Parsing Schemata for Grammars with 
Variable Number and Order of Constituents 
Karl-Michael Schneider 
Departinent of General Linguistics 
University of Passau 
Imlstr. 40, 94032 Passau, Germany 
schneide(@l)hil.uni-t)assau.de 
Abstract 
We define state transition grammars (STG) as 
an intermediate tbrmalism between grammars 
and parsing algorithms which is intended to 
separate the description of a parsing strategy 
from the grammar tbrmalism. This allows to de- 
fine more general parsing algorithms for larger 
classes of grammars, including gramnmrs where 
the nunfl)er and order of subconstituents de- 
tined by a production may not be tlxed. Various 
grammar formalisms are characterized in terms 
of prol)erties of STG's. We define an Earley 
parsing schema tbr S'rC's and characterize the 
wflid l)arse items. We also discuss the usabil- 
ity of STG's tbr head-(:orner parsing and direct 
1)arsing of sets of tree constraints. 
1 Introduction 
This t)aper addresses the qllestion of how l;o de- 
fine (talmlar) parsing algorithms on a greater 
level of al)straction, in order to apply them 
to larger (:lasses of grammars (as compared 
to parsing algorithms tbr context-Dee gram- 
lllars). SllCtl an abstraction is useflll beCallSe 
it; allows to study l)rot)erties of parsing algo- 
rithms, and to compare different parsing algo- 
rithms, independently of tile prot)erties of an 
mtderlying grammar formalism. While previ- 
ous atteml)ts to define more general parsers 
have only aimed at expanding the domain of 
the nontenninal symbols of a grammar (Pereira 
and Warren, 1983), this paper aims at a gen- 
eralization of parsing in a difl'erent dimension, 
namely to include grammars with a flexible con- 
stituent sI;ructure, i.e., where tile sequence of 
subconstituents specified by a grammar produc- 
tion is not fixed. We consider two grammar 
tbrmalisms: Extended context-ii'ee grammars 
(ECFG) and ID/LP granllllars. 
ECFG's (sometimes called r(~.q'ular righ, t part 
grammars) are a generalization of context-free 
grammars (CFG) in which a grammar produc- 
tion specifies a regular set of sequences of sub- 
constituents of its left-haM side instead of a 
fixed sequence of subconstituents. The right- 
hand side of a production can 1)e represented 
as a regular set, or a regular expression, or a 
finite automaton, which are all equivalent con- 
cepts (Hopcroft and Ulhnan, 1979). ECFG's 
are often used by linguistic and programming 
 grammar writers to represent a (pos- 
sibly infinite) set of context-free productions as 
a single production rule (Kaplan and Bresnan, 
1982; Woods, 1973). Parsing of ECFG's has 
been studied t br example ill Purdom, Jr. and 
Brown (1981)and l;~','r,nakers (1989). 'rab,ll~r 
parsing teclmiques tbr CFG's can be generalized 
1;o ECFG's in a natural way by using the con> 
putations of the tinite automata in the grammar 
productions to guide the recognition of new sub- 
constituents. 
ID/LP grammars are a variant of CFG's that 
were introduced into linguistic tbrmalisms to en- 
code word order generalizations (Gazdar et al., 
1985). Her(',, the number of snbconstituents of 
the left-hand side of a production is fixed, but 
their order can w~ry. ID rules (immediate dom- 
inance rules) speci(y the subconstituents of a 
constituent but leave their order unspeeitied. 
The adnfissible order|rigs of subeonstituents are 
specified separate, ly by a set of LP constraints 
(linear precedence constraints). 
A simple approach to ID/LP parsing (called 
indirect parsing) is to tully expand a gram- 
mar into a CFG, but this increases the nmnber 
of productions significantly. Therefore, direct; 
parsing algorithms for ID/LP grammars were 
proposed (Shieber, 1984). It is also possible to 
encode an ID/LP grammar as an ECFG by in- 
terleaving the ID rules with LP checking with- 
733 
out increasing the number of productions. How- 
ever, tbr unification ID/LP grammars, expan- 
sion into a CFG or encoding as an ECFG is 
ruled out because the information contained in 
the ID rules is only partial and has to be instan- 
tiated, which can result in an infinite number 
of productions. Moreover, Seiffert (1991) has 
observed that, during the recognition of sub- 
constituents, a subconstituent recognized in one 
step can instantiate t~atures on another subcon- 
stituent recognized in a previous step. There- 
tbre, all recognized subconstituents must remain 
accessible fbr LP checking (Morawietz, 1995). 
We define an intermediate tbrmalism be- 
tween grammars and parsers (called state tran- 
sition 9rammars, STG) in which different gram- 
mar fbrmalisms, including CFG's, ECFG's, and 
ID/LP grammars can be tel)resented. More- 
over, admissible sequences of subconstituents 
are defined in a way that allows a parser to 
access subconstituents that were recognized in 
previous parsing steps. Next, we describe an 
Earley algorithm tbr STG's, using the parsing 
schemata ibrmalism of Sikkel (1993). This gives 
us a very high level description of Earley's algo- 
rithm, in which the definition of parsing steps 
is separated from the properties of the grammar 
tbrmalism. An Earley algorithm for a grammar 
may be obtained tiom this description by rep- 
resenting the grammar as an STG. 
The paper is organized as tbllows. In Sec- 
tion 2, we define STG's and give a characteri- 
zation of various grammar tbrmalisms in terms 
of properties of STG's. In Section 3 we present 
an Earley parsing schema for STG's and give a 
characterization of the wflid parse items. In Sec- 
tion 4, we introduce a variant; of STG's tbr head- 
corner parsing. In Section 5, we discuss the us- 
ability of STG's to define parsers for grammars 
that define constituent structures by means of 
local tree constraints, i.e., formulae of a (re- 
stricted) logical . Section 6 presents 
final conclusions. 
2 State Transition Grammars 
Wc denote nonterminal symbols with A, B, ter- 
minal symbols with a, terminal and nonterminal 
symbols with X, states with F, strings of sym- 
bols with/3, % and the empty string with c. An 
STG is defined as tbllows: 
Definition 1 (ST(\]). Art STG G is a tuple 
( N, E, A~, AJ l;', He, P, S) where 
• N is a finite set of nonterminal symbols, 
• E is a finite set of terminal symbols, 
• A/I is a finite set of states, 
,, A.4\];, c_ .A4 is a set of final states, 
,, Ha c (.A4 x V) 2 is a binary relation of the 
form (r,/3) ~-a (F',/3X), where V = NUE, 
• P C_ N × AJ \ .A41,~ is a set of productions 
written as A -+ F, and 
• S E N is a start symbol. 
Note thai; we do not allow final states in the 
right-hand side of a production. A pair (F,/3) is 
called a configuration. If F is a fnal state then 
(P,/3) is called a final configuration. The reflex- 
ive and transitive closure of \[-c, is denoted with 
H~. The state projection of Hc is the binary 
relation 
 (Ho) = {(r, r')l  /3x: (p,/3) (p',/3x)}. 
Ha is called context:free iff a transition from 
(P,/3) does not del)end on fl, tbrmally: for all 
/3, fl', r, r', x: (r,/3) Ha (r', fiX) iff (r,/3') He; 
(F',/3'X). The set of terminal states of G is the 
set 
w(C) = {PlVP' : (1 ~, P') ~ ~(Ha)}. 
The  defined by a state P is the set 
of strings in the final configurations reachable 
t'rom (r, e): 
L(r) = {/313 r' My: (r, (r',/3)}. 
Note that if A --> F is a production then e 
L(P) (i.e., there are no ~-productions). The 
derivation relation is defined by 7A5 ==> 7fl5 
itf for some production A ~ P: /3 C L(P). The 
 defined by G is the set of strings in E* 
that are derivable fi'om the start symbol. 
We denote a CFG as a tuple (N,E,P,S) 
where N, E, S are as betbre and P C_ N x V + is 
a finite set of productions A -+/~. We assume 
that there are no e-productions. 
An ECFO can be represented as an extension 
of a CFO with productions of the tbrm A -+ A, 
where .A = (V, Q, qo, 5, Of) is a nondeterministic 
finite automaton (NFA) without e-transitions, 
734 
54 
ECFG Q 
 D/LP {MI A+M'cP: MC_M'} 
{c} 
Qf 
F = XF' 
(r, X, r') ~ 
r = r'u {x},/~x < LP 
Tnble 1: Encoding of grmnmars in STG's. 
with input alphalmt V, state set Q, initial state 
q0, final (or accepting) states Q f, m~(t tr~msi- 
tion relation 5 C_ Q x V x Q (I{opcroft and Ull- 
man, 1979). A accepts ~ string fl ill tbr some 
final st;;~l;e q C Q f, (qo,/'-\], q) ~ 5". Furl;hermore, 
we assume that q0 ~ Q f, i.e., ..4 does nol; ac- 
(:ept the emi)l;y word. We can assmne wit;hour 
loss of generalizal;ion thai, the mfl;omal;a in the 
right-lmnd sides of a grammar are nll disjoint. 
Then we cml rel)resent ml ECFG as a tul)le 
(N, E, Q, Q f, 5,1 ), S) where N, E, Q, Q f, 5, S m'e 
as befbre and P C N x (2 is ~t finite set of produc- 
tions A -> q0 (q0 is ml initial st~te.). For rely pro- 
duction p = A ~ q0 let A p = (17, Q, q0, (t, Oj.) 
l)e the NFA with initiM state q0. The, deriwd;ion 
relation is detined by 7A5 ~ 7/35 itf fbr some 
1)roduction p = A ---> q0, A p accet)ts fl. 
An ID/LP grnmm~tr is represented as a l;u- 
pie (N~ E, \] , LP, S) whoa'e. N, E, S are as before 
nnd P is a finite set of productions (ID rules) 
A --+ M, where. A C N ;uid ~4 is ~ multiset 
over V, and LP is a set ()f line~r l)re(:edence 
constraints. We are not concerned with de.tails 
of the LP constra.ints here. We write fl ~ LP 
to denote that the sl;ring fi s~d;isties all the con- 
straints in l,P. The derivation r(;l~|;ion is de- 
fined by 7A5 ~ 7\[3d i1\[ fl = X~...X~ and 
a > {X~,...,Xk} ~ 1" mM fl ~ LI'. 
CFG's, ECFG's and ID/LP grmnlnars (:;m 
t)e chara(:l;erized by al)t)rol)ri~te restrictions on 
the transition relation and the fired st;~l;es of an 
STG: ~ 
• CFG: \]-o is context-free and deterministic, 
cy(t-6,) is acyelic, 2~4F = T(G). 
• ECFG: t-a is context-free. 
• ID/LP: or(t-(;) is aeyclic: J~41,' = T(G), for 
all F: iffl, 7 C L(F) then 7 is ~t permutal,iolt 
These conditions define normal-forms of STG's; that 
is, for STG's that do not, satist~y the conditions for some 
type there can nevertheless lm strongly equivalent gram- 
mars of that; t;ype. These STG's are regarded as degen- 
erate mM are not fllrther considered. 
of ft. 
For instance, if G is an STG that satisfies the 
conditions tbr CFG's, then a CFG G / can be 
constructed as follows: l,br every production 
A -~ q0 in G, let A -~ fl be a production in 
G' whe.re L(qo) = {/3}. Then the deriw~tion re- 
lations of G mid G' coincide. Similarly tbr the 
other grammar tyl)es. Conversely, if ~t grammar 
is of a given type, l;hen it (:ml be rel)resented as 
ml STG satist~ying the conditions tbr that type, 
by spe(:it~ying the states and transition relation, 
as shown in Table 1 (tO denotes nmltiset lnlion). 
3 Earley Parsing 
Parsing schemat~ were proposed by Sikkel 
(1.993) as a framework for the specific~tion 
0rod comparison) of tabular parsing algorithms. 
Parsing schemata provide n well-detined level of 
abstra(:l;ion by al)stra(:ting fi'om (:ontrol struc- 
tures (i.e., or(lering of operations) and (later 
structures. A parsing schem;t cmJ \])e imple- 
mented as n tabulm: parsing ;flgorithm in ~ 
em~onical w;~y (Sikkel, 1998). 
A \])re:sing schema for n gr;tllllll;~r cla,ss is & 
function that assigns ('.~mh grmnmar and each 
input string a deduction system, called a parsing 
sy.ste.m. A parsing schema is usmdly defined by 
pre.senting a parsing system. A parsing system 
consists of ~ finite set Z of pars(; items, a finite 
set "H of hyt)otheses , whi(:h ell(:()(\](; the input 
string, mxd ~ finite set 29 of deduction stel)s of 
the fbrm x~,...,x, t- a: where xi C 2; U ~ and 
x E Z. The hypotheses can be represented as 
deduction steps with empty prenfises, so we can 
assume that, all xi m'e it;eros, and represent a 
parsing system as a pair (Z, 29). 
Correctness of a l)~rsing system is defined 
with respect to some item senmntics. Every 
item denotes a particub~r deriw~tion of some 
substring of the input string. A parsing sys- 
te.m is correct if an item is deducible precisely if 
it denotes an admissible deriw~tion. Items that 
denote admissible derivations are called coffee/,. 
735 
Z={\[A~/3.F,i,j\]IAEN, flEV*, £EM, 1/31 <.,, O<i<j<n} 
D Init =- S--~ P E P \[S ~ .r, 0,0\] 
Dpredi~t = \[A --+ ft. P, i, j\] 
T)Comp I =_ 
\[B --+ .P0,j,j\] 
\[A + fl.P,i,j\] 
\[A -+ \[3aj+l. F', i, j + 1\] 
r': (r,/3) No (r',/~B), B -~ ro e p 
(r,/5) \[-G (r',/3aj+l) 
\[A ~ /3.r,i,j\] 
\[B ~ ,>r:, j, k\] 
\[A -+ fiB. £', i, t~\] r: E M,, (r,/3) >a (r',/3~) 
Figure 1: The Earley parsing schema for an STG G and input string w = al ... an. 
STG's constitute a level of abstraction be- 
tween grammars and parsing schemata because 
they can be used to encode various classes of 
grammars, whereas the mechanism for recog- 
nizing admissible sequences of subconstituents 
by a parsing algorithm is built into the gram- 
mar. Thereibre, STG's allow to define the pars- 
ing steps separately fiom the mechanism in a 
grmnmar that specifies admissible sequences of 
subconstituents. 
A generalization of Earley's algorithm ibr 
CFG's (Earley, 1970) to STG's is described by 
the parsing schema shown in Fig. 1. An item 
\[A -~/3.P, i, j\] denotes an A-constituent that is 
partially recognized fi'om position i through j 
in tile input string, where/3 is the sequence of 
recognized subconstituents of A, and a sequence 
of transitions that recognizes ~ can lead to state 
F. Note that the length of/5 can be restricted 
to the length of the int)ut string because there 
are no g-productions. 
In order to give a precise definition of the se- 
mantics of the items, we define a derivation re- 
lation which is capable of describing the partial 
recognition of constituents. This relation is de- 
fined on pairs (7, A) where 7 E V* and A is a 
finite sequence of states (a pair (% A) could be 
called a super configuration). 7 represents the 
fi'ont (or yield) of a partial derivation, while A 
contains one state for every partially recognized 
constituent. 
Definition 2. The Earley derivation relation 
is defined by th, e clauses: 
• (TA, A) ~ (7/5, FA) iff 3A --+ P' E P: 
(r', e) e5 (r,/3). 
• (TAa, A) p (7/3a, A) /ff 7Aa ~ 798. 
The first clause describes the I)artial recog- 
nition of an A-constituent, where/3 is the rec- 
ognized part and tile state P is reached when 
/3 is recognized. The second clause describes 
~he complete recognition of an A-constituent; 
in this case, the final state is discarded. Each 
step ill the derivation of a super configuration 
(% A) corresponds to a sequence of deduction 
steps in the parsing schema. As a consequence 
of the second clause we have that w E L(G) iff 
(S, c) ~* (w, c). Note that ~-, is too weak to de-- 
scribe the recognition of the next subconstituent 
of a partially recognized constituent, but it is 
sufficient to define the semantics of the items in 
Fig. 1. The fbllowing theorem is a generaliza- 
tion of the definition of the semantics of Earley 
items for CFG's (Sikkel, 1993) (al... an is the 
input string): 
Theorem 1 (Correctness). 
F* \[A --+/3.F, i,j\] iff the conditions are satisfied: 
• for some A, (S, c) \]'--,* (al... aiA, A). 
• (A, e) b" (/3, F). 
• /3 ::==>* ai+ 1 . .. aj. 
The first and third condition are sometimes 
called top-down and bottom-up condition, re- 
spectively. The second condition refers to the 
partial recognition of the A-constituent. 
736 
\[~ -~. ~,, 0, 0\] (re, ~) >* (r,, ~), (E, ~) \[~ (~, q~) 
IT -~ .q3,0,0\] (m~) ~ (T,<~), (T,~) p (~, ¢~) 
(m ~) ~ (T, ~) 
\[F -+ .q,~, 0, 0\] (E,E) t..,(T,q,jI--,(F, q4q2), (F,~) ~ (E, qs) (Z, 
~) \[- (T, q2) b (F, ~t2) 
(z, ~) > (T, ~) b (F, <s4) 
(z, ~) b (m, ~) b (F, ~) 
\[F --> a.qu, 0, 1\] (E,e) b(T,q,jb(F,q,lq,2), (F,e)~,,(a, qo) 
(m ~) b (T, w) b (r, w) 
(m ~) P (T, ~) b (r, q4) (E, ~) b (T, ~) b (F, ~) 
\[T -+ S~.~s4,O,q (E,~)b(T, q2), (T,~)p(S<,s4), F~*a (E, ~) b (T, 
~) 
\[\]'~ --+ a. q6, 2, 3\] (E,m) b(T, q2) b(17*F, qaq2) b(a*F, q4q2), (/P, ¢) b (a, q6) 
(m ~) b (T, q~) b (~ * s< v',) b (o,. F, w) 
(z, ~) \[~ (T, ~) \[~ (F • F, q4) \[~ (o,. ~, w) 
(m ~) I--. (T, ~) I--' (F • sV, ~) t-' (<, * S< ~) \[E--+T*T.q,2,0,3\] 
(E,g) V*(E,e), (E,g)~,,(T*T, q2), T*T==>*a*a 
~ihble 2: Valid parse items and derbable super configurations for a * a. 
Example 1. Consider the following STG: 
G = ({z, T, r}, to,, +, .}, {m,..., <~6}, 
{q~, q4, q~}, FG, P, E), 
P = {E --> q~, T --> qa, F -+ q,~} 
with the following transitions (for all fi): 
(m, l~) i-(~ (<s~, I~T), (,s~, i~) i-c~ (m, i~+), 
(qa, i3) t-c (q4,/~S~), (<S4, h ~) i-c; (<S:~, iJ*), 
(q,~, f~) i-c (<so,/Ja). 
Table 2 shows soule valid parse items fbr the 
recognition of the string a * a, together with the 
conditions according to Theorem 1. 
4 Bidirectional Parsing 
STG's describe the recognition of admissi- 
ble sequences of subconstituents in unidirec- 
tional parsing algorithms, like Earley's algo- 
rithm. Bidirectional parsing strategies, e.g., 
head-conic< strategies, start the recognition of 
a sequence of subconstituents at sonic position 
in the middle of the sequence and proceed to 
both sides. We can define appropriate STG's 
for 1)idirectional parsing strategies as follows. 
Definition 3. A h, eaded, bidirectional STG G 
is like an STG excq~t that P is a finite set of 
productions of the form A --+ (P,X, A), 'where 
A c N and X E V and F, A c .M. 
The two states in a production accOullt for the 
bidirectional expansion of a constituent. The 
derivation relation for a headed, bidirectional 
STG is defined by 7A6 ~ 7fllXfl"6 if\[ for some 
production A -+ (P, X, A): (fit)-* c L(P) and 
fi' C L(A) ((S) -1 denotes the inversion of fit). 
Note that P defines the left part of an adnfissible 
sequence Doul right to left,. 
A t)ottom-up head-conmr parsing schema 
uses items of the tbrm \[A -+ F. fl. A, i, j\] (Schnei- 
der, 2000). The semantics of these items is given 
by the tbllowing clauses: 
• tbr some production A ~ (P0, X, A0), 
some fll,fl,.: fl = flZXflr and (P0,e) t-G (r, (/~)-~) dud (A0,~)~o (a,/~"). 
,, /3 ~* ai+l.., aj. 
5 Local Tree Constraints 
In this section we discuss the usability of STG's 
for the design of direct parsing algorithms for 
grammars that use a set of well-fonnedness 
conditions, or constraints, expressed in a logi- 
cal , to define the admissible syntac- 
tic structures (i.e., trees), in contrast to gram- 
mars that are based on a derivation mechanism 
737 
(i.e., production rules). Declarative characteri- 
zations of syntactic structures provide a nlealiS 
to tbrmalize grammatical frameworks, and thus 
to compare theories expressed in different for- 
malisms. There are also applications in the- 
oretical explorations of the complexity of lin- 
guistic theories, based on results which relate 
 classes to definability of structures in 
certain logical s (Rogers, 2000). 
From a model-theoretic point of view, such 
a grammar is an axiomatization of a class of 
structures, and a well-formed syntactic struc- 
ture is a model of the grammar (Blackt)urn et 
al., 1993). The connection between models and 
strings is established via a yield function, which 
assigns each syntactic structure a string of ter- 
minal symbols. The parsing problem can then 
be stated as the problem: Given a string w and 
a grammar G, find the models .A4 with A.4 ~ G 
and yieId(./V4) = w. 
In many cases, there are eft~ctive methods to 
translate logical fornmlae into equivalent tree 
automata (Rogers, 2000) or rule-based gram- 
mars (Pahn, 1997). Thus, a possible way to 
approach the parsing problem is to translate a 
set of tree constraints into a grammar and use 
standard parsing methods. However, depending 
on the expressive power of the logical , 
the complexity of the translation often limits 
this approach in practice. 
In this section, we consider the possibility to 
apply tabular parsing methods directly to gram- 
mars that consist of sets of tree constraints. The 
idea is to interleave the translation of tbrmu- 
lae into production rules with the recognition 
of subconstituents. It should be noted that this 
approach suffers from the same complexity lim- 
itations as the pure translation. 
In Schneider (1999), we used a fragment of 
a propositional bimodal  to express lo- 
cal constraints on syntactic structures. The two 
modal operators ($} and (-~) refer to the left- 
most child and the right sibling, respectively, of 
a node in a tree. Furthermore, the nesting of 
($) is limited to depth one. A so-called modal 
grammar consists of a formula that represents 
the conjunction of a set of constraints that must 
be satisfied at every node of a tree. In addition, 
a second formula represents a condition tbr the 
root of a tree. 
In Schneider (1999), we have also shown 
how an extension of a standard nlethod tbr 
automatic proof search in modal logic (so- 
called analytic labelled tableauz) in conjmm- 
tion with dynamic progrmnming techniques can 
be employed to parse input strings according 
to a modal grammar. Basically, a labelled 
tableau procedure is used to construct a la- 
belled tableau, i.e., a tree labelled with tbrmn- 
lae, by breaking tbrmulae up into subtbrmulae; 
this tableau may then be used to construct a 
model tbr the original formula. The extended 
tableau procedure constructs an infinite tableau 
that allows to obtain all admissible trees (i.e., 
models of the grammar). 
The approach can be described as tbllows: An 
STG is defined by using certain formulae that 
appear on the tableau as states, and by defining 
the transition relation in terms of the tableau 
rules (i.e., the operations that are used to con- 
struct a tableau). The states are formulae of 
the form 
x A A<,>o A AI. \]o' A A A\[q ' 
where X is a propositional variable and \[$\], \[-->\] 
are the dnal operators to (.\[), (~). X is used 
as a node \]abe\] in a tree model. The transition 
relation can be regarded as a silnnlation of the 
application of tableau rules to fbrmulae, and a 
tabular parser tbr this STG can be viewed as a 
tabulation of the (infinite) tal)leau construction. 
In particular, it should be noted that this con- 
struction makes no reference to any particular 
parsing strategy. 
6 Conclusion 
We have defined state transition grammars 
(STG) as an intermediate formalism between 
grammars and parsing algorithnls. They com- 
plement the parsing schemata formalism of 
Sikkel (1993). A parsing schema abstracts 
from unimportant algorithmic details and thus, 
like STG's, represents a well-defined level of 
abstraction between grammars and parsers. 
STG's add another abstraction to parsing 
schemata, namely on the grammar side. There- 
fore, we argued, a t)arsing schenla defined over a 
STG represents a very high level description of 
a tabular parsing algorithm that can be applied 
to various gralnlnar tbrmalisms. In this paper 
we concentrated on grammar formalisms with 
a flexible constituent structure, i.e., where the 
738 
mmfl)er and order of subconstituents st)e(:ified 
by a grammar i)roduction may not \[)e fixed. In 
particular, we have discussed extended context- 
free grammars (ECFG), II)/LP grammars, and 
grammars in which admissible trees are delined 
by means of local tree ('onstraints cxI)resscd in 
a simple logical . 

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