1 Introduction
Abstract categorial grammars (ACG) [2] are a formalism for generating formal languages, similar to wellknown 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 Lambekstyle categorial grammars, ACG (and their siblings from [15], [11]) are not restricted to wordbyword 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 contextfree languages [16]
, which is probably too weak for a natural language, then ACG, in general, can generate NPcomplete 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 contextfree languages [20], and it is questionable if using terms in the notation adds something really interesting to the much simpler original formalism of multiple contextfree 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 noncommutative Lambekstyle 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 noncommutativity of the calculus. In the context of noncommutative 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 nonassociative. 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 contextfree [5], just as in the bracketfree 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 bracketmodality 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 (bracketfree) second order ACG, hence to an MCFG (just as any Lambek grammar is reduced to a contextfree 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 contextfree 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 contextfree 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 wellsuited 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
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.
Lemma 1
The system is cutfree.
Proof by routine and lengthy induction on derivation.
Proving cutelimination by induction on derivation amounts essentially to specifying a cutelimination 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.
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 leftassociative (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 bracketfree. We denote the set of bracketfree 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 calculus
Definition 3
The system is defined by the following rules
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:
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
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 onetoone 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 onestep reducibility on terms is the smallest relation satisfying the properties
,
and if then
and
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 cutelimination
From the point of view of sequent calculus, normalization corresponds to cutelimination, and term assignment is a way to define an equivalence relation identifying a sequent proof with its cutfree 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 cutelimination.
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 cutfree form.
The correspondence between sequent calculus proofs and natural deduction proofs is onetoone up to equivalence.
Note 9
Given a labeling of a configuration , there is a onetoone 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
(1) 
be derivable in .
Then and we have the following possibilities:

is a variable and , or

with , , and for some type we have derivable sequents
or

, , and the sequent
is derivable, or

, , and the sequent
is derivable, or

, and the sequent
is derivable, or

, , and the sequent
is derivable, or

, and the sequent
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
and for some type the sequents
are derivable.
If , the statement follows from the induction hypothesis applied to .
Otherwise, from the induction hypothesis applied to we get
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
and for some type the sequents
are derivable.
If , the statement follows from the induction hypothesis applied to .
Otherwise, , and by the induction hypothesis we have
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
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
The bare size norm is defined as
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 cutfree 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 cutfree 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
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
such that is equivalent to a proof obtained from proofs using the Cut rule as follows:
Proof Induction on a cutfree form of . The proof is obtained as
corresponding to the derivable typing judgement
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
over and formulas (interpolants)
where for each , either , are free or , such that is equivalent to a proof obtained from proofs of, respectively,
using only the Cut rule.
5.1.2 Interpolation in the 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 cutfree 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 bracketfree.
Let be a free formula with .
If is a proof of , where , then

there exist a configuration with , a representation
and a free formula (interpolant) with
such that, up to equivalence, is obtained by the Cut rule from proofs of the sequents and ;

moreover, if there is a first order formula in , then can be chosen such that .
Proof (sketch): Induction on a cutfree 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
It follows that there are representations
(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
and a free formula (interpolant) with
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
(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,
using only the Cut rule.
Now we are prepared to discuss abstract categorial grammars.
6 Abstract categorial grammars
In this section we recall standard (bracketfree) 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
is a type assignment map.
We denote the set of linear bracketfree terms built from and as . That is .
Given a signature , the signature axioms of are the labeled sequents
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
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