File Information
File: 05-lr/acl_arc_1_sum/cleansed_text/xml_by_section/metho/00/c00-1071_metho.xml
Size: 17,569 bytes
Last Modified: 2025-10-06 14:07:09
<?xml version="1.0" standalone="yes"?> <Paper uid="C00-1071"> <Title>LANGUAGES OF ANALOGICAL STRINGS</Title> <Section position="1" start_page="0" end_page="0" type="metho"> <SectionTitle> LANGUAGES OF ANALOGICAL STRINGS Yves Lepage ATR Spoken Language Translation Research Labs, </SectionTitle> <Paragraph position="0"> yves. lepage~slt, atr. co. jp</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="metho"> <SectionTitle> 1 Introduction </SectionTitle> <Paragraph position="0"> Analogies between strings of symbols, noted j A:B = C: \]), put four strings of symbols into &quot;proportions.&quot; 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 \['undamental, common-sense hypotheses (axioms) about analogies in general; then to draw from them basic results (theorems)on analogies between strings of symbols in particula.r; so as to propose a possible definition tbr \]~ngua.ges of analogical strings; and to prove that some famous bmguages of particular interest to the language processing community are very simple languages in this respect. We further argue that the fact that the property of bounded growth is verified by ~ny such language is in favour of modelling part of natural language using such languages.</Paragraph> <Paragraph position="1"> 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 different methods, no mathematical formalisation has ever been proposed tbr anaJogies between strings of symbols.</Paragraph> <Paragraph position="2"> lln the sequel, A, B, C and D are variablcs denoting objects.</Paragraph> </Section> <Section position="3" start_page="0" end_page="488" type="metho"> <SectionTitle> 2 General Properties of Analogy </SectionTitle> <Paragraph position="0"> We start with results which \]told independently of the set to which the terms of the ~malogy belong.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 2.1 Fundamental Hypotheses </SectionTitle> <Paragraph position="0"> In the Nicoma.chean Ethics (Book V), Aristotle wrote: For proportion is equality of ra.tios, and involves four terms at least \[...\] As the term A, then, is to \]1, so will C be to D, a.nd therefore, alternando, as A is to C, B will be to D. \[Translation by W. 1). Ross\] As a consequence, we shall hypothesize the following property: Axiom 1 (Exchange of the means)</Paragraph> <Paragraph position="2"> Another equivalence is also used by Aristotle in his Poetics. It is based on the symmetry of the equality (the word &quot;as,&quot; 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.</Paragraph> <Paragraph position="3"> Axiom 2 (Symmetry of equality) A:B= C:D ~ C:D=A:B</Paragraph> </Section> <Section position="2" start_page="0" end_page="488" type="sub_section"> <SectionTitle> 2.2 Equivalent Forms of Analogy </SectionTitle> <Paragraph position="0"> By successive application of the previous hypotheses, 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.</Paragraph> <Paragraph position="1"> Theorem 1 (Equivalent forms) The eight following analogies are equivalent:</Paragraph> <Paragraph position="3"> 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.</Paragraph> </Section> </Section> <Section position="4" start_page="488" end_page="488" type="metho"> <SectionTitle> 3 Analogy on Strings of Sylnbols </SectionTitle> <Paragraph position="0"> 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.</Paragraph> </Section> <Section position="5" start_page="488" end_page="490" type="metho"> <SectionTitle> 3. l Examples </SectionTitle> <Paragraph position="0"> hi order to SUl)l)orl, 1;lie next hyl)othesis we will n~alC/e on analogies Oll strings of symbols, let us list a snlall nuinl)er of a,nalogies i:n l';nglisli:</Paragraph> <Paragraph position="2"> and some counl;er-examl)les (noted wil;h C/): aaaa : bbbb ~ cccc : dddd 4 dJTU,:a : bzvmbz C/ 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 inclusion axioni because the l!klua,lity of length sums on a&quot; :a'&quot; =a. 2&quot;:a 2&quot;' would yield n+2'm = m./2.n, i.c it = 'In, for any 'It, ill, C \]N, which is abslird.</Paragraph> <Paragraph position="3"> :~\]{efrain frolil thinking in l!higlisli, and recall tll~Lt we work Oll tile sole level of synll)ols: i just \[)e('allle (I, or (.'d has just been added.</Paragraph> <Paragraph position="4"> ~ln absence of a.ny knowledge about the world. Ilere, only tlle equalil, y belAveell synlliols caJi be t.ested, liecause the a.lphabetica\] orde.r is not. known, this anMogy Cilnno|, l)e verified.</Paragraph> <Section position="1" start_page="488" end_page="488" type="sub_section"> <SectionTitle> 3.2 Symbol Inclusion </SectionTitle> <Paragraph position="0"> \]~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 Appendix in the p:roofs that some well-known lang u ages a,re la.n g u ages of an alogical stria gs ('.l'heorems 5 and 6 C' Section 5.1).</Paragraph> <Paragraph position="1"> Axioln 3 (Symbol inclusion) Let 12 be an v(Jl, n,(',/)) < (12&quot;)&quot;,</Paragraph> <Paragraph position="3"> l!'or strings redu(;ed to olle symbol, lhis trivially inll)lies: a: b= b: a Cb a=b.</Paragraph> <Paragraph position="4"> lncidently, al)l)lie(I on the eight equiw~,lent forms of an a,tta,logy, the Sytnl)ol inclusion axiom 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) C/> 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</Paragraph> </Section> <Section position="2" start_page="488" end_page="489" type="sub_section"> <SectionTitle> 3.3 Similarity Constraint </SectionTitle> <Paragraph position="0"> 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.</Paragraph> <Paragraph position="1"> 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:.</Paragraph> <Paragraph position="2"> = sire(A, ~) + si,n(~,C) - ~(A, ~, C, U) The Equivalent forms theorem yields:</Paragraph> <Paragraph position="4"> Because all 7(.,.,-,-) are equal in all the equalities above, and by the symmetry of shnilarity, 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).</Paragraph> <Paragraph position="6"/> </Section> <Section position="3" start_page="489" end_page="489" type="sub_section"> <SectionTitle> 3.4 Equality of length sums </SectionTitle> <Paragraph position="0"> A remarkable theorem is easily derived from the Similarity constraint theorem by addition and substraction and by commutatitivity of similar- null ity.</Paragraph> <Paragraph position="1"> 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</Paragraph> </Section> <Section position="4" start_page="489" end_page="490" type="sub_section"> <SectionTitle> 3.5 Disjoint Analogies </SectionTitle> <Paragraph position="0"> 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.</Paragraph> <Paragraph position="1"> lIowever, the previous intuition seems to hold anyway when the two anMogies to be concatenated do not have any symbol in common. We cMl such analogies, disjoint analogies. The intuition 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.</Paragraph> <Paragraph position="2"> One gets 2 4 = 16 analogies by enumerating all possibilities of exchanging or :not exchanging the substrings indexed by I a.:nd 2 in A1A2 : B1B2 = CIC2 : DID2. By numbering 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 hereafter those analogies with A1A2 as a first term.</Paragraph> <Paragraph position="4"> Of these tbur possible analogies, the second one, (0001), where only one exchange is performed, 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 alphabct, and V:, C \]), \])2 C F, ~'uch that</Paragraph> <Paragraph position="6"> This axiom will be used in Appendix in tile proof of Theorems 5 and 6 of Section 5.1.</Paragraph> </Section> </Section> <Section position="6" start_page="490" end_page="490" type="metho"> <SectionTitle> 4 Languages of analogical strings </SectionTitle> <Paragraph position="0"/> <Section position="1" start_page="490" end_page="490" type="sub_section"> <SectionTitle> 4.1 Analogical Derivation </SectionTitle> <Paragraph position="0"> liT, order to show how s(mle languages, 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.</Paragraph> <Paragraph position="1"> rsl! i < Definition I Let 12 be (t'lz alphabet. ~ n ~. an, alogical de'rivatio~t, ,zoled t~-~ , ~nod'ulo a set ;bt C 12&quot; x 12&quot;, whose elcmc'nt.~&quot; (v, v') ave noted v -+ v',</Paragraph> <Paragraph position="3"> Altllough we use the notation -~ \['or the elenlents of jr4, it is not to l)e interl)reted in the way it wout(l be in classica.l rewriting systen:m.</Paragraph> <Paragraph position="4"> This notaPSion is just to make a parallel with classical i)resentations of gramma.rs~ where the elements of j~ are ca.lle(1 rules. \]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) &quot;niat('hes&quot; ~11 o'lzd .~;i tit the sa~nc ti~nc.</Paragraph> </Section> <Section position="2" start_page="490" end_page="490" type="sub_section"> <SectionTitle> 4.2 Derivational Systems Definition 2 A dcrivatio,zal .sy.slcm of a,talog- </SectionTitle> <Paragraph position="0"> i(:al stving.~ i.s a triple (; = (Y,A,;t4), where 12 i.';' a .finitc alphabet, A C 12&quot; (jinite) is the sct of axioms~ or, bcttcv, the set of attested strings, a'lzd ~t4 C 12&quot; x 12&quot; (.\[inilc) i.~&quot; lhc sc.t of rulc.s, 0% bcttcr, the set of too(lois.</Paragraph> </Section> <Section position="3" start_page="490" end_page="490" type="sub_section"> <SectionTitle> 4.3 Languages </SectionTitle> <Paragraph position="0"> Definition 3 Let 12 be art alphabet. Let A C F*</Paragraph> <Paragraph position="2"> ~l.'he previous definition conforms to the usual 1)rese.ntation of formal languages. It aims at the generation of a language, q'hus, as usual, standard structural induction is used to genera.te all of the members of a language of analogical strings. Starting with the elements of .4, all possible analogies with the elements of ~td as models a.re applied.</Paragraph> <Paragraph position="3"> ~.iPhe reciprocal problem of generation is that of reco.qnitio'n. With an analogical system, the grammaticality of a given string, i.e., its membership in a language, is tested against the set of a.ttested strings of that language, a.fter tile reduction of that given string, by aa|a.logy, using the set of models. For recognition, the strings in the pa.irs ot'fi4 a.re used in the reverse order they appea.r in jr4, and the ana.\]ogies are solved in the other direction tha.n for generation. ~Phis is possible thanks to form (iii) of the Equivalent forth s theorem.</Paragraph> <Paragraph position="4"> q'he &quot;linguistic&quot; hlterpl:etation of a language 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, conj 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.</Paragraph> </Section> </Section> <Section position="7" start_page="490" end_page="491" type="metho"> <SectionTitle> 5 Some Properties </SectionTitle> <Paragraph position="0"> s.x {,,,?,,4....',~} and {(,,&quot;~b&quot;~&quot;~m ,} hi al)l)en(lix , we give l)roof~q that the followiTlg \['a, tnOllS regular, context-free a,lld~ Col,textsensitive la.nguages are all langua.ges of analogical strings:</Paragraph> <Paragraph position="2"> 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.</Paragraph> <Paragraph position="3"> This language is famous for being the basis of two counter-examples against the context-freeness of natural language: in the morphology of Bambara (Culy 85), and in the syntttx of the Zurich dialect of Swiss German (Shieber 85).</Paragraph> <Section position="1" start_page="491" end_page="491" type="sub_section"> <SectionTitle> 5.2 Bounded Growth </SectionTitle> <Paragraph position="0"> Following the discussion about the non-context-freeness of natural language, the family of tbrreal languages that can be used to formalise natural language has been thought to be necessarily larger than the family of context-free languages, but it does not have to cover all context-sensitive languages, as some context-sensitive languages are obviously not relevant for natural languages. Mild contcxt, sensitivity was thus proposed by (Joshi 85) to characterise the family of languages captured by tree-adjoining grammars (larger than context-free, but strictly smaller than context-sensitive).</Paragraph> <Paragraph position="1"> However, this is a characterisation by a recognition device, and some have proposed other intrinsic characterisations. (Marcus & al. 96) have been advocating that, the key point in &quot;mild context-sensitivity&quot; is the property of bounded growth: for each sentence in a language, we can always find another sentence in the same language whose length differs t!l:om the length of the first sentence by at most a given con stan t.</Paragraph> <Paragraph position="2"> Definition 4 (Bounded growth) A la'nguage PS has the bounded growth property if (and only ~') PS is a singleton or 3k E IN / Now, it is easy to prove (see Appendix ) that: Theorem 7 Any language of analogical strings verifies the bounded growth property.</Paragraph> <Paragraph position="3"> Consequently, a language like {a2'~/n C IN} is not a language of analogical strings, as it does not have the bounded growth property. Lucldly thus, some &quot;unnatural&quot; languages are out of the reach of languages of analogical strings.</Paragraph> </Section> </Section> class="xml-element"></Paper>