Extending Lambek grammars:
a logical account of minimalist grammars
Alain Lecomte
CLIPS-IMAG
Universit´e Pierre Mend`es-France,
BSHM - 1251 Avenue Centrale,
Domaine Universitaire de St Martin d’H`eres
BP 47 - 38040 GRENOBLE cedex 9, France
Alain.Lecomte@upmf-grenoble.fr
Christian Retor´e
IRIN, Universit´e de Nantes
2, rue de la Houssini`ere BP 92208
44322 Nantes cedex 03, France
retore@irisa.fr
Abstract
We provide a logical definition of Min-
imalist grammars, that are Stabler’s
formalization of Chomsky’s minimal-
ist program. Our logical definition
leads to a neat relation to catego-
rial grammar, (yielding a treatment
of Montague semantics), a parsing-as-
deduction in a resource sensitive logic,
and a learning algorithm from struc-
tured data (based on a typing-algorithm
and type-unification). Here we empha-
size the connection to Montague se-
mantics which can be viewed as a for-
mal computation of the logical form.
1 Presentation
The connection between categorial grammars (es-
pecially in their logical setting) and minimalist
grammars, which has already been observed and
discussed (Retor´e and Stabler, 1999), deserve a
further study: although they both are lexicalized,
and resource consumption (or feature checking)
is their common base, they differ in various re-
spects. On the one hand, traditional categorial
grammar has no move operation, and usually have
a poor generative capacity unless the good prop-
erties of a logical system are damaged, and on
the other hand minimalist grammars even though
they were provided with a precise formal defini-
tion (Stabler, 1997), still lack some computational
properties that are crucial both from a theoreti-
cal and a practical viewpoint. Regarding appli-
cations, one needs parsing, generation or learning
algorithms, and, considering more conceptual as-
pects, such algorithms are needed too to validate
or invalidate linguistic claims regarding economy
or efficiency. Our claim is that a logical treat-
ment of these grammars leads to a simpler de-
scription and well defined computational proper-
ties. Of course among these aspects the relation
to semantics or logical form is quite important;
it is claimed to be a central notion in minimal-
ism, but logical forms are rather obscure, and no
computational process from syntax to semantics
is suggested. Our logical presentation of mini-
malist grammar is a first step in this direction:
to provide a description of minimalist grammar
in a logical setting immediately set up the com-
putational framework regarding parsing, genera-
tion and even learning, but also yields some good
hints on the computational connection with logi-
cal forms.
The logical system we use, a slight extension
of (de Groote, 1996), is quite similar to the fa-
mous Lambek calculus (Lambek, 1958), which is
known to be a neat logical system. This logic has
recently shown to have good logical properties
like the subformula property which are relevant
both to linguistics and computing theory (e.g. for
modeling concurrent processes). The logic under
consideration is a super-imposition of the Lam-
bek calculus (a non commutative logic) and of
intuitionistic multiplicative logic (also known as
Lambek calculus with permutation). The context,
that is the set of current hypotheses, are endowed
with an order, and this order is crucial for obtain-
ing the expected order on pronounced and inter-
preted features but it can also be relaxed when
necessary: that is when its effects have already
been recorded (in the labels) and the correspond-
ing hypotheses can therefore be discharged.
Having this logical description of syntactic
analyses allows to reduce parsing (and produc-
tion) to deduction, and to extract logical forms
from the proof; we thus obtain a close connection
between syntax and semantics as the one between
Lambek-style analyses and Montague semantics.
2 The grammatical architecture
The general picture of these logical grammars
is as follows. A lexicon maps words (or, more
generally, items) onto a logical formula, called
the (syntactic) type of the word. Types are de-
fined from syntactic of formal features a0 (which
are propositional variables from the logical view-
point):
a1 categorial features (categories) involved in
merge: BASE a2a4a3a6a5a8a7a10a9a11a7a10a12a13a7a15a14a11a7a17a16a18a7a20a19a20a19a20a19a22a21
a1 functional features involved in move:
FUN a2a4a3 a23a18a7 a24a13a7 a25a27a26a18a7a20a19a20a19a20a19a28a21
The connectives in the logic for constructing
formulae are the Lambek implications (or slashes)
a29
a7a31a30 together with the commutative product of lin-
ear logic a32 . 1
Once an array of items has been selected, a sen-
tence (or any phrase) is a deduction of IP (or of the
phrasal category) under the assumptions provided
by the syntactic types of the involved items. This
first step works exactly as Lambek grammars, ex-
cept that the logic and the formulae are richer.
Now, in order to compute word order, we pro-
ceed by labeling each formula in the proof. These
labels, that are called phonological and seman-
tic features in the transformational tradition, are
computed from the proofs and consist of two parts
that can be superimposed: a phonological label,
denoted by a30a34a33a36a35a38a37a6a39a40a30 , and a semantic label2 de-
noted by a41a42a33a36a35a38a37a6a39a40a43 — the super-imposition of both
1The logical system also contains a commutative impli-
cation, a44a6a45 , and a non commutative product a46 but they do not
appear in the lexicon, and because of the subformula prop-
erty, they are not needed for the proofs we use.
2We prefer semantic label to logical form not to confuse
logical forms with the logical formulae present at each node
of the proof.
label being denoted by a33a36a35a38a37a6a39 . The reason for hav-
ing such a double labeling, is that, as usual in
minimalism, semantic and phonological features
can move separately. It should be observed that
the labels are not some extraneous information;
indeed the whole information is encoded in the
proof, and the labeling is just a way to extract the
phonological form and the logical form from the
proof.
We rather use chains or copy theory than move-
ments and traces: once a label or one aspect (se-
mantic or phonological) has been met it should be
ignored when it is met again. For instance a label
a47a49a48a51a50a52a48
a37a53a41a55a54a4a56a57a37a59a58a60a43a62a61a63a35a38a64
a48a34a65
a54a66a56a27a37a59a58 corresponds to a se-
mantic label a41 a47a49a48a51a50a52a48 a37a67a43a68a41a55a54a66a56a27a37a59a58a69a43a70a41a63a61a63a35a38a64 a48 a43 and to the
phonological form a30 a47a49a48a20a50a10a48 a37a57a30a71a30a38a61a63a35a34a64 a48a72a65 a30a73a30a74a54a66a56a27a37a59a58a69a30 .
3 Logico-grammatical rules for merge
and phrasal movement
Because of the sub-formula property we need
not present all the rules of the system, but only
the ones that can be used according to the types
that appear in the lexicon. Further more, up to
now there is no need to use introduction rules
(called hypothetical reasoning in the Lambek cal-
culus): so our system looks more like Com-
binatory Categorial Grammars or classical AB-
grammars. Nevertheless some hypotheses can be
cancelled during the derivation by the product-
elimination rule. This is essential since this rule
is the one representing chains or movements.
We also have to specify how the labels are car-
ried out by the rules. At this point some non
logical properties can be taken into account, for
instance the strength of the features, if we wish
to take them into account. They are denoted by
lower-case variables. The rules of this system in
a Natural Deduction format are:
a75a77a76a79a78a81a80a27a82
a30a38a83 a84
a76
a58
a80
a83 a85
a30a38a86a68a87
a75a89a88
a84
a76a90a78
a58
a80a57a82
a84
a76
a58
a80
a83
a75a77a76a79a78a81a80
a83
a29 a82
a85
a29
a86a70a87
a84
a88a15a75a91a76
a58
a78a92a80a57a82
a75
a85
a41a55a84a79a93
a88
a84a95a94a51a43a96a87
a76a97a82
a48a20a98a99a50
a37a59a35a31a100a101a58
a75
a85
a41a55a84a79a93a51a7a102a84a95a94a51a43a96a87
a76a97a82
a75a77a76a97a103a104a80a67a82
a32a105a83 a84a97a7
a78a81a80a57a82
a7a10a58
a80
a83a90a7a102a84a95a106
a76a90a107a108a80a40a109
a85
a32a73a86a70a87
a84a97a7
a75
a7a102a84 a106
a76a110a107
a85
a103
a30a57a3
a78
a7a10a58a11a21a20a87
a80a40a109
This later rule encodes movement and deserves
special attention. The label a107 a85a103 a30a57a3 a78 a7a10a58a11a21a20a87 means
the substitution of a103 to the unordered set a3 a78 , a58a11a21
that is the simultaneous substitution of a103 for both
a78 and
a58 , no matter the order between
a78 and
a58
is. Here some non logical but linguistically mo-
tivated distinction can be made. For instance ac-
cording to the strength of a feature (e.g. weak
case a23 versus strong case a24 ), it is possible to de-
cide that only the semantic part that is a41 a103 a43 is sub-
stituted with a78 .
In the figure 1, the reader is provided with an
example of a lexicon and of a derivation. The re-
sulting label is a41a63a56a1a0 a35a59a35a3a2a60a43a18a37 a48 a56a40a39 a65 a56a4a0 a35a38a35a3a2 phonologi-
cal form is a30a34a37 a48 a56a27a39 a65 a30a70a30a38a56a5a0 a35a38a35a3a2a60a30 while the resulting
logical form is a41a63a56a6a0 a35a59a35a3a2a60a43 a41a42a37 a48 a56a40a39 a65 a43 .
Notice that language variation from SVO to
SOV does not change the analysis. To ob-
tain the SOV word order, one should sim-
ply use a24 (strong case feature) instead of a23
(weak case feature) in the lexicon, and use the
same analysis. The resulting label would be
a56a5a0 a35a59a35a3a2a53a37
a48
a56a27a39
a65
a56a5a0 a35a38a35a3a2 which yields the phonolog-
ical from a30a38a56a7a0a31a35a59a35a3a2a60a30a73a30a34a37 a48 a56a40a39 a65 a30 and the logical form
remains the same a41a63a56a6a0 a35a59a35a3a2a60a43 a41a42a37 a48 a56a40a39 a65 a43 .
Observe that although entropy which sup-
presses some order has been used, the labels con-
sist in ordered sequences of phonological and log-
ical forms. It is so because when using [/ E] and
[a29 E], we necessarily order the labels, and this or-
der is then recorded inside the label and is never
suppressed, even when using the entropy rule: at
this moment, it is only the order on hypotheses
which is relaxed.
In order to represent the minimalist grammars
of (Stabler, 1997), the above subsystem of par-
tially commutative intuitionistic linear logic (de
Groote, 1996) is enough and the types appearing
in the lexicon also are a strict subset of all possi-
ble types:
Definition 1 a8a10a9 -proofs contain only three kinds
of steps:
a1 implication steps (elimination rules for / and
a29 )
a1 tensor steps (elimination rule for
a32 )
a1 entropy steps (entropy rule)
Definition 2 A lexical entry consists in an axiom
a76
a33
a80a12a11 where a11 is a type:
a41a10a41a14a13 a94
a29
a41a14a13a16a15
a29
a19 a19 a19a22a41a14a13a18a17
a29
a41a20a19a53a93a74a32a6a19a73a94a72a32 a19 a19 a19a32a21a19a23a22a70a32
a82
a43a10a43a10a43a10a43a10a30a24a13a89a93a74a43
where:
a1 m and n can be any number greater than or
equal to 0,
a1 F
a93 , ..., Fa17 are attractors,
a1 G
a93 , ..., Ga22 are features,
a1 A is the resulting category type
Derivations in this system can be seen as T-
markers in the Chomskyan sense. [/E] and [a29 E]
steps are merge steps. [a32 E] gives a co-indexation
of two nodes that we can see as a move step. For
instance in a tree presentation of natural deduc-
tion, we shall only keep the coindexation (corre-
sponding to the cancellation of a82 and a83 : this is
harmless since the conclusion is not modified, and
makes our natural deduction T-markers).
Such lexical entries, when processed with
a8a10a9 -rules encompass Stabler minimalist gram-
mars; this system nevertheless overgenerates, be-
cause some minimalist principles are not yet sat-
isfied: they correspond to constraints on deriva-
tions.
3.1 Conditions on derivations
The restriction which is still lacking concerns the
way the proofs are built. Observe that this is an
algorithmic advantage, since it reduces the search
space.
The simplest of these restriction is the follow-
ing: the attractor F in the label L of the target a25
locates the closest F’ in its domain. This simply
corresponds to the following restriction.
Definition 3 (Shortest Move) : A a8a10a9 -proof is
said to respect the shortest move condition if it is
such that:
a1 the same formula never occurs twice as a hy-
pothesis of any sequent
a1 every active hypothesis during the proof pro-
cess is discharged as soon as possible
The consequences of this definition are the fol-
lowing:
Figure 1: reads a book
a37
a48
a56a27a39
a65 a80 a80
a2
a76
a37
a48
a56a40a39
a65a49a80
a41a10a41 a23
a29
a12a1a0a11a43a10a30a38a14a8a43
a56
a80 a80
a2
a76
a56
a80
a41a10a41a63a14a70a32 a23a60a43a10a30a51a16a11a43
a0a31a35a59a35a3a2
a80 a80
a2
a76
a0 a35a59a35 a2
a80
a16
a76
a56
a80
a41a10a41a63a14a68a32 a23a69a43a10a30a51a16a11a43
a76
a0 a35a59a35a3a2
a80
a16 a85
a30a38a86a70a87
a76
a56a6a0 a35a59a35a3a2
a80
a14a70a32 a23
a58
a80
a23
a76
a58
a80
a23
a76
a37
a48
a56a40a39
a65a53a80
a41a10a41 a23
a29
a12a2a0a11a43a10a30a38a14a8a43
a78a81a80
a14
a76a79a78a81a80
a14 a85
a30a38a86a68a87
a78a92a80
a14
a76
a37
a48
a56a27a39
a65 a78a81a80
a41 a23
a29
a12a2a0a101a43 a85
a29
a86a70a87
a58
a80
a23
a88a10a78a81a80
a14
a76
a58a70a37
a48
a56a40a39
a65 a78a92a80
a12a1a0 a85
a48a51a98a99a50
a37a6a35a31a100a101a58a27a87
a58
a80
a23a18a7
a78a81a80
a14
a76
a58a70a37
a48
a56a40a39
a65 a78a92a80
a12a1a0 a85
a32a70a86a70a87
a76
a41a63a56 a0 a35a59a35a3a2a60a43a99a37
a48
a56a40a39
a65
a56a21a0a31a35a59a35a3a2
a80
a12a2a0
1. a3a5a4 ... a3a6a4 ... a3a51a94 ... a76 C is forbidden
2. a1 if there is a sequent ... a3 ... a76 a3 a106 a29 C
a1 if there is a type
a3 a106 such that
a75a77a76
a3 a106 a32a7a3
is a (proper or logical) axiom,
a1 then a hypothesis
a3 a106 must be intro-
duced, rather than any constant a3 a106 , in
order to discharge a3
We may see an application of this condition in the
fact that sentences like:
*Whoa94 do you know [whoa93 ea94
likes ea93 ]
*Whoa94 do you know [whoa93 ea93
likes ea94 ]
are ruled out. Let us look at the beginning of their
derivation (in a tree-like presentation of natural
deduction proofs): at the stage where we stop the
deduction on figure 2, we cannot introduce a new
hypothesis a8a38a94 a80 a23 a32 a14 because there is already an
active one (a8a67a93 ), the only possible continuation is
to discharge a58 a94 and a78 a94 altogether by means of a
”constant”, like a9 a56a57a37a59a58 , so that, in contrast:
You know [whoa93 Mary likes
ea93 ]
is correct.
3.2 Extension to head-movement
We have seen above that we are able to account
for SVO and SOV orders quite easily. Neverthe-
less we could not handle this way VSO language.
Indeed this order requires head-movement.
In order to handle head-movement, we shall
also use the product a32 but between functor types.
As a first example, let us take the very sim-
ple example of: peter loves mary. Starting from
the following lexicon in figure 3 we can build
the tree given in the same figure; it represents a
natural deduction in our system, hence a syntac-
tic analysis. The resulting phonological form is
a30
a47a49a48a51a50a52a48
a37a57a30a6a30a38a61a42a35a38a64
a48a72a65
a30a6a30a74a54a66a56a27a37a59a58a69a30 while the resulting log-
ical form is a41 a47a49a48a51a50a52a48 a37a67a43a74a41a55a54a4a56a27a37a38a58a60a43a74a41a63a61a63a35a38a64 a48a72a65 a43 — the possi-
bility to obtain SOV word order with a a24 instead
of a a23 also applies here.
4 The interface between syntax and
semantics
In categorial grammar (Moortgat, 1996), the pro-
duction of logical forms is essentially based
on the association of pairs a10 a65a72a50 a37a5a11 a98a13a12 a7 a50 a58a34a100 a48a15a14
with lambda terms representing the logical form
of the items, and on the application of the
Curry-Howard homomorphism: each (a30 or a29 ) -
elimination rule translates into application and
each introduction step into abstraction. Compo-
sitionality assumes that each step in a derivation
is associated with a semantical operation.
In generative grammar (Chomsky, 1995), the
production of logical forms is in last part of the
derivation, performed after the so-called Spell Out
point, and consists in movements of the semanti-
cal features only. Once this is done, two forms
can be extracted from the result of the derivation:
a phonological form and a logical one.
These two approaches are therefore very differ-
Figure 2: Complex NP constraint
a58a67a94
a80
a23
a41a10a41 a23
a29
a9a60a43a10a30a34a12a1a0a11a43
a78
a94
a80
a14
a0
a4
a80
a23a49a32a105a14
a1
a58 a93
a80
a23
a61 a11 a2
a48a34a65a53a80
a41a10a41 a23
a29
a41a63a14
a29
a12a2a0a101a43a10a43a10a30a38a14a69a43
a1a78
a93
a80
a14
a61 a11 a2
a48a72a65 a78
a93
a80
a41 a23
a29
a41a63a14
a29
a12a2a0a11a43a10a43
a58a8a93 a61 a11 a2
a48a72a65 a78
a93
a80
a41a63a14
a29
a12a1a0a11a43
a8a67a93 a61 a11 a2
a48a34a65a53a80
a41a63a14
a29
a12a2a0a101a43
a78
a94 a8 a93 a61 a11 a2
a48a72a65a53a80
a12a1a0
a78
a94 a8a6a93 a61 a11 a2
a48a34a65a53a80
a41 a23
a29
a9a60a43
a58a67a94
a78
a94 a8a6a93 a61 a11 a2
a48a72a65a53a80
a9
Figure 3: Peter loves Mary
a61a63a35a38a64
a48a34a65 a80 a80
a2
a76
a61a63a35a34a64
a48a72a65a49a80
a41a10a41 a23
a29a3a2
a0a11a43a10a30a34a12a1a0a11a43a62a32 a41a10a41 a23
a29
a41a63a14
a29
a12a1a0a13a43a10a43a10a30a38a14a69a43
a100
a48a51a50a52a48
a37
a80 a80
a2
a76
a100
a48a20a50a10a48
a37
a80
a23a49a32a105a14
a9 a56a27a37a59a58
a80 a80
a2
a76
a9 a56a57a37a59a58
a80
a23a68a32 a14
a2
a0
peter
a23
a93
a41 a23
a29a3a2
a0a101a43
lovesa4
a41a10a41 a23
a29a3a2
a0a101a43a10a30a34a12a2a0a11a43 a12a1a0
a14
a93
a41a63a14
a29
a12a2a0a11a43
(mary)
a23
a94
a41 a23
a29
a41a63a14
a29
a12a2a0a11a43a10a43
(to love)
a41a10a41 a23
a29
a41a63a14
a29
a12a1a0a11a43a10a43a10a30a38a14a8a43
a15
mary
a14
a94
ent, but we can try to make them closer by replac-
ing semantic features by lambda-terms and using
some canonical transformations on the derivation
trees.
Instead of converting directly the derivation
tree obtained by composition of types, something
which is not possible in our translation of mini-
malist grammars, we extract a logical tree from
the previous, and use the operations of Curry-
Howard on this extracted tree. Actually, this ex-
tracted tree is also a deduction tree: it represents
the proof we could obtain in the semantic compo-
nent, by combining the semantic types associated
with the syntactic ones (by a homomorphism a0
to specify). Such a proof is in fact a proof in im-
plicational intuitionistic linear logic.
4.1 Logical form for example 3
Coindexed nodes refer to ancient hypotheses
which have been discharged simultaneously, thus
resulting in phonological features and semantical
ones at their right place3.
By extracting the subtree the leaves of which are
full of semantic content, we obtain a structure that
can be easily seen as a composition:
(peter)((mary)(to love))
If we replace these ”semantic features” by a1 -
terms, we have:
a41a2a1a4a3 a19a5a3 a41a100
a48a51a50a52a48
a37a57a43a31a7a72a41a2a1a4a3 a19a5a3 a41 a9 a56a27a37a38a58a60a43a31a7a6a1
a78
a19a7a1a60a58a11a19a61a63a35a38a64
a48
a41a42a58a11a7
a78
a43a10a43a10a43
This shows that necessarily raised constituants in
the structure are not only ”syntactically” raised
but also ”semantically” lifted, in the sense that
a1a4a3 a19a5a3 a41a100
a48a51a50a52a48
a37a57a43 is the high order representation of
the individual peter4.
4.2 Subject raising
Let us look at now the example: mary seems to
work From the lexicon in figure 4 we obtain the
deduction tree given in the same figure.
3For the time being, we make abstraction of the repre-
sentation of time, mode, aspect... that would be supported
by the inflection category.
4It is important to notice that if we consider
a8a10a9a12a11a9a14a13a16a15a18a17a20a19a6a21a23a22
a typed lambda term, we must only assume it is of some
type freely raised from a24 , something we can represent by
a13a25a13a16a24a27a26a29a28a30a22a31a26a29a28a30a22 , where X is a type-variable, here X =
a13a16a24a32a26a34a33a35a22 becausea8a10a36a12a11a8a10a21a37a11a38a40a39a42a41a20a43a44a13a16a21a46a45a47a36a37a22 has type a13a16a24a32a26a48a13a16a24a32a26a34a33a49a22a25a22
This time, it is not so easy to obtain the logical
representation:
a65a38a48a72a48
a9 a41
a50
a35 a33a36a35a38a37 a2a99a41 a9 a56a57a37a59a58a60a43a10a43
The best way to handle this situation consists in
assuming that:
a1 the verbal infinitive head (here to work) ap-
plies to a variable a78 which occupies the a14 -
position,
a1 the semantics of the main verb (here to
seem) applies to the result, in order to obtain
a65a34a48a34a48
a9 a41
a50
a35 a33a36a35a38a37 a2a99a41
a78
a43a10a43 ,
a1 the
a78 variable is abstracted in order
to obtain a1 a78 a19a65a34a48a34a48 a9 a41 a50 a35 a33a36a35a38a37 a2a99a41 a78 a43a10a43 just be-
fore the semantic content of the specifier
(here the nominative position, occupied by
a1a4a3 a19a5a3 a41 a9 a56a27a37a38a58a60a43 ) applies.
This shows that the semantic tree we want to
extract from the derivation tree in types logic is
not simply the subtree the leaves of which are se-
mantically full. We need in fact some transforma-
tion which is simply the stretching of some nodes.
These stretchings correspond to a50 -introduction
steps in a Natural deduction tree. They are al-
lowed each time a variable has been used before,
which is not yet discharged and they necessarily
occur just before a semantically full content of a
specifier node (that means in fact a node labelled
by a functional feature) applies.
Actually, if we say that the tree so obtained repre-
sents a deduction in a natural deduction format,
we have to specify which formulae it uses and
what is the conclusion formula. We must there-
fore define a homomorphism between syntactic
and semantic types.
Let a0 be this homomorphism.
We shall assume:
a1
a0 (a2 a0 )=t, a0 (a12a1a0 )a51 a3 t,a41a47a52a53a50a55a54a38a43a102a21 , a0 (a14 )=e,
a1
a0a77a41a2a56
a29a20a57
a43 =a0a77a41
a57
a30a46a56a27a43 = a41 Ha41a2a56a40a43a58a50 Ha41
a57
a43a10a43 ,
a1a60a59 a61 H
a41
a61
a43a62a51a91a3a27a41a10a41a47a52a63a50a65a64a97a43a66a50a29a64 a43a31a7a72a41a47a64a67a50a29a64 a43a102a21
5
5X is a variable of type. This may appear as non-
determinism but the instantiation of X is always unique.
Moreover, when a68 is of type a13a16a69a70a26a29a69a71a22 , it is in fact en-
dowed with the identity function, something which happens
everytime a68 is linked by a chain to a higher node.
Figure 4: Mary seems to work
a65a38a48a72a48
a9
a65 a80 a80
a2
a76a97a65a34a48a34a48
a9
a65a49a80
a41a10a41 a23
a29a3a2
a0a11a43a10a30a34a12a1a0a11a43a62a32 a41a42a12a2a0a11a30a34a12a1a0a101a43
a9 a56a27a37a59a58
a80 a80
a2
a76
a9 a56a57a37a59a58
a80
a14a73a32 a23
a50
a35 a33a71a35a34a37 a2
a80 a80
a2
a76a79a50
a35 a33a36a35a38a37 a2
a80
a41a63a14
a29
a12a1a0a11a43
a2
a0
mary
a23
a93
a41 a23
a29a3a2
a0a11a43
seemsa0
a41a10a41 a23
a29a3a2
a0a11a43a10a30a34a12a2a0a101a43 a12a1a0
(to seem)
a41a42a12a1a0a11a30a34a12a1a0a11a43
a94
a12a1a0
a14
a93
to work
a41a63a14
a29
a12a1a0a11a43
With this homomorphism of labels, the transfor-
mation of trees consisting in stretching ”interme-
diary projection nodes” and erasing leaves with-
out semantic content, we obtain from the deriva-
tion tree of the second example, the following ”se-
mantic” tree:
seem(to work(mary))
a54
a8a10a9a12a11a9a14a13a16a15a30a17a20a19a42a21a23a22
a41a10a41a47a52a53a50a55a54a38a43a58a50a55a54a38a43
a8a10a36 a11a1a49a43a35a43a49a15 a13
a2
a39 a3 a39a6a19a5a4 a13a16a36a46a22a25a22
a41a47a52a63a50a55a54a38a43
a93
t
a8a10a41 a11a1a49a43a42a43a49a15 a13a16a41 a22
a41 a54a30a50a29a54a38a43
to work(x)
a54
a8a10a21a46a11
a2
a39 a3 a39a42a19a5a4 a13a16a21a23a22
a41a47a52a63a50a29a54a38a43
x
a52
a93
where coindexed nodes are linked by the dis-
charging relation.
Let us notice that the characteristic weak or strong
of the features may often be encoded in the lexi-
cal entries. For instance, Head-movement from V
to I is expressed by the fact that tensed verbs are
such that:
a1 the full phonology is associated with the in-
flection component,
a1 the empty phonology and the semantics are
associated with the second one,
a1 the empty semantics occupies the first one6
Unfortunately, such rigid assignment does not
always work. For instance, for phrasal movement
(say of a a14 to a a23 ) that depends of course on the
particular a23 -node in the tree (for instance the sit-
uation is not necessary the same for nominative
and for accusative case). In such cases, we may
assume that multisets are associated with lexical
entries instead of vectors.
4.3 Reflexives
Let us try now to enrich this lexicon by consid-
ering other phenomena, like reflexive pronouns.
The assignment for himself is given in fig-
ure 5 — where the semantical type of himself
is assumed to be a41a10a41 a48 a50 a41 a48 a50 a50 a43a10a43 a50 a41 a48 a50
a50
a43a10a43 . We obtain for paul shaves himself
as the syntactical tree something similar to the
tree obtained for our first little example (peter
loves mary), and the semantic tree is given in
figure 5.
5 Remarks on parsing and learning
In our setting, parsing is reduced to proof search,
it is even optimized proof-search: indeed the re-
6as long we don’t take a semantical representation of
tense and aspect in consideration.
Figure 5: Computing a semantic recipe: shave himself
a65a1a0
a56a27a64
a48a72a65 a80 a80
a2
a85
a65a2a0
a56a57a64
a48a72a65a53a80a4a3 a80
a41a10a41 a23
a29a3a2
a0a101a43a10a30a34a12a2a0a101a43a96a87a69a32
a85a6a5
a80
a1
a78
a19a7a1a60a58a13a19
a65a2a0
a56a57a64
a48
a41a42a58a13a7
a78
a43
a80
a41a10a41 a23
a29
a41a63a14
a29
a12a1a0a11a43a10a43a10a30a38a14a69a43a96a87
a0
a11 a9
a65a34a48
a61a8a7
a80 a80
a2
a85a6a5
a80
a1 a3 a19a7a1 a8a69a19a5a3 a41 a8a69a7 a8a27a43
a80
a23a27a87a8a32
a85
a0
a11 a9
a65a38a48
a61a9a7
a80a6a78a81a80
a14a67a87
shave(paul,paul)
a54
a8a10a9a12a11a9a14a13a6a10 a17a20a9a46a38a16a22
a41a10a41a47a52a63a50a29a54a59a43a58a50a55a54a38a43
a8a12a11a23a11a1a14a13a23a17 a41a20a43a44a13a15a11a23a45a8a11a44a22
a41a47a52a53a50a55a54a38a43a17a16
shave(z,z)
a54
z
a52a18a16
a8a12a11a23a11a1a19a13a10a17a20a41a20a43a20a13a15a11 a45a9a11a44a22
a41a47a52a53a50a55a54a59a43
a8a10a9a12a11a8a12a11a23a11a9a14a13a15a11 a45a20a11a44a22
a41a10a41a47a52 a50 a41a47a52 a50a55a54a59a43a10a43 a50 a41a47a52a53a50a55a54a38a43a10a43
a8a10a36a12a11a8a10a21a46a11a1a14a13a23a17 a41a20a43a44a13a16a21a46a45a47a36a46a22
a41a47a52 a50 a41a47a52 a50a55a54a59a43
a4
a43
a8a10a21a46a11a1a14a13a23a17 a41a20a43a44a13a16a21a46a45a47a36a46a22
a41a47a52a53a50a55a54a38a43
a8a10a36 a11a8a10a21a46a11a1a19a13a10a17a20a41a20a43a44a13a16a21a37a45a36a37a22
a41a47a52a63a50 a41a47a52a63a50a29a54a38a43a10a43
a36
a52
a4
striction on types, and on the structure of proof
imposed by the shortest move principle and the
absence of introduction rules considerably reduce
the search space, and yields a polynomial algo-
rithm. Nevertheless this is so when traces are
known: otherwise one has to explore the possible
places of theses traces.
Here we did focus on the interface with se-
mantics. Another excellent property of categorial
grammars is that they allow — especially when
there are no introduction rules — for learning al-
gorithms, which are quite efficient when applied
to structured data. This kind of algorithm applies
here as well when the input of the algorithm are
derivations.
6 Conclusion
In this paper, we have tried to bridge a gap be-
tween minimalist program and the logical view
of categorial grammar. We thus obtained a de-
scription of minimalist grammars which is quite
formal and allows for a better interface with se-
mantics, and some usual algorithms for parsing
and learning.

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