Journal of Universal Computer Science, vol. 1, no. 7 (1995), submitted: 14/6/95, accepted: 14/6/95, appeared: 28/7/95 Springer Pub. Co.

Transcription

1 Jounal of Univesal Comute Science, vol., no. 7 (995), submitted: 4/6/95, acceted: 4/6/95, aeaed: 28/7/95 Singe Pub. Co. LCF: A Lexicogahic Binay Reesentation of the Rationals Pete Koneu (Det. of Mathematics and Comute Science Odense Univesity DK-5230 Odense, Denmak koneu@imada.ou.dk) David W. Matula (Det. of Comute Science and Engineeing Southen Methodist Univesity Dallas, TX matula@seas.smu.edu) Abstact: A binay eesentation of the ationals deived fom thei continued faction exansions is descibed and analysed. The concets \adjacency", \mediant" and \convegent" fom the liteatue on Faey factions and continued factions ae suitably extended to ovide a foundation fo this new binay eesentation system. Wost case eesentation-induced ecision loss fo any eal numbe by a xed length eesentable numbe of the system is shown to be at most 9% of bit wod length, with no ecision loss whatsoeve induced in the eesentation of any easonably sized ational numbe. The eesentation is suoted by a comute aithmetic system imlementing exact ational and aoximate eal comutations in an on-line fashion. Categoy: G..0 [Numeical Analysis]: Comute Aithmetic. B.5. [Registe Tansfe Level Imlementation]: Aithmetic and logic units. E.2 [Data Stoage Reesentations]. Key Wods: Comute aithmetic, continued factions, lexicogahic, numbe systems, numbe theoy, ational numbes. Intoduction. The foundations of a binay eesentation of the ationals ae esented, and many of the eesentation system's featues ae descibed. Evidence is ovided indicating that a comute aithmetic system emloying this eesentation would ovide a facility fo exact ational and aoximate eal aithmetic not cuently available in any single system. Ou oosed binay eesentation system deives fom the continued faction eesentation of the ationals. A self delimiting bitsting encoding of the integes is emloyed to eesent each atial uotient. Paticula featues of the intege encoding and the subseuent concatenation ocess allow us to obtain bit sting eesentations of the ationals, which ae shown lexicogahically ode eseving ove eal ode. Ou bitsting eesentation is thus temed the lexicogahic continued faction (LCF) eesentation of a ational numbe. The LCF eesentation can be consideed an encoding of the individual stes of the Euclidean algoithm efomed in binay, whee the detemination of the individual emaindes ae comuted using a non-estoing division algoithm. As such, it is deived fom algoithms efoming aithmetic oeations 484

2 uon ational oeands in faction fom, i.e. a numeato/denominato eesentation, [see Koneu and Matula 83] whee the LCF eesentation was st descibed. Howeve, as a numbe eesentation it natually leads to a kind of on-line aithmetic whee oeands ae consumed bit-seuential, and the esult is oduced bit-seuential, most signicant bit st. Such an on-line aithmetic unit has been descibed in [Koneu and Matula 88], caable of efoming all the basic aithmetic oeations in a unied manne as cases of the bihomogahic function axy + bx + cy + d z(x; y) = exy + fx+ gy + h secied by eight intege coecients a; b; ; h. By factoing cetain tansfomations (matices coesonding to the individual atial uotients of a continued faction) into simle \binay" matices, the algoithm can be ealized by simle shift-and-add oeations. Howeve, we shall not futhe usue the aithmetic hee, but concentate on oeties of the LCF eesentation. In [Section 2] we fomally dene the LCF exansion as a bitsting. We intoduce backgound mateial fom the theoy of continued factions to guide the develoment of a theoy fo LCF exansions. In aticula we extend the notion of the seuence of convegents (often temed \best ational aoximations") of a eal numbe to a sue-seuence of biconvegents (binay convegents) detemined by the LCF exansion of x. The biconvegents ae shown to fom a somewhat base deendent seuence of ational aoximations to x. The biconvegent seuence is shown to contain, on aveage, about 3.5 times the numbe of tems of the subseuence of canonically dened convegents. In [Section 3] we study the hieachy of ational numbes as detemined bitwise by thei LCF exansions though the constuct of the LCF binay tee. The LCF tee ovides fo enumeating biconvegent seuences as aths down the tee, and also ovides fo enumeating all xed length LCF exansion values by tavesal acoss the LCF tee tuncated at xed deth. One can visualize in the stuctue of the tee the ode eseving oety of LCF exansions, and the fact that LCF eesentation is one-to-one between nite bit stings and ositive ationals. We develo tools fo investigating the set Q k of ieducible factions in [0,] whose LCF exansions have ode k (euivalently: length k + bits o deth at most k in the LCF tee). Ou incial esults ae that the fundamental oeties fom the theoy of Faey factions [Hady and Wight 79] egading adjacency, mediant and ecusive constuction of the tee of Faey factions, can be extended to comaable concets of bijacency, binay mediant and ecusive constuction of the LCF tee. Poeties of the LCF tee ae then available as tools fo both the investigation of the ate of convegence of biconvegent seuences, and fo the study of the ga sizes between successive membes of the sets Q k. The latte esult dictates the ecision obtainable fo aithmetic emloying such xed length eesentations. Utilizing these tools the extemes of ga size vaiability ove Q k ae then discussed in [Section 4]. The main esult is that the maximum ga size in Q k is of the ode 2,ak fo a =0:84 :::. This imlies at most a 9% ecision loss (stoage sace loss) in the wost case aoximation eo by xed length LCF bit stings, being the ice to be able to accommodate the exact eesentation of a set of simle ationals at a faily egula sacing, and suoting a ational aithmetic in an on-line fashion. Moe detailed esults of exhaustive and samled distibution of ga sizes ae available in [Koneu and Matula 85]. 485

3 2 Continued Faction and Lexicogahic Continued Faction Exansions. The lexicogahic continued faction exansion of a ational numbe is a bitsting whose inteetation will be based on some fundamental oeties of continued faction exansions of ationals. Fo these uoses ou notation should make exlicit the aticula numeato and denominato comonents of a faction, as well as the aticula seuence of atial uotient values of a continued faction exansion, as these tems ae not necessaily uniuely detemined. Fo ou uoses it is sucient to teat only eesentation of nite nonnegative ational numbes, as signs can be aended fo the negative values. Fomally, a faction, denoted = o, is heein an odeed ai comosed of a nonnegative intege numeato, and a ositive intege denominato. The uotient of= is the ational numbe detemined by the atio of to. The numeato and denominato of an ieducible faction must have a geatest common diviso (gcd) of unity, othe factions being temed educible. Emloying the euality symbol between vaious foms of ational eesentation will heein denote the weake inteetation of euality between thei ational values with the following excetion. Euality between factions denoted with the hoizontal ba fomat shall imly eual numeato and denominato values. Thus i = and = s; wheeas = s = = =s i = s: Notationally, the symbol < is used to denote the simle than elation between factions, and is dened ove all ais of factions by < s i 6= s and both ; s: Fo examle 0 <, and 2 <2 3 < 2 4. We utilize the notation [a 0 =a =a 2 = =a n ] fo the n-th ode (simle) continued faction exansion a 0 + a + a whee the atial uotients a i ae assumed to be integal with a 0 0, a i fo i n.itisknown fom the theoy of continued factions [Khinchin 35, Hady and Wight 79] that any ositive ational numbe, denoted by the ieducible faction, has exactly two nite exansions heein temed the canonical and long exansions, as given and elated by: a n canonical ( = [a0 =a = =a n, =a n ] [a 0 =a = =a n, =a n, =] long whee a n 2 fo n fo n =0 486

4 with 0 having the uniue and canonical exansion [0]. It follows that any ositive ational numbe has both a uniue even ode continued faction exansion [a 0 =a = =a 2m ] togethe with a uniue odd ode continued faction exansion [a 0 =a = =a 2m+ ]. It is the uniue even ode exansion [a 0 =a = =a 2m ] that will late be emloyed fo the denition of the lexicogahic continued faction exansion. The ieducible factions i i = [a 0 =a = =a i ] detemined fo 0 i n by tuncating the continued faction [a 0 =a = =a n ] constitute a seuence of ieducible faction aoximations to = n =[a n 0 =a = =a n ] temed convegents to n,. The econvegent shall denote the convegent immediately n, eceding n n in the canonical exansion fo. Note that the long exansion of 6= 0 includes one additional convegent temed the aent in addition to those fo the canonical exansion of.fo examle, the canonical exansion 277 = 642 =[0=2=3=6==3=3] has the convegents 0 ; 2 ; 3 7 ; 9 44 ; 22 5 ; ; 277 with econvegent The aent of is then =[0=2=3=6==3=2] The convegents have many imotant oeties, some of which ae cited hee fo efeence fom [Hady and Wight 79, Khinchin 35]: Theoem. The convegents i =[a i 0 =a = =a i ] of any (canonical o long) continued faction =[a =a 2 = =a n ] fo i =0; ; ;n satisfy the following oeties: i) Recusive ancesty: With,2 =0;, =;,2 =, and, =0, i = a i i, + i,2 ; i = a i i, + i,2 ; o in matix fom: i,2 i, i,2 i, 0 a i = i, i i, i and 0 0 i Y j=0 0 a i = i, i ; i, i ii) Ieducibility: iii) Adjacency: iv) Simlicity: v) Altenating convegence: gcd( i ; i )=; i i,, i i, =(,) i ; i i < i+ i+ fo i n, ; 0 < 2 < 2i < 0 2 2i < 2i, < < ; 2i, 487

5 vi) Best ational aoximation: vii) Quadatic convegence: s < i i =) i ( i+ + i ) < i s, i, > i, i ; fo i n, ; i i+ viii) Real aoximation: x, < fo ieducible 2 2 imlies that is a convegent of a (ossibly innite) continued faction exansion of x. Fom Theoem (v) we see that the even ode convegents aoach fom below and the odd ode convegents aoach fom above, with the inteval between any two successive convegents containing. [Tab. ] illustates the ate at which the seuence of convegent values fo 277 =[0=2=3=6==3=3] give bette aoximations to 277/642. Note fom the 642 table that the accuacy attained by successive convegents has a lage incemental imovement when the next atial uotient is lage, as anticiated by Theoem (vii). Continued Faction Decimal Relative faction eesentation eo [0] 0/ 0:0 [0/] /2 0: [0/2/3] 3/7 0: [0/2/3/6] 9/44 0: [0/2/3/6/] 22/5 0: [0/2/3/6//3] 85/97 0: [0/2/3/6//3/3] 277/642 0: Table : The canonical convegents to 277 in continued faction, faction, and decimal 642 fom, and the elative eos of these convegents as best ational aoximations. Conside fom Theoem and the examle of [Tab. ] that the notion of an i'th ode best ational aoximation by itself is not useful in nite ecision comutational actice as the esulting accuacy deends without bound on the size of the aticula atial uotients involved. The lexicogahic continued faction exansion we now intoduce will be shown uite analogously to identify a seuence of ational aoximations temed "biconvegents". The biconvegents contain and sulement the seuence of convegents of the canonical continued faction suitably ganulaized at the bit level to allow a measue of accuacy in tems of bit length. Fo the uose of dening a binay based lexicogahic continued faction exansion, we will emloy a binay eesentation of the ositive integes which 488

6 is both "self delimiting", i.e. which imlicitly contains an end-make when ead fom left to ight, and lexicogahically ode eseving ove the integes. Fomally, if the intege has the (n + )-bit binay adix eesentation b n, b b 0, with n 0 and denoting sting concatenation, the (2n + )-bit bitsting `() n 0 b n, b n,2 b b 0 () will be temed the lexibinay fom of. `() isthus comosed on a (ossibly vacuous) unay at n delimited by the switch-bit 0, followed by the (ossibly vacuous) binay at b n, b P n,2 b 0. The value of the lexibinay intege n 0b n, b n,2 b 0 is then 2 n n, + i=0 b i2 i. This eesentation is ode eseving in that the lexicogahic odeing (leftmost bit st) of lexibinay bitstings is seen to coesond to the numeic odeing of thei values. [Tab. 2] illustates the lexibinay fom of seveal integes. Fo a discussion of altenative lexicogahic ode eseving binay encodings of the integes [see Knuth 82], whee simila eesentations ae analysed. Intege Binay Lexibinay Table 2: Right-adjusted standad binay eesentation and left-adjusted lexibinay bitsting eesentation of cetain integes. Note fom the denition of a continued faction exansion that [a 0 =a = =a n ] is an inceasing function of any even ode atial uotient, and a deceasing function of any odd ode atial uotient. Thus to obtain an ode eseving eesentation of the ationals we simly eesent the odd ode uotients in comlemented lexibinay intege fom befoe concatenation. To be able to comae bit stings lexicogahically fom left to ight it is assumed that any (nite length) eesentation is extended to the ight with an abitay numbe of exta zeoes. This coesonds with the obsevation that, suitably inteeted and extending the even ode continued faction exansion = =[a 0 =a = =a 2m ]=[a 0 =a = =a 2m =]: (2) 489

7 If we dene `() to be an innite sting of ones, then since occus in an odd ode osition in (2), `() will always aea in comlemented fom yielding `() = 00. This ovides a teminal innite sting of zeoes, which may eithe be denoted by 0 o taken as assumed. Fomally, imlicitly handling the case a 0 =0(0 = < ) by a leading zeo bit and a 0 ( )by a leading unit bit, the lexicogahic continued faction (LCF) exansion of 0 is the (innite) bitsting detemined emloying () and (2) by: LCF = ( `(a0 ) `(a ) `(a 2m ) `() fo 0 `(a ) `(a 2m, ) `(a 2m ) `() fo 0 < (3) The ositive valued LCF exansion b 0 b b k, 0 is said to have ode k (the index of the least signicant unit), with 0 =0 having ode zeo. The LCF exansion denoted with the concatenation symbol at each coesonding atial uotient bounday as in (3) is said to be in ased fom. The leading bit b 0 of the LCF exansion b 0 b b k, 0 is temed the eciocal bit. Note that all subseuent bits may uniuely be identied as membes of the unay, switch, o binay otions of the i'th ode atial uotient in the even ode continued faction exansion in (3). The nite bit sting b 0 b b n, with o without tailing zeos, is taken as an altenative nite LCF exansion euivalent to b 0 b b n 0. The minimal LCF exansion b 0 b b k, of > 0 is tuncated at the last unit bit, and thus including the eciocal bit has length one geate than the ode of the LCF exansion. Zeo is taken to have its minimal LCF exansion comosed of the single 0 eciocal bit. Examle. 22 = 22 ieducible faction fom 7 7 =[3=6=] even ode continued faction fom =[3=6==] innite extension 9 =`(3) `(6) `() `() = = euivalent ased LCF exansions ; = = 0000 minimal LCF exansion (ode 8) The ieducible faction k s k = b 0 b b k, fo 0 k n detemined by tuncating the lexicogahic continued faction s = n s n = b 0 b b n, at index k, and aending a unit bit, is temed the k'th ode biconvegent (binay convegent)of 0. Each biconvegent ; s s 0 s n s n in seuence ovides then eithe an imoved ue o lowe bound on s detemined by ( k = b 0 b b k, if s b k =0; s k s if b k =: To comae biconvegent aoximation with the k-bit binay adix aoximation, we can comute the \ecision" of the k'th ode biconvegent aoximation in bits by the negative base two logaithm of the bounding intevals, 490

8 as illustated by the following table of biconvegents of 277 though ode 2: 642 k k =s k / /2 /4 /3 2/5 4/9 3/7 7/6 3/30 25/58 9/44 22/5 4/95, log 2 (ga) Some useful facts about biconvegents follow fom the denition and cetain oeties of convegents, and we will summaize these in some obsevations. Noting that the eciocal of =[a 0=a = =a n ] is =[0=a 0=a = =a n ], so then the convegents to ae 0 and the eciocals of the convegents to. Fom (3) then Obsevation2. The LCF exansion of the eciocal of = b 0b b k, is the 2's comlement, = b 0b b k,. Thus the eciocal of a k'th ode LCF numbe is also of k'th ode. Obsevation 3. The biconvegents to the eciocal = b 0b b k, of = b 0 b b k, ae the eciocals of the biconvegents to. It is immediate fom (3) that the even ode convegents of the canonical continued faction fo ae also biconvegents to. The odd ode convegents to have eciocals that ae even ode convegents to and thus ae biconvegents. Using [Obsevations 2,3] it follows that to Obsevation 4. Evey convegent of the canonical continued faction exansion of is also a biconvegent to. Although the ode of the lexicogahic continued faction can be abitaily lage comaed to the ode of the odinay continued faction fo a given ational, on the aveage the odes can be elated. Fom classical mateial on continued factions it is known that the atial uotients in the continued faction exansion of a andomly chosen 2 [0; ] (see [Knuth 8] o [Blachman 84] s fo details) will have value i with obability essentially given by whee then i = log 2 + i(i +2) =0:45; 2 =0:70; 3 =0:093; 4 =0:059; : ; (4) With the distibution of atial uotient size given by (4), we note that 4.5% of all atial uotients ae unity and ae encoded by a single bit in the LCF exansion. Anothe 26.3% of the atial uotients have values two o thee and contibute 3 bits each to the LCF exansion, and an aveage atial uotient fom (4) has exected length X (2blog 2 ic + ) log 2 + =3:5 : i(i +2) i 49

9 Thus in summay, Obsevation 5. Fom the known distibution (4) of atial uotient size, it follows that the canonical continued faction exansion and LCF exansion of a ational =[a =a 2 = =a n ]=b 0 b b k, yield an exected biconvegent to convegent atio of Ex( k n )=3:5. Obsevation 6. Fom the known distibution (4) of atial uotient size, it follows that =3:5 =28:5% of the biconvegents to will also be convegents to, i.e. the so-called best ational aoximations which ae chaacteized without any deendence on the binay eesentation emloyed fo the LCF exansion. LCF exansions and biconvegent aoximations have many oeties of theoetical inteest and/o which nd use in emloying LCF eesentation as a basis of comute aithmetic unit design. We shall aticulaly usue heein issues elated to assessing the accuacy of nite ecision comutation emloying xed length LCF exansions. A summay of elated toics [see Koneu and Matula 83, 88] that will not be futhe usued in this ae ae listed hee fo efeence: Regading uniueness of LCF eesentation of the nonnegative ationals: { Thee is a one-to-one coesondence between all minimal LCF exansions and the nonnegative ational numbes. Regading aithmetic with LCF eesented numbes: { LCF exansions may be used bit-by-bit in a left-to-ight scan as inut to on-line algoithms fo the diect comutation of aithmetic exessions uon LCF oeands yielding LCF esults [Koneu and Matula 88]. In this context the LCF bitsting may be inteeted as an encoding of the individual stes (tansitions) in a nite automaton efoming the Euclidean gcd algoithm (in binay) on and. Regading the eciency in bit-length of LCF exansions: { The minimum edundancy encoding, Human encoding, and LCF exansion aveage bit-length e atial uotient can be comuted emloying classical esults [Knuth 8] on the distibution of the size (4) of atial uotients, yielding: Aveage Bits e Patial Quotient Minimum edundancy encoding , Human encoding , LCF exansion Thus the comutationally useful fomat of the LCF exansion is achieved with an encoding length only about 2% geate than that which could be obtained by any minimal edundancy encoding, and only about % geate than that achievable by a Human encoding, whee the latte two encodings would most likely be of no actical value fo aithmetic comutation. 492

10 H H, J, J X X X,, X X J X H H H, J J h h h h Regading multilicative and additive inveses: { LCF eesentation may be extended with a sign as follows: ( LCF ( SLCF = ) fo 0; 0 LCF( g ) fo < 0: whee g LCF denotes the 2's comlementofany minimal nite LCF bitsting, and the 's comlementof any innite bitsting. SLCF eesentation is then ode eseving ove the eals. Note then that SLCF eesentation has leading sign and eciocal bits that teat both the additive and multilicative inveses of eal numbes in an analogous manne. 3 The LCF Rational Numbe Hieachy A hieachy is imosed on the ationals by the ode (length) of thei LCF exansions. The enumeation of this ational hieachy is conveniently illustated by associating the ositive ieducible factions with the nodes of an innite binay tee, temed the LCF tee, whee the LCF bitsting denotes the ath to the node containing the associated ieducible faction. The faction = b 0b b k, of ode k is assigned to the node at deth k eached by oceeding to the left child when b i = 0 and to the ight child when b i = fo i =0; ; 2; ;k,. The left half of the LCF tee tuncated at deth 5 is illustated in [Fig. ], whee we note that the values in the nodes of the ight half of the LCF tee ae simly the eciocals of those in the left half eached by the comlemented bitsting. 0 J J 0 J J 0 J 0 J 0 J J 0 J 0 J H ( ( ( ( ( ( ( ( ( ( ( ( ( Figue : The left half of the LCF tee though deth ve. The LCF tee ovides a convenient efeence fo inteeting the accuacy of nite ecision LCF eesentation both "vetically" and "hoizontally": h 493

11 Vetically: The ath fom the oot down to any nodeenumeates the biconvegents to the ieducible faction at that node, e.g. ; 2 ; 4 ; 3 ; 2 5 ; 4 is the 9 seuence of biconvegents to 4. 9 Hoizontally: The (nonzeo) elements of the set Q k of ieducible factions of [0; ] whose LCF exansions have ode at most k can be enumeated by an inode tavesal of the left half of the LCF tee tuncated at deth k, e.g. including 0 ; : 0 Q2 = 2 ; 4 ; 2 ; 3 ; ; and Q 4 = 0 ; 6 ; 8 ; 6 ; 4 ; 2 7 ; 3 ; 2 5 ; 2 ; 5 9 ; 3 5 ; 5 8 ; 2 3 ; 3 4 ; 4 5 ; 8 9 ; ; : Thee ae then 2 k + membes of Q k in the inteval [0; ] yielding an aveage ga size of 2,k. The denition of LCF eesentation ovides no immediate clue as to the extent ofvaiation of these ga sizes fo xed k, knowledge of which is essential to assess the ecision obtainable by k-bit LCF aoximation. In this section we develo tools fo investigating ga sizes in Q k and thei elation to ga sizes in Q k+.fo uoses of analysis we shall be aticulaly concened with chaacteizing elations between LCF eesented numbes in tems of thei moe familia ieducible faction and continued faction eesentations. Fo the factions = < =s, the size of the inteval [=; =s] is given by the exession (, s)=s and will be a minimum elative to the size of the denominato s when j, sj =.Wesay the factions and s ae adjacent wheneve j, sj =.Fo adjacent to it follows immediately that both s and s ae ieducible, and that eithe < s o s <. Note that some, but not all, successive ais of factions of Q k ae adjacent. The notion of adjacency also identies an imotant elation among continued factions that will fom a bidge to undestanding the neighbo elations fo membes of Q k. The following theoem ovides altenative chaacteizations. Theoem 7. Fo the factions < s (i.e. 6= s and, s) each of the following fou oeties imlies the othe thee and seves as an euivalent denition of adjacency. i) Deteminant fom: j, sj =, ii) Inteval fom:, ae both ieducible and both simle than any othe s faction within the inteval bounded by and s iii) Continued faction fom: and s ae both ieducible and elated by s =[a 0=a = =a n, =a n ] (canonical fom of s ) and ( [a0=a= =an,] (econvegent of fo n ) = s o [a 0 =a = =a n, =a n, ] (aent of fo any n): s iv) Convegent fom: s is ieducible and is eithe its econvegent o aent. 494

12 Poof. The euivalence of (i) and (ii) is known fom the classical theoy of Faey Factions, e.g. [Hady and Wight 79]. The fact that (iii) imlies (i) follows fom Theoem. (iv) is essentially a estatement of (iii). To comlete the oof we need only show that the two altenatives given fo in (iii) yield the only factions simle than and adjacent to s. Let i =[a s 0 =a = =a i i ] fo i =0; ; ;n denote the convegents to the canonical continued faction fo s. The set of linea euations x + y = n, xs + y = s n, will have a uniue solution (x; y) since js,j =. Note that jyj = jyjjs,j = j n, s, s n, j =.Soy =, and x must be integal since s is ieducible. Fo y =,0 = n,, x, with n, <, imlies x = 0, hence = n, s n, =[a 0 =a = =a n, ]: Fo the case y =, then = x, n, imlies x = since. Hence = x(a n n, + n,2 ), n, = (a n, ) n, + n,2 and similaly we nd =(a n, )s n, + s n,2,thus =[a 0=a = =a n, ]: 2 Fom the LCF tee it is clea that membeshi of a faction in Q k deends in some manne on a tuncated binay eesentation of the nal atial uotient of a continued faction eesentation of the faction. This intoduces a base deendence henomenon simila to that obtained in nite length binay adix eesentation. We seek both to model and undestand the amications of this base deendency in LCF eesentation. Imotantly, the adjacency elation itself as inteeted on factions and/o continued factions sues no base deendence on the eesentation of the individual atial uotients. The following extension of adjacency intoduces a deendency on the binay eesentation only in the last atial uotient. This "binay adjacency" elation will be shown sucient tochaacteize all neighbo ais in Q k. We st note that two nite bitstings ; ae temed lexicogahically adjacent (with lexicogahically eceding ) if = 0 j = 0 j fo some ex and some j 0. Lexicogahically adjacent bitstings thus may euivalently be said to die by a unit in the last lace (ul). This is ecisely the elation between neighbos in Q k wewant to exess in tems of euivalent elations among ationals in faction o continued faction fom. Howeve, it tuns out to be most convenient to dene the elation wanted in tems of continued faction exansions. 495

13 The factions < s ae bijacent (of ositive, zeo o negative degee i) i and s ae both ieducible, and =[a 0=a = =a n ] s =[a 0=a = =a n +2 i ]; (5) whee a n = k2 i 2 when i. The denition includes situations coesonding to i being ositive, zeo, o negative which may be seaately inteeted as follows: have canonical exansions dieing only in the last atial uo- i 0: tient and s =[a 0=a = =a n, =k2 i ] s =[a 0=a = =a n, =(k + )2 i ]: i 0: In this case we inteet a n +2 i as two atial uotients s =[a 0=a = =a n +2 i ]=[a 0 =a = =a n =2,i ]; and futhe fo i negative, s must have a canonical exansion with last atial uotient a ositive owe of two, i.e. 2,i 2. Note that fo i being zeo both inteetations aly and is the aent of s.fo i being negative, is its econvegent. When i is ositive, is not a convegent to s, but is a biconvegent to s. The notion of bijacency may be euivalently chaacteized in tems of eithe LCF bitstings o ieducible factions as summaized in the following theoem. Theoem 8. Each of the following thee oeties imlies the othe two and seves as an euivalent denition of bijacency. i) Continued faction fom: < s ae bijacent of degee i as secied in (5), ii) LCF fom: The ieducible factions and have lexicogahically adjacent s LCF exansions (and thus ae neighbos in Q k fo some k), iii) Faction fom: < s ae bijacent of degee i i fo i 0: js, j =2 i = gcd(, ; s, ) and s <2 2, i 0: js, j =and 2,i 2,i < s <(2,i +) (2,i +) when 6= 0, and s = 2,i when = 0. Poof. It is staightfowad to ove ii) and iii) fom i). We st ove i)given iii). Let us stat with the case whee i, fo 6= 0, then thee exists a b < such that =2,i + a and s =2,i + b. Since j, s j = s < 2 it follows 2 fom Theoem (viii) that is a convegent of s. Since 2, cannot be the aent of s,so s =[a 0=a = =a n =2,i ] and =[a 0=a = =a n ]. Fo i =0 496

14 the same agument alies, excet that now =[a 0=a = =a n ] is the aent of s =[a 0=a = =a n +2 0 ]. Now, assume i. Then thee exists a and b such that, =2 i a and s, =2 i b.fom 2 i ja, bj = j(, ), (s, )j = j, sj =2 i we obtain ja, bj = and similaly jas, bj =. Hence a b and adjacent to as well as to s.now s < 2, imlies 2 s, = 2i a 2 i b < hence a b < and as above we nd that a b is a convegent of as well as of s. With is ieducible =[a 0=a = =a n ], a n 2, eithe a b is the econvegent o the aent of. Assume a b is the aent, then = a + n,2 b + n,2 whee n,2 is the econvegent of n,2, hence <2a 2b Thus we obtain both a b =[a 0=a = =a n, ]; = 2i a + s 2 i b + =[a 0=a = =a n, =a n +2 i ]; which contadicts 2i a 2 i b <. which comletes the oof of (iii) ) (i). Finally we have to ove that (ii) imlies (i), hence assume fo some j 0 that = s 0 j and = 0j.Two automatons decoding the bitstings 0 j and 0 j afte eading the sting, will both be in the same state, having ased and decoded an initial seuence of atial uotients fa 0 ;a ; ;a n, g, being in the state of decoding the n'th atial uotient a n, when encounteing the 0 o following. The automatons will be in one of fou ossible states, deending on n being even o odd (eading in tue o comlemented fom), and eithe eading the initial (unay including switch bit) at o the tailing (binay) at of `(a n ). We will stat with n being even, and let us st assume the automatons ae eading the unay at of a n,having aleady seen k ones of the unay at. The 0 in the LCF exansion of thus imlies that the unay at is comleted, and =[a 0=a = =a n, =2 k+ ]; since a suitable amount of zeoes just comletes a n =2 k+ as the nal atial uotient (n is even). The inteetation of LCF( ), howeve, deends on the s elation between k and j, since the zeo bings the automaton into the state of decoding the binay at. If j k we nd s =[a 0=a = =a n, =(2 j+, )2 k,j ]; 497

15 wheeas when j>kthee ae moe ones than needed to comlete the binay at, and the emaining ones then ae inteeted as two additional atial uotients: s =[a 0=a = =a n, =2 k+, ==2 j,k ]: so that in both cases, and s ae found to be bijacent of degee k, j. Now assume the automatons ae eading the binay at, still having to ead k bits to comlete a n, when has been ead. If k j +, then fo some a we obtain s =[a 0=a = =a n, =a 2 k,j, ] and =[a 0=a = =a n, =(a + )2 k,j, ]: If k j, fo some a n we obtain =[a 0=a = =a n + ] and s =[a 0=a = =a n ==2 j,k ]; thus in both cases and s ae bijacent. Fo n odd, the bits have tobeinveted when eading a n, and the continued factions have to be witten in even ode exansion fom, i.e. an (n + )' atial uotient has to be consideed. The esults then follow similaly. 2 By Theoem 8, the bijacency elation is ecisely the elation that holds among neighbos in the inode tavesal of the LCF tee given to any deth k, secically between consecutive factions of Q k.fo the uose of detemining the membes of Q k+ given Q k, again intuitively what wewantistochaacteize, in faction o continued faction fom, is the ational numbe fom Q k+ whose LCF eesentation is obtained by one-bit extensions of the LCF eesentations of membes of Q k. Secically, given two bijacent neighbos fom Q k with LCF eesentations 0 j and 0 j,wewanttochaacteize the uniue ational 0 j+ fom Q k+, falling between these. Let us then fo a moment digess to the classical concet of Faey factions. Recall (e.g. fom [Hady and Wight 79]) that given the Faey-set n F n = o j 0 n; n; cd(; )= a membe of F 2n can be constucted as the mediant + of two successive ationals < s aleady in F n, which can always be shown to be adjacent as dened +s eviously.we futhe obtain < + +s < s.howeve, when say is a \vey simle faction", wheeas is not (e.g. and s), thei mediant will be of s numeic value vey close to s, and athe distant fom. In geneal, the sacing between consecutive membes of F n is uite eatic, vaying between n, and n,2 [Matula and Koneu 80]. Now if and above both wee multilied by some common facto c, chosen such that c and c wee both of the same ode of magnitude as and s esectively, then the ational value of the exession c+ c+s would slit the inteval between = and =s in twointevals of moe nealy eual widths. 498

16 This obsevation leads us to the intoduction of an altenative fom of mediant, moe nealy bisecting each ga between consecutive membes of Q k, which tuns out to be ecisely the way we can geneate membes of Q k+ fom Q k. u v The binay mediant of the bijacent factions < s is the ieducible faction with value u=v =(2 j + )=(2 j + s); whee j is the lagest intege such that 2j 2 j < s. The binay mediant u v is dened to be in educed fom, howeve gcd(2 j + ; 2 j + s) can only have value one o two. The following altenative chaacteizations and oeties of the binay mediant ae eadily obtained by extending the oof of Theoem 8. Theoem 9. The binay mediant of two bijacent factions ; s is the ieducible faction u v euivalently detemined by eithe of the following thee conditions: i) Continued faction fom: If ii) LCF exansion fom: If < s iii) Faction fom: If < s =[a 0=a = =a n, =a n ] s =[a 0=a = =a n, =a n +2 i ] with a n = k2 i 2 when i ; then u v =[a 0=a = =an, =a n +2 i, ]: and fo some bit sting and intege j 0, = 0 j ; s = 0j ; then u v = 0 j+ : ae bijacent of degee i, then u v = 8 >< >: 2,i + 2,i + s ( + )=2 ( + s)=2 fo i 0; fo i>0: Lemma 0. The binay mediant u v of the bijacent factions ; s is bijacent to both and s. Theoems 8, 9 and Lemma 0 ovide us the comutational means of geneating sets of bijacent factions atitioning any inteval secied by two bijacent factions given in eithe the faction, continued faction o lexicogahic continued faction fom of eesentation. Regading the LCF tee we then immediately obtain the following \LCF Tee Labelling Lemma", hee emloying in an obvious way the \innite" faction

17 Lemma. In the LCF tee, the faction is assigned to the oot, and the faction assigned to any othe node is the binay mediant of the faction assigned to the neaest left ancesto node (o 0 if none exists) and the faction assigned to the neaest ight ancesto node (o if none exists) in the LCF tee. Futhemoe, with edges labelled 0 fo left banch and fo ight banch, the bitsting 0 comosed fom the labels on the edges of the ath fom the oot to a aticula node, with a teminal unit aended, ovides the minimal LCF exansion of the faction assigned to that node. 4 Extemes and Distibution of Ga Sizes ove Q k Taditional xed oint binay eesentation with k bits to the ight of the adix oint 0:b b 2 b k allows fo the eesentation of 2 k + values ove the unit inteval [0; ], with all gas of unifom size 2,k. The LCF exansion 0b b 2 b k with k bits beyond the eciocal zeo bit, detemine an eual sized 2 k + membeed set Q k of eesentable values ove the unit inteval. The ga sizes ove Q k necessaily vay about the mean 2,k to accommodate exact eesentations of the \simle" ational factions. In this section we shall discuss bounds on the vaiations on the size of gas ove Q k fo lage k. We st conm the existence of cetain elatively lage and small gas in Q k, that we contend ae indicative asymtotically of the maximum and minimum ga sizes in Q k. Lemma 2. Given >0, then fo suciently lage k, the maximum ga size in Q k will be atleast 2,(a+)k fo a = 4 log 2( ) = 0:82682 ; and the minimum ga size will be no bigge that 2,(b,)k fo b = log ! = :38848 : Poof. Fo the minimum ga esult, conside that z = 5, =[0=== ]= = 0:68033 has the seuence of convegents 0 ; ; 2 ; 2 3 ; 3 5 ; 5 8 ; which (deleting 0 ) ae also seen to be the seuence of biconvegents to z. The numeatos (and denominatos) ae the well known Fibonacci numbes and gow asymtotically at the ate ( + 5)=2. The bounding intevals on z detemined by the seuence of biconvegents ae then 2 ; 23 ; 35 ; ;, and decease 58 at a ate aoaching 4=(+ 5) 2 =2=(3+ 5), veifying the minimum ga size bound. Fo the maximum ga size esult note that y = 6, 2=[0=2=4=2=4= ]= = 0: has a seuence of convegents, whee the ate of incease of numeatos (and denominatos) in two stes, i+2 i, aoaches 5+2 6=9: Note then that afte eight bits in the LCF exansion coesonding to encoding anothe ai of atial uotients 2; 4, the bounding inteval only detemined by the biconvegents will decease at a ate aoaching =( ) 2, fom which the lowe bound on the maximum bound follows

18 Fo the uose of bounding the aoximation eo in ounding to a biconvegent in Q k,we ae concened with detemining an ue bound on the maximal ga size in Q k. By diect comutation we st obtain the maximum ga size in Q k fo modeate odes, given in tems of a k =, k log 2 (max ga in Q k); the values of which ae listed fo k =; 2; ; 20 in the following table: k a k k a k In these comutations we found fo the lage k that the maximum size gas in Q k always had a bounday oint 2 Q k whose LCF eesentation contains elications of the bit atten 0000, consistent with the examle of Lemma 2. This lemma futhe gives us an ue bound on any limiting value fo a k, and we suggest in the following that this is indeed the coect value. Conjectue 3. lim k! a k = 4 log 2(5+2 6) = 0:82682, whee futhemoe the gas in Q k containing the eal numbe 6, 2=[0=2=4=2=4= ] decease in size asymtotically in k as fast as the maximum size gas in Q k. The oof of the conjectue at this oint aeas uite tedious. We uote hee without oof fom [Koneu and Matula 85] a somewhat weake esult, which in conjunction with Lemma 2 ovides easonable tight bounds on a k. Theoem 4. Given >0, then fo suciently lage k, the maximum ga size in Q k is no geate than 2,(a,)k fo a = log 4 2 (6=53) = 0:84347 : Also fom [Koneu and Matula 85] we uote ndings on some comutations and simulations on the ga size distibution. Fo values of k u though 24 the distibutions wee comuted exhaustively. In each case it was also found that the minimum ga in Q k fell between two consecutive ationals of the fom fn,2 f n, and fn, f n, whee f n denotes the nth Fibonacci numbe, in coesondence with the obsevation about the LCF exansion of [0=== =]. The main uose of the comutations was, howeve, to obtain gahs of the distibution of ga sizes. Exhaustive comutations u to k = 24 and simulations fo k =32; 64; 28 showed the distibution of the negative base 2 logaithm of gas in Q k to be bell-shaed between aoximately 0.8 and.3, and centeed aound.0, the bell-shae getting naowe and highe eaked fo inceasing values of k. 50

19 Oveall ou study shows that the gas between LCF eesentable values ae subject to a vaiation in size coesonding to a wost case 9% ecision loss (o euivalently a 9% stoage caacity loss), and a best case 38% ecision gain, in comaison with an euivalent xed oint binay system with unifom ga size ove the unit inteval. This aeas to be a small ice to ay fo achieving exact eesentation of all simle ationals. 5 Conclusions A binay eesentation of the ationals has been descibed and analyzed. It is caable of eesenting in nite ecision a set of ationals faily egulaly saced on the unit inteval. It suots an online aithmetic unit fo ational aithmetic, which can altenatively be consideed an aoximative eal aithmetic with embedded exact comutations on simle ationals. The LCF eesentation is non-edundant, which fo an on-line (digit seial, most signicant digit st) aithmetic has the imlication that the delay between inut and outut can vay unboundedly. To be able to bound and educe such delays, it is necessay to intoduce edundancy in the eesentation. It is staightfowad to intoduce edundancy in the continued faction eesentation of ationals (allowing atial uotients suitably esticted to become negative), and futhemoe to intoduce edundancy in the binay encoding of the individual atial uotients. Aithmetic units suoting such edundant continued faction eesentations have also been investigated and eoted [see Koneu and Matula 90], howeve an analysis of these eesentations has neve been conducted and deseves a simila study. Acknowledgements This ae eots some eviously unublished esults esented as at of an invited talk entitled "Rational Numbe Systems and Aithmetic", given by the st autho at the symosium \Real Numbes and Comutes", in Saint Etienne, Ail 4-6, 995. The wok has been suoted by The Danish Reseach Councils, gants no and , and by The National Science Fondation unde gant DCR Refeences [Blachman 84] N. M. Blachman. The Continued Faction as an Infomation Souce. IEEE Tansactions on Infomation Theoy, IT-30(4):67{674, July 984. [Hady and Wight 79] C. H. Hady and E. M. Wight. An Intoduction to the Theoy of Numbes. Oxfod Univesity Pess, London, fth edition, 979. [Khinchin 35] A. Y. Khinchin. Continued Factions. The Univesity of Chicago Pess, 935. Tanslated fom Russian by P. Wynn, P. Noodho Ltd., Gooningen, 963. Also by H. Eagle, The Univesity of Chicago Pess, 964. [Koneu and Matula 83] P. Koneu and D. W. Matula. Finite Pecision Rational Aithmetic: An Aithmetic Unit. IEEE Tansactions on Comutes, C- 32(4):378{387, Ail

20 [Koneu and Matula 85] P. Koneu and D. W. Matula. Finite Pecision Lexicogahic Continued Faction Numbe Systems. In Poc. 7th IEEE Symosium on Comute Aithmetic, ages 207{24, 985. Also einted in the collection Comute Aithmetic, Vol II, E.E. Swatzlande, ed., IEEE Comute Society Pess, Washington, 990, ages [Koneu and Matula 88] P. Koneu and D. W. Matula. An On-line Aithmetic Unit fo Bit-Pielined Rational Aithmetic. Jounal of Paallel and Distibuted Comuting, 5(3):30{330, May 988. [Koneu and Matula 90] P. Koneu and D. W. Matula. An Algoithm fo Redundant Binay Bit-Pielined Rational Aithmetic. IEEE Tansactions on Comutes, C-39(8):06{5, August 990. [Knuth 8] D. E. Knuth. Seminumeical Algoithms,volume 2 of The At of Comute Pogamming. Addison Wesley, 2 edition, 98. [Knuth 82] D. E. Knuth. Suenatual Numbes. in Mathematical Gadne, D.A. Klane, ed., 982. [Matula and Koneu 80] D. W. Matula and P. Koneu. Foundations of Finite Pecision Rational Aithmetic. Comuting, Sul. 2, ages 88{, Febuay 980. [Matula and Koneu 83] D. W. Matula and P. Koneu. An Ode Peseving Finite Binay Encoding of the Rationals. Poc. 6th IEEE Symosium on Comute Aithmetic, ages 20{209,