## A Modal Perspective on the Computational Complexity of Attribute Value Grammar (1992)

Venue: | Journal of Logic, Language and Information |

Citations: | 42 - 7 self |

### BibTeX

@ARTICLE{Blackburn92amodal,

author = {Patrick Blackburn and Edith Spaan},

title = {A Modal Perspective on the Computational Complexity of Attribute Value Grammar},

journal = {Journal of Logic, Language and Information},

year = {1992},

volume = {2},

pages = {129--169}

}

### Years of Citing Articles

### OpenURL

### Abstract

Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value descriptions unify amounts to testing for modal satisfiablity. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalisations, and the ability to express recursive constraints. Two main techniques are used: either Kripke models with desirable properties are constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic. Further possibilities for the application of modal logic in computational linguistics are noted. Attribute Value Structures (AVSs) are probably the most widely used means of representing linguistic structure in current computational linguistics, and the process of unifying...

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Citation Context ...ional Dynamic Logic (PDL) are used. While these are explained, some readers maysnd the additional background provided by [21] helpful. For further information on modal logic the reader is referred to =-=[2-=-3]; and for more on unication based grammar, [44] is useful. Finally, it's worth remarking that there is a hidden agenda: although we emphasize the use of modal logic as tool for grammar specication, ... |

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Citation Context ...bers in their usual order, as follows: X holds at i if holds at i + 1, and G holds at i i for all i 0 i, holds at i 0 . Using the fact that satisability for this language is PSPACE-complete [45], it is easy to prove that the satisability problems for the languages with only hai and [] as modalities are PSPACE complete as well. Using similar methods, we get the same results for the correspo... |

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Citation Context ...roof that follows we make use of a certain 0 1 complete tiling problem. Theorem 3.1 If jLj 2, and A is countably innite then the satisability problem for L KR2 is 0 1 hard. Proof: As shown in [3] [39], the following problem is 0 1 complete: NN tiling: Given asnite set T of tiles, can T tile NN? That is, does there exist a function t from NN to T such that: right(t(n; m)) = left(t(n + 1; ... |

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Citation Context ... computational linguistics, and the process of unifying descriptions of AVSs lies at the heart of many parsers. As a number of people have recently observed (see for example [28], [6], [30], [38] and =-=[43-=-]) the most common formalisms for describing AVSs are notational variants of propositional modal languages, AVSs themselves are Kripke models, and unication amounts to looking for a satisfying model f... |

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Citation Context ...straint. Usually they are multimodal versions of K, the modal logic which puts no constraints on accessibility relations. As is well known, the satisability problem for this logic is PSPACE complete [=-=29-=-]. 3 The universal modality In this section we are going to examine the complexity of the satisability problems for three stronger modal languages, L 2 , L N2 and L KR2 These languages are, respective... |

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Citation Context ...he AVM are satised, including the demand that 1 picks out a unique node. How can we make modal languages referential in this way? The key idea needed can be traced back to early work by Arthur Prior [=-=37]-=-, and Robert Bull [7]: it is to introduce a second sort of atomic symbol constrained to be true at exactly one node. These new symbols 6 `name' the unique node they are true at. In this paper these sy... |

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Citation Context ...xplicitly espouses such views, and its work on feature co-occurrence restrictions remains one of the best examples of the approach in action. More recently, Head Driven Phrase Structure Grammar (HPSG)=-=[3-=-4], has taken this approach even further. Whereas in both GPSG and LFG the idea of unication was only one component (albeit an important one) of the systems, in HPSG the unicational apparatus complete... |

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Citation Context ...ts' business is to state general constraints about which AVSs are admissible. Such views are discernible in some of the earliest work in attribute value grammar, namely Lexical Function Grammar (LFG) =-=[24]-=-. Generalised Phrase Structure Grammar (GPSG) [14], explicitly espouses such views, and its work on feature co-occurrence restrictions remains one of the best examples of the approach in action. More ... |

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Citation Context ...ed in giving a precise account of AVMs that employed disjunction or negation. The distinction wassrst introduced by Pereira and Sheiber [33], and it underpins the in uential work of Kaspar and Rounds =-=[26]-=-[41][42]. Thus the move towards full Boolean expressivity marked an important turning point in the development of Attribute Value formalisms. What do computational linguists do with AVMs? The answer i... |

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Citation Context ...rectly deals with occurrences of the universal modality: ST (2) = 8y([y=x]ST ()): For a detailed discussion of the logical consequences of enriching modal languages with the universal modality, see [1=-=7-=-]. The authors know of only one explicit application of the universal modality to linguistic theorising, namely Evan's [8] analysis of the feature specication defaults of GPSG, which we shall consider... |

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Citation Context ...heorem 3.3, it follows immediately that the satisability problems for L 2 and L N2 are both decidable in nondeterministic exponential time. But we can improve these results. Using methods similar to [=-=3-=-6] and [18] we sketch a construction of a deterministic exponential time algorithm for both L 2 and L N2 satisability. 18 Theorem 3.4 The satisability problems for L 2 and L N2 are decidable in EXPTIM... |

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Citation Context ...g nominals. In addition to the early work by Prior and Bull already mentioned, see [31], [12] and [32] for an examination of nominals in the setting of Propositional Dynamic Logic (PDL); see [11] and =-=[13-=-] for nominals in the setting of modal logic; andsnally see [4] and [5] for nominals in tense logic. The language L KR is L augmented by two new symbols, 0 and . The symbol 0 acts as a name for the nu... |

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Citation Context ...3, it follows immediately that the satisability problems for L 2 and L N2 are both decidable in nondeterministic exponential time. But we can improve these results. Using methods similar to [36] and [=-=1-=-8] we sketch a construction of a deterministic exponential time algorithm for both L 2 and L N2 satisability. 18 Theorem 3.4 The satisability problems for L 2 and L N2 are decidable in EXPTIME. Proof:... |

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Citation Context ...n giving a precise account of AVMs that employed disjunction or negation. The distinction wassrst introduced by Pereira and Sheiber [33], and it underpins the in uential work of Kaspar and Rounds [26]=-=[41]-=-[42]. Thus the move towards full Boolean expressivity marked an important turning point in the development of Attribute Value formalisms. What do computational linguists do with AVMs? The answer is, t... |

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Citation Context ...semantics, but with an algebraic semantics, and duality theory, the study of the connections between the algebraic semantics and the Kripke semantics, is a highly developed branch of model logic; see =-=[16]-=- for a detailed recent account. While some use of the algebraic semantics has been made in connection with Attribute Value structures (Reape [38] for example, uses it to make connections with Smolka's... |

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Citation Context ...ing re-entrancy. There have been a number of logical investigations of intensional languages containing nominals. In addition to the early work by Prior and Bull already mentioned, see [31], [12] and =-=[32]-=- for an examination of nominals in the setting of Propositional Dynamic Logic (PDL); see [11] and [13] for nominals in the setting of modal logic; andsnally see [4] and [5] for nominals in tense logic... |

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Citation Context ...undary; as we shall now show its satisability problem is undecidable. To prove the undecidability result it suces to give a reduction from a 0 1 hard problem to L KR2 satisability. As is shown in [19=-=-=-], tiling problems provide a particularly elegant method of proving lower bounds for modal logics, so we'll use such an approach here. A tile T is a 11 squaresxed in orientation with coloured edges ri... |

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Citation Context ...onjunctive, but many linguists feel it is necessary to be able to express both disjunctive and negative information in their Attribute Value grammars. To give two well known examples due to Kartunnen =-=[2-=-5], one might write number plural case fnominative, genitive, accusativeg an AVM which states that the attribute case takes one of the values nominative, genitive, or accusative, but doesn't say whi... |

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Citation Context ...found in [1], for example. Further, later in the paper some ideas from Propositional Dynamic Logic (PDL) are used. While these are explained, some readers maysnd the additional background provided by =-=[21-=-] helpful. For further information on modal logic the reader is referred to [23]; and for more on unication based grammar, [44] is useful. Finally, it's worth remarking that there is a hidden agenda: ... |

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Citation Context ...or making this distinction was motivated by the diculties involved in giving a precise account of AVMs that employed disjunction or negation. The distinction wassrst introduced by Pereira and Sheiber =-=[33]-=-, and it underpins the in uential work of Kaspar and Rounds [26][41][42]. Thus the move towards full Boolean expressivity marked an important turning point in the development of Attribute Value formal... |

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Citation Context ...ncluding the demand that 1 picks out a unique node. How can we make modal languages referential in this way? The key idea needed can be traced back to early work by Arthur Prior [37], and Robert Bull =-=[7]-=-: it is to introduce a second sort of atomic symbol constrained to be true at exactly one node. These new symbols 6 `name' the unique node they are true at. In this paper these symbols are called nomi... |

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Citation Context ...larly interesting one concerns the organisation of computational lexicons. An important task in this application is the developed of formalisms for representing and manipulating lexical entries. DATR =-=[9-=-] is such a formalism, and an examination of its syntax and semantics suggests that it is open to modal analysis. What sort of benets might result from such an analysis? Complexity results and inferen... |

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Citation Context ...structure in current computational linguistics, and the process of unifying descriptions of AVSs lies at the heart of many parsers. As a number of people have recently observed (see for example [28], =-=[6-=-], [30], [38] and [43]) the most common formalisms for describing AVSs are notational variants of propositional modal languages, AVSs themselves are Kripke models, and unication amounts to looking for... |

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Citation Context ... complete. In the third section we examine modal languages in which recursive constraints on linguistic structure can be expressed, namely systems built using the master modality [] of Gazdar et al. [15] and Kracht [28]. We augment our base languages L, L N and L KR with [], forming L [] , L N [] and L KR[] respectively, and investigate the complexity of their satisability problems. We show that... |

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Citation Context ...tioned, see [31], [12] and [32] for an examination of nominals in the setting of Propositional Dynamic Logic (PDL); see [11] and [13] for nominals in the setting of modal logic; andsnally see [4] and =-=[5-=-] for nominals in tense logic. The language L KR is L augmented by two new symbols, 0 and . The symbol 0 acts as a name for the null transition. In what follows we shall assume without loss of general... |

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Citation Context ...r indicating re-entrancy. There have been a number of logical investigations of intensional languages containing nominals. In addition to the early work by Prior and Bull already mentioned, see [31], =-=[12]-=- and [32] for an examination of nominals in the setting of Propositional Dynamic Logic (PDL); see [11] and [13] for nominals in the setting of modal logic; andsnally see [4] and [5] for nominals in te... |

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Citation Context ...are explained, some readers maysnd the additional background provided by [21] helpful. For further information on modal logic the reader is referred to [23]; and for more on unication based grammar, [=-=44-=-] is useful. Finally, it's worth remarking that there is a hidden agenda: although we emphasize the use of modal logic as tool for grammar specication, it is our belief that modal techniques have a wi... |

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Citation Context ...containing nominals. In addition to the early work by Prior and Bull already mentioned, see [31], [12] and [32] for an examination of nominals in the setting of Propositional Dynamic Logic (PDL); see =-=[11-=-] and [13] for nominals in the setting of modal logic; andsnally see [4] and [5] for nominals in tense logic. The language L KR is L augmented by two new symbols, 0 and . The symbol 0 acts as a name f... |

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Citation Context ...ture in current computational linguistics, and the process of unifying descriptions of AVSs lies at the heart of many parsers. As a number of people have recently observed (see for example [28], [6], =-=[30-=-], [38] and [43]) the most common formalisms for describing AVSs are notational variants of propositional modal languages, AVSs themselves are Kripke models, and unication amounts to looking for a sat... |

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Citation Context ...n current computational linguistics, and the process of unifying descriptions of AVSs lies at the heart of many parsers. As a number of people have recently observed (see for example [28], [6], [30], =-=[38-=-] and [43]) the most common formalisms for describing AVSs are notational variants of propositional modal languages, AVSs themselves are Kripke models, and unication amounts to looking for a satisfyin... |

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Citation Context ...al approach of this paper with other approaches to Attribute Value logic which may more familiar to the reader. Note in particular that the standard translation links our approach with that of Smolka =-=[46]-=-. Smolka was perhaps thesrst person to make explicit the connection between AVSs andsrst order models, and he has proved a number of results concerning a certainsrst order language of AVSs, namely the... |

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Citation Context ...ving a precise account of AVMs that employed disjunction or negation. The distinction wassrst introduced by Pereira and Sheiber [33], and it underpins the in uential work of Kaspar and Rounds [26][41]=-=[42]-=-. Thus the move towards full Boolean expressivity marked an important turning point in the development of Attribute Value formalisms. What do computational linguists do with AVMs? The answer is, they ... |

4 |
Sag 1985. Generalized Phrase structure Grammar
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Citation Context ... which AVSs are admissible. Such views are discernible in some of the earliest work in attribute value grammar, namely Lexical Function Grammar (LFG) [24]. Generalised Phrase Structure Grammar (GPSG) =-=[14]-=-, explicitly espouses such views, and its work on feature co-occurrence restrictions remains one of the best examples of the approach in action. More recently, Head Driven Phrase Structure Grammar (HP... |

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v.: 1984, Correspondence Theory, in Handbook of Philosophical Logic 2, edited by
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Citation Context ...spondence theory. This subject is the 8 systematic study and exploitation of the relationships that exist between modal languages and various classical languages; an excellent overview is provided by =-=[2]-=-. The correspondence between L, L N and L KR andsrst order logic arises as follows. Note that AVSs (that is, Kripke models) can equally well be regarded as models for a certainsrst order language, nam... |