# Abstract categorial grammars with island constraints and effective decidability

A well-known approach to treating syntactic island constraints in the setting of Lambek grammars consists in adding specific bracket modalities to the logic. We adapt this approach to abstract categorial grammars (ACG). Thus we define bracketed (implicational) linear logic, bracketed lambda-calculus, and, eventually, bracketed ACG based on bracketed λ-calculus. This allows us modeling at least simplest island constraints, typically, in the context of relativization. Next we identify specific safely bracketed ACG which, just like ordinary (bracket-free) second order ACG generate effectively decidable languages, but are sufficiently flexible to model some higher order phenomena like relativization and correctly deal with syntactic islands, at least in simple toy examples.

## Authors

• 8 publications
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## 1 Introduction

Abstract categorial grammars (ACG) [2] are a formalism for generating formal languages, similar to well-known Lambek grammars [10], but based on the ordinary (commutative) linear logic [4] and linear -calculus. Variants of this formalism are also known as -grammars [15] and linear grammars [11].

Unlike more traditional Lambek-style categorial grammars, ACG (and their siblings from [15], [11]) are not restricted to word-by-word processing of continuous strings and can easily manipulate discontinuous syntactic elements (i.e. tuples of strings). This gives them a remarkable flexibility and expressivity. From a certain point of view, ACG seem more simple and natural, being based on a more familiar and intuitive commutative logic.

However, as far as natural language modeling is concerned, ACG turn out to be too flexible and too expressive. If Lambek grammars generate precisely the class of context-free languages [16]

, which is probably too weak for a natural language, then ACG, in general, can generate NP-complete languages

[19], which is a catastrophe. It seems that the only large class of ACG known today to generate effectively decidable languages is second order ACG. But in essence, second order ACG simply do not use any logic or any -calculus at all. These grammars generate precisely the class of multiple context-free languages [20], and it is questionable if using -terms in the notation adds something really interesting to the much simpler original formalism of multiple context-free grammars (MCFG) [21].

Unfortunately, potentially explosive parsing complexity is not the only drawback of ACG. For example, it was noticed that such grammars behave rather poorly when modeling coordination [12]. Hybrid type logical grammars [9], which combine commutative constructions of ACG with non-commutative Lambek-style operations, might be a promising improvement (but see also [18]).

Another issue, and it is what will be discussed in this work, is how to deal with syntactic islands, typical for natural languages.

From the point of view of logic and -calculus, syntactic island constraints are restrictions on introducing -abstraction in terms and implication in types. Islands present a problem for Lambek grammars as well, although in Lambek grammars -abstraction is already restricted by very non-commutativity of the calculus. In the context of non-commutative calculus, an approach to treating islands was proposed in [13], [14]. It consists in adding to the underlying logic specific bracket modalities in types and bracketed structure in sequents, which essentially make the calculus partly non-associative. Types in a bracketed sequent are not allowed to move out of brackets and this precludes derivations introducing unwanted implications. Languages generated by such bracketed Lambek grammars turn out to be context-free [5], just as in the bracket-free case. As for parsing such grammars, known algorithms so far are exponential [8].

There are also proposals for modeling island constraints in the ACG setting. In particular, see [17], where dependent types are used, and [7], where a general technique for encoding different language phenomena is discussed.

In this paper, we adapt to ACG the bracket-modality approach of [13], [14]. Thus we define bracketed (implicational) linear logic, then bracketed -calculus, which is typed with bracketed linear logic, and, eventually bracketed ACG based on bracketed -calculus. This allows us modeling at least simplest island constraints, typically, in the context of relativization, essentially mimicking structures of [13].

Next we identify specific safely bracketed and second order safely bracketed fragments of the logic, which satisfy certain

bounded interpolation

property. In particular, any proof of a second order safely bracketed sequent with formulas of some bounded complexity is equivalent to a one obtained from proofs of smaller sequents with formulas of the same complexity using only the Cut rule. This allows us reducing a second order safely bracketed ACG to a weakly equivalent ordinary (bracket-free) second order ACG, hence to an MCFG (just as any Lambek grammar is reduced to a context-free grammar in [16]). It follows that second order safely bracketed ACG of this paper generate effectively decidable languages. Yet, unlike ordinary second order ACG, they turn out to be sufficiently flexible to model some higher order phenomena like relativization and correctly deal with syntactic islands, at least in simple toy examples.

This effective decidability is the main interest of our approach compared to other proposals. We find quite remarkable that it is precisely the presence of island constraints that blocks explosive complexity of generated languages. Cannot this give a hint to the origin of island constraints (at least, some of them) in the natural language?

We should stress though that second order safely bracketed grammars are still very similar to ordinary second order ACG. For example, they require an excessive amount of atomic types compared to higher order formalisms. It can be said that, “morally”, second order ACG is not so much a categorial (logical) grammar as a generalized context-free formalism (see [6]), close to MCFG. From such a point of view, second order safely bracketed grammar to a large extent, also, is a generalized context-free formalism, but extended with some logical constructions. Well, why not?

Also we make no attempt to approach coordination, which is problematic in ACG. We hope, however, that bracket modalities and safe bracketing eventually can be combined with some hybrid constructions in the style of [9].

What is crucially missing at the moment is some concrete (denotational) model of bracketed logic and bracketed

-calculus that would give good understanding of the system. All results so far are obtained by purely syntactic manipulations on terms and derivations, copying, whenever possible, constructions of bracketed Lambek calculus from [13]. It is not clear if the given axiomatic (basically copied from [13]) is indeed well-suited for the ACG setting and cannot be improved or what its possible extensions to other formalisms like [9] should be like. Understanding denotational semantics of bracketed linear logic is a subject of current work.

Finally, we do not propose any direct parsing algorithm. Brutal reduction of a second order safely bracketed ACG to an ordinary ACG and, eventually, to an MCFG is certainly exponential in the size of the original grammar. This subject is left for future study.

## 2 Bracketed linear logic

In this section we define bracketed linear logic that eventually will be the typing system for our grammars.

Given a set of atomic types or atomic formulas the set of bracketed types or bracketed formulas is defined by induction:

• If then ;

• if then

 A⊸B,□A,□−1A∈Tp[](N).

Formulas (types) not containing the and connectives are familiar (linear) implicational formulas (types). We denote the set of implicational formulas (types) as .

From now on we use the words “type” and “formula” as completely synonymous, preferably saying “type” when there are some -terms around, and “formula” otherwise.

We will consider specific bracketed sequents, which are defined using configurations of formulas.

A configuration (over a given set ) is defined by induction:

• if then is an elementary configuration;

• if , are elementary configurations, then the multiset is a configuration;

• if is a configuration, then is an elementary configuration.

According to the above definition, the empty multiset is a configuration. In order to have consistent notation, we introduce the convention .

A configuration without brackets is called a context.

A sequent (over ) is an expression of the form , where is a configuration (over ), and is a formula (from ).

In the sequel, a Latin letter always stands for a configuration consisting of a single formula.

In order to formulate sequent calculus rules, we introduce notation for substituting a subconfiguration.

The expression denotes a configuration with a selected occurrence of the formula , and is the result of substituting the configuration for .

In details:

• if , then ;

• if is the multiset , then ;

• if , then .

###### Definition 1

Bracketed implicational linear logic is defined by the following sequent calculus rules.

 A⊢A(Id),Γ⊢AΓ′(A)⊢BΓ′(Γ)⊢B(Cut),
 Γ,A⊢BΓ⊢A⊸B(⊸R),Γ⊢AΓ′(B)⊢CΓ′(Γ,A⊸B)⊢C(⊸L),
 Γ⊢A[Γ]⊢□A(□R),Γ([A])⊢BΓ(□A)⊢B(□L),
 [Γ]⊢AΓ⊢□−1A(□−1R),Γ(A)⊢BΓ([□−1A])⊢B(□−1L).
###### Lemma 1

The system is cut-free.

Proof by routine and lengthy induction on derivation.

Proving cut-elimination by induction on derivation amounts essentially to specifying a cut-elimination algorithm. In the sequel we assume that such an algorithm is indeed specified.

### 2.1 Natural deduction

We will consider bracketed logic as a typing system for a term calculus extending linear -calculus. Since it is traditional to formulate -calculus in the natural deduction format, we develop a natural deduction system for .

###### Definition 2

The system is defined by the following rules.

 A⊢A(Id),
 Γ,A⊢BΓ⊢A⊸B(⊸I),Γ⊢AΓ′⊢A⊸BΓ,Γ′⊢B(⊸E),
 Γ⊢A[Γ]⊢□A(□I),Γ⊢□AΓ′([A])⊢BΓ′(Γ)⊢B(□E),
 [Γ]⊢AΓ⊢□−1A(□−1I),Γ⊢□−1AΓ′(A)⊢BΓ′([Γ])⊢B(□−1E).
###### Note 1

The set of sequents derivable is closed under the Cut rule.

Proof Induction on derivation.

###### Note 2

A sequent is derivable in the natural deduction system iff it is derivable in the sequent calculus .

There are translations of sequent calculus proofs to natural deduction proofs, and of natural deduction proofs to sequent calculus proofs.

Proof Right introduction rules of sequent calculus are the same as introduction rules of natural deduction. Left introduction rules of the sequent calculus are emulated in natural deduction using the Cut rule, which is admissible by the preceding note. Elimination rules of natural deduction are emulated in the sequent calculus similarly.

It should be noted though that the two above translations are not mutual inverses.

In the next section we label bracketed sequents with terms and develop a typed term calculus.

## 3 Term assignment

We use a term language extending linear -calculus. We assume that the reader is familiar with basic notions of -calculus, in particular, -conversion, - and - reductions and normalization. See [1] for a reference.

Terms of the extended language are built using application, abstraction and four special constants .

Let a countable set of variables not containing special constants be given. The set of bracketed linear -terms over is defined by induction:

• any is a term, and ;

• if are terms and , then is a term, and ;

• if is a term and is a special constant then is a term, and ;

• if is a term and , then is a term, and .

For a term , elements of the set are free variables of . Variable occurrences in which are not free are bound. We identify terms differing by renaming bound variables.

We use the usual notational conventions: the outermost pair of brackets is dropped, the application symbol is omitted (i.e. ) application is left-associative (i.e. ), nested abstractions are abbreviated , and the body of abstraction extends to the right as much as possible.

Terms without special constants are familiar linear -terms, we call them bracket-free. We denote the set of bracket-free terms as .

Labeled configurations are defined recursively, similarly to ordinary configurations.

• An expression of the form , where is a type and is a variable, is an elementary labeled configuration;

• if are elementary labeled configurations not having common variables, then the multiset is a labeled configuration;

• if is a nonempty labeled configuration, then is an elementary labeled configuration.

A labeled configuration without brackets is a labeled context.

A typing judgement is an expression of the form , where is a labeled configuration, is a term and is a type.

If is a labeled configuration and is a configuration obtained by erasing from all variables, we say that is a labeling of .

Similarly, if a labeled configuration is a labeling of , then any typing judgement is a labeling of the sequent .

We now develop a calculus of typing judgements, which is a labeling of bracketed linear logic proofs.

### 3.1 Bracketed λ-calculus

###### Definition 3

The system is defined by the following rules

 x:A⊢x:A(Id),
 Γ,x:A⊢M:BΓ⊢λx.M:A⊸B(⊸I),Γ⊢M:AΓ′⊢N:A⊸BΓ,Γ′⊢NM:B(⊸E),
 Γ⊢M:A[Γ]⊢bM:□A(□I),Γ⊢M:□AΓ′([x:A])⊢N:BΓ′(Γ)⊢N[x:=b−1M]:B(□E),
 [Γ]⊢M:AΓ⊢uM:□−1A(□−1I),Γ⊢M:□−1AΓ′(x:A)⊢N:BΓ′([Γ])⊢N[x:=u−1M]:B(□−1E).

The fragment of the above system not involving and -connectives in types, special constants in terms and brackets in configurations is the familiar linear typed -calculus. We denote it as . Note, however, that, unlike , the full system does not satisfy the familiar property that any derivable typing judgement has unique derivation.

Now let be a natural deduction proof of a sequent , and let be some labeling of . The following is immediate.

###### Note 3

There is a unique term and a derivation of the labeled sequent in such that is obtained from by erasing all terms.

In notation as above we say that is the labeling of and is the labeling of the conclusion of induced by .

Observe that Note 1 lifts to the labeled setting.

###### Note 4

The system is closed under the Substitution rule:

 Γ⊢M:AΓ′(x:A)⊢N:BΓ′([Γ])⊢N[x:=M]:B.

### 3.2 In the sequent calculus format

We now assign terms to sequent calculus proofs as well.

###### Definition 4

The system is defined by the following rules

 x:A⊢x:A(Id),Γ⊢M:AΓ′(x:A)⊢N:BΓ′(Γ)⊢N[x:=M]B(Cut),
 Γ,x:A⊢M:BΓ⊢λx.M:A⊸B(⊸R),Γ⊢M:AΓ′(x:B)⊢N:CΓ′(Γ,f:A⊸B)⊢N[x:=fM]:C(⊸L),
 Γ⊢M:A[Γ]⊢bM:□A(□R),Γ([x:A])⊢M:BΓ(x′:□A)⊢M[x′=b−1x]:B(□L),
 [Γ]⊢M:AΓ⊢uM:□−1A(□−1R),Γ(x:A)⊢M:BΓ([x′:□−1A])⊢M[x:=u−1x′]:B(□−1L).

Obviously, just as in the case of natural deduction, given a sequent calculus proof , we can define a labeling of in .

Then Note 2 lifts to the labeled setting.

###### Note 5

A typing judgement is derivable in iff it is derivable in .

We want, however, to have a one-to-one translation between natural deduction and sequent calculus formats. Thus, we are going to introduce an equivalence relation on derivations. The equivalence comes from -equivalence of terms, which we define next.

### 3.3 Normalization

We define -reductions of bracketed -terms, extending familiar -reduction of ordinary -calculus.

###### Definition 5

Binary relation of one-step -reducibility on terms is the smallest relation satisfying the properties

 (λx.M)N↦βN[x:=M]

,

 u−1(uM)↦βM,b−1(bM)↦βM,

and if then

 MN↦βM′N,NM↦βNM′,λx.M↦βλx.M′,

and

 cM↦βcM′

for any special constant .

The relation of -reducibility is the reflexive transitive closure of .

We say that a term reduces to if .

A term is normal if it does not reduce to any term other than itself.

A term is a normal form of if and is normal.

###### Lemma 2

If a typing judgement is derivable in then has a unique normal form and the typing judgement is derivable in as well.

Proof by induction on derivation.

We say that a typing judgement is normal if the term is normal.

###### Lemma 3 (Subformula property)

If a normal typing judgement is derivable in , then all types occurring in its derivation are subformulas of types occurring in and .

Proof By induction on derivation we establish that if

• and , or

• and , or

• and ,

then is a subformula of a type occurring in .

Using the above, we prove the lemma, again by induction on derivation.

We say that two terms are -equivalent if they have the same normal form

Similarly, we say that two derivable typing judgements , , are -equivalent if is -equivalent to .

Now let , be natural deduction proofs of the same sequent .

Let be some labeling of . We say that natural deduction proofs , are equivalent if labelings of their conclusions induced by are -equivalent. (See Note 3).

Obviously the above definition does not depend on a choice of .

### 3.4 Relationship with cut-elimination

From the point of view of sequent calculus, normalization corresponds to cut-elimination, and term assignment is a way to define an equivalence relation identifying a sequent proof with its cut-free form.

###### Note 6

A typing judgement derivable in without the Cut rule is normal.

###### Note 7

Let a sequent calculus proof of the sequent reduce to a proof by cut-elimination.

If is a labeling of , and , are labelings of, respectively, , induced by , with conclusions, respectively, and , then -reduces to .

We define equivalence of sequent calculus proofs, just as for natural deduction proofs.

Namely two sequent proofs of are equivalent if some (hence any) labeling of induces -equivalent labelings of their conclusion.

Then we easily observe the following.

###### Note 8

Any sequent calculus proof is equivalent to its cut-free form.

The correspondence between sequent calculus proofs and natural deduction proofs is one-to-one up to equivalence.

###### Note 9

Given a labeling of a configuration , there is a one-to-one correspondence -equivalence classes of derivable typing judgements of the form and equivalence classes of sequent calculus proofs of .

## 4 Commutation of rules

The material of this section is a digression and will not be used in the rest of the paper. Yet the question we are considering here certainly deserves attention.

Unlike the case of usual typed linear -calculus, in the system a derivable typing judgement may have different derivations. It seems natural to select some standard form of a derivation, if possible.

This is indeed possible by the following lemma.

###### Lemma 4

Let the typing judgement

 Γ⊢M:A (1)

be derivable in .

Then and we have the following possibilities:

1. is a variable and , or

2. with , , and for some type we have derivable sequents

 Γ′⊢K:X,Γ′′⊢N:X⊸A,

or

3. , , and the sequent

 Γ,x:A1⊢M′:A2

is derivable, or

4. , , and the sequent

 Γ′⊢M′:A′

is derivable, or

5. , and the sequent

 [Γ]⊢M′:A′

is derivable, or

6. , , and the sequent

 Γ′⊢M′:□−1A

is derivable, or

7. , and the sequent

 Γ⊢M′:□□−1A

is derivable.

Proof by induction on .

A nontrivial case is when .

Then the last rule in the derivation of (1) must be () or (E).

Assume that the last rule is ().

Then we have representations

 Γ=Γ1([Γ2]),M′=N[x:=u−1K],

and for some type the sequents

 Γ1⊢K:□−1X,Γ2(x:X)⊢N:A

are derivable.

If , the statement follows from the induction hypothesis applied to .

Otherwise, from the induction hypothesis applied to we get

 X=A,Γ2(x:X)=x:A.

Hence , , and possibility () holds. (Note that, by the induction hypothesis applied to , we have .)

Assume now that the last rule in the derivation of (1) is (E).

Then we have representations

 Γ=Γ1(Γ2),M′=N[x:=b−1K],

and for some type the sequents

 Γ1⊢K:□X,Γ2([x:X])⊢u−1N:A

are derivable.

If , the statement follows from the induction hypothesis applied to .

Otherwise, , and by the induction hypothesis we have

 X=□−1A,Γ2([x:X])=[x:A].

Hence and . Thus possibility () holds.

A pleasant (and probably not surprising) corollary is that in a derivable typing judgement (or, by Note 9, in a sequent proof) all brackets can be erased without loss of any information.

If is a (labeled) configuration, let us say that the underlying (labeled) context is the (labeled) context obtained from by erasing all brackets.

Conversely, if is a (labeled) context we say that is a bracketing of , if is the underlying context of .

Lemma 4 immediately yields the following.

###### Corollary 1

If is a term, is a type, and is a labeled context, such that for some bracketing of the typing judgement is derivable, then this bracketing is unique.

## 5 Safe bracketing and bounded interpolation

We now discuss a specific safely bracketed fragment, which will be used later to define effectively decidable safely bracketed grammars.

Let be a -free formula.

We define the order of by induction:

• , if is atomic;

• ;

• .

For a general formula , we say that is safely bracketed if

• any subformula of is -free;

• for any subformula of , the antecedent is -free.

We define the order of a safely bracketed formula as the maximal order of its -free subformula.

We say that a configuration is safely bracketed if all formulas occurring in are safely bracketed.

We define the order of a safely bracketed configuration as

 ord(Γ)=maxZ∈Γord(Z).

We will be interested in the second order safely bracketed fragment. Its crucial property is specific bounded interpolation, which we are gong to discuss now.

### 5.1 Bounded interpolation

We will need several parameters measuring complexity of formulas and sequents.

For a formula , the bare size of is defined inductively by

• for atomic;

• ;

• .

The -rank and -rank of , respectively, and are defined by

• for atomic;

• ,

;

• , ;

• , .

For a configuration the bare size and -, -ranks are defined, respectively, as

 |Γ|=∑Z∈Γ|Z|,rk□(Γ)=maxZ∈Γrk□(Z),rk□−1(Γ)=maxZ∈Γrk□−1(Z).

The bare size norm is defined as

 ||Γ||=maxZ∈Γ|Z|.

The cardinality (i.e. number of formula occurrences) of is denoted as .

Finally, given a configuration with an occurrence of a formula , the degree of in is defined by induction:

• ;

• , if occurs in ;

• .

#### 5.1.1 Interpolation in the safely bracketed case

###### Note 10

Let be safely bracketed formulas, and let be a safely bracketed configuration with an occurrence of , such that the sequent is derivable.

Then .

Proof by induction on a cut-free derivation.

###### Corollary 2

Let be safely bracketed formulas with , and let be a safely bracketed configuration with an occurrence of .

If is a proof of , then, up to equivalence of proofs, the last rule in is introducing .

Proof by induction on a cut-free form of . (Using Note 10 when the last rule in is .)

###### Corollary 3

Let be safely bracketed formulas with , and let be a safely bracketed configuration with an occurrence of .

If is a proof of , then there exists a representation , and , up to equivalence, is obtained from proofs of, respectively, the sequents and using the Cut rule.

Proof Follows immediately from the preceding corollary. We take as the proof

 B⊢B[B′]⊢B(□−1R)

corresponding to the derivable typing judgement .

###### Corollary 4

Let be safely bracketed formulas with , and let be a safely bracketed configuration with an occurrence of .

Assume that is a proof of .

If is not of the form , then is of the form , and there exists a representation

 Γ=Γ′([Γ0,B])

such that is equivalent to a proof obtained from proofs using the Cut rule as follows:

 π0Γ0⊢X~π[X,B]⊢Y[Γ0,B]⊢Y\rm({Cut})π′Γ′(Y)⊢AΓ⊢A\rm({Cut}).

Proof Induction on a cut-free form of . The proof is obtained as

 X⊢XY⊢Y[□−1Y]⊢E(□−1E)[X,B]⊢Y(⊸L)

corresponding to the derivable typing judgement

 [x:X,f:B]⊢u−1(fx):Y.

For a configuration , let us denote the set of all subformulas occurring in as .

###### Lemma 5

Let be a proof of a safely bracketed sequent , where is -free.

Then there exists a finite sequence of configurations

 Γ1,…,Γn

over and formulas (interpolants)

 X1,…,Xn∈Sf(Γ,A),

where for each , either , are -free or , such that is equivalent to a proof obtained from proofs of, respectively,

 Γ1⊢X1,…,Γn⊢Xn

using only the Cut rule.

Proof by induction on the number of occurrences in , using Corollaries 3 and 4. (When applying Corollary 4, it should be remembered that if a formula is safely bracketed, then is -free.)

#### 5.1.2 Interpolation in the □−1-free second order case

###### Lemma 6

Let be a -free configuration and , a -free formula, such that .

If the sequent is derivable, then .

Proof by induction on a cut-free derivation.

###### Corollary 5

Let be -free formulas with , .

If is a configuration whose underlying context is , and the sequent is derivable, then .

###### Corollary 6

Let be -free formulas with .

Let be a -free configuration with an occurrence of , such that all formulas in except possibly are of second order.

If the sequent is derivable then .

Proof Observe that any second order formula has bare size greater than 1, so .

###### Corollary 7

Let be a first order -free configuration with , and be a -free formula with .

If the sequent is derivable, then and .

###### Lemma 7

Let be -free configurations with , , and bracket-free.

Let be a -free formula with .

If is a proof of , where , then

1. there exist a configuration with , a representation

 Γ2=Γ′(Γ0)

and a -free formula (interpolant) with

 ord(X)≤2,rk□(X)≤rk□(Γ,A),

such that, up to equivalence, is obtained by the Cut rule from proofs of the sequents and ;

2. moreover, if there is a first order formula in , then can be chosen such that .

Proof (sketch): Induction on a cut-free form of .

In any case, if , then we write and put , .

So assume that

When the last rule in is , or , the statement immediately follows from the induction hypothesis.

Assume that the last rule in is .

Then there is a representation and has the form

 πlΓl⊢SπrΓr(T)⊢AΓ⊢A(⊸L).

It follows that there are representations

 Γl=Γl1,Γl2,Γr=Γr1,Γr2(T), and Γ2=Γr2(Γl2,S⊸T).

(Where can be nonempty only if .)

Let us prove claim of the Lemma.

We assumed that . This means that or .

If , then, by Corollary 7, we have . Then we can use the induction hypothesis applied to , and the claim follows.

If , the reasoning is similar.

Now let us prove claim .

Assume that there is a first order formula in .

If , then , hence , and, again we use the induction hypothesis applied to and deduce the claim.

If and , the reasoning is similar.

Finally, if and , so that contains no other formula, then , and, by Corollary 7, the configuration is empty. So .

Then we take and .

###### Lemma 8

Let be a -free configurations with , , and be a -free formula with .

If is a proof of the sequent , then there exist a configuration , a representation

 Γ=Γ′(Γ0),#(Γ0)=2

and a -free formula (interpolant) with

 ord(X)≤2,rk□(X)≤rk□(Γ,A),|X|≤||Γ,A||,

such that, up to equivalence, is obtained from proofs of the sequents and by the Cut rule.

Proof By Corollary 7 we have .

We put , , and apply Lemma 7.

This gives us a subconfiguration with a representation and an interpolant .

Moreover, if there is a first order formula in , we choose such that . Then we apply Corollary 5 to the derivable sequent .

And if there is no first order formula in , we apply Corollary 6 to the derivable sequent .

#### 5.1.3 Interpolation in the second order safely bracketed case

Let be a configuration or a finite set of formulas.

We say that is a generalized subformula of , if

 |Z|≤||Γ||,rk□(Z)≤rk□(Γ),rk□−1(Z)≤rk□−1(Γ), (2)

and all atomic formulas occurring in are occurring in .

Let us denote the set of generalized subformulas of as .

###### Lemma 9

Let be a proof of a safely bracketed sequent , where is -free and .

Then there exists a finite sequence of safely bracketed configurations over and safely bracketed interpolants , where for all the formula has the order not greater than 2, and the configuration has both the order and the cardinality not greater than 2, such that is equivalent to a proof obtained from proofs of, respectively,

 Γ1⊢X1,…,Γn⊢Xn

using only the Cut rule.

Proof immediate from Lemmas 5 and 8.

Now we are prepared to discuss abstract categorial grammars.

## 6 Abstract categorial grammars

In this section we recall standard (bracket-free) abstract categorial grammars [2].

### 6.1 Linear signatures

###### Definition 6

A linear signature, or, simply, signature is a tuple , where is a set of atomic types, is a set of variables, is a set of constants, and

 τ:C→Tp(N)

is a type assignment map.

We denote the set of linear bracket-free -terms built from and as . That is .

Given a signature , the signature axioms of are the labeled sequents

 ⊢c:τ(c), for c∈C.

A typing judgement is derivable in (notation: ) if it is derivable from signature axioms using rules of linear -calculus.

We say that a term is typeable in if there is a type such that . In this case we say that is the type of in .

###### Note 11

In notation as above, a typing judgement is derivable in iff there exist constants