A UNIFICATION-BASED PARSER FOR RELATIONAL 
GRAMMAR 
David E. Johnson 
IBM Research Division 
P.O. Box 218 
Yorktown Heights, NY 10598 
dj ohns @ war son. ibm. com 
Adam Meyers 
Linguistics Department 
New York University 
New York, NY 10003 
meyers@acf2.nyu.edu 
Lawrence S. Moss 
Mathematics Department 
Indiana University 
Bloomington, IN 47401 
lmoss@indiana.edu 
Abstract 
We present an implemented unification-based 
parser for relational grammars developed within 
the stratified feature grammar (SFG) frame- 
work, which generalizes Kasper-Rounds logic to 
handle relational grammar analyses. We first in- 
troduce the key aspects of SFG and a lexicalized, 
graph-based variant of the framework suitable for 
implementing relational grammars. We then de- 
scribe a head-driven chart parser for lexicalized 
SFG. The basic parsing operation is essentially 
ordinary feature-structure unification augmented 
with an operation of label unification to build the 
stratified features characteristic of SFG. 
INTRODUCTION 
Although the impact of relational grammar 
(RG) on theoretical linguistics has been substan- 
tial, it has never previously been put in a form 
suitable for computational use. RG's multiple syn- 
tactic strata would seem to preclude its use in the 
kind of monotonic, unification-based parsing sys- 
tem many now consider standard (\[1\], \[11\]). How- 
ever, recent work by Johnson and Moss \[2\] on a 
Kasper-Rounds (KR) style logic-based formalism 
\[5\] for RG, called Stratified Feature Grammar 
(S FG), has demonstrated that even RG's multiple 
strata are amenable to a feature-structure treat- 
ment. 
Based on this work, we have developed a 
unification-based, chart parser for a lexical ver- 
sion of SFG suitable for building computational 
relational grammars. A lexicalized SFG is sim- 
ply a collection of stratified feature graphs (S- 
graphs), each of which is anchored to a lexical 
item, analogous to lexicalized TAGs \[10\]. The ba- 
sic parsing operation of the system is S-graph 
unification (S-unification): This is essentially 
ordinary feature-structure unification augmented 
with an operation of label unification to build the 
stratified features characteristic of SFG. 
RELATED WORK 
Rounds and Manaster-Ramer \[9\] suggested en- 
coding multiple strata in terms of a "level" at- 
tribute, using path equations to state correspon- 
dences across strata. Unfortunately, "unchanged' 
relations in a stratum must be explicitly "car- 
ried over" via path equations to the next stra- 
tum. Even worse, these "carry over" equations 
vary from case to case. SFG avoids this problem. 
STRATIFIED FEATURE GRAM- 
MAR 
SFG's key innovation is the generalization of 
the concept \]eature to a sequence of so-called re- 
lational signs (R-signs). The interpretation of 
a stratified feature is that each R-sign in a se- 
quence denotes a primitive relation in different 
strata. 1 
For instance, in Joe gave Mary tea there are, 
at the clause level, four sister arcs (arcs with the 
same source node), as shown in Figure h one 
arc labeled \[HI with target gave, indicating gave 
is the head of the clause; one with label \[1\] and 
target Joe, indicating Joe is both the predicate- 
argument, and surface subject, of the clause; one 
with label \[3,2\] and target Mary, indicating that 
l We use the following R-signs: 1 (subject), 2 (direct 
object), 3 (indirect object), 8 (chSmeur), Cat (Category), 
C (comp), F (flag), H (head), LOC (locative), M (marked), 
as well as the special Null R-signs 0 and/, explainedbelow. 
97 
\[Ca~\] s 
\[1\] Joe 
\[Hi save 
\[3, 2\] Mary 
\[2, 8\] tea 
Figure 1: S-graph for Joe gave Mary tea. 
Mary is the predicate-argument indirect object, 
but the surface direct object, of the clause; and 
one with label \[2,8\] and target tea, indicating tea 
is the predicate-argument direct object, but sur- 
face ch6meur, of the clause. Such a structure is 
called a stratified feature graph (S-graph). 
This situation could be described in SFG logic 
with the following formula (the significance of the 
different label delimiters (,), \[, \] is explained be- 
low): 
RI:-- \[Hi:gave A \[1):Joe 
A \[3, 2): Mary A \[2, 8): tea . 
In RG, the clause-level syntactic information 
captured in R1 combines two statements: one 
characterizing gave as taking an initial 1, initial 
2 and initial 3 (Ditransltive); and one character- 
izing the concomitant "advancement" of the 3 to 
2 and the "demotion" of the 2 to 8 (Dative). In 
SFG, these two statements would be: 
Ditransitive :: 
\[Hi:gave A \[1):T A .\[2):T A \[3):T ; 
Dative :---- (3, 2): T ~ (2, 8_): T. 
Ditransitive involves standard Boolean con- 
junction (A). Dative, however, involves an opera- 
tor, &, unique to SFG. Formulas involving ~ are 
called e~tension formulas and they have a more 
complicated semantics. For example, Dative has 
the following informal interpretation: Two dis- 
tinct arcs with labels 3 and 2 may be "extended" 
to (3,2) and (2,8) respectively. Extension formulas 
are, in a sense, the heart of the SFG description 
language, for without them RG analyses could not 
be properly represented. 2 
2We gloss over many technicalities, e.g., the SFG notion 
data justification and the formal semantics of stratified fea- 
tures; cf. \[2\]. 
RG-style analyses can be captured in terms of 
rules such as those above. Moreover, since the 
above formulas state positive constraints, they can 
be represented as S-graphs corresponding to the 
minimal satisfying models of the respective formu- 
las. We compile the various rules and their com- 
binations into Rule Graphs and associate sets of 
these with appropriate lexical anchors, resulting 
in a lexicalized grammar, s 
S-graphs are formally feature structures: given 
a collection of sister arcs, the stratified labels are 
required to be functional. However, as shown in 
the example, the individual R-signs are not. More- 
over, the lengths of the labels can vary, and this 
crucial property is how SFG avoids the "carry 
over" problem. S-graphs also include a strict par- 
tial order on arcs to represent linear precedence 
(cf. \[3\], \[9\]). The SFG description language in- 
cludes a class of linear precedence statements, 
e.g., (1\] -4 (Hi means that in a constituent "the 
final subject precedes the head". 
Given a set 7Z,9 of R-signs, a (stratified) fea- 
ture (or label) is a sequence of R-signs which may 
be closed on the left or right or both. Closed sides 
are indicated with square brackets and open sides 
with parentheses. For example, \[2, 1) denotes a la- 
bel that is closed on the left and open on the right, 
and \[3, 2, 1, 0\] denotes a label that is closed on both 
sides. Labels of the form \[-.-\] are called (totally) 
closed; of the form (...) (totally) open; and 
the others partially closed (open) or closed 
(open) on the right (left), as appropriate. 
Let B£ denote the set of features over 7Z *. B£ 
is partially ordered by the smallest relation C_ per- 
mitting eztension along open sides. For example, 
(3) ___ (3,2) U \[3,2,1) C \[3,2, 1,0\]. 
Each feature l subsuming (C) a feature f provides 
a partial description of f. The left-closed bracket \[ 
allows reference to the "deepest" (initia~ R-sign of 
a left-closed feature; the right-closed bracket \] to 
the "most surfacy" (fina~ R-sign of a right-closed 
feature. The totally closed features are maximal 
(completely defined) and with respect to label uni- 
fication, defined below, act like ordinary (atomic) 
features. 
Formal definitions of S-graph and other defini- 
tions implicit in our work are provided in \[2\]. 
s We ignore negative constraints here. 
98 
AN EXAMPLE 
Figure 2 depicts the essential aspects of the S- 
graph for John seemed ill. Focus on the features 
\[0,1\] and \[2,1,0\], both of which have the NP John 
as target (indicated by the ~7's). The R-sign 0 is 
a member of Null, a distinguished set of R-signs, 
members of which can only occur next to brackets 
\[ or \]. The prefix \[2,1) of the label \[2,1,0\] is the 
SFG representation of RG's unaccusative analysis 
of adjectives. The suffix (1,0\] of \[2,1,0\]; the prefix 
\[0,1) of the label \[0,1\] in the matrix clause; and the 
structure-sharing collectively represent the raising 
of the embedded subject (cf. Figure 3). 
Given an S-graph G, Null R-signs permit the 
definitions of the predicate-argument graph, 
and the surface graph, of G. The predicate- 
argument graph corresponds to all arcs whose la- 
bels do not begin with a Null R-sign; the rele- 
vant R-signs are the first ones. The surface graph 
corresponds to all arcs whose labels do not end 
with a Null R-sign; the relevant R-signs are the 
final ones. In the example, the arc labeled \[0,1\] 
is not a predicate-argument arc, indicating that 
John bears no predicate-argument relation to the 
top clause. And the arc labeled \[2,1,0\] is not a 
surface arc, indicating that John bears no surface 
relation to the embedded phrase headed by ill. 
The surface graph is shown in Figure 4 and 
the predicate-argument graph in Figure 5. No- 
tice that the surface graph is a tree. The tree- 
hood of surface graphs is part of the defini- 
tion of S-graph and provides the foundation for 
our parsing algorithm; it is the SFG analog to 
the "context-free backbone" typical of unification- 
based systems \[11\]. 
LEXICALIZED SFG 
Given a finite collection of rule graphs, we could 
construct the finite set of S-graphs reflecting all 
consistent combinations of rule graphs and then 
associate each word with the collection of derived 
graphs it anchors. However, we actually only con- 
struct all the derived graphs not involving extrac- 
tions. Since extractions can affect almost any arc, 
compiling them into lexicalized S-graphs would be 
impractical. Instead, extractions are handled by 
a novel mechanism involving multi-rooted graphs 
(of. Concluding Remarks). 
We assume that all lexically governed rules such 
as Passive, Dative Advancement and Raising are 
compiled into the lexical entries governing them. 
\[Cat\] vP 
\[0,11 
\[HI seemed 
\[Cat\] AP 
\[c\] \[2,1,0\] 
\[n\] in 
Figure 2: S-graph for John seemed ill 
\[o,1) 
(1,o\] m \[el 
Figure 3: Raising Rule Graph 
\[cat\] 
(1\] 
\[H\] 
\[c\] 
VP 
~John 
seemed 
\[Cat\] AP 
\[HI in 
Figure 4: Surface Graph for John seemed ill 
\[Cat\] VP 
\[H\] seemed 
\[c t\] AP 
\[c\] \[2) John 
\[H\] iJ.J. 
Figure 5: Predicate-Argument Graph for John 
seemed ill 
99 
Thus, given has four entries (Ditransitive, Ditran- 
sitive + Dative, Passive, Dative + Passive). This 
aspect of our framework is reminiscent of LFG 
\[4\] and HPSG \[7\], except that in SFG, relational 
structure is transparently recorded in the stratified 
features. Moreover, SFG relies neither on LFG- 
style annotated CFG rules and equation solving 
nor on HPSG-style SUBCAT lists. 
We illustrate below the process of constructing 
a lexical entry for given from rule graphs (ignor- 
ing morphology). The rule graphs used are for 
Ditransitive, Dative and (Agentless) Passive con- 
structions. Combined, they yield a ditransitive- 
dative-passive S-graph for the use of given occur- 
ring in Joe was given ~ea (cf. Figure 6). 
Dltransitive: 
\[H\] given 
\[3) 
\[2) 
\[I) 
DATive: 
(2, 8) 
(3,2) 
DI tl DAT: 
\[H\] given 
\[3, 2) 
\[2, 8) 
\[1) 
PASsive: 
(2,1) 
\[1, 8, 0\] 
\[Cat\] s 
\[0,11 m Joe 
\[H\] was 
\[c\] 
\[Cat\] vP 
\[H\] given 
\[3,2,1,0\] m 
\[2, 8\] tea 
\[1,8,0\] 
Figure 6: S-graph for Joe was given iea. 
D113 DAT) U PAS: 
\[H\] given 
\[3,2, i) 
\[2, 8) 
\[1, s, 0\] 
The idea behind label unification is that 
two compatible labels combine to yield a label 
with maximal nonempty overlap. Left (right) 
closed labels unify with left (right) open labels to 
yield left (right) closed labels. There are ten types 
of label unification, determined by the four types 
of bracket pairs: totally closed (open), closed only 
on the left (right). However, in parsing (as op- 
posed to building a lexicalized grammar), we stip- 
ulate that successful label unification must result 
in a ~o~ally closed label. Additionally, we assume 
that all labels in well-formed lexicalized graphs 
(the input graphs to the parsing algorithm) are at 
least partially closed. This leaves only four cases: 
Case 1. \[or\] Ll \[o~1 = \[Or\] 
Case 2. \[~) u \[~#\] = \[~#1 
Case 3. (o~\] LI \[~\] : \[~c~\] 
Case 4. \[+#) u (#+\] = \[+#+\] 
Note: c~, fl, 7 @ T~S+ and/3 is the longest com- 
mon, nonempty string. 
100 
The following list provides examples of each. 
1. \[1,0\] U \[1,0\] = \[1,0\] 
2. \[1) U \[1,0\] = \[1,0\] 
3. (~,0\] U \[2,1,0\] = \[2,1,0\] 
4. \[2,1) U (1,0\] = \[2,1,0\] 
Case 1 is the same as ordinary label unifica- 
tion under identity. Besides their roles in unifying 
rule-graphs, Cases 2, 3 and 4 are typically used 
in parsing bounded control constructions (e.g., 
"equi" and "raising") and extractions by means 
of "splicing" Null R-signs onto the open ends of 
labels and closing off the labels in the process. We 
note in passing that cases involving totally open 
labels may not result in unique unifications, e.g., 
(1, 2) U (2, 1) can be either (2,1,2) or (1,2,1). In 
practice, such aberrant cases seem not to arise. 
Label unification thus plays a central role in build- 
ing a lexicalized grammar and in parsing. 
THE PARSING ALGORITHM 
S-unification is like normal feature structure 
unification (\[1\], \[11\]), except that in certain cases 
two arcs with distinct labels 1 and l' are replaced 
by a single arc whose label is obtained by unifying 
1 and l'. 
S-unification is implemented via the procedures 
Unify-Nodes, Unify-Arcs, and Unify-Sets-of- 
Arcs: 
1. Unify-Nodes(n,n') consists of the steps: 
a. Unify label(n) and label(n'), where node 
labels unify under identity 
b. Unify-Sets-of-Arcs(Out-Arcs(n), Out- 
Arcs(n')) 
2. Unify-Arcs(A,A') consists of the steps: 
a. Unify label(A) and label(A') 
b. Unify-Nodes(target (A),target (A')) 
3. Unify-Sets-of-Arcs(SeQ, Set2), 
where Sett = {Aj,...,A~} and Set2 = 
{Am,..., An}, returns a set of arcs Set3, de- 
rived as follows: 
a. For each arc Ai • SeQ, attempt to find 
some arc A~ • Set2, such that Step 2a 
of Unify-arcs(Ai,A~) succeeds. If Step 
2a succeeds, proceed to Step 2b and re- 
move A~ from Sets. There are three pos- 
sibilities: 
i. If no A~ can be found, Ai • Set3. 
ii. If Step 2a and 2b both succeed, then 
Unify-arcs(Ai, A~) • Set3. 
iii. If Step 2a succeeds, but Step 2b 
fails, then the procedure fails. 
b. Add each remaining arc in Set2 to Set3. 
We note that the result of S-unification can be a 
set of S-graphs. In our experience, the unification 
of linguistically well-formed lexical S-graphs has 
never returned more than one S-graph. Hence, 
S-unification is stipulated to fail if the result is 
not unique. Also note that due to the nature of 
label unification, the unification procedure does 
not guarantee that the unification of two S-graphs 
will be functional and thus well-formed. To insure 
functionality, we filter the output. 
We distinguish several classes of Arc: (i) Sur- 
face Arc vs. Non-Surface, determined by absence 
or presence of a Null R-sign in a label's last 
position; (ii) Structural Arc vs. Constraint Arc 
(stipulated by the grammar writer); and (iii) Re- 
lational Arc vs. Category Arc, determined by the 
kind of label (category arcs are atomic and have 
R-signs like Case, Number, Gender, etc.). The 
parser looks for arcs to complete that are Sur- 
face, Structural and Relational (SSR). 
A simplified version of the parsing algorithm 
is sketched below. It uses the predicates Left- 
Precedence , Right-Precedence and Com- 
plete: 
. Precedence: Let Q~ = \[n~,Li, R~\], F 
• SSR-Out-Arcs(n~) such that Target(F) 
= Anchor(Graph(n~)), and A • SSR-Out- 
Arcs(ni) be an incomplete terminal arc. 
Then: 
A. Left-Precedence(A, n~) is true iff: 
a. All surface arcs which must follow 
F are incomplete. 
b. A can precede F. 
c. All surface arcs which must both 
precede F and follow A are com- 
plete. 
B. Right-Precedence(A, n~) is true iff: 
a. All surface arcs which must precede 
F are complete. 
b. A can follow F. 
c. All surface arcs which must both 
follow F and precede A are com- 
plete. 
101 
2. Complete : A node is complete if it is either 
a lexical anchor or else has (obligatory) out- 
going SSR arcs, all of which are complete. An 
arc is complete if its target is complete. 
The algorithm is head-driven \[8\] and was in- 
spired by parsing algorithms for lexicalized TAGs 
(\[6\], \[10\]). 
Simplified Parsing Algorithm: 
Input: A string of words Wl,..., w~. 
Output: A chart containing all possible parses. 
Method: 
A. Initialization: 
1. Create a list of k state-sets 
$1,..., Sk, each empty. 
2. For c = 1,...,k, for each 
Graph(hi) of Wc, add \[ni, c - 1, c\] 
to Se. 
B. Completions: 
For c = 1,..., k, do repeatedly until no 
more states can be added to Se: 
1. Leftward Completion: 
For all 
= ¢\] Se, 
Qj = \[nj, Lj, L~\] E SL,, such that 
Complete(nj ) and 
A E SSR-Out-Arcs(ni), such that 
Left-Precedence(A, hi) 
IF Unify-a~-end-of-Path(ni, nj, A ) n~, 
2. 
THEN Add \[n~,Lj,c\] to So. 
Rightward Completion: 
For all 
Qi = \[n/, L~, R~\] E SR,, 
Qj = \[nj,Pq, c\] 6 Sc such that 
Complete(nj ), and 
A E SSR-Out-Arcs(ni), such that 
Right-Precedence(A, hi) 
IF Unify-at-end-of-Path(n~, nj, A) 
THEN Add \[n~, Li, el to So. 
To illustrate, we step through the chart for John 
seemed ill ( cf. Figure 7). In the string 0 John 1 
seemed 2 ill 3, where the integers represent string 
positions, each word w is associated via the lexi- 
calized grammar with a finite set of anchored S- 
graphs. For expository convenience, we will as- 
sume counterfactually that for each w there is only 
one S-graph G~ with root r~ and anchor w. Also 
in the simplified case, we assume that the anchor 
is always the target of an arc whose source is the 
root. This is true in our example, but false in 
general. 
For each G~, r~ has one or more outgoing 
SSR arcs, the set of which we denote SSR-Out- 
Arcs(r~). For each w between integers x and y 
in the string, the Initialization step (step A of the 
algorithm) adds \[n~, x, y\] to state set y. We de- 
note state Q in state-set Si as state i:Q. For an 
input string w = Wl,...,w,~, initialization cre- 
ates n state-sets and for 1 < i < n, adds states 
i : Qj,1 _< j < k, to Si , one for each of the k 
S-graphs G~. associated with wi. After initializa- 
tion, the example chart consists of states 1:1, 2:1, 
3:1. 
Then the parser traverses the chart from left to 
right starting with state-set 1 (step B of the algo- 
rithm), using left and right completions, according 
to whether left or right precedence conditions are 
used. Each completion looks in a state-set to the 
left of Sc for a state meeting a set of conditions. 
In the example, for c = 1, step B of the algorithm 
does not find any states in any state-set preced- 
ing S1 to test, so the parser advances c to 2. A 
left completion succeeds with Qi = state 2:1 = 
\[hi, 1, 2\] and Qj = state 1:1 = \[nj, 0, 1\]. State 2:2 
= \[n~, 0, 2\] is added to state-set $2, where n~ = 
Unify-at-end-of-Path(n,, nj, \[0, 1)). Label \[0, 1) is 
closed off to yield \[0, 1\] in the output graph, since 
no further R-signs may be added to the label once 
the arc bearing the label is complete. 
The precedence constraints are interpreted as 
strict partial orders on the sets of outgoing SSR 
arcs of each node (in contrast to the totally or- 
dered lexicalized TAGs). Arc \[0, 1) satisfies left- 
precedence because: (i) \[0, 1) is an incomplete ter- 
minal arc, where a terminal arc is an SSR arc, 
the target of which has no incomplete outgoing 
surface arcs; (ii) all surface arcs (here, only \[C\]) 
which must follow the \[H\] arc are incomplete; (iii) 
\[0 1) can precede \[H\]; and (iv) there are no (incom- 
plete) surface arcs which must occur between \[0 1) 
and \[H\]. (We say can in (iii) because the parser 
accomodates variable word order.) 
The parser precedes to state-set $3. A right 
completion succeeds with Q~ = state 2:2 = \[n~, 0, 2\] 
and Q~ = state 3:1 = \[n~,2,3\]. State 3:2 - 
\[n~', 0, 3\] is added to state set $3, n~' = Unify-at- 
102 
1..11 
LP=0 RP=I L.P=I RP=2 VP 
\[H\] • \[o, 
/ seemed AP 
John seemed 
=:=J 
LP:0 RP:2 
VP" OlJl , 
\[H\] ~. 
John 
John seemed 
3:1J 
LP:2 RP=3 
AP" '~ \[."\] 
NP ill 
ill 
3:2\] 
LP=0 RP=3 
VP" 
John 
John seemed ill 
Figure 7: Chart for John seemed ill. 
end-of-Path(n~, n~, \[C\]). State 3:2 is a successful 
parse because n~' is complete and spans the entire 
input string. 
To sum up: a completion finds a state Qi = 
\[hi, L,, R~\] and a state Qj = \[nj, Lj, Rj\] in adja- 
cent state-sets (Li = Rj or P~/ = Lj) such that 
ni is incomplete and nj is complete. Each success- 
ful completion completes an arc A E SSR-Out- 
Arcs(n~) by unifying nj with the target of A. Left 
completion operates on a state Qi = \[ni,Li, c\] 
in the current state-set Sc looking for a state 
Qj = \[nj, Lj, L~\] in state-set SL, to complete some 
arc A E SSR-Out-Arcs(ni). Right completion is 
the same as left completion except that the roles 
of the two states are reversed: in both cases, suc- 
cess adds a new state to the current state-set So. 
The parser completes arcs first leftward from the 
anchor and then rightward from the anchor. 
CONCLUDING REMARKS 
The algorithm described above is simpler than 
the one we have implemented in a number of ways. 
We end by briefly mentioning some aspects of the 
VP 
//~\[LOC\] 
V pp 
\[FJJ~\[MI / \ 
in NP 
\[/,Q\] 
Figure 8: Example: in 
P ~ 
\[c\] 
d 
what 
Figure 9: Example: What 
general algorithm. 
Optional Arcs: On encountering an optional 
arc, the parser considers two paths, skipping the 
optional arc on one and attempting to complete it 
on the other. 
Constraint Arcs These are reminiscent of 
LFG constraint equations. For a parse to be good, 
each constraint arc must unify with a structural 
arc. 
Multi-tiered S-graphs: These are S-graphs 
having a non-terminal incomplete arc I (e.g., the 
\[LOC\] arc in Figure 8. Essentially, the parser 
searches I depth-first for incomplete terminal arcs 
to complete. 
Pseudo-R-signs: These are names of sets of 
R-signs. For a parse to be good, each pseudo-R- 
sign must unify with a member of the set it names. 
Extractions: Our approach is novel: it uses 
pseudo-R-signs and multirooted S-graphs, illus- 
trated in Figure 9, where p is the primary root and 
d, the dangling root, is the source of a "slashed 
arc" with label of the form (b,/\] (b a pseudo- 
R-sign). Since well-formed final parses must be 
103 
single-rooted, slashed arcs must eventually unify 
with another arc. 
To sum up: We have developed a unification- 
based, chart parser for relational grammars based 
on the SFG formalism presented by Johnson and 
Moss \[2\]. The system involves compiling (combi- 
nations) of rules graphs and their associated lexi- 
cal anchors into a lexicalized grammar, which can 
then be parsed in the same spirit as lexicalized 
TAGs. Note, though, that SFG does not use an 
adjunction (or substitution) operation. 

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