{
    "paper_id": "J78-3005",
    "header": {
        "generated_with": "S2ORC 1.0.0",
        "date_generated": "2023-01-19T03:04:56.604683Z"
    },
    "title": "American Journal of Computational Linguistics A CRITICAL LOOK AT A FORMAL MODEL FOR STRATIFICATIONAL LINGUISTICS",
    "authors": [
        {
            "first": "Alexander",
            "middle": [
                "T"
            ],
            "last": "Borgida",
            "suffix": "",
            "affiliation": {
                "laboratory": "",
                "institution": "University of Toronto",
                "location": {
                    "postCode": "M5S 1A7",
                    "settlement": "Toronto",
                    "region": "Ontario"
                }
            },
            "email": ""
        }
    ],
    "year": "",
    "venue": null,
    "identifiers": {},
    "abstract": "We present here a formalization of the straiificational model of linguistics proposed by Sampson C131 and investigate its generative power. In addition to uncovering a number of counterintuitive properties, the results presented here bear on meta-theoretic claims found in the linguistic literature. For example, Postal [ l l j claimed that stratificational theory was equivalent to context-free phrase-structure grammar, and hence not worthy of further interest. We show, however, that Sampson's model, and several of its restricted versions, allow a far wider range of generative powers. In the cases where the model appears to be too powerful, we suggest possible alterations which may make it more acceptable.",
    "pdf_parse": {
        "paper_id": "J78-3005",
        "_pdf_hash": "",
        "abstract": [
            {
                "text": "We present here a formalization of the straiificational model of linguistics proposed by Sampson C131 and investigate its generative power. In addition to uncovering a number of counterintuitive properties, the results presented here bear on meta-theoretic claims found in the linguistic literature. For example, Postal [ l l j claimed that stratificational theory was equivalent to context-free phrase-structure grammar, and hence not worthy of further interest. We show, however, that Sampson's model, and several of its restricted versions, allow a far wider range of generative powers. In the cases where the model appears to be too powerful, we suggest possible alterations which may make it more acceptable.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Abstract",
                "sec_num": null
            }
        ],
        "body_text": [
            {
                "text": "Linguistic theories are at least partially interested in presenting the regularities found in natural languages. Given the current dominance of the Transformational Generative (TG) school in the field of linguistics, it seems necessary for theories competing for attention to possess a formal model, In addition to the advantages normally derived from presenting results through a formalism, such as precision, succinctness and verifiability, one can also comment on the veracity of metatheoretic claims. It was using such formal arguments that Chomsky and his collaborators demonstrated the inability of finite automata and of context-free grammars to describe all natural language constructs. Similarly, the formal work of Peters and Ritchie [ 8 , 9 1 was important in uncovering inadequacies of two notions of TG theory namely, the \"recoverability of deletions condition\" and the \"universal base hypothesis\".",
                "cite_spans": [
                    {
                        "start": 176,
                        "end": 180,
                        "text": "(TG)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "Finally, since many generative linguists want grammatical theories which characterize natural languages, they fault any theory which is .too powerful\" in the sense of being able to describe languages which clearly cannot be natural languages, such as nonrecursive sets. Furthermore, computer scientists working on natural languages will have to give in the future more consideration to the work of linguists, especially on \"exotic\" languages, in order to be able to observe a wider range of phenomena. Such access will be facilitated if the formalismsin which the grammaTs are prese'nted lend themselves to computer implementation for purpose$ such as parsing, testing, etc. This entails, among other things,that linguists should avoid as much as possible features which make their grammars generate non-recursive sets, and hence it is one of the purposes of the present paper to point out such features and discuss possible ways of avoiding them.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "In this paper we will discuss one model proposed for the stratificational theory of linguistics. This theory, advanced by S. Lamb, H . A . Gleason Jr. and their collaborators (C51,[61,C71), advocates that langdages be described in terms of several subsystems, known as strata. Each stratum has its own set of units and a tactics specifying the tlcorrectfl (\"all~wable'~) structures that stratum. specific grammar might for example have strata corresponding roughly to semantics, syntax-morphology and phonology, although this is by no means standard. Furthermore, the strata are linearly ordered as levels, and there is a realization relation which connects adjacent strata by attaching to every well-formed structure on one stratum, zero or more accompanying structures on the adjacent strata. Note therefore that a particular utterance has simultaneous expression on each stratum.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "In this paper we examine the formal model for stratificational linguistics proposed by Sampson ( L 1 3 1 ) . This model uses rewrite grammars G1,G2, ... to describe the tactics, while the realization relation is essentially a rewrite system R acting as a transducer between the languages of-the tactics. More specifically, realization connects adjacent tactics G and G j + l j by matching sentences u in the language generated by G with those sentences j v in the language of G j + l which can be derived from u by using rules from R. An important property of the linguistic realization relation is the fact that' every structure on some stratum can have only a finite number of llrealizates\" on the next stratum.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "This means that the rewrite system R must be constrained so that it has no recursive symbols. Such a rewrite system will be called acyclic.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "We investigate here the effect of acyclic rewrite systems acting as transaucers on axiom sets, varying the type of the derivations and rules allowed.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "We prove in this paper that regular languages are closed under transduction by acyclic rewrite systems, but that the linear context-free languages are mapped onto the recursively enumerable sets. This implies that stratificational grammars with non-selfembedding ta~ctics would be too weak while those with even one context-free tactics would be too strong. If the realization derivation is restricted to be in some sense \"leftmost\", then we show that the transduction can be performed by a finite,state device known as an a transducer. Furthermore, if productions with null right-hand sides are not allowed in an acyclic rewrite system then all the derivations can be made leftmost. This provides one possible method of restricting the generative power of acyclic rewrite systems.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "By deriving a recursive characterization of the languages generated with n-strata in terms of (n-1)-stratal languages, we can show that if the realization is restricted to being leftmost, then the languages described are homomorphic images of the intersections of the languages generated by the tactics. In particular, this means that we can find natural families of stratificational grammars which generate far example the sets recognized in real time by nondeterministic multitape Turing machines. This result partially confirms a hitherto unproven claim by Sampson, and discredits Postal's Clll classiciation of stratlficational grammars as just another variant of context-free phrase-structure grammars.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "Finally, we investigate the use of ordered rules in linguistic grammars and prove that in several models they allow the generation of sets which are not even recursively enumerable a clearly unsatisfactory situation.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "The remainder of the paper is structured as follows,",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "In Section 2, we present the formal definitions and notation to be used, including the formal model for stratificational grammars.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "In Section 3, we examine the properties of \"acyclic rewrite systemsw, which form the principal novel component in our definition of stratificational grammars. We then return in Section 4 to examine the generative power of st'ratificational grammars and relate the results to linguistics. e -o u t p u t f r e e i f f o r an$t ( r , u , v , s ) i n T , t h e s t r i n g v cannot be n u l l . ",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "To b e g i n w i t h , we remark t h a t i n G t h e n A => dD i n Gi+l i n c a s e bc -+ d is i n R ( s t e p 5 ) . where we used r u l e Xw A'Yz i n s t c p 0. and l a n g u a g e s L1,. . . , L s u c h t h a t f o r i = 1,. . . ,n L(G.) t h e r e e x i s t s a -t r a n s d u c e r O n v l s u c h t h a t L-RSTRAT(RST) i s e q u a l t o *",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 219,
                        "end": 239,
                        "text": "i = 1,. . . ,n L(G.)",
                        "ref_id": "FIGREF12"
                    }
                ],
                "eq_spans": [],
                "section": ". Generative power of a c y c l i c r e w r i t e systems",
                "sec_num": "3"
            },
            {
                "text": "+ v . \u20acVG $Rev ( v ) = v i -1 f o r i = 1,. . . , m -1 3 1 m + i -1 and --a' --IUN N V + L 2 = { % w l a . . .w % $ % w~+~ %...p ln>O, w . a V",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": ". Generative power of a c y c l i c r e w r i t e systems",
                "sec_num": "3"
            },
            {
                "text": "On+ 1 (L-RSTRAT(TOP(RST)) n L ( G ) ) n V , . n+ 1 ( 4 )",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": ". Generative power of a c y c l i c r e w r i t e systems",
                "sec_num": "3"
            },
            {
                "text": "1 Meaning t y p e 3 , t y p e 2 , t y p e 1, t y p e 0 , l i n e a r l a n g u a g e .",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": ". Generative power of a c y c l i c r e w r i t e systems",
                "sec_num": "3"
            },
            {
                "text": ".,. , L t t such t h a t L-RSTRAT(TOP(RST)) = hTt(L!i n . . . n -Lt\\3 . n n S u b s t i t u t i n g t h i s i n (4) and appllying n o t e s ( a ) , (b) and (c) we the realization derivation must be k-leftmost and a homomorphism must be applied to the intersection of the languages. C81 P e t e r s , P .S. and &W. ~i w h i e ( 1 9 7 1 ) . \"On ~e s t r i c t i n g the base component of T r a n s~o r m a t i d n a l grammar^'^. Information and \"",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "L : ,",
                "sec_num": null
            },
            {
                "text": "Control -18. -C91 P e t e r s , P.S. and R.W. Ritchiea ('1973) . \"On the g e n e r a t i v e power of Transformational ~r a r n m a r~\" 1.nformati06 Sciepces 6 49-83. E 1 2 1 Salomaa , A . (1973) . Formal Languages, Academic Press,New",
                "cite_spans": [
                    {
                        "start": 55,
                        "end": 62,
                        "text": "('1973)",
                        "ref_id": null
                    },
                    {
                        "start": 189,
                        "end": 195,
                        "text": "(1973)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "L : ,",
                "sec_num": null
            },
            {
                "text": "York.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "L : ,",
                "sec_num": null
            },
            {
                "text": "[ 1 3 J Sampson, G . ( 1 9 7 0 ) . S t r a t i f i c a t i o n a l grammar: 9 D e f a n i -t i o n and a n Example, Janua Linguarum, S e r i e s Minor: 8 8 , The Hague Mouton.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "L : ,",
                "sec_num": null
            }
        ],
        "back_matter": [],
        "bib_entries": {
            "BIBREF0": {
                "ref_id": "b0",
                "title": "Reversal Bounded, Mu1 ti-pushdoyn Machinestf",
                "authors": [
                    {
                        "first": "B",
                        "middle": [],
                        "last": "Baker",
                        "suffix": ""
                    },
                    {
                        "first": "R",
                        "middle": [],
                        "last": "Book",
                        "suffix": ""
                    }
                ],
                "year": 1974,
                "venue": "J. Computer Systems Sc jence",
                "volume": "1",
                "issue": "9",
                "pages": "315--332",
                "other_ids": {},
                "num": null,
                "urls": [],
                "raw_text": "Baker, B . and R. Book (1974) . \"Reversal Bounded, Mu1 ti-pushdoyn Machinestf, J. Computer Systems Sc jence --8 , 1 9 7 4 , 315-332 ..",
                "links": null
            },
            "BIBREF1": {
                "ref_id": "b1",
                "title": "Quasi-realtime Languages",
                "authors": [
                    {
                        "first": "R",
                        "middle": [
                            "V"
                        ],
                        "last": "Book",
                        "suffix": ""
                    },
                    {
                        "first": "S",
                        "middle": [
                            "A"
                        ],
                        "last": "Greibach",
                        "suffix": ""
                    }
                ],
                "year": 1970,
                "venue": "~a t h ; Systems Theory --4 , 1 9 7 0",
                "volume": "9",
                "issue": "",
                "pages": "",
                "other_ids": {},
                "num": null,
                "urls": [],
                "raw_text": "Book, R.V. and S .A. Greibach (1970) . \"Quasi-realtime Languages\", ~a t h ; Systems Theory --4 , 1 9 7 0 , ' 9 7 -1 1 1 .",
                "links": null
            },
            "BIBREF2": {
                "ref_id": "b2",
                "title": "D i s s e r t a t i o n , U n i v e r s i t y of Toronto",
                "authors": [
                    {
                        "first": "A",
                        "middle": [
                            "T"
                        ],
                        "last": "Borgida",
                        "suffix": ""
                    }
                ],
                "year": 1977,
                "venue": "l Grammars",
                "volume": "",
                "issue": "",
                "pages": "",
                "other_ids": {},
                "num": null,
                "urls": [],
                "raw_text": "Borgida, A.T. (1977) : \"l Grammars\", Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of Toronto, a l s o Technical Report No.112.",
                "links": null
            }
        },
        "ref_entries": {
            "FIGREF0": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "2 . D e f i n i t i o n s W e r e p e a t h e r e some important d e f i n i t i o n s from ( 1 2 1 , and assume t h a t t h e r e a d e r i s f a m i l i a r w i t h the o t h e r b a s i c n o t i o n s of formal language t h e o r y . + A vocabulary V i s a f i n i t e s e t o f symbols, and we use V t o denote t h e s e t of a l l n b n -n u l l s t r i n g s c o n s i s t i n g of symbols * from V; u s i n g e t o denote t h e n u l l s t r i n g , we a l s o d e f i n e V t o be A r e w r i t e system RW i s a p a i r (V, R) where V i s a vocabulary and R i s a f i n i t e s e t of r u l e s ( p r o d u c t i o n s ) o f t h e form u + v , + * where u E V and v E V ; u is known as t h e l e f t hand s l d e of t h e p r o d u c t i o n ( I h s .) and v i s i t s r i g h t hand s i d e ( r h s .) . + A word x E V i s s a i d t o d i r e c t l y d e r i v e o r g e n e r a t e -i n R * a n o t h e r word y E V (denoted by x =>R Y) i f f t h e r e e x i s t words U , V , W , Z such t h a t x = wuz, y = wvz and u + v belongs t o R. Let = + > R be t h e t r a n s i t i v e c l o s u r e of =>R, and =*> i t s t r a n s i t i v e r e f l e x i v e R c l o s u r e . A sequence of words w 1 , w 2 , . . . ,~, such t h a t w1 =>R w2 => . . . => w i s s a i d t o b e a ( f r e e ) .R-derivation ( Given a r e w r i t e system RW = (V,R) and a s u b s e t AX of V t h e language g e n e r a t e d by R from axiom s e t AX w i t h f r e e d e r i v a t i o n s i s d e f i n e d t o be the s e t ~( A X , R W ) = { w l u e A x , u=*> w ) . R Given t h e r e w r i t e system RW = (V,R) , d e f i n e t h e domjnance r e l a t i o n < on V x V by: d<b i f f xby + udv i s one of t h e p r o d u c t i o n s i n R ( f o r some s t r i n g s x , y , u , v ) o r i f t h e r e e x i s t s some c i n V such t h a t d < c and c < b . Then RW i s d e f i n e d t o be a c y c l i c ( a b b r e v i a t e d a . i f f t h e r e l a t i o n < i s a n t i -s y m m e t r i c and a n t i -r e f l e x i v e . I f u + v i s a p r o d u c t i o n i n a f e w r i t e s y s t e m , i t w i l l be c a l l e d a n u l l r u l e i f v i s t h e n u l l s t r i n g e , and i t w i l l b e c a l l e d c o n t e x t -f r e e i f 11.11, t h e l e n g t h of u , i s 1. A r e w r i t e grammar G i s a q u a d r u p l e ( N , T , S , P ) where N and T a r e t h e s e t s o f n o n t e r m i n a l s and t e r m i n a l s r e s p e c t i v e l y , d i s t i n g u i s h e d n o n t e r m i n a l and f = ( N U T ) P) i s a r e w r i t e system. I n t h i s c a s e , if =*? w t h e n t h i s i s c a l l e d a G -d e r i v a t i o n , o r a d e r i v a t i o n i n G , and t h e language g e n e r a t e d by,G,denoted by * L(G), i s d e f i n e d t o be t h e s e t ( t l~ =*> t i n G , t 6 T . We assume t h e r e a d e r i s f a m i l i a r w i t h t h e t e r m i n o l o g y o f t y p e 0 ( r e c u r s i v e l y enumerable o r R E ) , t y p e 1 ( c o n t e x t s e n s i t i v e ) , t y p e 2 ( c o n t e x t free) and type 3 ( r e g u l a r ) l a n g u a g e s , and c o r r e s p o n d i n g f a m i l i e s o f grammars and automata. A t y p e 2 grammar w i l l b e c a l l e d l i n e a r i f a l l i t s p r o d u c t i o~l~ a r e o f t h e form A + aBb, where A , B E N , a , b E T u { e l , and w i l l b e c a l l e d s e l f e m b e d d i n g i f f o r some A E N t h e r e i s a G -d e r i v a t i o n A =*> uAv where u and v a r e n o t n u l l . New languages c a n b e o b t a i n e d from o l d ones through such s e t o p e r a t i o n s a s u n i o n , i n t e r s e c t i o n and c o n c a t e n a t i o n . One can a l s o d e f i n e -mappings over s t r i n g s and t h e n e x t e n d them t o s e t s of s t r i n g s i n t h e obvious way. One such mapping i s t h e s u b s t i t u t i o n s which a s s o c i a t e s w i t h e v e r y symbol b o f some a l p h a b e t T , a s e t o f words ~( b ) o v e r a n o t h e r a l p h a b e t T t ; & f i n i n g s (xy) = s ( x ) s ( y ) and s ( e ) = e , a s u b s t i t u t i o n c a n be e x t e n d e d t o s t r i n g s . I f t h e s e t s s ( b ) , a r e r e g u l a r , f i n i t e o r e -f r e e then s i s s a i d t o be r e g u l a r , f i n i t e o r e -f r e e r e s p e c t i v e l y ; i f s ( b ) c o n t a i n s a s i n g l e word then s i s c a l l e d a homomorphism, and t h e b r a c e s f o r s e t s a r e dropped. A homomorphism h can a l s o be e -f r e e , o r i t can b e l e n g t h -p r e s e r v i n g , i f l h ( b ) 1 = 1 f o r a l l symbols b . I f L i s a f a m i l y of languages t h e n we u s e H ( L ) and 0 H ( L ) t o r e p r e s e n t t h e f a m i l i e s of languages o b t a i n e d from elements of L through e -f r e e homomorphisms and ~omomorphisms r e s p e c t i v e l y . One f i n a l o p e r a t i o n on s t r l n g s i s r e v e r s a l d e f i n e d by &v(b) = b i f lbl ( 2 and Rev(xy) = Rev(y)Rev(x). Ohe c a n a l s o use automata t o perform mappings between 0 s t r i n g s . The a -t r a n s d u c e r M = [K,T ,T ,k F , ) i s a n e x t e n s i o n 1 2 of t h e f i n i t e automaton, where T1 and T2 a r e t h e i n p u t and o u t p u t ' dr * a l p h a b e t s , and T i s a f i n i t e s u b s e t o f K x T1 x T 2 x K ( t h e t r a n s i -* * t i o n s e t ) . The r e l a t i o n Ii s d e f i n e d on K x TI x T 2 by t h e r u l e (k,uv,z) I-(kf,v,zx) i f ( k , u , v , k T ) E T. The o u t p u t of M f o r i n p u t -w o r d w i s one of t h e s t r i n g s i n t h e s e t 0 * { z l ( k , b e ) I-( k , e , z ) , keF). An a -t r a n s d u c e r i s s a i d t o be"
            },
            "FIGREF1": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "t i o n of languages A i s s a i d t o be c l o s e d under t h e o p e r a t i o n o i f a(L) E A whenever L E A. A ( f u l l ) t r i o i s a f a m i l y of languages c o n t a i n i n g a t l e a s t one nun-empty s e t , c l o s e d under ef r e e homomorphism ( a r b i t r a r y homomorl?hism) , i n v e r s e homomorphism, and i n t e r s e c t i o n w i t h r e g u l a r languages."
            },
            "FIGREF2": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "l l y , o m i t t i n g d e t a i l e d j u s t i f i c a t i o n ( s e e C 33) , t h e f o l l o w i n g formal d e f i n i t i o n c a p t u r e s t h e e s s e n t i a l a s p e c t s o f t h e n o t i o n of s t r a t i f i c a t i o n a l grammar, a s p r e s e n t e d by Sampson C 131: D e f i n i t i o n An n -s t r a t a l r e w r i t e grammar (n-RSTRAT) i s a 5 -t u p l e RST = (n,TCT,RLZ,V V ) , where VC and VE a r e t h e s e t of \" c o n t e n t C ' E u n i t s 1 ! and t l e x p r e s s i o n u n i t s t 1 r e s p e c t i v e l y , TCT = (G19G2, . . ,Gn) i s a v e c t o r of n r e w r i t e grammars, and RLZ = (RO,R1, ..., R ) is a n v e c t o r of ~+ 1 a c y c l i c r e w r i t e s y s t e m s . The t r a n s d u c t i o n p e r f o~m e d + by such a grammar w i l l be d e f i n e d by T-RSTRAT(RST) = { (u,v) lw0=u~vC, * * W n + l =vcVE, t h e r e e x i s t w c L(G.) s u c h t h a t w = i v a t i o n s f o r j = 0,1,. .., n l . I t s language i s d e s c r i b e d by I n t h i s formal model, t h e grammar i s t h o u g h t of a s t r a n s d u c i n g \"meaningu i n t o vtsound\" i n t h e f o l l o w i n g manner : s t a r t i n g w i t h a s t r i n g o f \" c o n t e n t u n i t s \" ( e x p r e s s i n g t h e meaning of *an u t t e r a n c e ) , t h e r e a l i z a t i o n r e w r i t e r u l e s a r e r e p e a t e d l y a p p l i e d u n t i l a s t r i n g o f \"expression u n i t s \" i s o b t a i n e d . The r e a l i z a t i o n d e r i v a t i o n i s c o n s t r a i n e d by t h e r e q u i r e m e n t t h a t f o r each t a c t i c s t h e r e e x i s t s an i n t e r m e d i a t e s t a g e i n the r e a l i z a t i o n d e r i v a t i o n which conforms t o t h e t a c t i c s s p e c i f i c a t i o n s (i .e."
            },
            "FIGREF3": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "belongs t b t h e language g e n e r a t e d by t h e t a c t i c s ).The above formalism i s based mainly on Lamb s v e r s i o n of s t r a t i E i c a t i o n a~ l i n g u i s t i c s ; an a l t e r n a t e approacki, c l o s e r i n s p i r i t t o Gleason's model, i s p r e s e n t e d i n C31."
            },
            "FIGREF4": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "allows i n t h e r e a l i z a t i o n s y s tern r e w r i t e r u l e s w i t h n u l l l e f t -h a n d s i d e s ( i . e m r u l e s of t h e form e + u ) . U n f o r t u n a t e l y , such r u l e s c o u l d b e a p p l i e d t o some s t r i n g an a r b i t r a r y number of t i m e s . I n o u r s t r a t i f i c a t i o n a l model, this would r e s u l t i n any s t r i n g having an i n f i n i t e number o f r e a l i z a t e s . Furthermore, r u l e s of t h e form e + u can a l s o be u s e d t o e s t a b l i s h c o n t e x t -f r e e dependencies i n s t r i n g s g e n e r a t e d even from s i n g l e t o n axiom s e t s . For example, i f R = ({c,d),{e+cd}) * t h e n ( e l R ) = { w~w E ( c ,~} , w has t h e same number o f \"c\" and \"dl! symbols}, which i s known t o be a n o n -r e g u l a r c e n t e x t -f r e e w a g e . The phenomena d e s c r i b e d above do n o t appear t o have l i n g u i s t i c e q u i v a l e n t s , and r u n c o u n t e r t o t h e s t r a t i f i c a t i o n a l p h i l o s o p h y which e n v i s a g e s only f l n i t e l y many r e a l i z a t i o n s f o r any s t r u c t u r e . A s it t u r n s o u t , i n p r a c t i c e r u l e s of t h e form e + u a r e only r e q u i r e d t o i n t r o d u c e i n t h e r e a l i z a t i o n d e r i v a t i o n s y n t a c t i c a l l y determined e l e m e n t s , such a s \"do\" i n q u e s t i o n s . Such i n s e r t i~n s need however be performed o n l y once, a t t h e end of every r e a l i z a t i o n d e r i v a t i o n between two t a c t i c s . T h e r e f o r e t h e y can be accomplished through normal a c y c l i c r u l e s i f each e + u i n R i s r e p l a c e d by r u l e s v + uw and v -+ wu f o r a l l v + w i n R . For t h i s r e a s o n , we w i l l c o n t i n u e t o use t h e d e f i n i t i o n of r e w r i t e systems which o n l y a l l o w s p r o d u c t i o n s w i t h p o n -n u l l l e f thand s i d e s ."
            },
            "FIGREF5": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "x t investigate t h e e f f e c t of a . r . on s i m p l e t y p e s of axiom s e t s . Theorem 3.1 Let AX be a r e g u l a r s e t o v e r a l p h a b e t T and l e t E b e some a l p h a b e t d i s j o i n t from T . If RW = (V,R) i s an a . r . t h e n * ~( A X , R W ) n E i s a l s o a r e g u l a r s e t .Proof L e t G = (N,T,S,P) be a t y p e 3 grammar g e n e r a t i n g AXy and w i t h o u t l o s s of g e n e r a l i t y assume t h a t N n VR i s empty. F u r t h e r -G more, n o r m a l i z e R s o t h a t a l l i t s r u l e s a r e o f t h e form a+bc, a+e o r bc+d. T h i s can be accomplished i n a 3 -s t e p p r o c e s s : f i r s t , * r e p l a c e r u l e s o f t h e form u + abv ( a , b e V , u , v d ) by r u l e s u + a a , a -+ bv where a i s a new symbol; r e p e a t t h i s until a l l r u l e s have r h s . no l o n g e r t h a n two symbols. Next, r e p l a c e r u l e s of t h e form -abu -+ v by ab + a , a u -+ v , u n t i l a l l l h s . o f r u l e s a r e a t most two symbols. F i n a l l y , e l i m i n a t e r u l e s of t h e form a + b by a d d i n g t o R a r u l e y + z b z l whenever y + z a z l i s i n R . Our g o a l i s t o produce a t y p e 3 grammar such t h a t Rd e r i v a t i o n s a r e Mprecomputedu i n i t s p r o d u c t i o n s , For example, i f t h e grammar G o r i g i n a l l y had p r o d u c t i o n s X+aY and Y+bZ, w h i l e R c o n t a i n e d t h e r u l e ab+d, t h e n t h e f i n a l grammar would c o n t a i n p r o d u c t i o n X+dZ. Fbr t h i s p u r p o s e , c o n s i d e r the f o l l o w i n g i t e r a t i v e c o n s t r u ot i o n : INITIALIZATION: E e t G I be G ; l e t T 9 = T u VR. CONSTRUCTION 1: For e v e r y i n t e g e r i , g i v e n grammar Gi = (Ni,Ti,SG,Pi), c o n s t r u c t from l't a t y p e 3 grammar G i + l --T ' 9SG ,Pi+l) a s f o l l o w s : f o r e v e r y aaTi, let P ( i . a ) be t h e s e t of a l l p r o d u c t i o n s i n Gi which have t h e symbol ??a1' on t h e r h s . ; t o b e g i n w i t h , l e t Pi+l c o n t a i n P., 1 and Ni+l c o n t a i n Ni; I F b+cd i s a production i n R y THEN f o r e v e r y A+bB i n P ( i , b ) , ADD t a N,+l a n o n t e r m i n a l [A;B;b+cd], and ADD t o P i + l p r o d u c t i o n s A+c[A;B;b+cd~-and [A;B;b+cd]+dB; I F b+e i s ih R , THEN f o r e v e r y A+bB i n P-(i ,b) ADD p r o d u c t i o n A+B t p Pi+l; I F bc+d i s i n R , THEN f o r e v e r y p a i r o f p r o d u c t i o n s A+bB i n P ( i , b ) , and C+cD i n P ( i , c ) , ADD t o Pi+l t h e new p r o d u c t i o n A+dD i f B=>*c i n Gi; END ; Suppose t h a t we were a b l e t o e s t a b l i s h t h a t From t h e c o n s t r u c t i o n i t i s e a s y t o s e e t h a t P i s always a s u bi s e t o f Pi+l (and hence L(G.) c L ( G i + l ) ) , and i f 1 G = G f o r some index m m m+ 1 ( i . e . no new p r o d u c t i o n s a r e added t o G i n C o n s t r u c t i o n I ) , t h e n m G would b e e q u a l t o G f o r e v e r y j > m.I m B u t , i f such a n m e x i s t s t h e n L ( L ( G ) ; R ) = U L ( G . ) = L(G ) i=l 1 m and G i s t h e t y p e . 3 grammar we a r e l o o k i n g f o r . T h e r e f o r e , i t m remains t o e s t a b l i s h e q u a l i t i e s (1) and ( 2 ) . To,prove ( I ) , we first d e f i n e a new type o f d e r i v a t i o n (\"singl&, l e f t -r i g h t pass\") r e l a t i o n \" = o >~ \" as f o l l o w s : = Q = '~ v i f f t h e r e e x i s t s i n t e g e r n such t h a t f o r j = l , . . . , n x + y . i s a r u l e i n R and z i s some s t r i n g w i t h t h e p r o p e r t y t h a t J J j u = z x z ' ..'x z and v = z y z . . . y z n = 0, t h e n u = v ) . We then c l a i m t h a t L ( G i + $ = I w ( 3 veL(G.) such t h a t v = o >~ W) 1 This, equaLi t y can be demonstrated by s t r a i g h t f o r w a r d i n d u c t i o n s on, r e s -a , S A~ ~hengt-r sd d~& v ; n . r i c~ 3% e, i+l ' and t h e i n t e g e r n appearing i n t h e d e f i n i t i o n o f I n both case% t h e I m p o r t a n t p o i n t s a r e t h a t i f A => bB i n G ( A , B \u20ac N i , b c T . ) i 1 n t h e n e i t h e r A => bB i n Y~+~ (by s t e p 2 i n C o n s t r u c t i o n 1) o r A => uB i f b + u i s i n R (by s t e p s 3 o r 4 ) ; and i f A => bB =*> b c => b c~"
            },
            "FIGREF6": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "We a r e now i n a p o s i t i o n t o prove (1). F i r s t , suppose t h a t wbelongs t o Z ( L ( G ) , R ) and w was o b t a i n e d from u E L(G) = L ( G~) i n an R -d e r i v a t i o n w i t h n s t e p s : u = u1 => --R U2 ->R ... -> R u n => W . I f w e n o t e t h a t f o r any strings x , y x =>R y i m p l i e s x =o> y R t h e n b y ( 3 ) we have f o r i = 1 , n t h a t u . c L ( G . 1 . But t h e n 1 00 = un must belong t o L (G ) , and hence t o U L ( G . ) . Conversely, \" L(G.) then t h e r e must e x i s t an index m s u c h t h a t 1-1 a w c L ( G ) Using ( 3 ) i t i s t h e n t r i v i a l t o prove by induction on m m t h a t t h e r e e x i s t sv E L(G) such t h a t v = vl = o >~ v 2 =o> . . * = o > R R Vm = W f o r some'v E L ( G . ) (1 = 1 ,m . But i n t h a t c a s e w E & ( L ( G ) , R ) i 1 because by d e f i n i t i o n x = o >~ y i m p l i e s t h a t x = * > y f o r any R strings x,y. T h i s concludes t h e p r o o f o f i d e n t i t y (1). To p r o v e (Z), one n i g h t t r y t o d e m o n s t r a t e t h a t t h e c o n s t r u c t i o n h a l t s a f t e r some precomputable number o f s t e p s . This approach u n f o r t u n a t e l y runs into t h e f o l l o w i n g problem: t h e a d d i t i o n o f a new p r o d u c t i o n t o G i n s t e p 4 , allows new p a i r s o f i v a r i a b l e s B ' and C ' t o be c o n n e c t e d by a d e r i v a t i o n B '=>*c 9 t h i s may aPlow new p r o d u c t i o n A'+dD1 t o be added t o Gi+14 in s t e p 5, which i n t u r n may e v e n t u a l l y a l l o w s t e p 4 t o add a new r u l e t o f o r some The above compels us t o look f o r an a l t e r n a t i v e p r o d o f ( 2 ) : e x h i b i t i n g a grammar GO such t h a t e v e r y Gi i s a subgrammar o f GO. This would mean t h a t t h e i n c r e a s i n g sequence o f grammars G1,GZ,is bounded above, and hence converges t o one o f i t s e l e m e n t s . To c o n s t r u c t G O , remember t h a t by d e f i n i t i o n o f R t h e r e i s an a n t i -s y m m e t r i c r e l a t i o n < on V R' Using t h i s , we a s s i g n t o e v e r y symbol i n V and every p r o d u c t i o n i n R a unique index number, a c c o r d i n g t o t h e f o l l o w i n g a l g o r i t h m : ) := 0 f o r e v e r y beV such t h a t t h e r e i s no daV and d > b ; 2 , FOR i = O t o I v I DO WHILE n o t a l l symbols have an i n d e x ; I F I ( b ) = i 6 b+cd is i n R, THEN I ( c ) : = I ( d ) : = i+l and I(b+cd) : = i+l; I F I ( b ) = i 6 I ( c ) s i 6 bc+d i s in R THEN I ( d ) := i+l and I(bc+d) : = i+l; PF I ( b ) = i 6 I ( c ) a i 6 cb+d i s i n R THEN I ( d ) : = i + l and I(bc+d) := i + l ; IF I(b) = i and b+e i s i n R THEN I(b+e) := i*l; END END By t h e a c y c l i c i t y o f R , t h e above a l g o r i t h m produces a unique value f o r e v e r y symbol and p r o d u c t i o n . Suppose t h e h i g h e s t index value a s s i g n e d i s n. Then G O w i l l be c~n s t r u c t e d from G by r e p e a t e d m o d i f i c a t i o n i n n p a s s e s through t h e f o l l o w i n g : CONSTRUCTION 2: Let G O = (N',T' ,s,P') be G i n i t i a l l y ; FOR i = l t o q n DO * i n t h e i -t h p a s s , add t o G O a l l p o s s i b l e p r o d u c t i o n s r e p r e s e n t i n g d e r i v a t i o n s by index i r u l e s */ I 1. For e v e r y symbol I d ' i n VR such t h a t I ( d ) = i, l e t P(d) b e the s e t of a l l p r o d u c t i o n s c u r r e n t l y 0 In G , w i t h ' d ' on t h e r h s . ; 2 . I F b-tdc ( o r b+e) 1 s in R and has index i ( i f f I (b) i ) , THEN a l t e r G O i n e x a c t . 1~ t h e same way as i n s t e p s 3 ( o r 4) of CONSTRUCTION 1; (except t h a t PO and NO a r e used i n s t e a d of Pi+l and N I F bc+d i s i n R and h a s index i , THEN f o r e v e r y p a i r o f p r o d u c t i o n s A-tbB i.1 P(b) and C+cD i n P ( c ) , ADD t o P O t h e new p r o d u c t i o n A+dD (whethero r n o t Note t h a t i n t h e i -t h pass t h e only p r b d u c t i o n s added t o G O have on t h e r h s : a t e r m i n a l symbol of index s t r i c t l y g r e a t e r t h a n i . T h e r e f o r e , i n s u c c e s s i v e p a s s e s through t h e loop a f t e r t h e i -t h one, P(d) remains unchanged f o r a l l symbols I1d\" w i t h L(d) i . Furthermore, t h e o u t p u t G O remains unchanged i f p a s s e d through CONSTRUCTION 2 a second time. Secondly, n o t e t h a t i f some grammar K remains unchanged by CONSTRUCTION 2 t h e n i t does s o through CONSTRUCTION 1 a s w e l l , because every r u l e i n R i s e v e n t u a l l y c o n s i d e r e d i n CONSTRUCTION 2 , and i n each c a s e a t l e a s t t h o s e p r o d u c t i o n s which would have been added by CONSTRUCTION 1 a r e added by CONSTRUCTION 2 . 0 T h e r e f o r e , s i n c e G i s a subgrammar of G , G i w i l l be s o f o r e v e r y i g r e a t e r t h a n 1, and t h e proof i s completed. 0 I n c r e a s i n g t h e range o f s e t s from which we choose t h e axiam s e t s , we o b t a i n t h e f o l l o w i n g : Theorem 3 . 2 L e t G = (N,T,S P ) be an a r b i t r a r y t y p e 0 grammar. G' G Then t h e r e e x i s t s a l i n e a r c o n t e x t -f r e e language LING+ and an a . r . R (which i s dependent only on t h e s e t N u T ) , such t h a t 0 P r o o f ( N o t a t i o n a l convention: l e t V = N u T, and i f E{-,-,v) t h e n u s e V t o r e p r e s e n t t h e s e t { & l a~V } , and -alw * a j if w = al..-.aj .) I t i s known (Cll) t h a t t h e r e e x i s t two l i n e a r c o n t e x t -f r e e languages L1 and L2, a s w e l l a s a homomorphism h , such t h a t L(G) = h(L1 n L 2 ) . W e have c o n s t r u c t e d (C3'1) p a i r s of new such laqguages:"
            },
            "FIGREF7": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "i = 0 , ..., n-11 n -i I n t h i s c a s e , t h e homomorphism h i s d e f i n e d a s h(;) = x i f xeT, null o t h e r w i s e . Observe t h a t L1 i s dependent s o l e l y on t h e v o c a b u l a r y V and i t o n l y checks whether t h e s t r i n g s around t h e c e n t r a l .I % s G % $ % ' a r e m i r r o r images of e a c h o t h e r . But t h e f a l l o w i n g r e w r i t i n g system does e x a c t l y t h e same j o b , and, i n a d d i t i o n , performs t h e homomorphism h : v Ro = (fS B $ F e , % F e y G + e f o r a l l X E V~, a+a f o r aeT) G * Then T n de ( L~, R~) = h(L1 n L2) = L(G) and by o b s e r v a t i o n i t i s 0, c l e a r t h a t R i s a c y c l i c . 0 T h i s r e s u l t is s u r p r i s i n g , e s p e c i a l l y from a l i n g u i s t i c p o i n t o f view, and d e m o n s t r a t e s t h e power o f a c y c l i c r e w r i t e s y s t e m s . S i n c e i t i s u n d e s i r a b l e t h a t l i n g u i s t i c mechanisms be s o p o w e r f u l , we w i l l a t t e m p t t o p u t bounds on them. One way t o do s o i s t o r e s t r i c t t h e p l a c e s where s t e p s i n d e r i v a t i o n s can occurl."
            },
            "FIGREF8": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "l y , i n a k -l e f t m o s t d e r i v a t i o n t h e r e i s a k-symbol wide \"windoww on t h e d e r i v a t i o n a l forms where r e w r i t i n g can o c c u r , and t h i s window i s o n l y allowed t o move t o t h e r i g h t . D e f i n i t i o n 3 . 1 L e t wo => w => ... 1 => W n be an R -d e r i v a t i o n , where f o r i = 1, ..., n p r o d u c t i o n s ui+vi a r e u s e d t o o b t a i n w = x . v . u i r i i from w~-~ x.u.y.,,(x 1 1 1 i ,yi,wicv--), For any i n t e g e r k , t h i s i s s a i d t o be a k -J e f t m o s t d e r i v a t i o n i f f o r a l l i = 1, . . n -1 t h e r e exist * s and t i n V such t h a t xi = t . s w i t h 1s.l 5 k a~d 1t.I.s iti+l i s d e f i n i t i o n of R -d e r i v a t i o n s g i v e s r i s e t o t h e new language * %(Ax,Rw,~-Id) = I w ~X E A X , x = >~w i n k -l e f t m o h t d e r i v a t i o n ) . Given a n a . r . RW = (V,R), t h e r e e x i s t s a n a -t r a n s d u c e r O R s u c h t h a t $ (AX,R,k-ld) = $$AX) . Proof By t h e a c y c l i c n a t u r e o f t h e r u l e s i n R , any s t r i n g o f t l e n g t h k can b e r e w r i t t e n i n t o a s t r i n g o f l e n g t h a t most kd where d 5s t h e l e n g t h o f t h e l o n g e s t r h s . o f a r u l e i n R , and t i s the. nufiber o f symbols i n V. T h e r e f o r e , i f we d e f i n e a T u r i n g machine t r a n s d u c e r O which s i m u l a t e s on i t s wofking t a p e k -l e f t m o s t R -R d e r i v a t i o n s , t h e n i t need have o n l y a bounded, f i n i t e -l e n g t h t a p e . B u t t h i s can o b v i o u s l y b e k e p t i n a f i n i t e memory, and hence O R c a n be made i n t o a n a -t r a n s d u c e r , 0 T h e r e f~r e , l e f t m o s t c a r l 3 t r a i n t s on R -d e r i v a t i o n 6 l e a d t o a much more r e s t , r i c t e d v e r s i o n caf r e w r i t e ~y s t e m s b e c a u s e a l l t r i o s ( i n p a r t i c u l a r LINEAR-CFL) a r e c l o s e d under a -t r a n s d u c t i o n . A second method o f bounding t h e power of a . r . i s t o r e s t r i c t t h e .form o f t h e p r o d u c t i o n s a l l o w e d i n R . Thporem 3 . 4 I f RW = (V,R) i s an a . r . w i t h no n u l l p r o d u c t i o n s , t h e n f o r e v e r y axiom s e t AX, ~( A X , R W ) = oe(AX,RW,k-ld) f o r some Proof Suppose R has r r u l e s i n v o l v i n g t symbols and l e t c be t h e l e n g t h o f t h e l o n g e s t l h s . of a p r o d u c t i o n . Now o b s e r v e * t t h a t i f v =>R w , t h e n no symbol i n w can have more t h a n k=c .~n c e s t o r s i n v , a l l of which must be a d j a c e n t i n v ( i . e . t h e p r e s e n c e o f a symbol i n w can depend o n l y on t h e p r e s e n c e of a t most k a d j a c e n t symbols i n v ) . Thi's v a l u e of k c a n be o b t a i n e d a s f o l l o w s : s i n c e t h e r u l e s a r e a c p c l l c , new symbols must appear a f t e r e v e r y a p p l i c a t i o n of a p r o d u c t i o n and hence e v e r y symbol i n w can be t h e r e s u l t of a p p l y i n g ae most t . p r o d u c t i o n s ; s i n c e each o f t h e s e u s e s a t most c t symbols as c o n t e x t , we g e t t h e v a l u e o f c . The adjacency r e q u i r ement comes ffom t h e c o n d i t i o n t h a t t h e r e be no e-r-ules i n R . * Consider now some R -d e r i v a t i o n tXv = * > t: ( t , ? ,~e~ , X E V ) , where no symbol i n t i s r e w r i t t e n , b u t X i s . IVe w i l l prove by i n d u c t i o n on t h e l e n g t h o f t h e d e r i v a t i o n t h a t t h e r e i s a n e q u i v a l e n t kl e f t m d s t d e r i v a t i o n . B a s i s . I f t h e d e r i v a t i o n h a s 0 6 r 1 s t e p s t h e n i t i s c l e a r l y k -l e f t m o s t . I n h c t i o n s t e p . Break up t h e d f i i v a t i o m i n t o s t e p s , t o s e e where X i s r e w r i t t e n :"
            },
            "FIGREF9": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "d t h e 1as.t production i n @ which produces o n l y n o n -a n c e s t o r s of Y . -(a) 11 t h s r e i s no s u c h p r o d u c t i o n , t h e n by wr opening remarks @ must be k -l e f t m o s t , and hence @ can be made k -l e f t m o s t by i n d u c t i o n .( b ) O t h e r w i s e , suppose t h a t t h e l a s t s u c h r u l e was a + p . Then we claim t h a t t h e d e r i v a t i o d i n f u r t h e r d e t a i l i s N tXv =*> tXu cry => tXu p u =*> tXh* 'pu => t Y zu pu =*> t? The s i g n i f i t a n t part o f t h i s c h i n i s t h a t no p r d d u c t i o n i n @ a f f e c t s t h e s t r i n g p u 2 , and t h i s i s t r u e hy o u r c h o i c e o f a + p a s t h e l a s t p r o d u c t i o n g e n e r a t i n g n o n -a n c r s t o r s o f Y , hence o f Yz, and t h e n e c e s s a r y c o n t i g u i t y o f a n c e s t o r s . But now n o t e t h a t s t e p @ can b c ~o s t p o n e d t o y i e l d t h e f o l l o w i n g r e o r d e r e d d e r i v a t i o n : By r e p e a t i n g t h e c o n s t r u c t i o n i n p a r t (b) on @ = @@ t h i s t i m e ( i n s t e a d o f @ @@ ) we w i l l e v e n t u a l l y (by a second induction, if d e s i r e d ) a c h i e v e c a s e ( a ) , and t h u s c o m p l e t e t h e p r o o f ."
            },
            "FIGREF10": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "Note t h a t i n t h e above p r o o f we had o n l y e x c l u d e d t h e u s e o f n u l l r u l e s (the symbol p c o u l d not b e n u l l ) s o t h a t o t h e r p r o d u c t i o n s w i t h l e f t -h a n d s i d e s l o n g e r t h a n r i g h t -h a n d s i d e s a r e s t i l l a l l o d i n R . F i n a l l y , w e i n c l u d e f o r c o m p l e t e n e s s t h e f o l l o w i n g r e s u l t whose p r o o f : i s trivial.. . P r o p o s i t i o r i 3 . 5 L e t RW = (V,R) be an a . r . which h a s only c o n t e x t f r e e r u l e s . Then t h e r e e x i s t s a f i n i t e s u b s t i t u t i o n s R such t h a t f o r every axiom s e t A X , &(AX,RW) = sR(AX)."
            },
            "FIGREF11": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "S t r a t i f i c a t i o n a l Grammars W e now r e t u r n t o t h e n o t i o n o f s t r a t i f i c a t i o n a l grammar which l e d us o r i g i n a l l y t o c o n s i d e r a c y c l i c r e w r i t e systems. To b e g i n w i t h , n o t e t h a t t h e o r i g i n a l d e f i n i t i o n of n-RSTRAT grammar h a s *o c o n s t r a i n t on t h e d e r i v a t i o n s o c c u r i n g on t h e t a c t i c s , w h i l e i n p r a c t i c e l i n g u i s t s appear t o view t h e d e r i v a t i o n s as-b e i n g l e f t m o s t"
            },
            "FIGREF12": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "e . t h e l e f t m o s t nonterminal i s t h e one r e w r i t t e n ) . T h e r e f o r e , throughout t h e f o l l o w i n g d i s c u s s i o n we w i l l examine t h e d i f f e r e n c e s a r i s i n g o u t of t h i s v a r i a t i o n .F i r s t , we p r e s e n t a r e c u r s i v e c h a r a c t e r i z a t i o n of the n-RSTMT languages. For t h i s p u r p o s e , d e f i n e t h e language g e n e r a t e d by a 0-RSTRAT grammar RST' = ( 0 , () , (R') ,VC ,VE) as L.-RSTRAT (RST') * &($,R*)n VE. Then t h e following theorem i s a n obvious conse-quence o f t h e d e f i n i t i o n o f L-RSTRAT: Theorem 4 . 1 If RST = ( n ( G G ) , ( R o , ..., n R-) ,VC,VE) i s an n n-RSTRAT gramman, and TOP (RST) i s t h e (n-1) -RSTRAT grammar (n-l,(G1, ..., G' ) , ( R o , . . . , R n-1 n-1 ) ,.V T ) , then L-RSTRATCRST) = C 9 n U s i n g Theorems 4 . 1 , 3 . 1 and t h e known c l o s u r e p r o p e r t i e s of t h e r e g u l a r languages, i t i s easy t o s e e t h a t i f a l l t h e t a c t i c s G1:..., GI of an n-RSTRAT grammar a r c non-sclfcmbcdding then t h e n, s t r a t i T i c -a t i o n a l grammar can g e n e r a t e o n l y a r e g u l a r language."
            },
            "FIGREF13": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "On t h c o t h e r h a n d , as s o o n as one o r t h e t a c t i c s i sa/llowed t o b c o f type 2 and selfembcdding, t h e n by Thaorcms 4 . 1 and 3.2 the.RSTRAT grammar can g e n e r a t e an a r b i t r a r y RE s e t . E v e r m o r e s u r p r i s i n g l y , t h i s can be accomplished u s i n g a \" u n i v e r s a l r e a l i z a t i o n r e l a t i o n M , mea-n-ing t h a t t o o b t a i n any RE s e t we need o n l y v a r y t h e t a c t i c s , n o t t h e r e a l i z a t i o n r e w r i t e system. This s i t u a t i o n i s s i m i l a r t o t h a t found f o r TG i n C91, where t h e t r a n s f o r m a t i o n a l component can b e v a r i e d w h i l e t h e b a s e grammar i s k e p t f i x e d . T h e r e f o r e , i n t h i s s t r a t i f i c a t i o n a l model t h e r e seems t o be no a l t e r n a t i v e between t h e i n s u f f i c i e n t d e s c r i p t i v e power of f i n i t e , a u t o r n a t a and t h e e x c e s s i v e poker o f a r b i t r a r y Turing machines. These r e s u l t s h o l d even i f t h e d e r i v a t i o n s on t h e t a c t i c s a r e cons t r a i n e d t o b e l e f t m o s t . W e must t h e r e f o r e s e a r c h f o r f u r t h e r l i m i t a t i o n s on t h e r e a l i z a t i o n p r o c e s s . I n s e c t i o n 3 we c o n s i d e r e d s e v e r a l p o s s i b l e ways of d6ing t h i s , namely e l i m i n a t i n g n u l l o r c o n t e x t -d e p e n d e n t r u l e s , o r making t h e r e a l i z a t i o n d e r i v a t i o n l e f t m o s t . I n l i n g u i s t i c grammars t h e r e i s a c l e a r need f o r c o n t e x t -d e p e n d e n t r e a l i z a t i o n ~u l e s , hence t h e s e c a n n o t b e e l i m i n a t e d . Although i n Sampsonts model nu11 r e a l i z a -A > t i o n r u l e s appear t o be needed (more on t h i s below), i t i s p o s s i b l e t o e n v i s a g e a l t e r n a t i v e models which a v o l d them. By"
            },
            "FIGREF14": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "4 , t h e absence o f n u l l r u l e s i s e q u i v a l e n t t o r e s t~i c t l %~ t h e r e a l i z a t i o n d e r i v a t i d n t o b e i n g k -l e f t m o s t . Furthermore, b a s e d on c u r r e n t l i n g u i s t i c l i t e r a t u r e t h e r e a p p e a r s to b e no o b j e c t i o f i t o l i m i t i n g t h e r e a l i z a t i o n t o b e i n g k -l e f t m o s t . T h e r e f o r e , we w i l l examine t h e g e n e r a t i v e power of n-RSTWT grammars under rhi-s cans t r a i n t ."
            },
            "FIGREF15": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "Theorem 4.2-I6 STR = (n,CG1,.. . , R n ) , ( R O , . . . , R n ) ,VC,VE] i s a n n-RSTRAT grammar w i t h r e a l i s a t i o n d e r i v a t i o n s r e s t r i c t e d t o b e k -l e f t m o s t f o r some, i n t e g e l r k, t h e n t h e r e e x i s t homombrphism h"
            },
            "FIGREF16": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "i s o f t h e n I same t y p e 1 a s . L i , and L-RSTRAT[STR) = h(L1 n . . . n L ) . n Proof The p r o o f i s b a s e d on a number o f r e s u l t s a b o u t j t r i o s , which we summarize h e r e from [ 4 ] : ( a ) F o r i =, 1 , n t h e f a m i l i e s o f l a n g u a g e s o f t h e same t y p e as L(G.) a r e t r i o s . 1 6 1 I f L i s a t r i o t h e n H ( L ) i s a t r i o and H'(L) i s a f u l l t r i o . (c) If L I P -m b , L n a r e t r i o s t h e n H ( H ( L~ n . . . n ) n L n ) i s a t r i o and i t i s e q u a l t o H (L1 n . . . n Ln-l n L 1; s i m i l a O ( H ( L~ n . * -n -1 nLnI = H IL1 n e b e n L,-l n L ) n i s a f u l l t r i o . (d) t r i o s a r e c l o s e d u n d e r i n t e r s e c t i o n w i t h r e g u l a r s e t s and e -o u t p u t bounded a -t r a n s d u c ' t i o n s , w h i l e f u l l t r i o s a r e a l s o cl-osed u n d e r a r b i t r a r y at r a n s d u c t i o n . We now T r o v e t h e . theorem by i n d u c t i o n on n . B a s i s . For n = l , by Theorems 4 . I and 3 . 3 t h e r e e x i s t a , * t r a n s d u c e r s O and O s u c h t h a t L-RSTRAT(RST) = o~( O~( V E )~I , ( G~~)~ 1 0 t h e n o u r theorcm h o l d s by n o t e s ( a ) and (d) above w i t h h b e i n g t h e i d e n t i t y map. induction s t e p . For t h e . c a m n + l , by T h e o r e m~r 4 . 1 and 3 . 3"
            },
            "FIGREF17": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "f i n d a homomarphism h and languages LI,. .-, L n + l o f t h e same type as L';, ..., L t t and L(Gn,+l) such t k a t L RSTRAT(RST) = h(Ll fi . . . nL ) . # n n+ 1 Remark t h a t by Theorems 34.4 and 3.5 t h e same r e s u l t h o l d s i n t h e c a s e when t h e r e a l i z a t i o n s do n o t c o n t a i n n u l l r u l e s , and by examining t h e above proof i t c a n be s e e n t h a t t h e homomoxphism h can b e r e s t r i c t e d t o b e i n g e f r e e i n t h i s c a s e . The f o l l o w i n g canverse t o Theorem 4 . 2 can be e a s i l y e s t a b l i s h e d : Theorem 4 . 3 Given homomorphism h from T t o T I . , and r e w r i t e grammars G . . . ,G w i t h t e r m i n a l alphabets T , t h e n f o r i = 0 , .. . , n 1' n t h e r e e x i s t c o n t e x t -f r e e a c y c l i c r e w r i t e systems Ri such t h a t Proof For j c n , d e f i n e R . t o be {a+al aaT] , by t h e d e f i n i t i o n o f 3 RSTRAT-derivations t h i s w i l l s i m u l a t e t h e i n t e r s e c t i o n of t h e languages g e n e r a t e d by t h e t a c t i c s . F i n a l l y , d e f i n e R t o be n {a+h(a) 1 a d ) , t h u s p e r f~r m~n g t h e horrtomorphism on t h e i n t e r s e c t i o n . 6 To b e g i n w i t h , t h e above theorems p a r t i a l l y confirm Sampson's h i t h e r t o unproven c l a i m (C 13 : page 111) t h a t s t r a t i f i c a t i o n a l languages a r e t h e ~e s u l t of i n t e r s e c t i n g t h e languages o f t h e t a c t i c s . Note however two i m p o r t a n t q u a l i f i c a t i o n s t o t h i s c l a i m :"
            },
            "FIGREF18": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "Theorems 4.2 and 4.3 show that with k-leftmost realization derivations, the type i languages (i = 1 , 2 ) can be obtained by using a type i grammar on one of the tactics, and making the other ones non-selfembedding. If all the tactics generate context-free languages (as in the case when tactic-derivations are leftmost) then n-RSTRAT grammar can generate the hornornorphic intersections of the CFLs. For 1122, this is known to equal the RE sets if null realizations are allowed; if null realization rules are not allowed then for ns3 the n-RSTRAT grammars generate the family QUASI of sets recognized by nondeterministic Turing machines in real or linear time ( C 21) . These observations demons t~a t e that n-RSTRAT grammars can be appropriately modif ied . so that they generate various language families intermediate between the regular and RE sets. Unfortunately, even when the realization derivation is restricted to being k-leftmost' 1-RSTRAT grammars with context-sensitive tactics and 2-RSTRAT grammars with context-free tactics can generate the RE sets, unless null realizations aro restricted. The b a s l c problem with restricting null rules lies in the pronounced bias o f this m o b 1 towards the realization of terminal units from one tactics to the next. In practice, in order to describe linguistic phenomena it is necessary to h w c information about the enti-rc derivation process on some tactics. Sampson accomplishes this by introducing \"pseudo-t e r m i n a l s v v i n t o s t r i n g s ; f o r example, if t h e a p p l i c a t i o n o f p r o d u c t i o n x-+y i s t o b e n o t e d f o r l a t e r u s e , t h e n e i t h e r r u l e x+py o r xtyp would be used i n t h e t a c t i c s t o i n t r o d u c e p a s a marker of t h e o c c u r r e n c e of x+y. The c h i e f drawback of t h i s approach i s t h a t t h e n p s e u d o -t e r m i n a l s \" such w p must e v e n t u a l l y be d e l e t e -d , making n h l l r u l e s n e c e s s a r y . One p o s s i b l e s o l u t i o n may be t o d i scover some bound on t h e number of null r u l e a p p l i t a t i o n s needed, r e s e m b l i n g t h e \" c p z l i n g f u n c t i o n t f proposed by P e t e r s and R i t c h i e (C91)."
            },
            "FIGREF19": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "Another s o l u t i o n i s t o c o n s i d e r a new form'al model which a l l o w s r g a l l z a t i o n t o a c c e s s u~i f o r m l y a l l p a r t s of t h e d e r i v a t i o n s on t a c t i c s ; t h i s approach i s c o n s i d e r e d i n C31. Before c o n c l u d i n g , w e t a k e a b r i e f look a t t h c prablams r a i s e d by one a d d i t i o n t o t h e b a s i c model d i s c u s s e d s o f a r , namely o r d e r e d y u l e s . I t has o f t e n been found u s e f u l i n l i n g u i s t i c d e s c r i p t i o n s t o u s e r u l e s af t h e form \"A+u i f some c , o n d i t i o n C h l d s , o t h e r w i s e A+vv; b a s i c a l l y , t h e s e t y p e s of r~l e S ' a v~i d s t a t i n g t h e n e g a t i o n of ~o n d i~t i o n C , which may b e cumbersome. I n c e r t a i n s t r a t i f i~a t~o n a l d e s c r i p t i o n s . t h i s h a s l a p s e d i n t o t h e u s e of r u l e s o f t h e form \"A+u i f t h i s can l e a d t o a completed d e r i v a t i o n , o t h e r w i s e A+vu. This n o t i o n is f o r m a l i z e d by Sampson through t h e assignment o f 'lwcightsr' o r \" p r e f e r e n c e v a l u e s \" t o c e r t a i n r u l e s . Thus A+u may be g i v e n v a l u e 1 w h i l e A+v r e c e i v e s v a l u e 0 , and t h e s e v a l u e s fire accbmulilt%d throughout t h e d e r i v a t i o n . A t t h e end, o n l y t h o s e e x p r e s s i o n s t r i n g s r e s u l t i n g from some c o n t e n t s t r l n g a r e t a k e n which have d e r i v a t i o n s w i t h maximal p r e f e r e n c e v a l u e s . The fundamental problem w i t h t h i s u s e o f f l o r d e r e d r u l e s v i s t h a t even i n c o n t e x t -s e n s -i t i v e grammars i t i s i n g e n e r a l r e c u r s i v e l y u n d e c i d a b l e whether a c e r t a i n d e r i v a t i o n can be s u c c e s s f u l l y completed o r n o t . In faat, we show t h a t u s i n g \"ordered r u l e s \" we c a n g e n e l a t e even n o n -r e c u r s i v e l y enumerable s e t s , an o b v i o u s l y u n d s i r a b l e s i t u a t i o n . Theorem 4 . 4 There e x i s t s a c o n t e x t -s e n s i t i v e grammar G w i t h one \"ordered r u l e u which g e n e r a t e s a non-RE language. Proof The p r o o f r e s t s on t h e we11 known r e s u l t t h a t t h e r e e x i s t s an RE language LO over some a l p h a b e t T , bhose complementi sn o t -3 RE and t h a t t h e r e i s a t y p e 1 grammar G O = (NO , T~, S ' , P~) , where TO = T u { b , # ) , such t h a t L(G') = {uf#b i (w) we^', i(w) i s some i n t e g e l ; depending on w 1 ( 1 . Consider t h e grammar N G' = ( N 1 , T ? , S f , P r ) where N f *= NO u TO u { Y , S v , Z ) , To' = T u T , P t c o n t a i n s P O and a d d i t i o n a l p r o d u c t i o n s a s d e s c r i b e d below, The grammar G' behaves i n f o r m a l l y as f o l l o w s : * ( a ) from S t , we g e n e~a t e some s t r i n g w S f such t h a t weT , u s i n g p r o d u c t i o n s from H f + a s 9 I BET-); (b) t h e n wc a p p l y thk o r d e r e d r u l e \" S t -+ YSO w l t h w e i g h t 1, S ' -+ ?Z with weight \"0\"; t h e p l a n i s that t h e new n o n t e r m l n a l Y can be r e w r i t t e n I n t o a t e r m i n a l , 7, i f and only i f Y a p p e a r s i n a L s t r i n g $clanging t o {wYwlf} { b } ( i .e. i f f r u l e s of G O can b e 3 u s c d t o ~e w r i t e SO i n t o somc w#b , where w i s t h c same a s t h e g u e s s 0 made i n f a ) ) . Once some d e~i v a t i o n from S i s completed, i t i s c l e a r t h a t c o n t e x t -s e n s i t i v e r u l e s can b e b u s e d t o check o u t t h e aoove c o n d i t i o n f o r Y. I n a d d i t i o n , t h e same r u l e s can place t v b a r s v over a l l t h e symbols thus checked, r e s u l t i n g , i f s u c c e s s f u l , ---j i n a s e n t e n c e of t h e form w#w#b ."
            },
            "FIGREF20": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "on t h e o t h e r hand simply t r a v e l s a c r o s s t h e s t r i n g w and p l a c e s f f d o t s v l on t o p o f e v e r y symbol, u s i n g r u l e s from {sZ + Z S~S~T } ---c The r e s u l t w i l l b e t h a t L ( G v ) = (w#wtbj lwcI,O) u {~I w~L ' )"
            },
            "FIGREF21": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "Suppose t h a t L ( G f ) i s RE, and l e t h be t h e homomorphism which d e l e t e s a l l symbols n o t i n T , and removes t h e d o t s from t h e o t h e r s . Then h ( C ( G 1 ) ) i s a l s o RE because t h e RE s e t s a r e c l o s e dunder homomorphism; b u t h ( L ( G t ) ) i s t h e complement o f L O , and t h u s n o t i n RE by o u r c h o i c e o f L O . T h e r e f o r e by c o n t r a d i c t i o n , L ( G 1 ) i s not RE. A similar pr0o.f can b e g i v e n fo'r s t r a t i f i c a t i o n a l grammars w i t h two o r more c o n t e x t -f r e e t a c t i c s . These r e s u l t s draw a t t e n t i o n t o t h e need t o r e d e f i n e t h e n o t i o n of \"ordered yule\" i n s t r a t i f i c a t i o n a l u s a g e , and p o i n t o u t t h a t c a r e must be t a k e n whenever f~r m a l i z i n g a s p e c t s o f l i n g u i s t i c p r a c t i c e .I n -conclusiofl, om i n v e s t i g a t i o n of t h e formal p r o p e r t i e s o f t h e s t r a t i f i c a t i o n a l model proposed by Sampson r e v e a l e d c e r t a i n u n i n t u i t i v e p r o p e r t i e s which make i t l e s s d e s i r a b l e as a t o o l f o r n a t u r a l languitge d e s c r i p t i o n . Thus, t h e u s e o f r e a l i z $ t i o n r u l c s w i t h n u l l l c f t h a f f t d s i d e s was shown t o a l l o w unbounded number o f r e a l i z a t i o n s f o r c e r t a i n s t r i n g s . More s i g n i f i c a n t l y , we showed t h a t n-RSTRAT grammars w i t h even one t a c t i c s a l l o w i n g s e l fembedding c o u l d g e n e r a t e a l l RE s e t s . S i n c e t h e r e a r e w e l l known problems r a i g e a by t h i s p o s s i b i l i t y , most s i g n i f i c a n t being t h e i n a b i l i t y t o decide g r a m m a t i c a l i t y , we i d e n t i f i e d a l i p g u i s t i c a l l y a c c e p t a b l e r e s t r i c t i o n on t h e r e a l i z a t i o n , namely k -l e f t m o s t d e r i v ations*, which l e d t o improvements i n some s i t u a t i o n s . Under t h i s addi t 3 o n a l c o n s t r a i n t , c l a s s e s of n-RSTRAT grammars were shown. t o v a r i o u s l y g e n e r a t e t h e c o n t e x t -f r e e languages, t h e QuasA-realtime languages and t h e c o n t e x t -s e n s i t i v e languages. U n f o r t u n a t e l y , even i n t h i s c a s e n-RSTRAT grammars could g e n e r a t e n o n -r e c u r s i v e s e t s , u n l e s s n u l l r e a l i z a t i o n s were r e s t r i c t e d , and we d i s c u s s e d t h e problems inhepcrrt i n t h i s approach. F i n a l l y , we examined t h e d e f i n i t i o~ o f \"ordered r u l e s f f used i n some s t r a t i f i c a t i c h a l gxammars, and f o r m a l i z e d by Sampson, showing t h a t i t allowed t h e g e n e r a t t o n o f even non-RE s e t s w i t h type, 1 t a c t i c s . The above formal r e s u l t s about t h e g e n e r a t i v e power o f s t r a t i f g c a t i o n a l grammars h o p e f u l l y answer t h e r e q u e s t s of c r i t i c s such as P i t t h a ([101), and demonstrate t h e i n a c c u r a c y of P o s t a l ' s c l a s s i f i c a t i o n of s t r a t i f i c a t i o n a l grammars a$ simply v a n i a n t s 01 c o n t e x t -f r e e p h r a s e s t r u c t u r e grammars-The ' r e s u l t s a l s o i n d i c a t e same o f t h e problcm a r e a s i n this formal model f o r s t r a t i f i c a t i o n~l l i n g u i s t i c s . W e emphasize though t h a t t h e problems a r c s p e c i f i c t o t h i s p a r t i c u l a r formalism, ahd s h o u l d n o t bc t a k e n a% condemnations o f s t r a t i f i c a t i b n a l l i n g u i s t i c s iq g e n e r a l , s i n c e t h e r e a r e o t h e r s t r a t i f i c a t i o n a l models which avoid t h e s e p i t f a l l s -C41 Ginsburg, S . (1975) . Algebraic and Automata-Theoretic P r o p e r t i e s of Formal Languages, North-Holland Publishing Co. C51 Gle-ason, H.A. Jr. ( 1 9 6 4 ) . \"The o r g a n i z a t i o n o f language: a s t r a t i f i c a t i o n a l viewIf , ~ono'graph S e r i e s on Language and* L i n g u i s t i c s -1 7 , p . 7 5 -9 5 , Georgetown Universi t y . C61 Lamb, S. (1966) . Outline of S t r a t i f i c a t i q n a l Grammar, Georgetown University P r e s s , Washington. ' C 71 L'ockwood. D.G. 11972). I n t r o d u c t i o n t o ~t r a t i f i -t a t h n a l -~i n~u i s t i c s , ~a r c o u r t Brace Jovanovich,*Inc."
            },
            "FIGREF22": {
                "num": null,
                "uris": null,
                "type_str": "figure",
                "text": "-101 'P'itfha, P . (1974). \"On a new form of Lamb.'$ s t r a t i f i c a t i o n a l grammarft, Slovo a a S.lovesnost -35,' p ,208-'218; t r a n s l a t e d . from t h e o r i g i h c C x c h l by D .G. LockiEod. [ l k ] P o s t a l , P . ( 1 9 6 4 ) . C o n s t i t u e n t S t r u c t u r e : A Study of Contemporary Models o f S y n t z c t i c D e w s c r i p t i o n , l n d i a n a U n i v e r s i t y , Bloomington, I n d . F i r s t a p p e a r e d i n I n t . J . Amer. L i n g u i s t i c s -3 0 . 1 , p a r t 3 ."
            }
        }
    }
}