Proceedings of the 21st International Conference on Computational Linguistics and 44th Annual Meeting of the ACL, pages 137–144,
Sydney, July 2006. c©2006 Association for Computational Linguistics
 
Polarized Unification Grammars 
Sylvain Kahane 
Modyco, Université Paris 10 
sk@ccr.jussieu.fr
 
 
Abstract 
This paper proposes a generic mathemati-
cal formalism for the combination of 
various structures: strings, trees, dags, 
graphs and products of them. The polari-
zation of the objects of the elementary 
structures controls the saturation of the 
final structure. This formalism is both 
elementary and powerful enough to 
strongly simulate many grammar formal-
isms, such as rewriting systems, depend-
ency grammars, TAG, HPSG and LFG. 
1 Introduction 
Our aim is to propose a generic formalism as 
simple as posible but powerful enough to write 
real grammars for natural language and to handle 
complex linguistic structures. The formalism we 
propose can strongly simulate most rule-based 
formalisms used in linguistics.
1
 
Language uterances are both strongly struc-
tured and compositional and the structure of a 
complex uterance can be obtained by combining 
elementary structures associated to the elemen-
tary units of language.
2
 The most simple way to 
                                                        
1
 A formalism A strongly simulates a formalism B if A has a 
beter strong generative capacity than B, that is, if A can 
generate the languages generated by B with the same struc-
tures asociated to the uterances of these languages. 
2
 Whether a natural language uterance contains one or 
several structures depends on our point of view. On the one 
hand it is clear that a sentence can receive various structures 
acording to the semantic, syntactic, morphological or 
phonological point of view. On the other hand these difer-
ent structures are not independent from each other and even 
if they are not structures on the same objects (for instance 
the semantic units do not corespond one to one to the syn-
tactic units, that is the words) there are links betwen the 
diferent objects of these structures. In other words, consid-
ering separately the diferent simple structures of the sen-
tence does not take into acount the whole structure of the 
sentence, because we lost the interelation betwen struc-
tures of diferent levels. 
combine two structures A and B is unification, 
that is, to build a new structure C by partially 
superimposing A and B and identifying a part of 
the objects of A with those of B. This idea recalls 
an old idea, used by Jespersen (1924), Tesnière 
(1934) or Ajduckiewicz (1935): the sentence is 
like a molecule whose words are atoms, each 
word bearing a valence (a linguistic term explic-
itly borrowed from chemistry) that forces or al-
lows it to meet some other words. Nevertheless, 
unification grammars cannot directly take into 
account the fact that some linguistic units are 
unsaturated in a sense that they must absolutely 
combine with other structures to form a stable 
unit. Saturation is ensured by additional mecha-
nisms, such as the distinction of terminal and 
non-terminal symbols in rewriting systems or by 
requiring some features to have an empty list as a 
value in HPSG. 
This paper presents a new family of formal-
isms, Polarized Unification Gramars (PUGs). 
PUGs extend Unification Grammars with an 
explicit control of the saturation of structures by 
attributing a polarity to each object. Using polari-
ties allows integrating the treatment of saturation 
in the formalism of the rules. Thus the processing 
of saturation wil pilot the combination of struc-
tures during the generation processing. Some 
polarities are neutral, others are not, but a final 
structure must be completely neutral. Two non-
neutral objects can unify (that is, identify) and 
form a neutral object (that is, neutralizing each 
other). Proper unification holds no equivalent. 
Polarization takes its source in categorial 
grammar and subsequent works on resource-
sensitive logic (see Lambek’s, Girard’s or van 
Benthem’s works). Nasr (195) is among the first 
to introduce a rule-based formalism using an 
explicit polarization of structures. Duchier & 
Thater (199) propose a formalism for tree de-
scription where they put forward the notion of 
polarity (and they uses the terms of polarity and 
neutralization). Perrier (200) is probably the 
137
 
first to develop a linguistic formalism entirely 
based on these ideas, the Interaction Gramar. 
PUG is both an elementary formalism (struc-
tures simply combine by identifying objects) and 
a powerful formalism, equivalent to Turing ma-
chines and capable of handling strings, trees, 
dags, n-graphs and products of such structures 
(such as ordered trees).
3
 But, above all, PUG is a 
well-adapted formalism for writing grammars 
and it is capable of strongly simulating many 
classic formalisms. 
Part 2 presents the general framework of PUG 
and its system of polarities. Part 3 proposes sev-
eral examples of PUG and the translation in PUG 
of rewriting grammars, TAG, HPSG and LFG. 
We hope that these translations shed light on 
some common features of these formalisms. 
2 Polarities and unification 
2.1 Polarized Unification Gramars 
Polarized Unification Grammars generate sets of 
finite structures. A structure is based on objects. 
For instance, for a (directed) graph, objects are 
nodes and edges. These two types of objects are 
linked, giving us the proper structure: if X is the 
set of nodes and U, the set of edges, the graph is 
defined by two maps π
1
 and π
2
 from U into X 
which associate an edge with its source and its 
target. 
Our structures are polarized, that is, objects 
are associated to polarities. The set P of polarities 
is provided with an operation noted “.” and called 
product. The product is commutative and gener-
ally associative; (P, . ) is called the system of 
polarities. A non-empty strict subset N of P con-
tains the neutral polarities. A polarized structure 
is neutral if all its polarities are neutral. 
Structures are defined on a colection of ob-
jects of various types (syntactic nodes, semantic 
nodes, syntactic edges …) and a colection of 
maps: structural maps linking objects to objects 
(such as source and target for edges), label maps 
linking objects to labels and polarity maps link-
ing objects to polarities. 
Structures combine by unification. The unifi-
cation of two structures A and B gives a new 
structure A⊕B obtained by “pasting” together 
these structures and identifying a part of the ob-
jects of the first structure with objects of the sec-
ond structure. When two polarized structures A 
                                                        
3
 A dag is a directed acyclic graph. An n-graph is a graph 
whose nodes are edges of a (n-1)-graph and a 1-graph is a 
standard graph. 
and B are unified, the polarity of an object of 
A⊕B obtained by identifying two objects of A 
and B is the product of their polarities; if the two 
objects bear the same map, these maps must be 
identified and their values, unified. (For instance 
identifying two edges forces us to identify their 
sources and targets.) 
A Polarized Unification Gramar (PUG) is 
defined by a finite family T of types of objects, a 
set of maps attached to the objects of each type, a 
system (P,.) of polarities, a subset N of P of neu-
tral polarities, and a finite subset of elementary 
polarized structures, whose objects are described 
by T; one elementary structure is marked as the 
initial one and must be used exactly once. The 
structures generated by the grammar are the neu-
tral structures obtained by combining the initial 
structure and a finite number of elementary struc-
tures. In the derivation process, elementary struc-
tures combine successively, each new elementary 
structure combining with at least one object of 
the previous result; this ensures that the derived 
structure is continuous. Polarities are only neces-
sary to control the saturation and are not consid-
ered when the strong generative capacity of the 
grammar is estimated. Polarities belong to the 
declarative part of the grammar, but they rather 
play a role in the processing of the grammar. 
2.2 The system of polarities 
In this paper we wil use the system of polarities 
P = {■,□,–,+,■} (which are called like this: ■ = 
black = saturated, + = positive, – = negative, □ = 
white = obligatory context and ■ = grey 
= absolutely neutral), with N = {■,■}, and a 
product defined by the folowing array (where ⊥ 
represents the imposibility to combine). Note 
that ■ is the neutral element of the product. The 
symbol – can be interpreted as a need and + as 
the corresponding resource. 
 . ■ □ – + ■ 
■ ■ □ – + ■ 
□ □ □ – + ■ 
– – – 
⊥ 
■ 
⊥ 
+ + + ■ 
⊥ ⊥ 
■ ■ ■ 
⊥ ⊥ ⊥ 
The system {□,■} is used by Nasr (195), 
while the system {■,■,–,+}, noted {=,↔,←,→}, 
is considered by Bonfante et al. (204), who 
show advantages of negative and positive polari-
ties for prefiltration in parsing (a set of structures 
bearing negative and positive polarities can only 
138
 
be reduced into a neutral structure if the sum of 
negative polarities of each object type is equal 
the sum of positive polarities). 
The system (P, . ) we have presented is 
comutative and associative. Commutativity 
implies that the combination of two structures is 
not procedurally oriented (and we can begin a 
derivation by any elementary structure, provided 
we use only once the initial structure). 
Asociativity implies that the combination of 
structures is unordered: if an object B must 
combine with A and C, there is no precedence 
order between the combination of A and B and 
the one of B and C, that is, A⊕(B⊕C) = 
(A⊕B)⊕C. If we leave polarities aside, our formalism is 
trivially monotonic: the combination of two 
structures A and B by a PUG gives us a structure 
A⊕B that contains A and B as substructures. We 
can add a (partial) order on P in order to make 
the formalism monotonic.
4
 Let ≤ be this order. In 
order to give us a monotonic formalism, ≤ must 
verify the folowing monotonicity property: 
∀x,y∈P x.y ≥ x. This provides us with the folow-
ing order: ■ < □ < +/– < ■. A PUG built with an 
ordered system of polarities (P, . ,≤) verifying the 
monotonicity property is monotonic. Monotonic-
ity implies god computational properties; for 
instance it allows translating the parsing with 
PUG into a problem of constraint resolution 
(Duchier & Thater, 199). 
3 Examples of PUGs 
3.1 Tree gramars 
The first tree grammars belonging to the para-
digm of PUGs was proposed by Nasr 195. The 
folowing grammar G
1
 allows generating all fi-
nite trees (a tree is a conected directed graph 
such that every node except one is the target of at 
most one edge); objects are nodes and edges; the 
initial structure (the box on the left) is reduced to 
a black node; the grammar has only one other 
elementary structure, which is composed of a 
black edge linking a white node to a black node. 
Each white node must unify with a black node in 
order to be neutralized and each black node can 
unify with whatever number of white nodes. It 
can easily be verified that the structures gener-
ated by the grammar are trees, because every 
node has one and only one governor, except the 
node introduced by the initial structure, which is 
the root of the tree. 
 
                                                        
4
 I was sugested this idea by Guy Perier. 
 
 
 
 
 
G
1
       G
2
     
The grammar G
1
 does not control the number 
of dependents of nodes. A grammar like G
2
 al-
lows controling the valence of each node, but it 
does not ensure that generated structures are 
trees, because two white nodes can unify and a 
node can have more than one governor.
5
 The 
grammar G
3
 solves the problem. In fact, G
3
 can 
be viewed as the superimposition of G
1
 and G
2
. 
Indeed, if P
0 
= {□,■}, P
1 
= P
0
×P
0
 = 
{(□,□),(□,■),(■,□),(■,■)} is equivalent to {□,+,–
,■}. The first polarity controls the tree structure 
as G
1
 does, while the second polarity controls the 
valence as G
2
 does. 
 
 
 
 
   G
3
 
With the same principles, one can build a de-
pendency gramar generating the syntactic de-
pendency trees of a fragment of natural language. 
Grammar G
4
, directly inspired from Nasr 195, 
proposes a fragment of grammar for English 
generating the syntactic tree of Peter eats red 
beans. Nodes of this grammar are labeled by two 
label maps, /cat/ and /lex/. Note  that the  root of 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
G
4
 (Dependency grammar for English) 
                                                        
5
 Nasr 195 proposes such a gramar in order to generate 
tres. He uses an external requirement, which forces, when 
two structures are combined, the rot of one to combine 
with a node of the other one. 
subj dobj 
cat: V 
lex: eat 
cat: V 
cat: N 
cat: Adj 
lex: red 
cat: N cat: N 
cat: N 
lex: Peter 
mod 
cat: N 
lex: beans 
139
 
a b c 
the elementary structure of an adjective is a white 
node, allowing an unlimited number of such 
structures to adjoin to a noun. 
3.2 Rewriting systems and ordered trees 
PUG can simulate any rewriting system and have 
the weak generative capacity of Turing machines. 
We folow ideas developed by Burroni 193 or 
Dymetman 199, themselves folowing van 
Kampen 193’s ideas. 
A sequence abc is represented by a string of 
labeled edges a, b and c: 
 
 
Intuitively, edges are intervals and nodes model 
their extremities. This is the simplest way to 
model linear order and precedence rules: X pre-
cedes Y iff the end of X is the begining of Y. 
The initial category S of the grammar gives us 
the initial structure: 
 
A terminal symbol a corresponds to a positive 
edge: 
 
A rewriting rule ABC → DE gives us the ele-
mentary structure: 
 
 
 
 
 
This elementary structure is a “cell” whose 
uper frontier is a string of positive edges corre-
sponding to the left part of the rule, while the 
lower frontier is a string of negative edges corre-
sponding to the right part of the rule. Each posi-
tive edge must unify with a negative edge and 
vice versa, in order to give a black edge. Nodes 
are grey (= absolutely neutral) and their unifica-
tion is entirely driven by the unification of edges. 
Cells wil unify with each other to give a final 
structure representing the derivation structure of 
a sequence, which is the lower edge of this struc-
ture. The next figure shows the derivation struc-
ture of the sequence Peter eats red beans with a 
standard phrase structure grammar, which can be 
reconstructed by the reader. In such a representa-
tion, edges represent phrases and correspond to 
intervals in the cuting of the sequence, while 
nodes are bounds of these intervals. 
 
 
 
 
 
 
 
 
 
 
 
For a context-free rewriting system, the gram-
mar generates the derivation tree, which can be 
represented in a more traditional way by adding 
the branches of the tree (giving us a 2-graph). 
 
 
 
 
 
Let us recall that a derivation tree for a context-
free grammar is an ordered tree. An ordered tree 
combines two structures on the same set of 
nodes: a structure of tree and a precedence rela-
tion on the node of the tree. Here the precedence 
relation is explicitly represented (a “node” of the 
tree precedes another “node” if the target of the 
first one is the source of the second one). It is not 
posible, with a PUG, to generate the derivation 
tree, including the precedence relation, in a sim-
pler way.
6
 Note that the usual representation of 
ordered trees (where the precedence relation is 
not explicit, but only deductible from the planar-
ity of the representation) is very misleading from 
the computational viewpoint. When they calcu-
late the precedence relation, parsers (of the CKY 
type for instance) in fact calculate a data structure 
like the one we present here, where beginings 
and ends of phrase are explicitly considered as 
objects. 
3.3 TAG (Tree Adjoining Gramar) 
PUG has a clear kinship with TAG, which is the 
first formalism based on combination of struc-
tures to be studied at length. TAGs are generally 
presented as grammars combining (ordered) 
trees. In fact, as a tree grammar, TAG is not 
                                                        
6
 The most natural idea would be to encode a rewriting rule 
with a tre of depth 1 and the precedence relation with edges 
from a node to its sucesor. The dificulty is then to propa-
gate the order relation to the descendants of two sister nodes 
when we aply a rewriting rule by substituting a tre of 
depth 1. The simplest solution is undeniably the one pre-
sented here, consisting to introduce objects representing the 
begining and the end of phrases (our grey nodes) and to 
indicate the relation betwen a phrase, its begining and its 
end by representing the phrase with an edge from the begin-
ning to the end. 
S 
A 
B 
C 
D 
E 
beans 
Peter 
S 
NP 
VP 
N 
eats 
red 
Adj 
NP 
V 
S 
NP VP 
 
a 
140
 
monotonic and cannot be simulated with PUG. 
As shown by Vijay-Shanker 192, to obtain a 
monotonic formalism, TAG must be viewed as a 
grammar combining quasi-trees. Intuitively, a 
quasi-tree is a tree whose nodes has been split in 
two parts and have each one an uper part and a 
lower part, between which another quasi-tree can 
be inserted (this is the famous adjoining opera-
tion of TAG). Formally, a quasi-tree is a tree 
whose branches have two types: dependency 
relations and dominance relations (respectively 
noted by plain lines and doted lines). Two nodes 
linked by a negative dominance relation are po-
tentially the two parts of a same node; only the 
lower part can have dependents. 
The next figures give an α-tree (= to be sub-
stituted) and a β-tree (= to be adjoined) with the 
corresponding PUG structures.
7
 A substitution 
node (like D↓) gives a negative node, which wil 
unify with the root of an α tree. A β-tree gives a 
white root node and a black foot node, which wil 
unify with the uper and the lower part of a split 
node. This is why the root and the foot node are 
linked by a positive dominance link, which wil 
unify with a negative dominance link conecting 
the two parts of a split node. 
 
 
 
 
 
 
 
 
 
 
 
An α tre and its PUG translation 
 
 
 
 
 
                                                        
7
 For sake of simplicity, we leave aside the precedence 
relation on sister nodes. It might be encoded in the same 
way as context-fre rewriting systems, by modeling semi-
nodes of TAG tres by edges. It does not pose any problem 
but would make the figures dificult to read. 
 
 
 
 
 
 
 
 
A β tre and its PUG translation 
 
At the end of the derivation, the structure 
must be a tree and all nodes must be recon-
structed: this is why we introduce the next rule, 
which presents a positive dominance link linking 
a node to itself and which wil force two semi-
nodes to unify by neutralizing the dominance link 
between them. 
 
 
 
This last rule again shows the advantage of 
PUG: the reunification of nodes, which is proce-
durally ensured in Vijay-Shanker 192 is given 
here as a declarative rule. 
3.4 HPSG (Head-driven Phrase Structure 
Gramar) 
There are two ways to translate feature structures 
(FSs) into PUG. Clearly atomic values must be 
labels and (embedded) feature structures must be 
nodes, but features can be translated by maps or 
by edges, that is, objects. Encoding features by 
maps ensures to identify them in PUG. Encoding 
them by edges allows us to polarize them and 
control the number of identifications.
8
 
For the sake of clarification of HSPG struc-
tures, we chose to translate structural features 
such as HDTR and NHDTR, which give the 
phrase structure and which never unify with other 
“features”, by edges and other features by maps 
(which wil be represented by hashed arrows). In 
any case, the result loks like a dag whose 
“edges” (true edges and maps) represent features 
and whose nodes represent values (e.g. Kesper & 
Mönich 203). We exemplify the translation of 
HPSG in PUG with the schema of combination 
                                                        
8
 Perier 200 uses a feature-structure based formalism 
where only features are polarized. Although more or les 
equivalent we prefer to polarize the FS themselves, i.e. the 
nodes. 
A 
A 
A*
 
C
 
D↓
 
A 
C
 
C
 A
 
D
 
A 
eat 
V 
Q 
H 
cat 
Q 
elist 
SC 
HD 
cat 
N 
H 
cat 
N 
A 
B
 C
 
A 
B
 
A 
B
 
C
 
D↓
 
C 
D 
 
141
 
SC 
HD 
H 
Q 
HD 
SC 
HD 
elist 
NHDTR 
HDTR 
SC 
of head phrase with a subcategorized sister 
phrase, namely the head-daughter-phrase:
9
 
HEAD: 1  
SUBCAT: 3 
HDTR: HEAD: 1 
  SUBCAT: 〈 2 〉 ⊕  3 
NHDTR: HEAD: 2 
  SUBCAT : elist 
This FS gives the folowing structure, where a 
list is represented recursively in two pieces: its 
head (value of H) and its queue (value of Q). 
 
 
 
 
 
 
 
 
 
 
A negative node of this FS can be neutralized 
by the combination with a similar FS represent-
ing a phrase or with a lexical entry. The next 
figure proposes a lexical entry for eat, indicating 
that eat is a V whose SUBCAT list contains two 
phrases headed by an N (for sake of simplicity 
we deal with the subject as a subcategorized 
phrase). 
 
 
 
 
 
 
 
The combination of two head-daughter-
phrases with the lexical entry of eat gives us the 
previous lexicalized rule, equivalent to the rule 
for eat of the dependency grammar G
4
 (/subj/ is 
the NHDTR of the maximal projection and /obj/ 
                                                        
9
 Numbers in boxes are values shared by several features. 
The value of SUBCAT (= SC) is a list (the list of subcatego-
rized phrases). The non-head daughter phrase (NHDTR) has 
a saturated valence and so neds an empty SUBCAT list 
(elist). The subcat list of the head daughter phrase (HDTR) 
is the concatenation, noted ⊕, of two lists: a list with one 
element that is the description of the non-head daughter 
phrase and the SUBCAT list of the whole phrase. The rest 
of the description of this phrase (value of HEAD) is equal to 
the one of the head daughter phrase. 
the NHDTR of the intermediate projection of 
eat). 
 
 
 
 
 
 
 
 
 
 
 
 
Polarization of objects shows exactly what is 
constructed by each rule and what are the re-
quests filed by other rules. Moreover it allows us 
to force SUBCAT lists to be instantiated (and 
therefore allows us to control the saturation of the 
valence), which is ensured in the usual formalism 
of HPSG by a botom-up procedural presentation. 
3.5 LFG (Lexical Functional Gramar) 
and synchronous gramars 
We propose a translation of LFG into PUG that 
makes LFG appear as a synchronous grammar 
approach (see Shieber & Schabes 190). LFG 
synchronizes two structures (a phrase structure or 
c-structure and a dependency/functional structure 
or f-structure) and it can be viewed as the syn-
chronization of a phrase structure grammar and a 
dependency grammar. 
Let us consider a first LFG rule and its trans-
lation in PUG: 
[1]  S  → NP  VP 
  ↓ = ↑ SUBJ ↓ = ↑ 
 
 
 
 
 
 
Equations under phrases (in the right side of [1]) 
ensure the synchronization between the objects of 
the c-structure and the f-structure: each phrase is 
synchronized with a “functional” node. Symbols 
↓ and ↑ respectively designate the functional 
node synchronized with the current phrase and 
the one synchronized with the mother phrase 
(here S). Thus the equation ↓=↑ means that the 
current phrase (VP) and its mother (S) are syn-
chronized with the same functional node. The 
eat 
V 
Q 
H 
cat 
lex 
HD 
HD 
SC 
HD 
SC 
SC 
HDTR 
HD 
HD 
NHDT
R 
HDTR 
NHDTR 
Q 
elist 
cat 
N 
H 
cat 
N 
SC elist 
SC 
elist 
eat 
V 
 
Q 
 
H 
 
cat 
 
Q 
 
elist 
 
SC 
 
HD 
 
cat 
 
N 
 
H 
 
cat 
 
N 
 
SUBJ 
S 
S 
NP 
VP 
142
 
expression ↑ SUBJ designates the functional node 
depending on ↑ by the relation SUBJ. 
In PUG we model the synchronization of the 
phrases and the functional nodes by synchroniza-
tion links (represented by doted lines with dia-
mond-shaped polarities) (see Bresnan 200 for 
non-formalized similar representations). The two 
synchronizations ensured by the two constraints 
↓=↑ SUBJ and ↓=↑ of [1], and therefore built by 
this rule, are polarized in black. 
A phrasal rule such as [1] introduces an f-
structure with a totally white polarization. It wil 
be neutralized by lexical rules such as [2]: 
[2]  V  → wants 
  ↑ PRED = ‘want 〈SUBJ,VCOMP〉’ 
  ↑ SUBJ = ↑ VCOMP SUBJ 
 
 
 
 
 
 
The feature Pred is interpreted as the labeling of 
the functional node, while the valence 
〈SUBJ,VCOMP〉 gives us two black edges and two 
white nodes. The functional equation ↑SUBJ = 
↑ VCOMP SUBJ introduces a white edge SUBJ 
between the nodes ↑ SUBJ and ↑VCOMP (and is 
therefore to be interpreted very differently from 
the constraints of [1], which introduce black syn-
chronization links.) 
PUG allows to easily split up a rule into more 
elementary rules. For instance, the rule [1] can be 
split up into three rules: a phrase structure rules 
linearizing the daughter phrases and two rules of 
synchronization indicating the functional link 
between a phrase and one of its daughter phrases. 
 
 
 
 
 
 
 
 
 
 
Our decomposition shows that LFG articulated 
two different grammars: a classical phrase struc-
ture generating the c-structure and an interface 
grammar between c- and f-structures (and even a 
third grammar because the f-structure is really 
generated only by the lexical rules). With PUG it 
is easy to join two (or more) grammars: it suf-
fices to ad on the objects by both grammars a 
white polarity that wil be saturated in the other 
grammar (and vice versa) (Kahane & Lareau 
205). 
Let us consider another problem, ilustrated 
here by the rule for the topicalization of an ob-
ject. The unbounded dependency of the object 
with its functional governor is an undetermined 
path expressed by a regular expression (here 
VCOMP* OBJ; functional uncertainty, Kaplan & 
Zaenen 1989). 
[3]  S'  → NP   S 
  ↓ = ↑ VCOMP* OBJ ↓ = ↑ 
  ↓ = ↑ TOP 
 
 
 
 
 
 
 
The path VCOMP* (represented by a dashed ar-
row) is expanded by the folowing regular gram-
mar, with two rules, one for the propagation and 
one for the ending. 
 
 
 
 
 
Again the translation into PUG brings to the 
fore some fundamental components of the for-
malism (like synchronization links) and some 
non-explicit mechanisms such as the fact that the 
lexical equation ↑ PRED = ‘want 〈SUBJ,VCOMP〉’ 
introduces both resources (a node ‘want’) and 
needs (its valence). 
4 Conclusion 
The PUG formalism is extremely simple: it only 
imposes that combining two structures involves 
at least the unification of two objects. Forcing or 
forbiding more objects to combine is then en-
tirely controled by polarization of objects. Po-
larization wil thus guide the process of 
combination of elementary structures. In spite of 
its simplicity, the PUG formalism is powerful 
enough to elegantly simulate most of the rule-
based formalisms used in formal linguistics and 
NLP. This sheds new light on these formalisms 
and allows us to bring to the fore the exact nature 
SUBJ 
S 
NP VP 
S 
NP 
⊕ 
S 
VP 
⊕ 
VCOMP* 
VCOMP VCOMP* 
V 
wants 
SUBJ 
  VCOMP 
SUBJ 
 
‘want’ 
S' 
S 
VCOMP* 
TOP 
 
NP 
OBJ 
VCOMP* 
143
 
of the structures they handle and to extract some 
procedural mechanisms hiden by the formalism. 
But above all, the PUG formalism allows us to 
write separately several modules of the grammar 
handling various structures and to put them to-
gether in a same formalism by synchronization of 
the grammars, as we show with our translation of 
LFG. Thus PUGs extend unification grammars 
based on feature structures by allowing a greatest 
diversity of geometric structures and a best con-
trol of resources. Further investigations must 
concern the computational properties of PUGs, 
notably restrictions allowing polynomial time 
parsing. 
Acknowledgements 
I thank Benoît Crabé, Denys Duchier, Kim Gerdes, 
François Lareau, François Métayer, Piet Mertens, Guy 
Perier, Alain Polguère and Benoît Sagot for their 
numerous remarks and enlightening comentaries. 
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