PROOF-NETS AND DEPENDENCIES 
Alain LECOMTE 
GRIL 
Universit6 Blaise Pascal 
34 Avenue Carnot 
63037- CLERMONT-FERRAND codex (France) 
Abstract 
Proof-Nets (Roorda 1990) are a good device for processing 
with eategorial grammars, mainly because they avoid 
spurious ambiguities. Nevertheless, they do not provide 
easily readable structures and they hide the true proximity 
between Categorial Grammars and Dependency Grammars. 
We give here an other kind of Proof-Nets which is much 
related to Dependency Structures similar to those we meet 
in, for instance (Hudson 1984). These new Proof-Nets are 
called Connection Nets. We show that Connection Nets 
provide not only easily interpretable structures, but also 
that processing with them is more efficient. 1 
1,Introduction 
Nowadays, two formalisms are very attractive in Natural 
Language Processing: 
- Categorial Grammars, and 
- Dependency Grammars. 
Numerous studies try to shed light on their similarities 
and differences. We may quote for instance works by 
Hudson (1984, 1990), Barry & Picketing(1990), Hausser 
(1990), Hepple (1991). One interesting particularity 
common to these two formalisms seems to be the 
capacity of leading to an incremental processing, which, 
in turn, leads to an on-line processing. 
Moreover, these formalisms are now very well known. 
Categorial Grammars have been much studied recently, 
particularly since the article of Ades and Steedman 
(1982) and the re-discovering of previous works done by 
Lambek (1958, 1961). The most comprehensive form 
taken by Categorial Grammars is the Lambek Calculus, 
studied by many authors like Moortgat (1988, 1990), 
Buszkowski (1986, 1988), Descl6s (1990)etc. Since the 
recent work by J-Y Girard (see for instance Girard 1987), 
which led to the framework of Linear Logic, it has 
become apparent that the Lambek Calculus amounts to a 
non-commutative version of a sub-system of Linear 
Logic, where a structural rule forbids seqnents with an 
empty antecedent. 
Semantic properties of this system have been studied by 
Buszkowski (1986, 1988) and Wansing (1990). Two 
models are often given: one consists of residuation 
semigroups spread over free semigroups, and another one 
is given by the directional typed lambda-calculus. 
Dependency Grammars are originating from earlier works 
by the French linguist Tesnitre (1965). They were 
theoretically studied by Gaifman, who demonstrated 
theorems on the Generative Capacity of Dependency 
Grammars. We will consider here that the formalism of 
"Word Grammar" (Hudson 1984, 1990) is representative 
of this trend. 
Our purpose in this communication is to show that 
building dependency structures gives an other kind of 
semantics for the Lambek Calculus and various 
subsystems. This semantics is useful in that it will 
allow us to conceive extensions of the Lambek 
Calculus. Moreover, the correspondance proposed 
between these two aspects provides us with a method of 
parsing related to the conception of "parsing as 
deduction", together with a method for avoiding spurious 
ambiguities. We will show that it is isomorphic to the 
method of proof-nets (Girard 1987, Danos and Regnier 
1989, Roorda 1990, 1991), but that it has the advantage 
over this last method of being more effieient and of 
providing more clarity on the result of processing. The 
devices we obtain are more readable, because they are 
interpretable in terms of dependency structures. 
Otherwi~, the parsing method can be an incremental 
one. 
2. The Method of Proof-Nets in the Lambek 
Calculus 
The problem of spurious ambiguities in Categorial 
Grammar is very often discussed (see for instance 
Hendriks and Roorda (1991)). A proof-net is a device 
which contains all the equivalent proofs of the same 
result. As Roorda (1990) says: "A proof-net can be 
viewed as a parallellized sequent proof \[...\] It is a 
concrete structure, not merely an abstract equivalence 
class of derivations, and surely not a special derivation 
with certain constraints on the order in which the rules 
must be applied." 
The principles of construction of proof-nets are related to 
the inference rules of the Lambek Calculus, when it is 
viewed as a sequent calculus. If we here omit the 
product, we have the following rules, which belong to 
two different types: 
1 1 am indebted to Dirk Roorda for fruitful discussions 
during a brief visit I made in Amsterdam in Spring 1991 
Ac'rv_s DE COL1NG-92, NANTES, 2.3-28 hofrr 1992 3 9 4 Pgoc. OF COLING.92, NANTES, AUG. 23-28, 1992 
Binary rules (or type-2 rules): 
(where O is a non-empty sequence of categories, and F 
and A are arbitrary sequences of categories) 2 
\[L/I: O--4B F, A, A -4 C 
..................... 
F, A/B, O, A --~ C 
\[L\\]: O -~ B F, A, A --7 C 
F, O, B\A, A --~ C 
Unary rules (or type-I rules): (F is non empty) 
\[R/l: 0, B --~ A \[R\\]: B, @ ~ A 
@~A/B O~BXA 
In these rules, contexts (O, F, A) are "static". That 
means that they can neither be contracted, nor expanded, 
nor permuted. They play no role in the application of 
rules. So, it is convenient to "forget" them and to 
represent the rules according to schemes similar to: 
A*B 
Such a scheme is called a link. Other types of links are 
provided by the identity axiom: \[axl A---~A, which 
becomes: 
r--------i + 
A A 
and by the cta-rule: 
O--)A A, A, F --~ C 
\[cut\]: .................................. which becomes: 
A, O, F-o C 
+ 
A A 
Links associawal to rules belong to two type.s: 
type-2 links corresponding to type-2 rules 
(depicted by lines) 
type-I links corresponding to type-1 rules 
(depicted by dashed lines) 
D. Reorda (1990) has shown the following theorem: 
Theorem: (Soundness and Completeness of Proof-Nets 
w.r.t.L.) 
2 We use here the notation introduced by J. Lambek, 
according to which the argument category is always under 
the slash, in such a way that Aft\] means a category which 
becomes an A if a B is met on file right, and BXA a category 
which becomes an A if a B is met on tlie left. 
If the rules \[L\\], \[L/J, JR\\], \[R/I are represented by the 
following links: 
+ + 
A B I3- A 
A~I3 B/A 
+ + 
B d d B • ° • ° 
A~B B/A 
then a sequent F --~ A is a theorem of the (Product free) 
Lambek Calculus if and only if we can build, starting 
with this sequent and applying links re, cursively, a 
connected planar graph having the following property: 
for each application of a typed link, every suppression 
of one of the two dashed lines leaves the graph 
connected. 
Examples: 
a -', b / (a \ b) is a theorem: see figure (1) below. 
b / (a \ b) ~ a is not a theorem: see figure (2) below. 
figure (1): 
b/&,) 
figure (2): 
r-- 
a 
b/(aXb) +a 
In this last ease, we see that file suppression of the edge 
2 leads to disconnection. 
3.Dependency Structures 
A dependency structure associated to a sentence is a tree 
on the words of this sentence. Edges represent 
dependency links such that the source of an edge is 
considered as the head and the target as a dependant. 
Hudson (1984) givcs criteria to distinguish heads and 
dependants. It is an open question whether a head can be 
vicwed as a functor, the dependant being viewed as an 
argumenL The facts that criteria involve agreement and 
ACTES DE COLING-92, NANTES, 23-28 AO'tYr I992 3 9 5 Paoc. OF COL1NG-92, NANTES, AUG. 23-28, 1992 
that according to the Keenan's thesis: "functors agree 
with their argument" seem in favour of identification. 
But other scholars disagree, like Moortgat and Morrill, 
who introduce in their recent works, four notions: head, 
dependant,functor and argument. Nevertheless, we will 
accept the first thesis in the following, adopting the 
conception of Barry and Pickering (1990) on this 
subjecL 
Another problem appears in the necessity of accounting 
for slructums with multiplicity of heads (in Ihe case of 
the control of infinitives for instance) because this 
necessity leads to graphs which are no longer trees, but 
dags. 
We assume that a dependency structure is a graph on 
words, In a first step, we will consider only trees. The 
approach will be that of a semantic interpretation in 
terms of wee.s, similar to what we do when we give a 
semantic interpretation of logic formulae in terms of 
sets. The usual operators like / and \ will be interpreted 
as connection operations in an algebra of trees. In a 
second step, we will have to modify this interpretation 
in order to obtain not only application and composition 
but division too. 
4.Operations on Trees 
We start with a set of directed trees associated to lexical 
entries. (see figure (3) below). 
figure (3) 
c o nnai t : A 
np np 
promet 
np sp\[A\] 
np 
trees are called initial trees. The initial state of a 
representation of the structure of a sentence consists in 
an ordered sequence of these initial trees, Then, at each 
~aep. we build a new tree obtained by connection of 
previous trees, These operations are: (cf Lecomte 1990) 
- left-linkage 
- right-linkage 
A tree GI is right (resp. left) linkable to a tree G2 iff: 
I) G 1 and G2 are adjacent, G2 being adjacent to the right 
(w.ap, left) ofG 1 
2) GI has a rightmost (resp leftmost) branch the first 
edge of which is right (resp left) directed and the 
maximal sub-tree attached to this first edge entirely 
covers a continuous subtree of G2. 
The by-product of the right-linkage (resp left-linkage) of 
GI with G2, when GI is right (resp left) linkable to G2 
is the tree G3 obtained as the union of G 1 and G2, 
modified in the following way: 
The rightmost dght-direeted (resp. leftmost left-directed) 
first-levd edge of G 1 is connected to the root of G2, and 
the subtree of G2 covered by the maximal subtree 
attached to this edge is said to be marked in G2. Left 
(res'p. right)-directed edges of G 2 which are not marked 
romainfree and take precedence in the left-to-right (resp 
right-to-left) order of first-level edges over those 
remaining free in G 1. 
We can introduce restrictions on these operations: 
we will call restriction-AB the following constraint: the 
subtree of G 2 covered by the subtre, e of G1 must be 
identical to the whole tree G2, 
restriction-C: at most the rightmost (rcsp leftmost) 
branch of G2 may be uncovered. 
restriction-Crec: a right (resp left) subtree of G2 may be 
uncove~d 
restriction-Cmix: at most the rightmost (resp leftmos0 
or the leftmost (resp rightmost) branch of G2 may be 
uncovered 
Definition: we call connection tree every initial tree 
and every tree obtained by the application of linkage 
operations on earlier connection trees (according to 
eventual restrictions). 
We claim that such a system gives an interpretation of 
very simple categorial grammars, depending on the 
restrictions we select. Like similar constructions 
(Stoedman 1991) where general principles such as 
Adjacency, Directional Consistency and Directional 
Inheritance arc explained in terms of a more detailed 
analysis of categories, this system is suited to express 
such generalities. Because of the structure of linkage 
operations, these principles are obvious. Adjacency and 
Directional Consistency are contained in the definition. 
Directional Inheritance comes from the fact that we never 
allow to change anything in the labels of edges (the fact 
that they are left or right directed). We only allow m 
change tile status of an edge (free to bound or marked). In 
so doing, we reach, like Steedman does. the conclusion 
that so-called Dysharmonic Composition Rules are 
consistent with these principles (even if they are not 
with the Lambek Calculus\[). 
A connection system eliminates spurious ambiguities 
because when they are bound, links are undefeasable : 
there is no way of re-doing something that was 
primilarly done with success. In this respect, the 
calculus on trees concurs with the well known method of 
chart-parsing. (see figure (4): there is only one tree for 
two reductions by means of Cancellation Schemes). 
Ac'rF.s DE COLING-92, NANTES, 23-28 AOt~'r 1992 3 9 6 PREC. OF COLING-92, NANTES, AUO. 23-28, 1992 
figure (4): 
a/ (c\b) (c\b) /d 
c\b 
d 
> 
a 
a/(c\b) (c\b)/d 
a/d 
d 
>B 
a 
/ 
Moreover, a connection system provides us with a 
semantics for Dependency-Constituency Grammars, in 
the tradition of Barry and Picketing (1990) and Mark 
Hepple (1991). 
5.Connection and Identification: an Extension 
of Connection Systems 
5.1.The Need for Division Rules 
It is obvious that the previous system does not include 
any kind of Division Rules or any kind of Type-Raising 
Rule. So, it cannot provide any analysis for sentences 
with extraction, as for instance: 
le livre dont je connais le titre est sur la table 
(the book the title of which I know is on the table) 
because in such an analysis, we have to transform a 
regular n (titre) into a functorial category which requires 
a nonn-tandifier on its right (n/(nkn)). 
We shall define a new connection system which is a 
conservative extension of the previous one (except for 
the admissibility of Dysharmonic Rules). We will call 
it: the Connection Net System. 
As for the proof-nets, we want to demonstrate theorems 
that have a sequent form like: F---~ X, where F is a non 
empty sequence of categories and X is a category. We 
distinguish two kinds of connection Irees: those which 
are on the right-hand side of the sequent we want to 
demonstrate, and those which are on the left-hand side. 
When we are viewing the problems in a natural- 
deduction way, we can say that the first are the trees to 
build and the second are those which are used in this 
task. We will call the firstright-trees and the second left- 
trees. The set of left-trees and right-trees at any stage 
will be called a Construction Net. 
Schematically, operations are not merely connections 
because connections can only expand elementary trees 
towards more complex ones. And we need operations to 
reduce the complexity of a tree. For instance, to show 
the usual rule of Type-Raising: a ~ b/(a~b) we have to 
show that the fight-tree associated to b/(a~b) reduces to 
something isomorphic to a. The fact that, generally, the 
converse (b/(a~)-oa) is not true results from the fact that 
the same reduction is not possible when the same tree is 
put on the left-hand side. This exemplifies the 
fundamental dissymetry of the calculus. 
5.2.Type-I Edges and Type-2 Edges 
We will then distinguish two sorts of edges and two 
sorts of nodes in a connection tree: typed edges and 
nodes and type-2 edges and nodes. 
Definition: A type-2 edge in a connection tree is: 
- an odd level edge in a left-tree, or 
- an even level edge in a right-tree 
A type-1 edge in a connection tree is: 
- an even level edge in a left-tree, or 
- an odd level edge in a right-tree 
A type-i (i =1.2) node is the target of a type-i 
edge. 
Roots are type-1 nodes if in a left-tree, and tyl~- 
2 nodes if in a right-tree. 
Two nodes are mid to be complementary if they have not 
the same type. 
Examples: figure (5) 
a) a new tree assigned to a lexical entry: 
promet 
np ,," s \[infl sp\[/l\] 
np 
b) a pair (L, R) associated to a sequent: 
a ~ b/ (a\b) 
b % , 
L R 
Definition: we call identification link either a non- 
directed edge which links two identical nodes which are 
complementary, one in a left-tree, the other in a right- 
tree, or a type-I directed edge linking two comple- 
mentary nodes having same label. 
We call connection link every link we shall be able to 
establish, according to the following conventions, 
between a typo-I node, which is the ending point of a 
ACRES DE COLING-92, NANTFm, 23-28 Aotrr 1992 3 9 7 PRec. OF COLING-92, NANTES, AUG. 23-28, 1992 
type-2 edge, and a type-2 node which does not belong to 
the same tree. 
5.3.Nodes-numbering 
Rule: each node of the initial construction net receives a 
number, called its degree, according to the following 
roles: 
-for a type-2 edge: 
if it is right directed, the degree of its source is less 
than the degree of all the nodes below it, 
if it is left directed, the degree of its source is 
greater than the degree of all the nodes below 
it, 
for two type-2 edges, children's degrees of the 
leftmost branch are less than those of the 
rightmost branch. 
-for a type-1 edge: 
if it is right directed, the degree of its source is 
greater than the degree of all the nodes below 
it, 
if it is left directed, the degree of its source is less 
than the degree of all the nodes below it, 
for two type-I edges, children's degrees of the 
rightmost branch are less than those of the 
leftmost branch. 
The lowest degree of the right successor of an initial 
tree is the successor of the greatest degree of this 
latter tree. 
Example of such a numbering: figure (6) 
.... 
©",% 
L R 
intervals: 
s'-s' : \[1_8\] 
s-s : \[3 _ 41 
np-np : \[2 _ 7\] 
s-s: \[5_61 
Each link is now associated to a pair of degrees, called 
its interval. 
From now on, L and R will denote respectively: the left 
hand side and the right hand side of a Construction Net. 
The Construction Net will be denoted by: <L I R>. 
5.4.Linking the Nodes 
Nodes will be linked according to the following 
principles: 
COMPLEMENTARITY: two nodes are linked only if 
they have the same label and they are of complementary 
types. 
NON-OVERLAP: the linking of all the nodes in the 
Construction Net must meet the non-overlap 
convention, which stipulates that given two arbitrary 
intervals, either one contains the other or they are 
disjoint. 
Theorem: (Conservativity of Connection Operations) 
The Non-Overlap condition is a conservative extension 
of the conditions on connection (restriction C rec) 
stipulated in ~4. That means: every connection system 
based on C rec, when translated in the Connection Net 
System, follows this convention. 
5.5.Building a Correct Net 
Definition: Given an ordered sequence of left-trees L 
and a right-tree R, we will say that L and R yield a 
correct net iff there is a linkage of all the nodes in the 
Construction Net <L I R>, which gives a connected 
graph, respects the complementarity principle and the 
non-overlap principle, and is such that: when all the 
rype-I edges are removed, the graph remains connected. 
The fundamental result is the following: 
Theorem: (Soundness and Completeness w.r.t. A 3) 
Let F --~ A be a sequent expressed in the 
Product-Free Lambek Calculus, where F is a non 
empty sequence of categories and A is a 
category, let L be the sequence of left connection 
trees associated to the elements of F and R be 
the right tree associated to A, the sequent is a 
theorem if and only if L and R yield a correct 
neL In other terms: the Connection Net System 
is sound and complete w.r.t, the Product-Free 
Lambek Calculus, 
Examples: figure (6) shows that: 
s/(s/np) s/((s/np)ks) I-- s 
is a theorem of the Lambek Calculus. 
Figure (7) below gives a correct net for the analysis of 
the sentence: 
le livre dont je connais le litre est sur la table 
3 A is the usual designation of the Product-Free Lambek 
Calculus (see Zielonka 1981). 
AcrEs DE COLING-92, NANTES, 23~28 AOUT 1992 3 9 8 PROC. OF COL1NG-92, NANTES, AUO. 23-28, 1992 
(figure 7): 
np npXnp/(s/(npXnp))np npks/np np np\s ~ s 
np~ s 
Theorem: (Categorization of Links) In a correct net, 
links are either identification links or connection links. 
Corollary: A net is correct if and only if all nodes are 
either identified or connected. 
Definition: we call Tree on words the tree obtained 
from a correct net by merging connected nodes and 
removing identification links. In the case of an 
identification link consisting in a type-1 edge, the link 
and the nodes linked by it are removed, and the adjacent 
type-2 edges are merged. 
Example: figure (g): Pierre promet d Marie de venir 
np s sp\[~t\] s 
___ ii s 
np 
6. Building Dependencies 
Obtaining a Dependency Structure from a tree on words 
amounts to doing little transformations on the correct 
net obtained by "equating" an ordered sequence of initial 
trees to a node representing a primitive category. These 
transformations involve indexing nodes in such a way 
that: 
- indices of two different initial trees constitute two 
disjoint sets. 
- indices inside an initial tree may be identical (if we 
want to express a coreference) 
- linking two nodes results in identifying their 
indices. 
Example: figure (9) 
promet 
np \[i\] O~#: s \[infl sp\[~\] \[j\] 
np \[i\] 
spJ~l lkl 
a "~ np\[kl 
After getting a tree on words, we identify two distant 
nodes having the same index: we call the new node 
obtained: a shared node. 
Finally, we can say that dependencies are obtained in the 
following stages: 
1 ~ indexing the nodes having the same label and 
belonging to different initial trees by different variables, 
taken in a set {i, j, ...} (the distribution of indices inside 
an initial tree being set by the lexicon) \[INDEXING- 
step\]. 
2-building the net corresponding to the 
assertion that the sequence is of type s \[NET-step\]. 
3- suppressing the nodes identified by type-1 
edges of the left-hand side and all the identification links 
\[COLLAPSING-step\]. 
4- if the same index appears on distinct nodes 
having the same label: merging them \[MERGING-step\] 
Example: figure (10) 
Pierre promet fi Marie de venir 
\[~ s \[infl 
np 
O=kJ \[ 
~\[11 li=l} 
yields: 
np \[i~~ln fl 
but, with permet instead of promet: 
permet e~~ ~ 
npli\] O,,~'s\[inf\] sp\[~\] \[j\] 
np\[Jl 
Pierre permet d Marie de venir: 
np~ 
ACRES DE COLING-92, NANTES, 23-28 AOOT 1992 3 9 9 PROC. OF COLING-92, NANTES, AUG. 23-28, 1992 
7.Remarks and Conclusion 
This method of Connection Nets has many advantages 
over other methods. 
Firstly, compared to classical strategies in the sequent 
calculus, it avoids spurious ambiguities and in so doing, 
it improves efficiency of searching the solution. 
Secondly, compared to the method of Proof-Nets, it 
gives more clarity to the resulting structures. It is more 
efficient too, because the stage of checking the coanexity 
when suppressing a branch of a type-1 link is replaced 
by a stage where the connexity is checked only once: 
when we have removed all the type-1 edges. The 
corresponding stage in Proof-Nets is usually named 
switching. In the early method by Girard which used the 
"long lrip condition", there was a switch for each link 
and that gave an exponential-time algorithm (in the 
number of links). In the method defined by Roorda, only 
type-1 links lead to switches. The reason lies in the 
necessity of checking that a type-1 link is not used to 
connect two subsets of the net, which would not be 
connected without it. (Let us recall that a type-I link 
refers to a unary rule). In our method, switches are 
completely avoided. 
Thirdly. it can be done incrementally. The reason is that 
the numbering of nodes is consistent with the order of 
initial trees. Thus, at each stage of the processing from 
left to right, we may have a beginning net which 
represents the present state of the processing. Here, the 
properties of left-associative grammars (Hausser 1990) 
are reeL 
Finally, a very few transformations are needed in order to 
obtain graphs on words which can be really interpreted as 
Dependency Structures. 
ACRES DE COLING-92., NANTES, 2.3-28 ^OUT 1992 4 0 0 PROC, OF COLING-92, NANTEs, AUG. 23-28, 1992 

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