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| "text": "From the Preface: \"The volume i s more or less a random sample of the great number of works done in the field of mathematical linguistics by Soviet scholars. The random character of the selection i s due t o the d i f f i c u l t i e s which an editor inevitably encounters i n compiling an anthology 1 i ke the present volume. I f I were t o s t a r t working on this volume now I would certainly choose more recent papers, perhaps ones i n one or anotaer aspect more representative than those i ncl uded in t h i s vol ume. None,$hel ess , these a r t i cl es are a t l e a s t in one respect representative. They clearly t e s t i f y t o the breadth of i n t e r e s t and variety of approaches in Soviet mathematical 1 inauisti c s . This antholoav i s intended t o convince", |
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| "section": "(Currently v i s i t i n g at the University of Colorado)", |
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| "text": "Matheina t i c a l Models of Language t h e readerwho has n o t mastered Russian and i s perhaps n o t f a m i l i a r w i t h works by S o v i e t \"mbiihematical 1 i n g u i s t s \" t h a t they deserve much more d t t e n t i on than they have r e c e i v e d up t o now. I I Tl?e papers i n t h i s volume are indeed beginning t o show t h e i r age. From i n t e r n a l evidence, p r i m a r i l y t h e b i b l i o g r a p h y o r notes a t t h e end o f each paper, these papers were w r i t t e n i n 1967-1972. As a \"random sample\", the o n l y way t o r e v i e w these p.apers i s t o take each i n t u r n .", |
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| "start": 274, |
| "end": 277, |
| "text": "I I", |
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| "section": "(Currently v i s i t i n g at the University of Colorado)", |
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| "text": "On the Complexity o f Government Trees. \"On the one hand, t h e government t r e e c o n t a i n s i n f o r m a t i o n about t h e structcnne o f t h e t e x t which must be taken account o f i n any model. On t h e o t h e r hand, i t i s a comparatively simple o b j e c t f o r which i t i s e a s i e r t o develop a su4 t a b l e mathematical appar a t u s . II The complexity depends o n the i n t e r n a l arrangement o f v e r t i c e s , thus f o r a sentence of l e n g t h n, t h e r e a r e government t r e e s w i t h n leaves which have m i n imal complexi ty. \"Here we s h a l l proceed from t h e assumption t h a t thoseestruct u r e s which have minimal o r c l o s e -t o minimal complexity are r e a l i zed i n n a t u r a l l a n o i a g e .\". Very reasonable.", |
| "cite_spans": [ |
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| "start": 366, |
| "end": 368, |
| "text": "II", |
| "ref_id": null |
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| "section": "M. V. Ardpov -E. N e Efimova", |
| "sec_num": "1." |
| }, |
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| "text": "The c o m p l e x i t y measure used i s developed as f o l l o w s : For each v e r t e x l e t ki be t h e out-degree of i, t h a t i s , t h e number of descendents o f i, and l e t i* be t h e f a t h e r o f i . L e t t h e r o o t o f t h e t r e e be node 0. The conip l e x i t y o f each v e r t e x i s d e f i n e d as: $(N) = ( A~) $ i s t h e c o m p l e x i t y measure s t u d i e d . T h i s does n o t d i r e c t l y f i n d t h e c o m p l e x i t y o f t h e minimum t r e e w i t h n leaves, which i s a more int e r e s t i ng q u e s t i o n g i v e n the p a p e r ' s s t a t e d o r i e n t a t i o~ toward 1 i n g u i s t i c s .", |
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| "section": "Mathematical ~o d e l s of Language", |
| "sec_num": null |
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| "text": "F ( i ) = ki + F ( i X ) f o r i>O. F o r tree h N w i t h N v", |
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| "section": "Mathematical ~o d e l s of Language", |
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| "text": "Nonetheless, t h e authors f i n d several suggestive r e s u 1 t s about the s t r u c t u r e o f minimal t r e e s .", |
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| "section": "Mathematical ~o d e l s of Language", |
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| "text": "Theorem 5 . I f AEM, then-ko5?", |
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| "section": "81", |
| "sec_num": null |
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| "text": "Assuming t h a t . t h i s m o t i o n o f m i ni.mal i t y i s indeed a p r i n c i p1.e o f econonly, then po sentence has more t h a n t h r e e main c o n s t i t u e n t s .", |
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| "section": "81", |
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| "text": "Theorem 11. I f AEM, f o r each v e r t e x 'i>O, ki*lki.", |
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| "section": "81", |
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| "text": "The deeper ofle goes i n a minimal t r e e , t h e lower the out-degree. here -t h & both copies of J dominate isomorphic subtrees. This enables one t o specify !'syntactically\" t h a t every variable in a programming language m u s t be declared before i t i s used.", |
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| "section": "81", |
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| "text": "However, there are better methods t o hand1 e these non-tree restrictions. I c a n ' t t h i n k o f any use f o r t h i s order-reversal i n carrying o u t the translation, r b u t i t i s an interesting idea !nonetheless. ( A t t h a t t i m e t h e young man was i n / a t t h e t h e a t e r ) I 81 C q n t r o l 8(1965'), 304-337*). 1 found 1 i t t l e o f i n t e r e s t i n t h i s s e l e c t i o n . all-too-brief discussion oy the poem \"Eugene Onegin\" in which the theory i s \"An accented syllable can be located or~ly on an even-numbered The study o f control led grammars arises from the psycho1 i ngui s t i c i dea t h a t derivations are control led by a \"generation program\" which determines t h e semantics of the phrases and their grammaticdl structure. Thereby, the re-. sul t s presented here presumab;ly expl ic'ate the pdtenti a1 graniatical structures which such a generation prog.ram could possibly produce. Whether or not on' e accepts the \"geqerpti ng pr,ogram\" hypothesis, these are nice resul ts i n formal", |
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| "section": "81", |
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| "text": "anguage theory.", |
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| "text": "pages 3-36. A gov.ernment t r e e i s a d e r i v a t i o n t r e e d e p r i v e d o f i ts l a b e l s . Thus t h e complexity r e l a t e s s o l e l y t o t h e s t r u c t u r e o f t h e t r e e w i t h o u t regard t o phrasenames (nonterminal s), 1 e x i c a l c o n s i d e r a t i o n s and t h e 1 i ke.", |
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| "text": "e r t i c e s , t h e c o m p l e x i t y i s : e s e t o f a1 1 t r e e s with N v e r t i c e s , MN and d e f i n e AEM t o be N the minimal i f and o n l y i f F ( A ) < -F ( A ' ) f o r a l l A'EM,,,. L e t M N c M N be t h e s e t o f minimal t r e e s w i t h # v e r t i c e s and l e t M = U R N be t h e s e t o f a l l minimal t r e e s . Then f o r each minimal t r e e A N~M N", |
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| "text": "..the monotonous decrease i n t h e number o f arrows i s s u i n g from a v e r t e x propotttionate t o i t s . . . d i s t a n c e From t h e r o o t o f t h e t r e e e s s e n ' t i a l l y agrees w i t h t h e e m p i r i c a l f a c t s . 'In f a c t , the numbero f camp1 e t e d ya1 ences f o r the' v e r b -p r e d i c a t e (which a r e usual l y p l a c e d i n t h e r o o t o f t h e t r e e ) i s on t h e average r a r g e r than For a noun which i s s u b o r d i n a t e t o i t , larger f o r the noun t h a n f o r ana d j e c t i v e s u b o r d i n a t e t o t h e noun, and t h i s number i s more o f t e n than n o t equal t o zero f o r an adverb governed by such an adject i v e . O f course, such a monotony i s i n r e a l i t y o n l y approximate. 1 I Theorem I 11. For-any hN&b! w i t h N>81, ko = 3. Theorem I V . $(N) i s o f t h e o r d e r N In ' N. t h e authors p o i n t o u t t h a t a d e t a i l e d comparison o f minimal government trees. w i t h ' c o n c r e t e ' s y n t a c t i c s t r u c t u r e s i s w i t h o u t much meaning. Nonethel e s s , t h i s i s the f i r s t paper t h a t I know o f which broaches t h e n o t i o n o f an .economy o f s y n t a c -t i c e f f o r t . Whether t h e theorems a r e indeed suggestive o f l i n g u j s t i c r e a l i t y i s a m a t t e r f o r f u t u r e research. 2. V. B. borscev -M. V. Xomjakov A x i o m a t i c approach t o a D e s c r i p t i o n o f Formal i zed Languages and Transl&.i.inn Netghborhood Languages. pages 37-1 14. Mathema t i c a l Models of Language 82 This l e n g t h y c o n t r i b u t i o n consi s t s o f f o u r chapters o f d e t a i l e d development. The b a s i c p l a n i s an i n t e r p r e t a t i o n o f P. M. Cohn's U n i v e r s a l A1gebr.a Row, New York, 1965) as r e l a t i o n a l systems t o t r e a t \" t e x t s 1 ' and grammars. Whi 1e I enjoyed reading Cohn's excel l e n t treatment o f u n i v e r s a l algebra, I d i d n o t e n j o y t h i s paper. I t tends t o wander, whereas I prefer papers which b u i l d t o a d e f i n i t e c l i m a x . F u r t h e r , most o f t h e a u t h o r s ' ideas have been presented i n t h e Western 1 i t e r a t u r e , so I found a t most two new nuggets o f wisdom. Nonetheless, here i s the substance o f t h e paper. Chapters I and I1 b u i l d a n o t i o n o f I 1 t e x t \" and grammar v i a systems o f r e l a t i o n s . One has r e l a t i o n s o f \" t o t h e l e f t o f \" and \"below\" i n t r e e s as w e l l as o t h e r r e l a t i -o n s , such as \"isomorphic subtree\". Even t h e n o t i o n s o f t e r m i n a l and non-termimal a1 phabets a r e t r e a t e d as r e 1 a t i ons . Thi s u n i formi ty m i g h t o f f e r some advantages f o r t h e a b s t r a c t devel.opment about classes o f s i g n systems, t e x t s and grammars, b u t makes t h e concrete cases and examples hard t o f o l l o w . I n f$&t t h e r e a r e o t h e r u n i f o r m treatments, mentioned below, which are undoubtedly b e t t e r f o r the p a r t i c u l a r cases i n q u e s t i o n . The authors t r e a t neighborhood grammars i n these two chapters. A neighborhood of a v e r t e x i n a t r e e c o n s i s t s o f some o f t h e connecting arcs and nearby nodes. For example, a neighborhood o f F i s (page 5.3): where t h e d i s t i n g u i s h e d node whose neighborhood i s i n q u e s t i o n i s marked by Given a c o l l e c t i o n o f neighborhoods, a t r e e i s i n t h e neighborhood language i f a1 1 t h e neighborhood c o n s t r a i n t s s p e c i f i e d i n fprmul a which c o n s t i t u t e the Mathematical Models of La qguage 8 3 grammar are s a t i s f i e d a1 1 nodes of the t r e e . The major virtue of thif approach i s i n e n a b l i i t h one t o specify other connections between the nodes o f trees other than the usual descendant relation. Thus i s a neighborhood specifying -via other information t o o complex t o describe", |
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| "text": "were o r i g i n a l l y w r i t t e n i n 1969-1970. This development was c l e a r l y r i p e a t t h a t t i m e i n Russia, Europe and t h e U. S.Thenominal ----neicjhborhoodgralnmars a r e an extension o f n e i ghborhood grammars which a1 low f a i r l y complex s t r u c t u r e s . For example, t h e f o l l o w i n g i s taken froni page 90: I t appears t h a t the language n n n Iala2c-q In21 1 can be generated by nominal neighborhood grammars i n an e s s e n t i a l l y c o n t e x tf r e e manner. The norriinal neighborhood grammars a r e new t o me and appear t o o f f e r considerable g e n e r a t i n g power a t t h k usual expense o f a complex. d e f i n i t i w .Chapter I V t r e a t s syntax-di r e c t e d t r a r~s l a t i o n s and c e r t a i n extensions t h e r e o f u s i n g t h e i d e a o f neighborhoodsMost o f t h e i r development i s now standard (Aho and U l lrnan , The Theory o f Par5i ng, Trans1 a t i on, and Cornpi 1 i ng:", |
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| "text": "3. S:--Ja. Fi tialovOncLhe E~L ! valence of IC Grammars and Depende-ncy Grammarspages 115-158. kccording t o ttlc author, b o t h the direction and nesting of syntactic Ma thema t i c a l Models of Language86 r e l a t i o n s h i p s s h o u l d be accounted f o r i n a s u f f i c i e n t l y adequate and complete 1 i n g u i s t i c d e s c r i p t i o n . As dependency grammars handle d i r e c t i o n and Irmediate Cohst-i t u e n t g,rammars handle nesting, t h e q u e s t i o n o f t h e r e l a t i o n s h i p between t h e two d e s c r i p t i ve mechanisms a r i s e s . As t h e .phrase names (non-terminal symbols) .can n o t be determined from the dependencies, the I C s t r u c t u r e considered c o n s i s t s s o l e l y o f t h e tree. T h i s i s b e s t i l l q s t r a t e d by the f o l l o w i n g example. The element groups i n t h e dependency s t r u c t u r e a r e enclosed i n parentheses, t h e dependent d i recti ons i s shown below t h e sentence and t h e I C t r e e i s shawn above. #From page 128:", |
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| "text": "Two dependency structdres can give r i ' < e t o a s i n g l e I C structure. Conl pare w i t hFig. 1 . (The student o r c h e s t r a gave a b i g concert i n the PaPCice of Cul t u r e ) -Ma thema t i c a l Models of Language8 Furthermore, F i t i a l o v c i u e s examples i n which the same sentence can have two d i f f e r e n t I C t r e e s . Thus t h e \"equivalence\" i s many-rto-many. The a u t h o r t h e n s e t s up an a l g o r i t h m t o con9tPuct a dependency grammar from c e r t a i n IC grammars. The I C grammar must have \" f i n i t e degree\", a t e c h n i c a l concept t h a t need n o t d e t a i n us. The f i n a l t o p i c i s c a r r y i n g the i d e a o f \"degree o f nesti ng\" f r o m I C s t r u c t u r e s w i th non-t e r m i n a l s over t o dependency s t r u c t u r e s . Much o f t h i s paper i s a p p a r e n t l y devoted t o c l a r i f y -i n g t h e ideas presented by Gaifman (De-penden~y Systems and Phrase-Structure Systems, Inform.", |
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| "text": "t tht: Formal D e f i t i i t i o n ~f Case and Gender o f t h e Noun. pages 159-204. With t h e r e c e n t i n t e r e s t 1.n case i n computati.ona1 l i n g u i s t i c s , t h i s paper by t h e foremost Russjan formal languages e x p e r t should appeal t o those who w i s h t o bui I d l o g i c a l 19 c o h e r e n t case s t r u c t u r e s . The vathematics i s minimal b u t suggestive. The Soc~as o f t h e wovk i s on a class4 f i c a t i a n o f (Russian) nouns. I a~ll 1 1 1 no p o s i t i o n t o comment on t'he qua1 i t y oP t h e c l a s~i f i c a t i o n system propose$ onet the less, hr-e i s a sketch o f t h e method. kt V.-be -the set. o f words. These ace c a l l e d segments by Gladki j t c s t r e s s t h e g r a p h i c a l sense o f w o r d he i s u-sing. Each. s u b s e t o f V h'aving \" i d e n t i c a l 1 e x i c a l meaning\" i s cal l e d a neighborhood; Thus : f mM, GOMA, DOMU, DOMOM, DOME, DOMA, DOMOV, DMAM, D' OMAMI, DOPIAX) (-house) LC% , a subset of V, be the wet o f nouns, \"The s e t S s h o u l d be. a union o f some neighbbrhoods.\" The next n o t i o n i s s u b o r d i n a t i o n or dependency. Say ~atBemati c a l Models of Language 88 t h a t x ( b o t e n t i a l l y ) subordinates y i f t h e r e i s a sentence i n which some occurrence o f segment x \" s y n t a c t i c a l ly d i r e c t l y subordinates\" some occurrence o f segment y. Now l e t 0 be any neighborhood. Say t h a t 0 subordinates y i f y i $ subordinate t o a t l e a s t one segment i n 0. L e t No be t h e s e t of a l l S-segments (noun words) which are subordinate t o 0.. A s e t No i s s a i d t o be minimal i f No i s n o t empty and t h e r e a r e no non-empty NOI which a r e proper subsets o f No. The minimal s e t s No a r e s a i d t o be cases. \" I f two d i f f e r e n t neighborhoods 0 and 0 ' o f t h e sets No and No, coincide, we w i l l n o t consider No and No' t o be d i f f e r e n t cases, b u t one and the same case. II Gladki j gives examples o f a l l these concepts, i n c l u d i n g the d i s t i n c t i o n between minimal and non-minimal neighborhoods. He goes on t o s'how t h a t the cases a r e n o t n e c e s s a r i l y m u t u a l l y d i s j o i n t , and then uses t h e devel'opment t o expl i c a t e t h e \" s p e c i a l posi t i o n \" occupied by t h e second p r e p o s i t i o n a l and secnpd g e n t i v e cases i n Russian grammar. G l a d k i j then shows, t o no one's s u r p r i s e , t h a t t h e r e a r e instances i n which meaning, even t h e meaning o f the p r i o r several sentences, must be taken i n t o c o n s i d e r a t i o n t o determine the case o f c e r t a i n words. I f one's purpose i s t o understand the t e x t , then i n these instances the case s t r u c t u r e w o n ' t he1 p. I n most sentencqs however, i t w i l l c l a r i f y the 'relationships o f t h e segments i n t h e sentence and thus a i d understanding. , Whether Gladki j ' s formul atiaon i s more u s e f u l than unaided i n t u i t i o n and knowledge o f t h e language i s f o r o t h e r s t o iudge.. The l a s t s i x t e e n pages o f the paper develop a s i m i l a r forma-lism f o r the concept ~f coordinated c l a s s , apparently as an d i d t~ d r r i v i n g , a t t h e very end. o f t h e paper, i n a d e f i n i t i o n o f gender. The mathematics i s very easy, b u t the Russian examples a r e not--for t h i s r d l e w e r , Mathema tical Models of Language 5. Ju. K. Lekomcev On Models f o r a Syntax w i t h E x p l i c i t l y D i f f e r e n t i a t e d Elements (D-Syntax) . pages 205-239. This paper i s , by Western standards, fussy and pedantic. One mustlsup, pose t h a t t h e editor's s e l e c t i o n was r a t h e r more random than l e s s . ~e s p l k e the f o l l o w i ng q u o t a t i on from the i n t r o d u c t i o n --\"Concerning the c h a r a c t e r i s t i c o f a D syntax model , i t should be noted t h a t o u r model i s a c o n t i n u a t i o n o f the glossematic v a r i a n t o f t h e Saussurian trend, p a r t l y complemented by Russian and American concepts. The notions o f syntagmati c-paradi gmati c re1 a t i o n s and o f d i s t i n c t i v e features l i e a t t h e h e a r t o f the concept.'' --I was d i s a o~o i n t e d . The mathematical model, s t a t e d i n t h e complete f o r m a l i t y of f i r s t -o r d e r p r e d i c a t e l o g i c , actual ly says very 1 i t t l e . The foundation o f t h e paper's development i s a n o t i o n o f d i f f e r e n t i a t i o n system (DS). A DS i s b a s i c a l l y a system o f l i s t s o f t h e values o f a t t r i b u t e s . Thus two element ( i .e., l i s t s ) d i f f e r i f some value o f some common a t t r i b u t e d i f f e r s . Actual l y the paper devel bps Somewhat more complex d i f f e r e n t i a t i o n systems, b u t t h e a d d i t i o n a l complexities are obvious, n o t r e q u i r i n g such an o v e r l y formal devel opment . This n o t i o n i s then a p p l i e d t o the q u e s t i o n o f generating (resp.,analyzing) words from phonemes, i n a f a s h i o n t h a t would have produced more i n s i g h t i f i t had been t r e a t e d i n automata-theoretic terms. The concluding remarks--on a p p l y i n g OS to semantics--seem t o t h i s reviewer t o be i r r e l e v a n t , o r e l se more c T e a r l y presented e l sewhere.Mathema tical Models of Language6. Ju. A. SrejderOn the Contrast between the Concepts 'Language Model' The concept 'Tanguage model \" i s widely used i n structural and mathematical 1 inguistics. In a certain serise, this concept is the cornerstone of these branches or 1 inguistics, where so called formalized or precise methods have taken root. I t i s of some use, therefore, t o gain an understanding of just w h a t i s meant with the words \"language model\".\"The author, evidently a mathematician, contrasts the notion of a mathematical theory and a language model. I n the terms o f mathemati-cal l o g i c , a theory consists of names for relations or functions, names for variables, a method of contructi ng we1 1 -formed formulae ( w f f ) ,the system of formal deduct i o n t o be used, and the axioms of t h e q r y . A podel o f a theory i s a system of 'sets and rel'ations such t h a t \" i f reqation Ri i s compared t o every name o f relation IRi in sudh a way that i f variables x , y , z , . . . are explained' as elements o f s e t IM a1 1 formulae of th-e given Thecry are true.\"After several examples o f mathematical theories and. model s--the theory of parti'al orders and a model of i t i n the natural numbers i s one--the a u t h o rgives a f a i r l y strong argument that what many linguistics \"call a model i s i n mathematics known a s a thoery.\" He then gives some examples o f ' language models' t o give substanence t o this t h e s i s . The most interesting i s an", |
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| "text": "d a c e from t h e beainnina o f the, link.\" Mathematical Mode-of Language 91 f o r which presumably a standard e d i t i o n o f \"Eugene Onegin' stands as a model. He continues by g i v i n g a s h o r t neighborhood grammar as t h e axioms o f t h e theory. I n an appendix he shows t h a t Chomsky's \" g e n e r a t i v e model. o f c o n t e x tf r e e grammars i s i n f a c t a p a r t i c u l a r mathematical. theory. However, l i n g u i s t i c s i s n o t mathematics and t h e models ( i .e., t h e a c t u a l utterances o r t e x t s ) f a i l t o s a t i s f y a l l d e t a i l s o f t h e theory. Thus. the a u t h o r suggests \"1) The quasi-model o f a Theory, i .e., t h e s e t ih which t h e t h e o r y i s almost f u l f i l l e d ( f o r t h i s i t i.s necessary tc. 'introduce a measure onto the Theory) and; 2) The measure on a cl'asg o f quasi-models o f g i v e n Theory, which a1 lows us say -that t h e Theory can be f u l f i l led-f o r almost a l l quasT -hodel s . I I U n f c r t u n a t e l Y , thesef i ne ideas . a r e n o t d e v e l o~e d . Nonetheless. t h i s paper doe+ he1 P e x p l a i n t h e t e r m i n o l o g i c a l d i f f e r e n c e s betweet, mathematicians and i n Q u i s t s . .7. E. D. S t o c k i J General i zed G~ammars and Thei r P r o p e r t i e s . pages 269-284. \" L e t us a'sume t h a t i n grammar r nett a i l l e r i v a t i o n s a r e permiss i bl0, b u t o n l y those which can themsel ves be generated by another qramrnar r',which Ts worltiflq as. t h e device f o r prgqrammin.0 o f t h e d e r i v a t i o n . W e s h a l l , i n v e s t i t l a t e t h e a u e s t i o n o f how t h t s e l e c t i o n o f a s t r a t e g y o f phrase generation i n grammar Mathema t i c a l Models of Language r ( i n o t h e r words, s e l e c t i n g grammar r ' ) a f f e c t s t h e g e n e r a t i v e c a p a c i t i e s o f grammar r.\" L e t XodXI-==.~ ... X, be a d e r i v a t i o n i n grammar r. L e t V ' be a s e t u f names f o r t h e r u l e s o f r . Thus t h e d e r i v a t i -m corresponds t o a word ~1~2 . . . p n c ( V t ) * where p i s t h e name o f t h e r u l e d o i n g t h e r e w r i t i n g i Ximl --Xi . Each word o v e r V 1 t h a t correspond: t o a d e r v i a t i o n i s c a l l e d a contr0.1word. I n general t h e r e i s no one-to-one correspondence between t h e d e r i v a t i o n s and t h e i r c o n t r o l words. A g e n e r a l i z e d grammar i s a p a i r o f grammars ( r r ) such t h a t th, second grammar i s used t o c o n t r n l t h e f i r s t , Speci f i c a l l y , XoZ=.X1-=-\\. ... =+-X i s \" n an a l l o w a b l e d e r i v a t i o n of r i f and o n l y i f t h e r e i s a t l e a s t one c o n t r o l word corresponding t o i t i n L ( r l ) . The language o f t h e g e n e r a l i z e d grammar i s t h a t subset o f L ( r ) f o r which each word has a t 1 e a s t one a1 lowable d e r i v a t i o n . Note t h a t t h e r e i s no requirement t h a t t h e d e r i v a t i o n s be canonical ( l e f tmost). Grammars i n t h i s paper a r e c l a s s i f i e d by t h e u s u d Chomsky H i e r a r c h y i n t o types 0, 7, 2, 3. Then g e n e r a l i z e d grammars have type ( i , j ) where i i s t h e t y p e o f t h e language-producing grammar and j i s t h e t y p e o f t h e c o n t r o li n g grammar. L e t Di be t h e c l a s s o f 1 anguages generated b y a1 l general i zed grammars o f t y p e ( i , j ) , and Di be t h e c l a s s o f languages generated by a l l ( o r d i n a r y ) grammars o f t y p e i . The main p o r t i o n o f t h e paper presents, w i t h o u t p r o o f ,. the r e 1 a t i p n s h i ps known among t h e D 3 s o f A p g i l , 1969, -e x c l u d i n g some American work Such as 113 G i nsburg and Spani e r ' s ( D e r i ~a t i an bounded 1 anguages, J . Comp . $ys . -P-Sci ---. 2: 3(1968), 228-250). Most bf the' r e f e r e n c e s c i tehr-whi ch c o o t a i n t h e pnoofs--are t o S t o c k i j 1 s own worK on rhese questions. Example r e s u l t s are: Mathematical Models of Language -D1O = D20 -D30 = Do (without null words) This l a s t i s a consequ~llce o f Stockij s disallowin-q rewritinqs t o the null word in grammars of4type 1 , 2 and 3 , Now consider the s e t o f control words, P(r), of an ordinary uncontrol led grammar, r. Let t b ( i ) denote the type of the langoage P ( r ) for grammaUr r o f type i . The rot 1 P~i n g are representaki ve resul t s .", |
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| "content": "<table><tr><td>Trans-</td></tr><tr><td>l a t i o n , and Compiling: Vol. 2, Compiling, Prentice-Hall,, Englewood Cliffs,</td></tr><tr><td>N.</td></tr></table>", |
| "text": "Fi scher, Grammars with macro-1 i ke productions. Ph. D. Thesi s , Harvard Uni versi ty, 1968), and mathematical semantics ( R . D. Tennent, The Lnotational Semantics of Programmi ng Languages, Comm. ACM 19: 8 (Aug. 1976), 437-453. ) Chapter I I I , \" c o n c r~t e s i gn sys terns\" devel ops phrase structure grammars 3nd nominal neighborhood grammars. The type 0 phrase structure grammars produce \"phrase structures\", as generalizations o f trees. iese phrase structures have appeared i n the Western 1 i terature i q W least the fbl lowing papers: J. Loeckx, The Parsing for General Phrase-Structure Grammars, Inform& Control 16:5 (~u l 1970), 443-464, H. W. Buttelrnann, Oa the .Syntactic-Structures of Un-", |
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| "text": "Mathematical Models of LanguageVdl . 1, Parsing, Prentice-Hall , Englewood Cl i f f s , N . J . , 1972) and has been advanced t o truely elegant abstractions by Alagic (Natural State Transformations, J . Gbmp: Sys. Sci . 10: 2 ( A p r 1475), 266-307.) However, the use of nei ghborhooas a1 lows f o r the extension of syntax-directed trans1 ations i n new directions, best indicated here by the authors l diagram o f a translation from t r e e TI to t r e e T2.", |
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