LANGUAGES OF ANALOGICAL STRINGS 
Yves Lepage 
ATR Spoken Language Translation Research Labs, 
Itikari-dai 2-2-2, Seika-tyS, Ssraku-gun, Kyoto 6\]9-0288, Japan 
yves. lepage~slt, atr. co. jp 
1 Introduction 
Analogies between strings of symbols, noted j 
A:B = C: \]), put four strings of symbols 
into "proportions." They render an account 
of, for instance, look: looked = walk : walked 
or fablc: fabulous = miracle : miraculous, on 
tlhe level of strings of symbols. They are not 
intended to deal directly with, for instance, 
bird: wings = fish :fins or work: worked = 
go: went which suppose knowledge about the 
world or the tongue (IIofl?nan 95). Analogies 
may be read as equalities, as well as equations 
to be solved, as in: 
to looh : I lookcd = to act : x ~ x= l actcd 
The goal of this paper is to establish some 
\['undamental, common-sense hypotheses (ax- 
ioms) about analogies in general; then to draw 
from them basic results (theorems)on analogies 
between strings of symbols in particula.r; so as 
to propose a possible definition tbr \]~ngua.ges 
of analogical strings; and to prove that some 
famous bmguages of particular interest to the 
 processing community are very simple 
s in this respect. We further argue that 
the fact that the property of bounded growth 
is verified by ~ny such  is in favour of 
modelling part of natural  using such 
s. 
Our feeling is that analogy between strings of 
symbols is an operation as Nndamental as, e.g., 
addition is to naturM numbers. Ilowever, to our 
knowledge, letting aside the Copycat project 
(Hofstadter et el. 94, Chap. 5-7, pp. 195 
318) which has no such goals and relies on dif- 
ferent methods, no mathematical formalisation 
has ever been proposed tbr anaJogies between 
strings of symbols. 
lln the sequel, A, B, C and D are variablcs denoting 
objects. 
2 General Properties of Analogy 
We start with results which \]told independently 
of the set to which the terms of the ~malogy 
belong. 
2.1 Fundamental Hypotheses 
In the Nicoma.chean Ethics (Book V), Aristotle 
wrote: 
For proportion is equality of ra.tios, and in- 
volves four terms at least \[...\] As the term 
A, then, is to \]1, so will C be to D, a.nd there- 
fore, alternando, as A is to C, B will be to D. 
\[Translation by W. 1). Ross\] 
As a consequence, we shall hypothesize the fol- 
lowing property: 
Axiom 1 (Exchange of the means) 
A:B= C:D ~ A:C=B:I) 
Another equivalence is also used by Aristotle 
in his Poetics. It is based on the symmetry of 
the equality (the word "as," here): if we can say 
that A is to B as C is to D, then we should also 
be able to say that C is to D as A is to B. 
Axiom 2 (Symmetry of equality) 
A:B= C:D ~ C:D=A:B 
2.2 Equivalent Forms of Analogy 
By successive application of the previous hy- 
potheses, we get eight equiva.lent forms of the 
same a.m~logy, listed hereafter in the alphabet 
order of the term variables A, B, C a.nd D. 
Theorem 1 (Equivalent forms) The eight 
following analogies are equivalent: 
488 
A:S =C:D (? 
A:C= (ii) 
11 : A = I) : C (iii) ~ ii+vi+ii 
B: D = A : (/ (iv) ~ iii+ii 
C: A = D: B (v) <= ii+iii (v,,:) 
/) : B = C: A (vii) <=- ii+vi+iii 
D : (7 = H: A ('viii) ~ iii-tvi 
SOllle interesting results may I)e el)rained on 
the number of different l)ossible analogy classes 
given four ol)jex:ts, llowever, we shall leave 
them aside :for lack of spa(;e. 
3 Analogy on Strings of Sylnbols 
VVe shall :now specialise on tim case where the 
IliOili bets of 1;he analogy being considered I)elong 
to a. set of strings of symbols. The st:i:u('ture of 
strings Jail)lies new l)roperties. 
3. l Examples 
hi order to SUl)l)orl, 1;lie next hyl)othesis we will 
n~al¢e on analogies Oll strings of symbols, let us 
list a snlall nuinl)er of a,nalogies i:n l';nglisli: 
hypoUw_s'i.s' : hyl.~th, ese,+ = thesis : U~,e,~c.w 
l(:af : lcave,~ = call': calves 
give: ga, ve = si'<q : sang 
i'ne:cacl : exact = incapable : capable 
plus SOli/e tl'tlO analogies but with I10 Illea.liillg 
iit \[aligtlag;o: 
2 aa : aaa, a -- aaaa : a, aaaa, a, 
give : gave = bid : bad 
walk :walkcd = go : gocd 3 
and some counl;er-examl)les (noted wil;h ¢): 
aaaa : bbbb ~ cccc : dddd 4 
dJTU,:a : bzvmbz ¢ bzwnbz : dfh, ka 
>l'his a.nah)gy holds indel)endently of the truth (or 
falsity of) aa : aaaa = aaaa : aaaoaaaa ( a ~ : a 4 = 
a 4 :a 8 ). In filet, hylmthesising A: 1\]= AA : 1111 for 
~iiy string A and B is incompatible with the Syml)ol in- 
clusion axioni because the l!klua,lity of length sums on 
a" :a'" =a. 2":a 2"' would yield n+2'm = m.÷2.n, i.c 
it = 'In, for any 'It, ill, C \]N, which is abslird. 
:~\]{efrain frolil thinking in l!higlisli, and recall tll~Lt we 
work Oll tile sole level of synll)ols: i just \[)e('allle (I, or (.'d 
has just been added. 
~ln absence of a.ny knowledge about the world. Ilere, 
only tlle equalil, y belAveell synlliols caJi be t.ested, lie- 
cause the a.lphabetica\] orde.r is not. known, this anMogy 
Cilnno|, l)e verified. 
3.2 Symbol Inclusion 
\]~y insl)ection of the previous eXaml)les , one can 
sta,te that there is no solution to a,n a,nalogy on 
the stl'illgS of syml)ols A : 13 = C: x if sonic 
symbols of A apl)ear neither in l\] nor in C. lhe 
contraposil;ive, is tllat~ for an ana.logy to hold, 
any syinl)o\] of A has to a.ppear in either 11 or 
C. Noting l)y A the set of symbols contained 
in A, we, restate the i)revious ol)se, rwition as the 
following hyl)othesis which will be used in Ap- 
pendix in the p:roofs that some well-known lan- 
g u ages a,re la.n g u ages of an alogical stria gs ('.l'he- 
orems 5 and 6 C' Section 5.1). 
Axioln 3 (Symbol inclusion) Let 12 be an 
v(Jl, n,(',/)) < (12")", 
m A: 11= 
C: \]) ~ A C IIUC 
l!'or strings redu(;ed to olle symbol, lhis trivially 
inll)lies: a: b= b: a Cb a=b. 
lncidently, al)l)lie(I on the eight equiw~,lent 
forms of an a,tta,logy, the Sytnl)ol inclusion ax- 
iom implies eight inclusions, of which, only four 
are distincl; I)y commutativil;y of union. 'l'hese 
four inclusions hill)ly , and are implied 1)y, two 
recil)rocal in elusion s: 
- ( A C IIU(; IJ C A U I) A U I) C IIU(; 
(-v c A U I) ¢> 11U(,' C A U /) 
I) C II U C' 
so thai;, one ca, n state: 
Theorem 2 
v o,,, V(A, c, D) c (V*) 
A :17= C: /) ~ AUI) = 11UC 
3.3 Similarity Constraint 
The Syml)o\] inclusion axiom ca,n be refined by 
saying that, the sum of the shnilarities s of A 
with II and C must I)e greater than or equal to 
Xtslength: sim(n,//)+si,~(n,c) >_ IAI 
Wheit the length of A is less than the sum of 
th e sim ila, rities, some symbols of A a,re corn men 
s'\['he similarity between two strings is defined 
a.s the length of their longest conimon subsequence 
(\]lirs(:hl)erg 75). A subscqucncc of a. string is any not 
necessa.rily connex sequence of symbols fronl tilat st, ring 
in the sanie order. 
489 
to all strings, A, B, and C in the same order, 
and these symbols are necessarily present in .D 
in the same order also. We call 7(A,B,C,D) 
the number of such symbols. A.s a result, 
A: c:. 
= sire(A, ~) + si,n(~,C) - ~(A, ~, C, U) 
The Equivalent forms theorem yields: 
IA.I = sire(A, C)-I- sim(A, U)- 7(A,C, B,D) I~Xl 
; siln(./J, A ) q- siln(.1\], .1)) - ~/(\]), A, \]), C) 
I\]~1 = Siln(/~, \])) q- siln(.B, A) - 7(\]\],\]),A,C) 
ICl = siln(C, A) + sim(C,D) - "7(C,A,B,I)) 
IC\[-- sire(c, D) + sire(C, A) - 7(C,D,A,H) 
IDI = aim(l), ~3) + sin,(D, C) - 7(D, v, c, A) 
IDI = si,n(l),C) + sin,(.D, \]3)- 7(D,C, V,A) 
Because all 7(.,.,-,-) are equal in all the 
equalities above, and by the symmetry of shn- 
ilarity, the substraction of pairs of lines yields 
the following theorem, which is necessary for 
the proof of our theorem on bounded growth 
property ('_l.'heorem 7 of Section 5.2). 
Theorem 3 (Similarity constraint) 
Let V ~,c ,,,, a~vha~ct. V(A,t~,C,~)) ~ (V*) ~, 
A : B= C: I) 
IAI-sim(A,~) -- ICl-s.im(C,\])) 
I/Sl- sin,(2~,D) -- IAI- s.~ (A,6) 
ICl- sim(C,A) = IDI- si,,~(~),\]~) 
S" " / 
3.4 Equality of length sums 
A remarkable theorem is easily derived from the 
Similarity constraint theorem by addition and 
substraction and by commutatitivity of similar- 
ity. 
Theorem 4 (Equality of length sums) 
Let 12 be an alphabet. V(A,.B,C,D) ~ (1)*) 4, 
A:\]~--C:D ~ IAI+I~)I=I~I+ICl 
3.5 Disjoint Analogies 
Another intuitive idea about analogies between 
strings of symbols is that two analogies could 
always be concatenated. Whether this is true 
remains an open problem. 
lIowever, the previous intuition seems to hold 
anyway when the two anMogies to be concate- 
nated do not have any symbol in common. We 
cMl such analogies, disjoint analogies. The intu- 
ition is that, disjoint analogies :m~y be applied 
one after another without any problem. \]3ut 
concatenating in the same order is not the only 
p ossibility. 
One gets 2 4 = 16 analogies by enumerat- 
ing all possibilities of exchanging or :not ex- 
changing the substrings indexed by I a.:nd 2 
in A1A2 : B1B2 = CIC2 : DID2. By number- 
ing these 16 a.ualogies using a binary notation 
reflecting the place where this exchange took 
place, numbers which are binm:y complements 
denote two equivalent analogies, of which one 
may be eliminated from the list. We list here- 
after those analogies with A1A2 as a first term. 
(0000) A1A2 : \]\]lB2 = CIC2 : D1D2 
(0001) A1A2 : B112,2 = C2C:1 :.l)2l)1 
(0010) A1A2:11:,112 = C2C' 1 :.Dj1)2 
(0011) AIA2 : 1\]1.112 = (/2C1 : \])2\])1 
(0100) A1A2 : B>B1 = CIC'2 : DID2 
(010l) AIA2 : \]\]2.111 = C'~C2 : D2D1 
(0110) A1A2 : .I)2.B1 = (./2(3'1 : D1 De 
(011\].) A1A2 : l)2B1 = C'2C1 :.l)2.D1 
The number of diIferent cases is further 
reduced using (i) ¢> (viii) of the Equiw> 
lent t or,ns: (ooo:,) ,~ (~ooo) ~, ((/~:,~) 
and (00\]0) ~ (0100)• The reduced set is: 
{ (0000), (000\] ), (00\]0), (00\] l ), (0101), (0110) }. 
Similarly, (i) ¢:> (ii) of the Equivalent tbrn,s 
yields the equivalences: (0010) ~ (0100) and 
(00ll) ~ (0101)• The reduced set becomes: 
{ (oooo), (ooo~), (oo:l :,), (o:1 :, o) }. 
Of these tbur possible analogies, the second 
one, (0001), where only one exchange is per- 
formed, is not true in genera.l. For instance, 
ay:az = by:x is not acceptable when x = 
zb. On the contrary, the three other possible 
analogies meet intuition, so that the following 
hypothesis m~y be laid• 
Axiom 4 (Concatenation) Let I) bc an al- 
phabct, and V:, C \]), \])2 C F, ~'uch that 
\]21 ffl \])2 = O, V(A1,\])I,C,,\])I) C (12.)4, 
V(A2, Jh, C2, J)2) c (\])2*)2 
A1 : BI : C'I : \])1 \] 
A2 :B2 C ~ = 2:D2 
AIA2 : B1B2 = C1C2 : D1D2 
A..tA2 : BIB2 = C2C1 : .l)2D1 
A.1A~ : B'eB1 = C.eC1 :.D1\])2 
This axiom will be used in Appendix in tile 
proof of Theorems 5 and 6 of Section 5.1. 
490 
4 Languages of analogical strings 
4.1 Analogical Derivation 
liT, order to show how s(mle s, i.e., some 
sets of symbol strings, (:a,n I)e c\]ta.T:acterised by 
~ device based on analogy, we first introduce 
analogical derivations. We intentionally use this 
term to make a. paralM with the vocabu\]a.ry of 
fol:m al gralnma.rs. 
rsl! i < Definition I Let 12 be (t'lz alphabet. ~ n ~. an, a- 
logical de'rivatio~t, ,zoled t~-~ , ~nod'ulo a set ;bt C 
12" x 12", whose elcmc'nt.~" (v, v') ave noted v -+ v', 
is deft'ned in lhc Jbllowi~zg way: 
V(w, w') ~ 12" x 12", 
w~w' ~ ~'v-+v'~Ad/ w:w'= v:v' 
Altllough we use the notation -~ \['or the ele- 
nlents of jr4, it is not to l)e interl)reted in the 
way it wout(l be in classica.l rewriting systen:m. 
This nota£ion is just to make a parallel with 
classical i)resentations of gramma.rs~ where the 
elements of j~ are ca.lle(1 rules. \]lowever, the 
meaning here is di:frerel,t. With standa.rd i:ules, 
w is exa(;tly ntal,(;ll(;(l against v to pro(lu(;e, in 
a second step, 'u/. Ilere, the result w' del)ends 
o:u the wa.y 'o (Hot w) "niat('hes" ~11 o'lzd .~;i tit the 
sa~nc ti~nc. 
4.2 Derivational Systems 
Definition 2 A dcrivatio,zal .sy.slcm of a,talog- 
i(:al stving.~ i.s a triple (; = (Y,A,;t4), where 12 
i.';' a .finitc alphabet, A C 12" (jinite) is the sct 
of axioms~ or, bcttcv, the set of attested strings, 
a'lzd ~t4 C 12" × 12" (.\[inilc) i.~" lhc sc.t of rulc.s, 0% 
bcttcr, the set of too(lois. 
4.3 Languages 
Definition 3 Let 12 be art alphabet. Let A C F* 
and Ad C F* x Y*, both .fi~zitc. flTte la~tguagc of 
.,,,.togi~,a ~t,.i,,,g~ a(A,a4) = < A,{ M~-} > 
is defined in the Jbllowing way: 
A(A,M) = ~tu{ ,,,' c- v* / ~,, c- A / ,,, M~- ,,,' } 
-l- with k~-, th, e t'ransitivc clo.surc of the analogical 
derivation I /t4 . 
~l.'he previous definition conforms to the usual 
1)rese.ntation of formal s. It aims at 
the generation of a , q'hus, as usual, 
standard structural induction is used to gener- 
a.te all of the members of a  of analog- 
ical strings. Starting with the elements of .4, 
all possible analogies with the elements of ~td 
as models a.re applied. 
~.iPhe reciprocal problem of generation is that 
of reco.qnitio'n. With an analogical system, the 
grammaticality of a given string, i.e., its mem- 
bership in a , is tested against the set 
of a.ttested strings of that , a.fter tile re- 
duction of that given string, by aa|a.logy, using 
the set of models. For recognition, the strings 
in the pa.irs ot'fi4 a.re used in the reverse order 
they appea.r in jr4, and the ana.\]ogies are solved 
in the other direction tha.n for generation. ~Phis 
is possible thanks to form (iii) of the Equivalent 
forth s theorem. 
q'he "linguistic" hlterpl:etation of a  
of a.na\]ogica\] strings A(A, jD/) is thus a.s follows: 
A is the set of a.tl;esl;ed strings, i.e., the set of 
strings aga.inst which any (:andidate element of 
the la:ngua.ge will be compa.red in fi~ze; ;td is the 
set or paradigmatic models (declensions, con- 
j u gati on s, m or\])ll ol ogi cal deri wl.tion s, sy n t a.cti c 
tra.nsforn,ations, etc.), a.ccording to which any 
candidate element of the la.nguage is re(luted ~; 
by an a,logy. 
5 Some Properties 
s.x {,,,?,,4....',~} and {(,,"~b"~"~m ,} 
hi al)l)en(lix , we give l)roof~q that the follow- 
iTlg \['a, tnOllS regular, context-free a,lld~ Col,text- 
sensitive la.nguages are all langua.ges of analog- 
ical strings: 
(,,''/. _> '} 
{,,'~b '~ I., > it} 
{anb'"c n /n > 1} 
= A({a}, {a --- aa}) 
~- a ((.~, }, ( .,b ~ ,,,,~,b }) 
= a((.V~}, (.l,~: + .,,bb~}) 
and th a,t, more generally: 
Theorem 5 {a'~a~ ~ . ('~ 1} • .~.~ /n_> = 
A((,,1,2 ...,,,,~}, ((,,,,,,2...,,,,, + ,,~,g. • • ,G)}) 
In a, similar way, I)y induction and use of the 
(,o ~catcna.t~on of disjoint analogies, it is easy to 
prove th at: 
Theorem 6 {ambncmd '~ / n >_ I A~)~ > 1} = 
A( {abcd}, {abcd ---+ abbcdd, abcd -0 aabccd} ) 
(;'l?he word rcducc is taken to mea.n ;t reduction to a. 
normal form, not in the sense that the strings become 
shorter. 
491 
This  is famous for being the basis 
of two counter-examples against the context- 
freeness of natural : in the morphology 
of Bambara (Culy 85), and in the syntttx of the 
Zurich dialect of Swiss German (Shieber 85). 
5.2 Bounded Growth 
Following the discussion about the non-context- 
freeness of natural , the family of tbr- 
real s that can be used to formalise nat- 
ural  has been thought to be necessar- 
ily larger than the family of context-free lan- 
guages, but it does not have to cover all context- 
sensitive s, as some context-sensitive 
s are obviously not relevant for nat- 
ural s. Mild contcxt, sensitivity was 
thus proposed by (Joshi 85) to characterise the 
family of s captured by tree-adjoining 
grammars (larger than context-free, but strictly 
smaller than context-sensitive). 
However, this is a characterisation by a recog- 
nition device, and some have proposed other 
intrinsic characterisations. (Marcus & al. 96) 
have been advocating that, the key point in 
"mild context-sensitivity" is the property of 
bounded growth: for each sentence in a lan- 
guage, we can always find another sentence in 
the same  whose length differs t!l:om the 
length of the first sentence by at most a given 
con stan t. 
Definition 4 (Bounded growth) A la'n- 
guage £ has the bounded growth property if (and 
only ~') £ is a singleton or 3k E IN / 
Now, it is easy to prove (see Appendix ) that: 
Theorem 7 Any  of analogical strings 
verifies the bounded growth property. 
Consequently, a  like {a2'~/n C IN} is 
not a  of analogical strings, as it does 
not have the bounded growth property. Lucldly 
thus, some "unnatural" s are out of the 
reach of s of analogical strings. 
6 Conclusion 
Only a slnall number of proposals have been 
made tbr the modelisation of analogy, the rare 
exceptions being (Itkonen 8: Ilaukioja 97) and, 
out of linguistics, (Hofsta.dter et al. 94), maybe 
because the dominant strea.m in linguistics for 
years, tile generative one, against works by 
the founders of modern linguistics (e.g. (Sans- 
sure J6, Part 1II, Chap. 4 & 5)), explicitly re- 
butted analogy as a possible object of research 
(see (Itkonen gz naukioja 97, 7132 and 136), for 
quotations from Chomsky) under the fallacious 
pretext that blind application of analogy may 
lead to falsity in logic and agra.mmaticality in 
syntax, ltowever, following recent results in ex- 
perimental psychology and refuta.tions of the in- 
nateness hypothesis (Itkonen 94), analogy nlay 
reasonably be argued to be a component in lan- 
guage (of course, surely not the only one). 
llaving posited only lbur fundanmntal hy- 
potheses on analogy, we have shown how to gen- 
erate a fmnily of formal s, called lan- 
guages of analogical strings. It is important to 
note that analogical string grammars, like sim- 
ple contextual grammars (llie 96), do not lnake 
any use of non-terminals. Crammaticality is 
sin\]ply tested against some attested strings, af- 
ter reduction according to some models. The 
a.pproach by reductiol~ to attested \['orms has al- 
ready heen advocated in natural  pro- 
cessing (Sager 81). 
The key  {a'%'~c"~d'~/n > l} against 
the context-freeness hypothesis of natural lan- 
guage is easily shown to be a  of ana- 
logical strings. Also, all s of analog- 
ical strings possess the bounded growth prop- 
erty, which attempts to capture mild context- 
sensitivity, a notion introduced to cope with the 
apparent power of human s. 
The fact that the regular  {a'~}, the 
context-free  {a%'~}, and the context- 
sensitive  { a ~ b ~ c '~ } are very similar 1 an- 
guages of analogical strings shows that analogy 
allows us "to get round" the Chomsky classiti- 
cation. 
7 Acknowledgements 
Thanks to Pr. Boitet for fl'uitfnl discussions 
about the content of this paper. 
492 
a : w I -~ (t : (t(t ~=} "w I : a ~ a(t : a 
W 1 : W 2 ~ a : aa ~ w 2 : w 1 ~ tl,(i, : a 
: 
'iv n : el) ~ a : (t(t ~ '~U : wn = aa : (t 
~c~,u~= {.} 
App endix 
Theorems 5 and 6 
Proof: for {a'~ln >_ 1}. 
Completion: A({a},{a ~ a,a}) C {(,/'/n > 
1}. Recall that N is the set ot" different symbols 
i,, s,;,:i,,g ,,~. S.pposo that ,,, ¢ A({..}, {. -~ 
aa}). This is equivalent to: a t--- w. llence, 
there exists a sequence of strings wl, 'w2, ..., 
'wn such tha.t the first column in the set of rela- 
tions at the top of this pa,ge holds; the second 
column is the equivalent form (iii); the third 
column is the applica.tion of t\]le Symbol inclu- 
rill • sion a.xiom. ,.m, last column implies: ~ C {a}, 
which means that w is of the type a '~ (note tha.t 
there is no empty string here). 
Co,,siste,,ce: {,,,'~/,,. >_ :l} C a({,d,{,~ -+ 
aa}). 13y induction on n, any string of the form 
a"" is obtained by analogy with an element of 
ex({,,,}, {,,, -~ ,-d). ~)ase: {'d < a({,d, {,, -~ 
aa}) I)y the definition o\[' a  of analogical 
st)'hlgs. \]ll(lu('tion: SUl)l)ose that a" is a, meta- 
l>e,' or A({d, {,,, -~ ,,,,,.}). The sob, tie., x or th(~ 
a.nalogy a/" : x = a'aa, is a ''+' E {a.'~/n, > \] }. 
QED. 
Proof: tbr {a'q/'/n > 1}. 
Completion: A({ab}, {ab--+ aabb}) C 
{a~%'~/n > 1}. A rat;ionale shnilar to the one 
~U, ovo gives ~,; ~ {.'w7,,. >_ 1} ~ ~ c {(~, b}. 
Con ca.tcnatmn of d is.join t lly induction, by the -' ' " 
analogies, a.l\] a's are beibre the b's, hence w = 
a~% m with n necessarily equal to ~n. 
Consistence: {a'q/~/n > :1 } < A({ab}, {ab 
aabb}). By induction on n. Base: ab E 
A({ab}, {ab -+ aabb}) ix true, by delinition of a. 
hmguage of a.nalogical strings. Induction: sup- 
pose that a'% '~ is a member of k({ab},{ab 
aabb}). Because a ~:a '~-1 = aa:a and 
b '~ : b '~-1 = bb : b are true analogies, and by the 
(,oncatenatio:~ of disjoint a.nalogies axiom, the 
sohttion x of the ana.logy a% '~ : x = ab : aabb 
ix a"~+lb *~+j C {a'*b'~/n > \]}. QED. 
Proof: re,: {."U',~'7,,. >_ ~ }. 
The p,'oor is the san~e as for {.,'~b'V,, >_ 1}, 
by decomposing 
a'q/~c '~ : a ~-1 b n-~ c '~-1 = aabbcc : abc 
into a~'~b '~ : a'~-l b '~-I = aabb : ab and 
c '~ : c '~-~ = co: c which both hold. QED. 
identical rationales prove .\[ heorelns 5 and 6. 
Tlmorem 7 
Proof: \],et A(A, M) be a.  of a.nalog- 
\]caJ strings not reduced to a singleton. For a 
given w in this la:ugua.ge, eitlmr w is ill A oi' 
not. In the tirst case, as ,,4 ix finite, t,:A the 
ma,xin,um eve,' a,U Iw'l- I~'q with 'w and w' in 
A, exists• In the ca.se w is not in A, by def- 
inition, there exists another element w' in the 
same la,ngua.ge, and there exists v' -+ v ~ .,t4 
such that w': w = v': v. The Similarity con- 
straint implies: 
I,,/I- si,.(.,,/, .w) = I~,'1- si,,,(v', ~,) 
Iwl - s~:,,,(w', ,,,,) = Ivl - sh,,( ,,/, v) 
Because sim(w,,d) < .~in(l'wl,l"'/I)is aJ- 
w,vs t,'~e: I*'/I - Iw\[ = m.~x(l'wl, I'/I) - 
"~'~(Iwl, I*~'1) -< ,~,a.x(Iwl, I~'1) - s~m(~,,J). 
Thus: Wv ff A(A,M),~w' E A(A, A4)\{w} / 
I~'l - 1,4 -< ma,x(b,/I- sim(w,~'), 
I~1 - sire(w, w')) 
_< max(l¢l- SilYI(V, V'), 
I~l - sire(v, d)) 
By taking/c the maximum over all v -+ v' of j~ 
and hA: 
vw < A(A,M),3~' e A(A,M)\b,} / 
iw'l- Iwl _< ~ 
QED. 
493 

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