq(k)917 2491 y Fu(\);)p eop %%Page: 18 18 18 17 bop 300 187 a Fu(18)454 b Fy(JEFFREY)12 b(SHALLIT)577 291 y Fp(\003)21 b Fs(r)12 b Fp(\021)g Fu(0)i(\(mo)q(d)e(2)859 276 y Fq(s)877 291 y Fu(\),)h(with)h(2)1034 276 y Fq(s)1063 291 y Fs(<)e(p)1128 276 y Fq(k)1160 291 y Fs(<)f Fu(2)1224 276 y Fq(s)p Fn(+1)1284 291 y Fu(.)300 341 y(This)j(construction)h (uses)g(at)e(most)g Fs(O)q Fu(\(\(log)7 b Fs(n)p Fu(\))1040 326 y Fn(3)1059 341 y Fs(=)p Fu(\(log)f(log)g Fs(n)p Fu(\)\))15 b(states.)375 391 y Fk(Open)h(Question)g(9.)k Ft(What)15 b(is)f(a)h(go)n(o)n(d)g(lower)e(b)n(ound)i(on)g(the)f (nondetermin-)300 441 y(istic)i(automaticity)i(of)f(the)g(set)g Fs(P)858 426 y Fq(R)885 441 y Ft(,)g(the)g(\(r)n(everse)n(d\))g(r)n (epr)n(esentations)f(of)h(primes)300 490 y(in)e(b)n(ase)g Fu(2)p Ft(?)375 563 y Fz(9.)24 b Fs(k)q Fz(-regular)c(sequences.)j Fu(The)c(last)g(topic)g(I)g(wish)g(to)f(consider)i(in)f(this)300 613 y(surv)o(ey)d(is)g Fs(k)q Fu(-regular)f(sequences.)26 b(These)16 b(are)g(generalizations)g(of)f(the)h(automatic)300 663 y(sequences)g(men)o(tioned)d(ab)q(o)o(v)o(e)h(in)f(Section)i(5.)375 713 y(While)c(there)i(are)f(man)o(y)e(examples)h(of)g(automatic)f (sequences)k(in)d(n)o(um)o(b)q(er)g(the-)300 763 y(ory)m(,)g(their)i (expressiv)o(e)h(p)q(o)o(w)o(er)e(is)g(somewhat)f(limited)f(b)q(ecause) k(of)d(the)i(requiremen)o(t)300 812 y(that)k(they)h(tak)o(e)f(only)g(a) g(\014nite)g(n)o(um)o(b)q(er)g(of)f(v)n(alues.)28 b(Ho)o(w)17 b(can)h(this)f(b)q(e)h(general-)300 862 y(ized?)g(As)c(w)o(e)g(ha)o(v)o (e)f(seen)i(ab)q(o)o(v)o(e)e(in)g(Section)h(5,)f(a)g(sequence)j(is)d Fs(k)q Fu(-automatic)f(i\013)h(its)300 912 y Fs(k)q Fu(-k)o(ernel)i(is) g(\014nite.)21 b(This)15 b(suggests)h(studying)f(the)g(class)h(of)e (sequences)j(where)f(the)300 962 y Fj(Z)p Fu(-mo)q(dule)g(generated)i (b)o(y)f(the)h Fs(k)q Fu(-k)o(ernel)f(is)g Ft(\014nitely)h(gener)n(ate) n(d)p Fu(.)27 b(W)m(e)17 b(call)f(suc)o(h)i(a)300 1012 y(sequence)f Fs(k)q Fu(-regular.)k(The)15 b(prop)q(erties)h(of)e(suc)o (h)i(sequences)h(and)e(man)o(y)e(examples)300 1062 y(w)o(ere)i(giv)o (en)e(in)h([6)o(].)375 1111 y(Here)h(are)f(some)f(examples)g(of)h Fs(k)q Fu(-regular)f(sequences)k(in)c(n)o(um)o(b)q(er)g(theory)m(.)300 1211 y Ft(Example)19 b(1.)29 b(The)19 b Fu(3)p Ft(-adic)f(valuation)h (of)f(a)h(sum)f(of)h(binomial)f(c)n(o)n(e\016cients.)29 b Fu(Let)300 1261 y Fs(r)q Fu(\()p Fs(n)p Fu(\))12 b(:=)444 1229 y Fr(P)488 1273 y Fn(0)p Fm(\024)p Fq(i