A decidable linear logic for transforming DRSs in context - CiteSeerX

A decidable linear logic for transforming DRSs in context Tsutomu Fujinami1

IMS, University of Stuttgart

Abstract

We present a decidable linear logic for encoding and transforming Discourse Representation Structures (DRSs) in context. The logic is a particular fragment of intuitionistic propositional linear logic (ILL), slimed down by not allowing for any occurrence of exponential, !, in the consequence and slightly enriched by allowing to combine commutative and non-commutative multiplicative connectives. Its model is given as processes with geometric structure, which can also be seen as proofnets explicit with respect to location and direction. We show that the model is rich enough to encode DRSs and that the proof search is bound to be nite and terminates.

1 Introduction The construction of Discourse Representation Structures (DRSs) has been studied by several researchers, e.g., [Muskens 1996]. The research aims to know how DRSs can be constructed from sentences. Inspired by the research, we study the issue of transforming DRSs. Our research is motivated by a practical problem. Discourse Representation Theory (DRT) [Kamp and Reyle 1993] is currently applied to building application systems such as speech translation system [Kay et al. 1994]. In the case of translation, the meaning of a source language sentence is represented as a DRS, which is transformed into another DRS appropriate to generate a target sentence. The question in building such a system is this: How can one be assured that the transformation will always terminate, however it succeeds or fails? Our goal is to answer the question positively. To answer the question, we formalise the transformation of DRSs in a logic. This gives us at least two advantages. First, the problem of termination is reduced to the decidability problem. Second, eliminating redundant transformation steps becomes equivalent to cut elimination. Given these incentives, we turn to linear logic among many logics for two reasons. For one reason, linear logic gives us a natural setup to model transformation owing to its resource sensitiveness. For anoher, its semantics is directly related to computation. The second point is important because we consider DRSs as proof or program to be typed in the logic. In this setting, we can regard the transformation as proof search. One can recognise in our work several sources that help to develop our ideas. To encode DRSs, we follow the approach advocated by [Copestake et al. 1995] in that we keep the description of DRSs at. The technique helps us to keep our logic propositional because we do not need to consider nested structures. 1 The work is funded by the German Federal Ministry of Education, Science, Research and Technology (BMBF) in the framework of the Verbmobil project (Grant No. 01 IV 701 N3).

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To represent a DRS as a set of terms, we follow the glue language approach proposed by [Dalrymple et al. 1993] in that we glue terms using the multiplicative connective, . The relation between terms and types is as with [Muskens 1996], but we employ as term a variant of the -calculus [Milner et al. 1992] to encode DRSs, which is typed by our fragment of linear logic. The relation between the -calculus and linear logic is as proposed by [Bellin and Scott 1994], but we simplify their encoding to t the -calculus terms into interaction nets [Lafont 1995]. Overview: We go through three tasks. First, we design a decidable fragment of linear logic, then specify its model, into which DRSs can be encoded. We nally prove the decidability. Due to the limited space, we cannot explain the speach-translation system we are developing and our approach to translation in Verbmobil project. Readers are referred to [Dorna and Emele 1996] and [Buschbeck-Wolf and Dorna 1997].

2 The decidable fragment It is known that multiplicative linear logic (MLL) is decidable. MLL is however too restrictive to specify the transformation in context because a formula can only be referred to once while we would like to refer to formulas more than once if they de ne part of contexts. By context, we mean a part of DRSs that is not transformed itself but aects the way some other parts are transformed. We have to therefore extend the fragment with the exponential, !, but then we face the problem that the decidability of multiplicative-exponential linear logic (MELL) is unknown [Lincoln 1995]. We observe that the source of diculty is in allowing for both unlimited supply and consumption of resources. For our purpose, however, the latter is not necessary because transformed DRSs do not need to be referred to repeatedly. Once we restrict the use of exponential to the antecedent, the decidability of the fragment is trivial. Reusable resources are referred to as many times as a demand arises and will be erased out when they are not needed anymore. The second elaboration is made on multiplicative conjunction, . The connective is usually commutative or non-commutative exclusively, but we allow to combine both kinds of connectives so that we can specify both set and list data structures. Table 1 shows our fragment of linear logic. The fragment is basically the (associative) Lambek Calculus as presented in [Abrusci 1996], but it is extended by combining both the commutative connective, , and the non-commutative one, . Note that the exchange rule is applicable to the commutative connective only. The weakening is as usual, but the contraction rule only allows to generate A without the exponential. With the side condition that A should be neither in ? nor in
, the rule prohibits the logic from generating a number of unused !As. We also exclude the rules for dereliction (Table 2). The same eect by the Dereliction Left rule can be derived in our logic with the Contraction, Left, and Weakening rules. The Dereliction Right rule does not conform to our idea of eliminating any occurrences of the exponential in the consequent.







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(, is de ned as the non-directional version of

The linear implication,

n

and

=, but the de nition is not included in the table to save the space. Note that

the logic is weaker than classical intuitionistic logic because no formulas in the form of !A B can be constructed as they lead to an occurrence of !A in consequence by the Left rule.

(

(

Exchange

?; A B;
C ?; B A;
C

Weakening

?;
B ?; !A;
B



`



`

?; !A A;
B ?; !A;
B

`

`

Identity

`

`

`

`

`

`

1

Left

?;
A ?; 1;
A



Left

?; A; B;
C ?; A B;
C

? A
B ?;
A B



Left

?; A; B;
C ?; A B;
C

? A
B ?;
A B

n

Left

? A ; B;
C ; ?; A B;
C

A; ? B ? AB

`

`

`

`



`

`

`

`

n

`

`

`

`

1

1

`

`

`



62

? A ; A;
B Cut ; ?;
B

A A



Contraction (A ?;
)





Right



Right

`

`

`


C = Left ?; AB=A;?; B; ;
C

n

Right

n

Right

?; A B = Right ? B=A Table 1: Sequent calculus formalisation of the logic `

`

`

`

`

A Dereliction Left ??;;!AA BB !? !? !A Dereliction Right Table 2: The dereliction rules (Not in our fragment) `

`

`

`

3 The model The model is essentially proofnets [Lafont 1995] explicit with respect to location and direction. It can also be seen as -calculus terms [Milner et al. 1992] enriched with geometric structure. Table 3 shows the relation between proofnets and -terms. The explanation is in order. We mark propositions in the antecedent with and assign them an action such that it imports a datum of the type indicated by the proposition. We also assume that any actions composing a process occur at a particular location and write the location together with the proposition, e.g., x : A, meaning that the action occurs at x. The propositions >

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in the consequence are on the other hand left unmarked and assigned exporting actions. In the -translation, x a means the action such as exporting the datum a at x and y(a) the action such as importing the datum a at y. (We assume in the table that a is a datum of the type A and b a datum of the type B .) Terms assigned to composite types are almost as proposed in [Bellin and Scott 1994], but we retain the geometric structure. The term assigned to y : A B is for example expressed as a tree, y x ; x a ; x b , where y x is the mother node, x a the left daughter, and x b the right daughter. The term can be seen as de ning a complex action such that a and b are exported through x, which is accessible through y. h i

i



h

h i

h

i

h i

h ii

h

i

h i

-translation

Proofnet

Axiom x : A

y : A> (x = y)

Cut

x : A x : A>

( )-Negative

x : B > x : A> y : (A B )>



-Negative



-Positive



-Negative

n

h

h

h



h



h

n

=-Negative

x : A> x : B y : (A=B )>

=-Positive

x : B> x : A y : A=B

!-Read(left)

[x(x a1

x : A> x : B > y : (A B )> x:A x:B y:A B x : A x : B> y : (A B )> x : B x : A> y:A B

!-Delete

h i






-Positive

n

[x a

6

h

n

h

h

y(a)]

i




x(a2 ))]

y(x); x(b); x(a) y(x); x(a); x(b)

i

i

y x ;x a ;x b h

i

h i

h ii

y(x); x a ; x(b) h i

y x ; x b ; x(a) h

i

h i

i

i

y(x); x(a); x b

h ii

y x ; x(b); x a h

i

x : (!A)>

h ii

0

x : A> y : (!A)> y : (!A)>

h

!y:y(a ); x(a ); !y:y(a +1 ) i

i

i

i

Table 3: The proofnet and -translation The enriched -calculus with geometry enables us to encode DRSs. Let us take up as an example a relation, r(a; b). The two place relation r is regarded as a process such that it takes two data, a and b, from left-hand side at s to 4

yield r to the right and is encoded as u t ; t(s); s(y); s(x) ; t r , where we assume that the interaction occurs at t accessible through u. We employ the non-commutative connective, , to type the process and express its type as R=(A B ), assuming that the relation r is of type R, x of type A, and y of type B . The objects a and b to ll the argument are regarded as an action such as exporting a or b. They are encoded as v a and w b and are typed as A and B , respectively. These three (sub)processes comprise the process encoding r(a; b), bound altogether. For typing the process, we employ the commutative connective, , to glue types and express the type as R=(A B ) A B . h

h i h

i

h ii





h i



h i







4 The proof search The above model is identical to proofnets when the direction and location are suppressed. For instance, the above process encoding r(a; b) can be seen as a proofnet connected by two id-links indicated by sharing the same symbols, a and b, when x is replaced by a and y by b. We can therefore employ the proofnet technology to construct proofs. The procedure is based on [Roorda 1991], consisting of four steps. We rst form a tree whose edges are decorated with formulas. We then connect every pair of leaf nodes whose edges are decorated by dual atomic formulas. At this step, one can duplicate exponential parts untill all the leaf nodes whose edges are decorated with a positive atomic formula, are paired with a node whose edge is decorated with a negative formula. In the next step, we ensure that there should be no crossing between pairs. If there is a cross, we try to uncross the graph by exchanging a left and right subpart where the connective is commutative. The graph is nally checked against a criteria, which is in essence the same as the colouring proposed in [Roorda 1991], but tuned up for refutation. For the sake of explanation, we call connected leaf nodes id-node. When colouring nets, subnets are coloured in the same colour as their parent if the parent is of conjunctive type, but either subnet has to be coloured dierently if it is of disjunctive type. The wrong con guration can then be detected if both edges of an id-node are coloured the same or if there exist two id-nodes whose edges are coloured identically. Because the number of pairs is nite and the graph can be coloured in nite steps, the proof search always terminates.

5 Conclusion We presented a fragment of linear logic and de ne a model for it. The model is the -calculus enriched with geometric structure and expressive enough to encode DRSs. The technique is to encode semantic objects bit by bit and to glue them together, using the commutative connective, . There is a proof search procedure which terminates, however the search succeeds or fails. In our setting, translation rules are encoded as a reusable process and the proof search can be seen as translation procedure. In our logical approach to trans

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lation, eliminating redundant transformation steps becomes equivalent to cut elimination. The structure found in proofnets will also be useful for selecting the best translation among many candidates, e.g. the atter the structure is, the better the translation is.

References Abrusci, V. M.: 1996, Lambek calculus, cyclic multiplicative-additive linear logic, noncommutative multiplicative-additive linear logic: Language and sequent calculus, in V. M. Abrusci and C. Casadio (eds.), Proceedings of 1996 Roma Workshop on Proofs and Linguistic Categoris, pp 21{48 Bellin, G. and Scott, P. J.: 1994, On the -calculus and linear logic, Theoretical Computer Science 135(1), 11{65 Buschbeck-Wolf, B. and Dorna, M.: 1997, Using hybrid methods and resources in semantic-based transfer, in Proceedings of the 2nd International Conference on Recent Advances in Natural Language Processing Copestake, A., Flickinger, D., Malouf, R., Riehemann, S., and Sag, I.: 1995, Translation using Minimal Recursion Semantics, in 6th International Conference on Theoretical and Methodological Issues in Machine Translation (TMI'95), Leuven, Belgium Dalrymple, M., Lamping, J., and Saraswat, V.: 1993, LFG semantics via constraints, in Proceedings of the 6th Meeting of the European Association for Computational Linguistics Dorna, M. and Emele, M. C.: 1996, Semantic-based transfer, in Proceedings of Coling '96, pp 316{321 Kamp, H. and Reyle, U.: 1993, From Discourse to Logic: Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory, Dordrecht: Kluwer Kay, M., Gawron, J. M., and Norvig, P.: 1994, Verbmobil - a translation system for face-to-face dialog, No. 33 in CSLI Lecture Notes Lafont, Y.: 1995, From proof-nets to interaction nets, in J.-Y. Girard, Y. Lafont, and L. Regnier (eds.), Advances in Linear Logic, pp 225{247, Cambridge University Press Lincoln, P.: 1995, Deciding provability of linear logic formulas, in J.-Y. Girard, Y. Lafont, and L. Regnier (eds.), Advances in Linear Logic, pp 109{122, Cambridge University Press Milner, R., Parrow, J., and Walker, D.: 1992, A calculus of mobile processes, parts I and II, Information and Computation 100, 1{40 and 41{77 Muskens, R.: 1996, Combining Montague semantics and discourse representation, Linguistics and Philosophy 19, 143{186 Roorda, D.: 1991, Resource Logics: Proof-theoretical Investigations, Ph.D. thesis, University of Amsterdam, Amsterdam

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