Formal Properties of Metrical Structure

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F o r m a l P r o p e r t i e s o f M e t r i c a l S t r u c t u r e . Marc v a n O o s t e n d o r p Werkverband Grammaticamodellen. Tilburg University P.O.Box 90153. 5000 LE Tilburg The Netherlands oostendo~kub.nl. Abstract. This paper offers a provisional mathemat- ical typology of metrical representations. First, a family of algebras corresponding to different versions of grid and bracketed grid theory is introduced. It is subsequently shown in what way bracketed grid theory differs from metrical theories using trees. Finally, we show that there are no sig- nificant differences between the formalism of bracketed grids (for metrical structure) and the representation used in the work of [Kaye, et al., 1985], [1990] for subsyllabic structure.. 1 Introduction. The most well-known characteristic of Non-linear Phonology is that it shifted its attention from the theory of rules (like in [Chomsky and Halle, 1968]) to the theory of representations. During the last decade phonologists have developed a theory of representa- tions that is sufficiently rich and adequate to describe a wide range of facts from the phonologies of various languages.. It is a fairly recent development that these repre- sentations are being studied also from a purely for- mal point of view. There has been done some work on autosegmental structure (for instance [Coleman and Local, 1991; Bird, 1990; Bird and Klein, 1990]) and also some work on metrical trees (like [Coleman, 1990; Coleman, 1992] in unification phonology and [Wheeler, 1981; Moortgat and Morrill, 1991] in cat- egorial logics). As far as I know, apart from the pio- neering work by [Halle and Vergnaud, 1987], hitherto no attention has been paid to the formal aspects of. the most popular framework of metrical phonolog~y nowadays, the b r a c k e t e d g r i d s framework.. Yet a lot of questions have to be answered with regard to bracketed grids. First of all, some authors (for instance [Van der Hulst, 1991]) have expressed the intuition that bracketed grids and tree structures (e.g. the [sw] labeled trees of [Hayes, 1981] and re- lated work) are equivalent. In this paper, I study this intuition in some formal detail and show that it is wrong.. Secondly, one can wonder what the exact rela- tion is between higher-order metrical structure (foot, word) and subsyllabic structure. In this paper I will show that apart from a fewempirically unimportant details, bracketed grids are equivalent to the kind of subsyllabic structure that is advocated by [Kaye, et al., 1985], [1990] t.. 2 T h e d e f i n i t i o n o f a b r a c k e t e d g r i d . Below I give a formal definition of the bracketed grid, as it is introduced by HV and subsequently elabo- rated and revised by these authors and others, most notably [Hayes, 1991]. HV have a m a j o r part of their book devoted to the formalism themselves, b u t there are numerous problems with this formalization. I will mention two of them.. First, their formalization is not flexible enough to capture all instances of (bracketed) grid theory as it is actually used in the literature of the last few years. They merely give a sketch of the specific im- plementation of bracketed grid theory as it is used in the rest of their book. Modern work like [Kager, 1989] or [Hayes, 1991] cannot be described within. 1In this paper, I will use H V as an abbreviation for [Halle and Vergnaud, 1987] and K L V as an abbreviation for [Kaye, e t al., 19851, [1990].. 322. this framework. Secondly, their way of formalizing bracketing grids. has very much a ' d e r i v a t i o n a l ' flavour. T h e y are m o r e interested in how grids can be built t h a n in w h a t t h e y look like. A l t h o u g h looking at t h e deriva- tional aspects is an interesting a n d worthwile enter- prise in itself, it makes their formalism less suitable for a c o m p a r i s o n with m e t r i c a l trees.. A grid in t h e linguistic l i t e r a t u r e is a set o f lines, each line defining a certain subgroup o f t h e stress b e a r i n g elements. T h u s , in 1 (HV's (85)), t h e as- terisks ( ' s t a r s ' ) on line 0 represent t h e syllables o f t h e word formaldehyde, the stars on line 1 seco n d ary stress a n d line 2 represents the syllable with p r i m a r y stress:. (1) (. *) line 2 (*) (* .) (*) line 1 for m a l de hyde. We can formalize the u n d e r l y i n g notion o f a line as follows:. D e f i n i t i o n 1 ( L i n e ) A line L i i s a pair < A t , -'4i> where A i = { a ~ , . . . , a ? } , where a ~ , . . . , a n are con- stants, n a fixed number -~i is a total ordering on A i such that the following axioms hold. a. Vot, 13, 7 E L i : ot ~ i 13 A 13 ~ i 7 ~" ot -41 7 (transi- tivity) b. Vow, 13 ELi : a -4i 13 ::~ -'(13 "~i a ) ( a s y m m e t r y ) c. V a G Li : ~ ( a -41 a ) Orreflezivity) We say that Li C L j i f A i C A#, a E L i i f a E A t . Other set theoretic expressions are extended in a like- wise fashion.. Yet this formalisation is not c o m p l e t e for bracketed grids. It has to be s u p p l e m e n t e d by a t h e o r y a b o u t the brackets t h a t a p p e a r on each line, i.e. by a the- ory o f c o n s t i t u e n c y and by a t h e o r y of w h a t e x a c t l y counts as a s t a r on a given line.. We have e x a c t l y one dot on top o f each c o l u m n o f stars2.Moreover, each constituent on a line has one s t a r in it plus zero, one or m o r e dots. T h e stars are heads. HV say t h a t these heads govern t h ei r c o m p l e m e n t s . T h i s g o v e r n m e n t relation can o n l y be a relation which is defined in t e r m s o f precedence. Suppose we m a k e this g o v e r n m e n t relation into t h e p r i m i t i v e n o t i o n instead o f the constituent. A m et ri - cal line is defined as a line plus a g o v e r n m e n t rel at i o n on t h a t line:. D e f i n i t i o n 2 ( M e t r i c a l l i n e ) A metrical line M L i is a pair < L i , P ~ > , where Li is a line R i is a relation on Li and an element of {-~i,>-i,--,i , ~ - ' i , ' ~ i } . ZThat is, if we follow the current tradition rather than HV.. W i t h t h e following definitions holding3:. D e f i n i t i o n 3 ( P r e c e d e n c e R e l a t i o n s ) a Ni 13 ¢~. We assume t h a t s o m e t h i n g like G o v e r n m e n t Require- m e n t 1 holds, j u s t as it is assu m ed in HV t h a t every el em en t is in a c o n s t i t u e n t , m o d u l o e x t r a m e t r i c a l i t y (which we will ignore here).. G o v e r n m e n t R e q u i r e m e n t 1 (to be revised be- low) A line L i m e e t s the government requirement iff all dots on Li are governed, i.e. a star is in relation R i to them.. Now a c o n s t i t u e n t can be defined as t h e d o m a i n t h a t includes a star, plus all t h e dots t h a t are governed by this st ar. We have t o be a little bit careful here, because we want t o m a k e sure t h a t t h e r e is o n l y one s t a r in each c o n s t i t u e n t . . In a s t r u c t u r e like t h e following we d o n o t want t o say t h a t t h e a p p o i n t e d d o t is governed b y t h e first star. It is governed b y t h e second st ar, which is nearer t o it:. (2) * . . . * . . . T. In o rd er t o ensure this, we a d o p t an idea f r o m m o d - ern G B s y n t a x , viz. Mi n i m a l i t y, which i n fo rmally says t h a t an el em en t is o n l y governed b y a n o t h e r el- e m e n t if t h ere is no closer governor. T h e definition o f Phonological M i n i m a l i t y could look as follows:. D e f i n i t i o n 4 ( p h o n o l o g i c a l g o v e r n m e n t ) a G i # (a governs 13 on line i) i f f a is a star and art13 A -~37, 7 a star : [7R/13 A a R / 7 ] . We will give t h e f o r m a l definition o f a s t a r l a t e r on in this chapter. T h e g o v e r n m e n t r e q u i r e m e n t is now t o be slightly modified.. G o v e r n m e n t R e q u i r e m e n t 2 to be revised below A line L i m e e t s the government requirement iff all dots on Li are governed, i.e. a star in L i i s in relation Gi to them.. We can now f o r m a l l y define t h e n o t i o n o f a constituent 4.. aActually, HV also use a fifth kind of constituent in their book, viz. one of the form (. * .). Because there has been a lot of criticism in the literature against this type of government, I will not not discuss it here.. 4The reviewer of the abstract for EACL notices that under the present definitions it is not possible to ex- press the kind of ambiguity that is current in (parts of) bracketed grid literature, where it is not sharply defined whether a dot is governed by the star to its left or by t h e star to its right. This is correct. It is my present purpose to define a version of bracketed grids that c o m e s c l o s e s t to trees because only in this way we can see which are t h e really essential differences between the two formalisms.. 323. D e f i n i t i o n 5 ( C o n s t i t u e n t ) A constituent on a line Li is a set, consisting of exactly one star S in Li plus all elements that are not stars but that are governed by S.. We now have a satisfying definition of a metrical line. We can define a grid as a collection of metrical lines, plus an ordering relation on them:. D e f i n i t i o n 6 ( G r i d ) A grid G is a pair < £ , 1>, where £ = {L1 .... , Ln}, where L I , ..., L , are metrical lines I is a total ordering on £ , such that VLi, Lj • £ : Li I L1 ¢~ [Li C L j A Va,~3 • Li fl L j : [a -~i Z ¢~. 8]] where C is intended to denote the proper subset re- lation, so Li C L 1 ~ ",(Li = L i ) . . It is relatively easy to see t h a t I by this definition is transitive, a s y m m e t r i c and irreflexive. We also define the inverse operator T such t h a t Li T L 1 iff L j I L i . . The m o s t interesting part of definition 6 is of course the 1 ('above')-relation. Look at the grid in (3) ( = HV's (77), p. 262):. (3) line 3 (. . ) line 2 ( . .) ( . ) line 1 T e n ne see. Each of the lines in this grid is shorter t h a n the one i m m e d i a t e l y below it, in t h a t it has fewer elements. This follows f r o m elementary pretheoretical reason- ing. Every stressed syllable is a syllable, every syl- lable with p r i m a r y stress also has secondary stress. We expressed this in definition 6 by s t a t i n g t h a t ev- ery line is a subset of the lines below it. By this s t a t e m e n t we also expressed the i d e a t h a t the ele- ments represented on the higher line are in fact the same things as those represented on the lower lines, not j u s t features connected of these. T h e second p a r t of the definition says t h a t the relative ordering of the elements in each line is the same as t h a t on the other lines.. Our present definition of a metrical grid already has some nice properties. For example, the Continu- ous C o l u m n Constraint, which plays a crucial role as an independent stipulation in [Hayes, 1991] can be derived from definition 6 as a theorem:. (4) Continuous Column Constraint (CCC): A grid containing a column with a m a r k on layer ( = o u r metrical line) n + l and no m a r k on layer n is ill-formed. Phonological rules are blocked when t h e y would create such a configuration.. T h e C C C excludes grids like (5), where b is present on the t h i r d line, b u t n o t on the second.. (5) a b a c d a b e d . We can formalize the C C C as a theorem in our sys- tem:. T h e o r e m 1 ( C C C ) V a V L i L j : ( a E Li A L i ]. e. P r o o f o f t h e o r e m 1 Suppose a E Li, suppose Li I L j . T h e n (by (13)) L, C L j . Now the stan- dard definition of C implies VX : X E L i -"* X E L j . I n s t a n t i a t i o n of X by c~, our first a s s u m p t i o n a n d Modus Ponens give a E L i . O We can also easily define the notion of a dot a n d a star, informally used in the above definitions of government.. D e f i n i t i o n 7 ( S t a r a n d d o t ) 1. V a E L i :. stari(ot)de--~f3Lj : ILl ~ L j A ( a E Lj)] def. e. V a E L i : d o t i ( o t ) = ~ s t a r i ( o t ) . G o v e r n m e n t Requirement 2 can now be fully for- malised and subsequently extended to the grid as a whole.. G o v e r n m e n t R e q u i r e m e n t 3 ( f o r l i n e s ) - - fi- nal version A metrical line Li meets the government requirement iff V a E L i : d o t i ( a ) =~ 3 8 E Li :. ^. G o v e r n m e n t R e q u i r e m e n t 4 ( f o r g r i d s ) - - t o be revised below A grid G meets the government re- quirement iff all lines in G meet the g o v e r n m e n t re- quirement.. We want t o introduce an e x t r a requirement on grids. Nothing in our present definition excludes grids con- sisting of infinitely m a n y lines. However, in our lin- guistic analyses we only consider finite construetions. We need to express this. First, we define the notions of a top line and a b o t t o m line. T h e n we say t h a t a finite grid always has one o f each.. D e f i n i t i o n 8 ( T o p l i n e a n d b o t t o m l i n e ) For a certain grid G, VLi E G. (LToP, G = Li)d--efVLj E G : [(Li = L j ) V (Li ~ Lj)]. (LBoTTOM,G -" LI)~'~fVLj • G : [(ni = L j ) V ( L j 1 Li)]. D e f i n i t i o n 9 ( F i n i t e g r i d ) A grid G is called a fi- nite grid i f 3 L i • G : [ L i = LTOP, G] A 3 L j • G : [L i = LBoTTOM,e]. Note t h a t we have to say s o m e t h i n g special with re- gard to the government relation in LTOP, G. By defi- nition, this line has only dots in it, so it always looks as s o m e t h i n g like (6).. (6) . . . . . . . . There can be no star on this level. A star by defini- tion has to be present at some higher line a n d there is no higher line above LTOP, G. T h i s m e a n s t h a t the LTOP, G c a n never be m e e t i n g the government. 324. requirement and t h a t in t u r n means t h a t no linguis- tic grid can ever meet the government requirement. In order to avoid this r a t h e r u n f o r t u n a t e situation, we have to slightly revise the definition o f meeting the government requirement for linguistic grids.. G o v e r n m e n t P ~ q u i r e m e n t 5 ( f o r g r i d s ) - - fi- nal version A grid G meets the government relation i f f all the lines Li E G - {LTop, a } meet the govern- ment requirement.. D e f i n i t i o n 10 ( L i n g u i s t i c g r i d ) A linguistic grid is a finite grid which meets the government relation. A last definition m a y be needed here. I f we look at the grids t h a t axe a c t u a l l y used in linguistic theory, it seems t h a t there is always one line in which there is j u s t one element. Furthermore, this line is the top line (the only line t h a t could be above it would be an e m p t y line, b u t t h a t one d o e s n ' t seem to have a n y linguistic significance).. This observation is phrased in [Hayes, 1991] as fol- lows: i f prominence relations are obligatorily defined on all levels, then no matter how many grid levels there are, there will be a topmost level with just one grid mark.. We can formalize this ~s follows:. D e f i n i t i o n 11 ( C o m p l e t e l i n g u i s t i c g r i d s ) A linguistic grid G is called a complete linguistic grid iff [LToP, G] = 1, i.e. 3 a : [a E LTOP, G A Vfl : L 8 e LTOP, G ::~ # ---- O~]]. We call this t y p e of grid complete because we can eas- ily construct a complete linguistic grid o u t o f every linguistic grid.. I f LTOP, G is non-empty, we construct a complete grid by projecting the r i g h t m o s t (or alternatively the leftmost) element to a new line L / a n d by adding the government requirement ~- (or -4) to LTOP, G. Fi- nally we a d d the relation L i l LTOP, G to the grid, i.e. we m a k e L i t o the new LTOP, G.. I f the top line o f the grid is empty, we remove this line from the grid a n d proceed as above. Most linguistic grids t h a t are known from the literature, are complete.. Some a u t h o r s impose even more restrictions on their grids. I believe most of those claims can be expressed in the formal language developed in this section. One example is [Kager, 1989], who claims t h a t all phonological constituents are binary. This Binary Constituency Hypothesis can be f o r m u l a t e d by replacing definition 2:. D e f i n i t i o n 12 A metrical line M L I is a pair < L i , R i > , where Li is a line Ri is a relation on Li and an element of { ~ i , ~ i } . 3 G r i d s a n d t r e e s . In this section, we will t r y to see in how much brack- eted grids and trees are really different formal sys-. terns, i.e. to w h a t extent one can say things in one formalism t h a t are impossible to s t a t e in the other.. First recall the s t a n d a r d definition o f a tree (we cite from [Partee et al., 1990])5:. D e f i n i t i o n 13 ( T r e e ) A (constituent structure) tree is a mathematical con- figuration < N, Q, D, P, L >, where N is a finite set, the set of nodes Q is a finite set, the s e t of labels D is a weak partial order in N × N , the dominance relation P is a strict partial order in N x N , the precedence relation L is a function from N into Q, the labeling function and such that the following conditions hold:. ( a ) 3 a E N : V/~ G N : [ < a , / ~ > E O] (Single root condition). (b) W , a ~ N : [ ( < ~ , ~ > ~ PV < a , ~ > E P ) ¢* (< a,/~ > ¢ D ^ </~, a > ¢ D)] (Exclusivity condi- tion). (c) V a , ~ , 7 , 6 : [(< ot,/~ > E P A < a , 7 > E DA < 8, 6 >E D) ~ < 7, 6 >E P] (Nontangling condi- tion). It is clear t h a t bracketed grids a n d trees have structures which cannot be compared immediately. Bracketed grids are pairs consisting o f a set of com- plex objects (the lines) a n d one t o t a l ordering rela- tion defined on those objects (the above relation). Trees on the other h a n d are sets of simple objects (the nodes) with two relations defined on t h e m (dom- inance a n d precedence). These s i m p l y appear to be two different algebra's where no isomorphism can be defined.. Yet if we decompose the algebraic s t r u c t u r e of the lines, we see t h a t there we have sets of simple objects (the elements of the line) plus two relations defined on t h e m . One of those relations ('~i) is a strict par- tial order, j u s t like P . T h e other relation, Gi, vaguely reminds us of dominance.. Yet a line clearly is n o t a tree. A l t h o u g h -4i has the right properties, it is n o t so sure t h a t Gi does. While this relation clearly is a s y m m e t r i c (because it is directional), it is n o t a p a r t i a l order.. First of all, it is not transitive. (7) is a counterex- ample.. (7) • • +-- a b c. Here aGib and bGie b u t n o t aGie, because of min- i m a l i t y (there is a closer governor, viz. b). Gi also. 5For the moment, we will not consider Q and L, be- cause these are relatively unimportant for our present aim and goal and there is nothing comparable to the la- beling function in our definition of bracketed grids. This is to say that for now we will study unlabeled trees. Notice however that the trees actually used in the phonological literature do use st least a binary set of labels { s, w }. 325. is irreflexive, o f course, because no element is t o t h e left or to the right of itself.. A m o r e interesting relation emerges if we consider the grid as a whole. Because trees are finite struc- tures, we need to consider linguistic grids only. T h e line LBOTTOM,G has t h e p r o p e r t y t h a t VaVLI ~ G : [a ~ L i =~ ~ ~ LBOTTOM,G]. T h i s follows f r o m t h e definitions o f LBOTTOM,G and of the ' a b o v e ' rela- tion.. T h i s m e a n s t h a t all basic elements of the grid are present o n LBOTTOM,G and, as we have seen above, we can equal P to "~BOTTOM,G. F u r t h e r m o r e , we can build up a ' s u p e r g o v e r n m e n t ' relation {7, which we define as the disjunction o f all g o v e r n m e n t rela- tions Gi in G.. D e f i n i t i o n 14 ( S u p e r g o v e r n m e n t ) . {70 d¢__f U { < a , f ~ > laG~3Adot~(~)} LiEG. Again, we exclude the g o v e r n m e n t relation o f the two stars in (7).. If we want to c o m p a r e {7 to dominance, we have t o m a k e sure it is a p a r t i a l order. However, {7 obviously still is irreflexive. It also is intransitive. Consider t h e following grid for example.. (8) •. a cd In this grid a{Tc A c{Td b u t --,a{Td. For this reason, we take t h e t r a n s i t i v e and reflexive closure o f {7, which we call 7"7¢{7.. LFrom this, we can define the superline of a lin- guistic grid 6.. D e f i n i t i o n 15 ( S u p e r l i n e ) The superline S/~ of a linguistic grid G is the tuple < ABOTTOM,G, "~BOTTOM,G, "Jf'T~{TG >.. T h e superline is an e n t i t y which we can for- m a l l y c o m p a r e to a tree, with -~BOTTOM,G = P, ABOTTOM,G = N , 7"T~{7 = D. O f m o s t interest are the c o m p l e t e linguistic grids, firstly because these are the ones t h a t seem t o have m o s t applications in lin- guistc t h e o r y and secondly because the r e q u i r e m e n t t h a t t h e y be complete (i.e. their LTOP, G should have e x a c t l y one e l e m e n t ) m i r r o r s the single r o o t condi- tion on trees. F r o m now on, we will use the abbrevi- ation C L G for ' c o m p l e t e linguistic grid'.. Note t h a t we also restrict o u r a t t e n t i o n t o grids which m e e t the g o v e r n m e n t requirement, i,e. t o lin- guistic grids. We are n o t so sure t h a t this restriction is equally well s u p p o r t e d by metrical t h e o r y as t h e restriction t o completeness. However, the restriction. 6The superfine itself has no specific status in linguistic theory. I also do not claim it should have one. The superfine is a formal object we construct here because it is the substructure of the bracketed grid that comes closest to a tree.. t o linguistic grids m ak es sure t h a t all elements in the grid p a r t i c i p a t e in t h e g o v e r n m e n t relation, because e v e r y t h i n g ends as a s t a r somewhere an d hence has t o be governed by a n o t h e r el em en t . In order t o s o m e w h a t simplify o u r proofs below, we i n t r o d u c e one new n o t a t i o n a l s y m b o l here: ~ . . D e f i n i t i o n l 6 ( T o p L i n e ) V ~ E AVLi ~ G : [c~Li ¢~ c~ ~ Li A -,~Lj [a ~ Lj A L~ ~ Li]] T h i s s y m b o l '_~' ' t o p line' d en o t es t h e highest line on which a cert ai n el em en t is present. I f a ~ Li, t h e n Liis t h e highest line a t which a can b e f o u n d . By definition, this m e a n s t h a t ~ is a d o t o n Li.. O f course, for every e l e m e n t in a linguistic grid t h ere is o n e specific t o p line.. We now prove:. T h e o r e m 2 For every linguistic grid G, i f G is com- plete, then SLG satisfies the Single Root Condition.. P r o o f o f t h e o r e m 2: Consider a c o m p l e t e linguis- tic grid G . We have t o p ro v e t h a t 3 a E A : V~ E A : [< a , / 3 > E "/-T~{TG] (for shortness, we will refer here an d in t h e following t o ABOTTOM,G as A a n d to "~BOTTOM,G as "~ where n o confusion arises). Con- sider t h e (single) e l e m e n t o f LTOP.a. We call this el- e m e n t 7 an d prove t h a t V/~ E A : I < 7 , / ~ > E TT~gG].. (Reductio ad absurdum.) S u p p o s e 3/~ E A : [< 7 , / ~ > ~ TT~{Ta]. Because this/~ is in A, 3 Li : ~_ELi]. We now take t h e h i g h e s t / 3 for which this co n dition is t ru e, i.e. V~ E A : [ < 7 , ~ > ~ TT~{TG =~ 3Lk : [~E_L~ALk T Li]]. /~_ELI b y definition m e a n s t h a t t h e r e is n o L j higher t h a n Li o f w h i c h / ~ is a m e m b e r . B u t this in t u r n m e a n s t h a t / 3 is a d o t on Li (or doti(/~)).. Li c a n n o t b e equal t o LTOP, G, because in t h a t case w e would h a v e / 3 -- 7 an d since 7"~{TGis reflexive, < 7,/3 > E TT~{TG, c o n t r a r y t o o u r a s s u m p t i o n . So LTOP, G ~ Li. Because doti(/~) a n d t h e grid m e e t s t h e govern- m e n t r e q u i r e m e n t 36 : [stari A 6Gi7]. F r o m t h e defi- nition o f s u p e r g o v e r n m e n t we t h e n get t h a t < 6 ,/~>E T ~{T G. 6 is a s t a r o n Li. T h i s m e a n s t h a t 3Lm : [6~Lm A Lm ~ Li]. We can conclude now t h a t < 7 , 6 > E TT~{TG holds, because delta is o n a higher line t h a n /3 an d we assu m ed /3 was t h e highest e l e m e n t for which this condition did not hold.. B u t now we have < 6, fl > E q'T~{TaA < 3', 6 > E TT~{TG an d because TT~{Tais t ran si t i v e, < %/3 > E "/'7~{7G. T h i s is a c o n t r a d i c t i o n w i t h o u r initial as- s u m p t i o n . []. So superlines have o n e i m p o r t a n t ch aract eri st ic o f trees. Yet exclusivity a n d nontangling still do n o t hold for superlines o f CLG s, even i f t h e y m e e t the g o v e r n m e n t r e q u i r e m e n t . . A co u n t er e x a m p l e t o exclusivity is (9), where a -~ bA < a , b > E TT~{TG.. 326. ( 9 ) • *-- a b. A counterexample to the nontangling condition is (10), where a -~ bA < a,c > • 7"TCGcA < b,c > • q'7~Ga but c ~ c.. (10) * ~- • * +--. a b c. The reason why these conditions do not hold is that, on lines as well as on superlines, elements can both govern and precede another element. Exclusivity and nontangling are meant to keep precedence and dom- ination apart.. Sometimes in the literature on trees (e.g. Samp- son 1975) we find some weakening of the definition of a tree, in which exclusivity and nontangling are replaced by the Single mother condition.. We first define the mother relation, which is im- mediate dominance:. D e f i n i t i o n 17 ( M o t h e r ) - - to be revised below For all T, T a tree Va/9 • T[aM/9 ¢~< a,/9 > • D[= T T ~ G ] A -~37 : [< a , 7 > • D[= 7"T~6G]A <%/9>6 D[= 7-TdgG]]] D e f i n i t i o n 18 ( S i n g l e m o t h e r c o n d i t i o n ) Va,/9, 7 : [ ( a M ~ 9 A 7M/9) ¢=> (a = 3')1. Because 7 - ~ G is the transitive closure of TONG, we can rephrase defintion 17) as definition 19 for super- lines of CLGs:. D e f i n i t i o n 19 ( M o t h e r ) - - final version For all snperlines 8 f.G Va/9 6_ S £ a [ a M t ~ ¢~< a , / 9 > 6 7~#a]. We can now prove:. T h e o r e m 3 For every grid G, i f G is a CLG then Sf.G satisfies the Single Mother Condition.. P r o o f o f t h e o r e m 3: (By R A A . ) Suppose a,/9, 7 • S £ G and aM/9 A 7M/9 A a # 7- If aM/9, then by definition 19, < a , / 9 > • ~ T a , and if 7M/9, similarily < 7 , / 9 > 6 TONG. Because a # 7, we have a # /9. For if a = / 9 , we would have 7 M a A aM~9. But by (19) we then cannot have 7M/9. A similar line of reasoning shows t h a t / 9 ~ 7. So a ¢ 7 A 7 ¢ / 9 . By (17) this means t h a t [< a , / 9 > 6 ~GA < 7, ~ > 6 Ca], because 7ZQG is t h e transitive closure of GG. We have reached the following proposition:. P r o p o s i t i o n 1 The mother relation equals ~ on su- perlines: V~/9 • SfG : [otMfl =~< a , / ~ > 6 ~G] Definition 14 says that if < a , / 9 > • fig there is a line L/such that aGi/gAdoti(~)}. Also, if < %/9 > • Ga there is a line Ljsuch that 7Gj/gAdob(#)}. Because. can by definition be a dot at exactly one line, L i = L j a n d otGi/9 and 7Gi/9. However, from the minimal- ity definition of government (4), it follows that in t h a t case (~ = 7. Which is a contradiction. []. 4 D e p e n d e n c y T r e e s . Let us summarize the results so far. We have seen t h a t from bracketed grids we can extract sup•tithes, on which the government relations of the normal lines are conflated.. These superlines are equivalent to some sort of un- labeled trees, under a very weak definition of the lat- ter notion. Whereas the minimal restrictions of the Single Root Condition and the Single Mother Condi- tion do hold, the same is not necessarily true for the Exclusivity Condition and the Non-Tangling Condi- tion.. It can be shown t h a t in the linguistic literature a form of tree occurs t h a t is exactly isomorphic to bracketed grids. These are the trees t h a t are used in Dependency Phonology.. We did not yet discuss what the properties of these trees are. This is what we will briefly do in the present section.. First let us take a look at the kind of tree we can construct from a given grid. We give the CLG in (11) as an example:. (11) LTOP, G * b-. e ? . /,From this grid we can derive a superline 8 £ a with {~G = { < a , b > , < c , d > , < e , f > , < e , a > , < e, c >}. If we interpret this as a dominance relation and if we draw dominance in the usual way, with the dominating element above the dominated one, we get the following tree:. (12) e. b d f. This tree looks rather different than the structures used in the syntactic literature or in metricM work like [Hayes, 1981].. Yet there is one type of structure known in the linguistic (phonological) literature which graphically strongly resembles (12). These are the Defendency Graphs (DGs) of Dependency Phonology ([Durand, 1986] a . o . ) . . According to [Anderson and Durand, 1986], DGs have the following structure. They consist of a set of primitive objects together with two relations, de- pendency and precedence. For example within the s y l l a b l e / s e t / t h e following relations are holding (no- tice some of the symbols we introduced above are used here with a slightly different interpretation):. D e p e n d e n c y s ,--- e --* t (i.e. / s / depends o n / e / a n d / t / d e p e n d s on ~el.. 327. P r e c e d e n c e s < e < t (i.e. / s / bears a relation of ' i m m e d i a t e strict precedence t o / e / w h i c h , in t u r n , bears the same relation t o / t / . . Anderson and D u r a n d also introduce the transi- tive closure of ' i m m e d i a t e strict precedence', 'strict precedence', for which t h e y use the symbol << and the transitive closure of dependency, 'subordination', for which t h e y use the double-headed arrow. More- over, well-formed dependency graphs conform to the following informal characterisation ( = A n d e r s o n and D u r a n d ' s (10)):. D e f i n i t i o n 20 ( D e p e n d e n c y g r a p h ) (---Anderson and Durand's (10)). 1. There is a unique vertex or root. 2. A l l other vertices are subordinate to the root. 3. A l l other vertices terminate only one arc. 4. No element can be the head of two different con- structions. 5. No tangling of arcs or association lines is al- lowed. 20.1 a n d 20.2 together form a redefinition of the Single R o o t Condition. T h e o r e m 2 states t h a t su- perlines of C L G s with the government requirement satisfy this Condition.. Because 'arcs' are used as graphic representations for dependency (which is intransitive), 20.3 seems a f o r m u l a t i o n of the Single Mother Condition. Theo- rem 3 states t h a t this condition also holds for super- lines of complete linguistic grids.. 20.4 needs some f u r t h e r discussion because it is the only requirement t h a t does not seem to hold for our grids. T h e condition says t h a t s o m e t h i n g c a n n o t be a head at more t h a n one level of representation, e.g. s o m e t h i n g cannot be the head of a foot and of a word. However, because of the CCC, in bracketed grid systems the head of a word always is present (as a star - hence as a head) on the foot level.. It is exactly this requirement t h a t is abandoned by all authors of at least Dependency Phonology. [An- derson and D u r a n d , 1986] (p.14) s t a t e t h a t one el- ement can be the head of different constructions, as indeed we have already argued in presenting a given syllabic as suecesively the head of a syllable, a foot and a tone group.. In order to represent this, a new t y p e of relation is introduced in their system, subjunction. A node a is subjoined t o / 3 iff (~ is dependent o n / 3 b u t there is no precedence relation between the two.. T h e word intercede t h e n gets the following repre- sentation:. (13) o. O O. b,, o o I I I. in ter cede. We once again cite [Anderson and D u r a n d , 1986] (p.15):. The node of dependency degree 0 is ungoverned (the group head). On the next level down, at de- pendency degree 1, we have two nodes governed by the DDO node representing respectively the first foot (inter) and the second foot (cede). The first node is adjoined to the DDO node, the second one is sub- joined. Finally, on the bottom level, at DD2, the nodes represent the three syllables of which this word is comprised. These latter nodes are in turn governed by the nodes at DD1 and once again related to them by either adjunction or snbjunction.. Lifting the restriction this way seems to be exactly w h a t is needed to fit the superline into the DG for- malism.. 20.5 holds trivially in the bracketed grid frame- work as well. It can be interpreted as: if ~ precedes /3 on a given line in the grid, there is no other line such t h a t / 3 precedes c~ on t h a t line. T h i s is included in the definition of the '~' relation,. The difference between a phonological DG and a bracketed grid is the same as the difference be- tween a superline and a bracketed grid: the DG is not f o r m a l l y divided into separate lines. Interest- ingly, [Ewen, 1986] analyses English stress shift, one of the m a i n empirical m o t i v a t i o n s behind the grid formalism, with s u b j u n c t i o n . In [Van Oostendorp, 1992b] I argue t h a t using s u b j u n c t i o n Ewen's way actually m e a n s an i n t r o d u c t i o n of lines into Depen- dency Phonology.. 5 G o v e r n m e n t P h o n o l o g y . Now let us t u r n over to a well-known phonological t h e o r y t h a t also employs the notion of government as well as ' a u t o s e g m e n t a l representations'. I refer of course to the syllable t h e o r y of KLV a n d [Charette, 1991].. This t h e o r y of the syllable in fact does n o t have a syllable constituent at all. In stead of such a con- stituent, KLV p o s t u l a t e a line of x-slots a n d par- alelly, a tier of representation which conforms to the pattern ( 0 R)* - that is an arbitrary number of rep- etitions of the pattern O R . ( K L V [1990]) T h e t e r m 'tier' suggests an a u t o s e g m e n t a l r a t h e r t h a n a met- rical (bracketed grid) approach to syllable structure, b u t KLV are never explicit on this point.. T h e fact t h a t O and R appear in a strictly regular p a t t e r n can be explained either by invoking t h e (met- rical) Perfect Grid requirement or, alternatively, the. 328. ( a u t o s e g m e n t a l ) OCP. T h e same applies to t h e 'la- bels' O and R: we can define t h e m a u t o s e g m e n t a l l y as the two values o f a t y p e 'syllabic c o n s t i t u e n t ' or in- directly as n o t a t i o n a l conventions for stars and dots on a 'syllable line', i.e. we could have the following representations for KLV's (O R)* line:. (14) a. tier:. [type: syll.const; vMue: O ; . . . ] [type: syll. c o a s t ; . v~lue: R ; . . . ], etc.. b. line: *, etc.. For (14b), we would have to show t h a t the r h y m e s or nuclei p r o j e c t to some higher line. We will r e t u r n to this below.. We still c a n n o t really decide between an autoseg- m e n t a l and a m e t r i c a l approach. If we look at m o re t h a n one single line, this s i t u a t i o n changes.. At first sight, it t h e n seems very clear t h a t KLV's syllables act as ARs, n o t as grids. For instance, we can have representations like (15) (from [Charette, 1991]), with a floating Onset constituent:. (15) O r t O R I I I. X X X. I I I a m 1. T h i s is a possible a u t o s e g m e n t a l chart, b u t n o t a possible grid (because the C o m p l e x C o l u m n Con- s t r a i n t is violated by the word initial onset). How- ever, empirical m o t i v a t i o n for (15) is h a r d to find. As far as I know, the s t r u c t u r e o f (15) is m o t i v a t e d o n l y by the assumption t h a t on the syllabic line we should find (OR)* sequences r a t h e r t h a n , say, ( R ) ( O R ) * . . T h e same s t a t e o f p o o r m o t i v a t i o n does no t hold, however, to the r e p r e s e n t a t i o n [Charette, 1991] as- signs to words with an 'h aspir6'7:. (16) O R I I. X X X. I I a §. As is well known, the two types o f words behave very differently, for e x a m p l e with regard to the def- inite article. While words with a lexical represen- t a t i o n as in (16) behave like words s t a r t i n g with a 'real', overt, onset, words with a representati o n like (15) behave m a r k e d l y different:. (17) a. le t a p i s - *l' tapis b. la hache - *l' hache c. *la amie - l' amie. It seems t h a t , while the e m p t y onset o f (15) is invisible for all phonological processes, the sam e is not t r u e for t h e e m p t y onset o f (16).. rI disregard the (irrelevant) syllabic status of the final [§] consonant.. So t h ere are two different ' e m p t y o n set s' in KLV's t h e o r y s. Notice t h a t t h e t y p e o f e m p t y onset for which there is some empirical evidence is e x a c t l y the one where t h e ( 0 R ) - x slot c h a r t does b eh av e like a grid (i.e. where it does n o t v i o l at e t h e C C C ) . . So whereas we have here a f o r m a l difference be- tween KLV's t h e o r y an d grid theory, this has n o real empirical repercussions.. A n o t h e r si m i l ari t y is o f course t h e n o t i o n 'govern- m e n t ' . For KLV, g o v e r n m e n t o n l y plays a role on t h e line o f x-slots. [ C h a r e t t e , 1991J(p. 27) gives the following s u m m a r y : . Governing relations m u st have the following prop- erties:. ( i ) Constituent government: the head is initial and government is strictly local.. ( i t ) Interconstituent government: the head is final and government is strictly local.. Government is subject to the following properties:. (i ) Only the head o f a constituent m a y govern.. ( i l ) Only the nuclear head m a y govern a constituent head.. T h e m o s t i m p o r t a n t g o v e r n m e n t relation is con- s t i t u e n t g o v ern m en t : this is t h e rel at i o n t h a t defines the phonological c o n s t i t u e n t . Moreover, t h e 'prin- ciples' given by C h a r e t t e are o n l y i n t r o d u c e d into t h e t h e o r y to constrain i n t e r c o n s t i t u e n t g o v ern ment. By definition, c o n s t i t u e n t g o v e r n m e n t r e m a i n s unaf- fected b y these. (As for (i), t h e definition o f t h e no- tion c o n s t i t u e n t implies t h a t it is o n l y t h e h ead t h a t governs and (it) does n o t a p p l y because we never find two c o n s t i t u e n t heads w i t h i n o n e c o n s t i t u e n t ) . . T h e two conditions on c o n s t i t u e n t g o v e r n m e n t ( t h a t the h ead be initial an d t h e governee ad j acent t o it) can b e expressed in o u r f o r m a l i s a t i o n of the grid in a v ery simple way:. D e f i n i t i o n 21 Rx-stot =~'-". According t o KLV, *-- is t h e o n l y possible con- s t i t u e n t g o v e r n m e n t relation. O t h e r c a n d i d a t e s like {---~, ~-,-~,,~} are explicitly rejected, so in fact we have (w i t h some r e d u n d a n c y ) : . D e f i n i t i o n 2 2 VLi : [ R / E {~-)] A R ~ - , l o t =*'--. 8[Piggot and Singh, 1985] propose a different distinc- tion, namely one in which the empty onset of ami is rep- resented as (in) and the one of hache as (ib) (0 is a null segment):. (i) a. 0 b. 0 I I. X X. I 0. Under this interpretation of Government Phonology, the syllable structure is formally even more similar t o grids, if we assume that the linking between segmental material and x-slots has to be outside the grid (treated as autosegmental association) anyway.. 329. T h i s is one o f the reasons why KLV do not accept the syllable as a constituent: under their definition o f g o v e r n m e n t , this would m a k e the onset into t h e head o f the syllabic constituent.. At least we can see f r o m these definitions t h a t t h e x-slot line in KLV's t h e o r y behaves like a n o r m a l m e t r i c a l line.. Yet there is one e x t r a condition defined on this line; this is called interconstituent government. Be- cause o f t h e restrictions in (15), K L V notice t h a t this t y p e o f g o v e r n m e n t only concerns the following con- texts. (Square brackets d e n o t e d o m a i n s for intercon- s t i t u e n t g o v e r n m e n t , n o r m a l brackets for c o n s t i t u e n t g o v e r n m e n t ) : . (18) a. N O. A I ( x Ix) (x]. b. N O N I I. [(x) (xl c. 0 R. I I Ix) (x]. B u t the fact t h a t there is an e x t r a conditio n on a line does not alter its being metrical, even if we call this e x t r a condition a government relation 9.. We now have reached the following r e p r e s e n t a t i o n (20) o f the grid v a r i a n t o f the x line in (19) (we use t h e s t a r - a n d - d o t n o t a t i o n and leave o u t the associa- tion o f the a u t o s e g m e n t a l m a t e r i a l to the skeleton):. (19) O R O R. I I A i X X X X X. I I I I I p a t r l. (20) O R O t t (*) (*) (* .) (*). B y definition, stars are present on a higher line. As we have seen above, there is no reason n o t t o consider the (O R)* tier to be this higher line. We t h e n get the following representation:. (21) (. *) (. *) line 2 (?) (*) (*) (* .) (*) line l ( ~ ) p a t r i. As we n o t e d above, KLV do not accept an y con- s t i t u e n t s on the higher line. One of their reasons was their s t i p u l a t i o n t h a t all c o n s t i t u e n t s are left-headed.. T h e r e are i n d e p e n d e n t reasons to a b a n d o n this re- striction. [Charette, 1991] argues for a prosodic anal- ysis o f French schwa/[e] alternations. In order t o do this, she has to build metrical (Is w] labeled) trees representing feet on top of t h e nuclei. She gives t h e . 9 I n [Van Oostendorp, 1992a] I sketch a way of trans- lating 'interconstituent government' to a bracketed grid theory of the syllable.. following crucial e x a m p l e [Ch aret t e, 1991] (p.180, ex- a m p l e ( l l c ) ) : . (22) F. W S I I. N N I I. 0 N O N I I I I. X X X X. I I I I m ~ n e. Here we have a clear case o f a r i g h t - h e a d e d p hono- logical c o n s t i t u e n t , n a m e l y t h e fo o t .. F u r t h e r m o r e , we see t h a t t h e nuclei are p r o j e c t e d f r o m t h e (O R)* line t o a i i n e where t h e y are the single elements. I f we change t h e t o p N's in this p i ct u re into ~'s we have s o m e t h i n g like a m e t r i c a l syllable line.. If we i n c o r p o r a t e these two i n n o v a t i o n s i n t o o u r theory, we can t r a n s l a t e t h s t r u c t u r e in 22 i nto a p erfect l y n o r m a l grid, in fact i n t o a complete linguis- tic grid:. (23) - h e a d - o f - f o o t ( - - LTOP, G) (. *) Soot ( - ) . (. * ) (. * ) s y l l a b , c o n s t . ( ~ ) (*) (*) (.) (*) x-nine(--). I I I I m O n e. Concludingly, we can say t h a t , a l t h o u g h KLV's syllable r e p r e s e n t a t i o n s are s o m e w h a t different f r o m linguistic grids, two m i n o r a d j u s t m e n t s can m a k e t h e m isomorphic:. • in stead o f (O R)* we assu m e ( R ) ( O R)*, i.e. t h ere can be onsetless syllables (KLV themselves n o t e t h a t m o s t o f t h e (O R)* s t i p u l a t i o n can be m a d e t o follow f r o m i n d e p e n d e n t s t i p u l a t i o n s like i n t e r c o n s t i t u e n t g o v e r n m e n t ) . T h i s follows a forteriori for t h e ( R ) ( O R)* s t i p u l a t i o n . . • in st ead o f 22 we assume VR~ : [R~ E {~--, }] ^ R ~ _ , ~ . =*-- T h e first c o n j u n c t o f this definition is s i m p l y m y t r a n s l a t i o n o f K a g e r ' s ([1989]) B i n a r y Con- s t i t u e n c y H y p o t h e s i s 12 an d t h e second con- j u n c t does t h e s a m e as t h e original definition of KLV: it gives t h e correct choice o f g o v e r n m e n t for t h e subsyllabic line.. As far as I can see, n o n e o f these m o d i fi cations alters t h e empirical scope o f KLV's t h e o r y in any i m p o r t a n t way. I conclude t h a t for all p r a c t i c a l pur- poses, KLV's r e p r e s e n t a t i o n o f t h e syllable equals m y definition o f a linguistic grid.. 6 C o n c l u s i o n In this p a p e r we have seen t h a t t h r e e m o r e or less p o p u l a r r e p r e s e n t a t i o n a l s y s t e m s in m o d e r n phonol- ogy are n o t a t i o n a l v ari an t s o f each o t h e r in m o s t . 330. important ways: these are bracketed grid theory, Dependency Phonology and Government Phonology. The basic ideas underlying each of these frameworks are government/dependency on the one hand and the division of a structure into lines on the other.. The similarity between the frameworks is obscured mainly by the immense differences in notation; but we have shown that the algebraic systems underlying these formalisms is basically the same.. In [Maxwell, 1992] it is shown that the differences between Dependency Graphs and X-bar structures as used in generative syntax are minimal. It remains to be shown whether there are any major formal differ- ences between the bracketed grids that are presented in this paper and the 'X-bar-structures-cure-lines' as they are represented in [Levin, 1985] and [Hermans, 1990].. A c k n o w l e d g e m e n t s . I thank Chris Sijtsma, Craig Thiersch and Ben Her- marls and the anonymous EACL reviewer of the ab- stract for comments and discussion. I alone am re- sponsible for all errors.. R e f e r e n c e s . [Anderson and Durand, 1986] John Anderson and Jacques Durand 'Dependency Phonology' In: Du- rand (1986). [Bird, 1990] Steven Bird Constraint-Based Phonol- ogy Doctoral dissertation, Edinburgh.. [Bird and Klein, 1990] Steven Bird and E. Klein 'Phonological events' Journal of linguistics 29.. [Charette, 1991] Monik Charette Conditions on phonological government Cambridge, Cambridge University Press, 1991.. [Chomsky and Halle, 1968] Noam Chomsky and Morris Halle The Sound Pattern of English. MIT Press, Cambridge (Mass.) 1968.. [Coleman, 1990] John Coleman 'Unification phonol- ogy: Another look at sythesis-by-rule.' In: H. Karlgren (ed.) Proceedings of COLING lggo. In- ternational Committee on Computational linguis- tics, Vol. 3:79-84.. [Coleman, 1992] John Coleman 'The phonetic inter- pretation of headed phonological structures con- taining overlapping constituents' Phonology 9.1.. [Coleman and Local, 1991] John Coleman and John Local The 'No Crossing Constraint in Autoseg- mental Phonology' Linguistics and Philosophy 295-338.. [Durand, 1986] Jacques Durand (ed.) Dependency and non-linear phonology Croom Helm, London, 1986.. [Ewen, 1986] Colin Ewen 'Segmental and superseg- mental structure' In: Durand (1986). [Halle and Vergnaud, 1987] Morris Halle and Jean- Roger Vergnaud An essay on stress MIT Press, Cambridge (Mass.) 1987.. [Hayes, 1981] Bruce Hayes A metrical theory of stress rules Doctoral dissertation, MIT.. [Hayes, 1991] Bruce Hayes Metrical stress theory: Principles and case studies. Manuscript, UCLA.. [Hermans, 1990] Ben Hermans 'On the fate of stray syllables' In: J. Masearo and M. Nespor (eds.) Grammar in progress: G L O W essays for Henk van Riemsdijk. Dordrecht, Foris, 1990.. [Van der Hulst, 1991] Harry van der Hulst 'The book of stress' Manuscript, Leiden University, 1991.. [Kager, 1989] Ren~ Kager A metrical theory of stress and destressing in English and Dutch Doctoral dissertation, Rijksuniversiteit Utrecht, 1989.. [Kaye, et al., 1985] Jonathan Kaye, Jean Lowen- stamm and Jean-Roger Vergnaud 'The internal structure of phonological elements: a theory of charm and government' Phonology yearbook 2:305- 328.. [Kaye, et al., 1990] Jonathan Kaye, Jean Lowen- stamm and Jean-Roger Vergnaud 'Constituent structure and government in phonology' Phonol- ogy 7.2. [Levin, 1985] Juliette Levin A metrical theory of syl- labicity Doctoral dissertation, MIT. Cambridge (Mass.), 1985.. [Maxwell, 1992] Dan Maxwell 'Equivalences be- tween constituency and dependency.' In: J. van Eijck and W. Meyer Viol (eds.): Computational linguistics in the Netherlands; Papers from the second CLIN-Meeting 1991. Utrecht, OTS, 1992.. [Moortgat and Morrill, 1991] Michael Moortgat and Glynn Morrill 'Heads and Phrases; Type calculus for dependency and constituent structure' To ap- pear in Journal of logic, language and information. [Van Oostendorp, 1992a] Marc van Oos- tendorp 'The grid of the French syllable' In R. van Hout and R. Bok-Bennema (eds.) Linguistics in the Netherlands Igge. Amsterdam, Benjamins.. [Van Oostendorp, 1992b] Marc van Oosten- dorp 'The isomorphism between bracketed grid and dependency phonology' Manuscript, Tilburg University.. [Piggot and Singh, 1985] Glynn Piggot and Rajen- dra Singh 'The phonology of epenthetie segments' Revue Canadienne de Linguisiiqne 30.4:415-453.. [Partee et al., 1990] Barbara Partee, Alice ter Meulen and Robert Wall Mathematical methods in linguistics Dordrecht: Kluwer Academic.. [Wheeler, 1981] Deirdre Wheeler Aspects of a cat- egorial theory of phonology Doctoral dissertation Amherst, 1981.. 331

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