{ "paper_id": "P06-1019", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:25:55.448031Z" }, "title": "Partially Specified Signatures: a Vehicle for Grammar Modularity", "authors": [ { "first": "Yael", "middle": [], "last": "Cohen-Sygal", "suffix": "", "affiliation": { "laboratory": "", "institution": "University of Haifa", "location": {} }, "email": "yaelc@cs.haifa.ac.il" }, { "first": "Shuly", "middle": [], "last": "Wintner", "suffix": "", "affiliation": { "laboratory": "", "institution": "University of Haifa", "location": {} }, "email": "" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "This work provides the essential foundations for modular construction of (typed) unification grammars for natural languages. Much of the information in such grammars is encoded in the signature, and hence the key is facilitating a modularized development of type signatures. We introduce a definition of signature modules and show how two modules combine. Our definitions are motivated by the actual needs of grammar developers obtained through a careful examination of large scale grammars. We show that our definitions meet these needs by conforming to a detailed set of desiderata.", "pdf_parse": { "paper_id": "P06-1019", "_pdf_hash": "", "abstract": [ { "text": "This work provides the essential foundations for modular construction of (typed) unification grammars for natural languages. Much of the information in such grammars is encoded in the signature, and hence the key is facilitating a modularized development of type signatures. We introduce a definition of signature modules and show how two modules combine. Our definitions are motivated by the actual needs of grammar developers obtained through a careful examination of large scale grammars. We show that our definitions meet these needs by conforming to a detailed set of desiderata.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Development of large scale grammars for natural languages is an active area of research in human language technology. Such grammars are developed not only for purposes of theoretical linguistic research, but also for natural language applications such as machine translation, speech generation, etc. Wide-coverage grammars are being developed for various languages Hinrichs et al., 2004; Bender et al., 2005; King et al., 2005) in several theoretical frameworks, e.g., LFG (Dalrymple, 2001 ) and HPSG (Pollard and Sag, 1994) .", "cite_spans": [ { "start": 365, "end": 387, "text": "Hinrichs et al., 2004;", "ref_id": "BIBREF8" }, { "start": 388, "end": 408, "text": "Bender et al., 2005;", "ref_id": "BIBREF1" }, { "start": 409, "end": 427, "text": "King et al., 2005)", "ref_id": "BIBREF11" }, { "start": 473, "end": 489, "text": "(Dalrymple, 2001", "ref_id": "BIBREF7" }, { "start": 501, "end": 524, "text": "(Pollard and Sag, 1994)", "ref_id": "BIBREF17" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Grammar development is a complex enterprise: it is not unusual for a single grammar to be developed by a team including several linguists, computational linguists and computer scientists. The scale of grammars is overwhelming: for example, the English resource grammar (Copestake and Flickinger, 2000) includes thousands of types. This raises problems reminiscent of those encountered in large-scale software development. Yet while software engineering provides adequate so-lutions for the programmer, no grammar development environment supports even the most basic needs, such as grammar modularization, combination of sub-grammars, separate compilation and automatic linkage of grammars, information encapsulation, etc.", "cite_spans": [ { "start": 269, "end": 301, "text": "(Copestake and Flickinger, 2000)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "This work provides the essential foundations for modular construction of signatures in typed unification grammars. After a review of some basic notions and a survey of related work we list a set of desiderata in section 4, which leads to a definition of signature modules in section 5. In section 6 we show how two modules are combined, outlining the mathematical properties of the combination (proofs are suppressed for lack of space). Extending the resulting module to a stand-alone type signature is the topic of section 7. We conclude with suggestions for future research.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "We assume familiarity with theories of (typed) unification grammars, as formulated by, e.g., Carpenter (1992) and Penn (2000) . The definitions in this section set the notation and recall basic notions. For a partial function F , 'F (x)\u2193' means that F is defined for the value x.", "cite_spans": [ { "start": 93, "end": 109, "text": "Carpenter (1992)", "ref_id": "BIBREF3" }, { "start": 114, "end": 125, "text": "Penn (2000)", "ref_id": "BIBREF16" } ], "ref_spans": [], "eq_spans": [], "section": "Type signatures", "sec_num": "2" }, { "text": "Definition 1 Given a partially ordered set P, \u2264 , the set of upper bounds of a subset S \u2286 P is the set S u = {y \u2208 P | \u2200x \u2208 S x \u2264 y}.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Type signatures", "sec_num": "2" }, { "text": "For a given partially ordered set P, \u2264 , if S \u2286 P has a least element then it is unique.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Type signatures", "sec_num": "2" }, { "text": "Definition 2 A partially ordered set P, \u2264 is a bounded complete partial order (BCPO) if for every S \u2286 P such that S u = \u2205, S u has a least element, called a least upper bound (lub).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Type signatures", "sec_num": "2" }, { "text": "Definition 3 A type signature is a structure TYPE, \u2291, FEAT, Approp , where:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Type signatures", "sec_num": "2" }, { "text": "1. TYPE, \u2291 is a finite bounded complete partial order (the type hierarchy)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Type signatures", "sec_num": "2" }, { "text": "2. FEAT is a finite set, disjoint from TYPE.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Type signatures", "sec_num": "2" }, { "text": "Approp : TYPE\u00d7FEAT \u2192 TYPE (the appropriateness specification) is a partial function such that for every F \u2208 FEAT:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3.", "sec_num": null }, { "text": "(a) (Feature Introduction) there exists a type Intro(F ) \u2208 TYPE such that Approp(Intro(F ), F ) \u2193, and for every t \u2208 TYPE, if", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3.", "sec_num": null }, { "text": "Approp(t, F ) \u2193, then Intro(F ) \u2291 t; (b) (Upward Closure) if Approp(s, F ) \u2193 and s \u2291 t, then Approp(t, F ) \u2193 and Approp(s, F ) \u2291 Approp(t, F ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3.", "sec_num": null }, { "text": "Notice that every signature has a least type, since the subset S = \u2205 of TYPE has the non-empty set of upper bounds, S u = TYPE, which must have a least element due to bounded completeness.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3.", "sec_num": null }, { "text": "Definition 4 Let TYPE, \u2291 be a type hierarchy and let x, y \u2208 TYPE. If x \u2291 y, then x is a supertype of y and y is a subtype of x. If x \u2291 y, x = y and there is no z such that x \u2291 z \u2291 y and z = x, y then x is an immediate supertype of y and y is an immediate subtype of x.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "3.", "sec_num": null }, { "text": "Several authors address the issue of grammar modularization in unification formalisms. Moshier (1997) views HPSG , and in particular its signature, as a collection of constraints over maps between sets. This allows the grammar writer to specify any partial information about the signature, and provides the needed mathematical and computational capabilities to integrate the information with the rest of the signature. However, this work does not define modules or module interaction. It does not address several basic issues such as bounded completeness of the partial order and the feature introduction and upward closure conditions of the appropriateness specification. Furthermore, Moshier (1997) shows how signatures are distributed into components, but not the conditions they are required to obey in order to assure the well-definedness of the combination. Keselj (2001) presents a modular HPSG, where each module is an ordinary type signature, but each of the sets FEAT and TYPE is divided into two disjoint sets of private and public elements. In this solution, modules do not support specification of partial information; module combination is not associative; and the only channel of interaction between modules is the names of types. Kaplan et al. (2002) introduce a system designed for building a grammar by both extending and restricting another grammar. An LFG grammar is presented to the system in a priority-ordered sequence of files where the grammar can include only one definition of an item of a given type (e.g., rule) with a particular name. Items in a higher priority file override lower priority items of the same type with the same name. The override convention makes it possible to add, delete or modify rules. However, a basis grammar is needed and when modifying a rule, the entire rule has to be rewritten even if the modifications are minor. The only interaction among files in this approach is overriding of information. King et al. (2005) augment LFG with a makeshift signature to allow modular development of untyped unification grammars. In addition, they suggest that any development team should agree in advance on the feature space. This work emphasizes the observation that the modularization of the signature is the key for modular development of grammars. However, the proposed solution is adhoc and cannot be taken seriously as a concept of modularization. In particular, the suggestion for an agreement on the feature space undermines the essence of modular design.", "cite_spans": [ { "start": 87, "end": 101, "text": "Moshier (1997)", "ref_id": "BIBREF14" }, { "start": 686, "end": 700, "text": "Moshier (1997)", "ref_id": "BIBREF14" }, { "start": 864, "end": 877, "text": "Keselj (2001)", "ref_id": "BIBREF10" }, { "start": 1246, "end": 1266, "text": "Kaplan et al. (2002)", "ref_id": "BIBREF9" }, { "start": 1953, "end": 1971, "text": "King et al. (2005)", "ref_id": "BIBREF11" } ], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "3" }, { "text": "Several works address the problem of modularity in other, related, formalisms. Candito (1996) introduces a description language for the trees of LTAG. Combining two descriptions is done by conjunction. To constrain undesired combinations, Candito (1996) uses a finite set of names where each node of a tree description is associated with a name. The only channel of interaction between two descriptions is the names of the nodes, which can be used only to allow identification but not to prevent it. To overcome these shortcomings, Crabb\u00e9 and Duchier (2004) suggest to replace node naming by colors. Then, when unifying two trees, the colors can prevent or force the identification of nodes. Adapting this solution to type signatures would yield undesired orderdependence (see below).", "cite_spans": [ { "start": 79, "end": 93, "text": "Candito (1996)", "ref_id": "BIBREF2" }, { "start": 239, "end": 253, "text": "Candito (1996)", "ref_id": "BIBREF2" }, { "start": 532, "end": 557, "text": "Crabb\u00e9 and Duchier (2004)", "ref_id": "BIBREF6" } ], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "3" }, { "text": "To better understand the needs of grammar developers we carefully explored two existing grammars: the LINGO grammar matrix (Bender et al., 2002) , which is a basis grammar for the rapid development of cross-linguistically consistent gram-mars; and a grammar of a fragment of Modern Hebrew, focusing on inverted constructions (Melnik, 2006) . These grammars were chosen since they are comprehensive enough to reflect the kind of data large scale grammar encode, but are not too large to encumber this process. Motivated by these two grammars, we experimented with ways to divide the signatures of grammars into modules and with different methods of module interaction. This process resulted in the following desiderata for a beneficial solution for signature modularization:", "cite_spans": [ { "start": 123, "end": 144, "text": "(Bender et al., 2002)", "ref_id": "BIBREF0" }, { "start": 325, "end": 339, "text": "(Melnik, 2006)", "ref_id": "BIBREF13" } ], "ref_spans": [], "eq_spans": [], "section": "Desiderata", "sec_num": "4" }, { "text": "1. The grammar designer should be provided with as much flexibility as possible. Modules should not be unnecessarily constrained.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Desiderata", "sec_num": "4" }, { "text": "2. Signature modules should provide means for specifying partial information about the components of a grammar.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Desiderata", "sec_num": "4" }, { "text": "3. A good solution should enable one module to refer to types defined in another. Moreover, it should enable the designer of module M i to use a type defined in M j without specifying the type explicitly. Rather, some of the attributes of the type can be (partially) specified, e.g., its immediate subtypes or its appropriateness conditions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Desiderata", "sec_num": "4" }, { "text": "4. While modules can specify partial information, it must be possible to deterministically extend a module (which can be the result of the combination of several modules) into a full type signature.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Desiderata", "sec_num": "4" }, { "text": "5. Signature combination must be associative and commutative: the order in which modules are combined must not affect the result.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Desiderata", "sec_num": "4" }, { "text": "The solution we propose below satisfies these requirements. 1", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Desiderata", "sec_num": "4" }, { "text": "We define partially specified signatures (PSSs), also referred to as modules below, which are structures containing partial information about a signature: part of the subsumption relation and part of the appropriateness specification. We assume enumerable, disjoint sets TYPE of types and FEAT of features, over which signatures are defined. We begin, however, by defining partially labeled graphs, of which PSSs are a special case. ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "q 1 , q \u2032 1 , q 2 \u2208 Q and F \u2208 FEAT, if (q 1 , F, q 2 ) \u2208 Ap and q 1 * q \u2032 1 , then there exists q \u2032 2 \u2208 Q such that q 2 * q \u2032 2 and (q \u2032 1 , F, q \u2032 2 ) \u2208 Ap; and (b) (Maximality) for all q 1 , q 2 \u2208 Q and F \u2208 FEAT, if (q 1 , F, q 2 ) \u2208 Ap then for all q \u2032 2 \u2208 Q such that q \u2032 2 * q 2 and q 2 = q \u2032 2 , (q 1 , F, q \u2032 2 ) / \u2208 Ap.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "A PSS is a finite directed graph whose nodes denote types and whose edges denote the subsumption and appropriateness relations. Nodes can be marked by types through the function T , but can also be anonymous (unmarked). Anonymous nodes facilitate reference, in one module, to types that are defined in another module. T is oneto-one since we assume that two marked nodes denote different types.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "The ' ' relation specifies an immediate subsumption order over the nodes, with the intention that this order hold later for the types denoted by nodes. This is why ' * ' is required to be a partial order. The type hierarchy of a type signature is a BCPO, but current approaches (Copestake, 2002) relax this requirement to allow more flexibility in grammar design. PSS subsumption is also a partial order but not necessarily a bounded complete one. After all modules are combined, the resulting subsumption relation will be extended to a BCPO (see section 7), but any intermediate result can be a general partial order. Relaxing the BCPO requirement also helps guaranteeing the associativity of module combination.", "cite_spans": [ { "start": 278, "end": 295, "text": "(Copestake, 2002)", "ref_id": "BIBREF5" } ], "ref_spans": [], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "Consider now the appropriateness relation. In contrast to type signatures, Ap is not required to be a function. Rather, it is a relation which may specify several appropriate nodes for the values of a feature F at a node q. The intention is that the eventual value of Approp(T (q), F ) be the lub of the types of all those nodes q \u2032 such that Ap(q, F, q \u2032 ). This relaxation allows more ways for modules to interact. We do restrict the Ap relation, however. Condition 3a enforces a relaxed version of upward closure. Condition 3b disallows redundant appropriateness arcs: if two nodes are appropriate for the same node and feature, then they should not be related by subsumption. The feature introduction condition of type signatures is not enforced by PSSs. This, again, results in more flexibility for the grammar designer; the condition is restored after all modules combine, see section 7.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "Example 1 A simple PSS S 1 is depicted in Figure 1, where We impose an additional restriction on PSSs: a PSS is well-formed if any two different anonymous nodes are distinguishable, i.e., if each node is unique with respect to the information it encodes. If two nodes are indistinguishable then one of them can be removed without affecting the information encoded by the PSS. The existence of indistinguishable nodes in a PSS unnecessarily increases its size, resulting in inefficient processing.", "cite_spans": [], "ref_spans": [ { "start": 42, "end": 57, "text": "Figure 1, where", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "Given a PSS S, it can be compacted into a PSS, compact(S), by unifying all the indistinguishable nodes in S. compact(S) encodes the same information as S but does not include indistinguishable nodes. Two nodes, only one of which is anonymous, can still be otherwise indistinguishable. Such nodes will, eventually, be coalesced, but only after all modules are combined (to ensure the associativity of module combination). The detailed computation of the compacted PSS is suppressed for lack of space.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "Example 2 Let S 2 be the PSS of Figure 2 . S 2 includes two pairs of indistinguishable nodes: q 2 , q 4 and q 6 , q 7 . The compacted PSS of S 2 is depicted in Figure 3 . All nodes in compact(S 2 ) are pairwise distinguishable. ", "cite_spans": [], "ref_spans": [ { "start": 32, "end": 40, "text": "Figure 2", "ref_id": null }, { "start": 160, "end": 168, "text": "Figure 3", "ref_id": null } ], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "q 6 q 7 b q 8 q 2 q 3 q 4 q 5 q 1 a F F F F", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Partially specified signatures", "sec_num": "5" }, { "text": "We now describe how to combine modules, an operation we call merge bellow. When two modules are combined, nodes that are marked by the same type are coalesced along with their attributes. Nodes that are marked by different types cannot be coalesced and must denote different types. The main complication is caused when two anonymous nodes are considered: such nodes are coalesced only if they are indistinguishable.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Module combination", "sec_num": "6" }, { "text": "The merge of two modules is performed in several stages: First, the two graphs are unioned (this is a simple pointwise union of the coordinates of the graph, see definition 7). Then the resulting graph is compacted, coalescing nodes marked by the same type as well as indistinguishable anonymous nodes. However, the resulting graph does not necessarily maintain the relaxed upward closure and maximality conditions, and therefore some modifications are needed. This is done by Ap-Closure, see definition 8. Finally, the addition of appropriateness arcs may turn two anonymous distinguishable nodes into indistinguishable ones and therefore another compactness operation is needed (definition 9).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Module combination", "sec_num": "6" }, { "text": "S 1 = Q 1 , T 1 , 1 , Ap 1 , S 2 = Q 2 , T 2 , 2 , Ap 2 be two PLGssuch that Q 1 \u2229 Q 2 = \u2205. The union of S 1 and S 2 , denoted S 1 \u222aS 2 , is the PLG S = Q 1 \u222a Q 2 , T 1 \u222a T 2 , 1 \u222a 2 , Ap 1 \u222a Ap 2 .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "Definition 8 Let S = Q, T, , Ap be a PLG. The Ap-Closure of S, denoted ApCl(S), is the PLG Q, T, , Ap \u2032\u2032 where:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "\u2022 Ap \u2032 = {(q 1 , F, q 2 ) | q 1 , q 2 \u2208 Q and there exists q \u2032 1 \u2208 Q such that q \u2032 1 * q 1 and (q \u2032 1 , F, q 2 ) \u2208 Ap} \u2022 Ap \u2032\u2032 = {(q 1 , F, q 2 ) \u2208 Ap \u2032 | for all q \u2032 2 \u2208 Q, such that q 2 * q \u2032 2 and q 2 = q \u2032 2 , (q 1 , F, q \u2032 2 ) / \u2208 Ap \u2032 }", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "Ap-Closure adds to a PLG the arcs required for it to maintain the relaxed upward closure and maximality conditions. First, arcs are added (Ap \u2032 ) to maintain upward closure (to create the relations between elements separated between the two modules and related by mutual elements). Then, redundant arcs are removed to maintain the maximality condition (the removed arcs may be added by Ap \u2032 but may also exist in Ap). Notice that Ap \u2286 Ap \u2032 since for all (q 1 , F, q 2 ) \u2208 Ap, by choosing q \u2032 1 = q 1 it follows that q \u2032 1 = q 1 * q 1 and (q \u2032 1 , F, q 2 ) = (q 1 , F, q 2 ) \u2208 Ap and hence (q \u2032 1 , F, q 2 ) = (q 1 , F, q 2 ) \u2208 Ap \u2032 . Two PSSs can be merged only if the resulting subsumption relation is indeed a partial order, where the only obstacle can be the antisymmetry of the resulting relation. The combination of the appropriateness relations, in contrast, cannot cause the merge operation to fail because any violation of the appropriateness conditions in PSSs can be deterministically resolved.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "Definition 9 Let S 1 = Q 1 , T 1 , 1 , Ap 1 , S 2 = Q 2 , T 2 , 2 , Ap 2 be two PSSs such that Q 1 \u2229 Q 2 = \u2205. S 1 , S 2 are mergeable if there are no q 1 , q 2 \u2208 Q 1 and q 3 , q 4 \u2208 Q 2 such that the following hold:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "1. T 1 (q 1 )\u2193, T 1 (q 2 )\u2193, T 2 (q 3 )\u2193 and T 2 (q 4 )\u2193 2. T 1 (q 1 ) = T 2 (q 4 ) and T 1 (q 2 ) = T 2 (q 3 ) 3. q 1 * 1 q 2 and q 3 * 2 q 4", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "If S 1 and S 2 are mergeable, then their merge, denoted", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "S 1 \u22d2S 2 , is compact(ApCl(compact(S 1 \u222a S 2 ))).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "In the merged module, pairs of nodes marked by the same type and pairs of indistinguishable anonymous nodes are coalesced. An anonymous node cannot be coalesced with a typed node, even if they are otherwise indistinguishable, since that will result in an unassociative combination operation. Anonymous nodes are assigned types only after all modules combine, see section 7.1.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "If a node has multiple outgoing Ap-arcs labeled with the same feature, these arcs are not replaced by a single arc, even if the lub of the target nodes exists in the resulting PSS. Again, this is done to guarantee the associativity of the merge operation. Figure 4 depicts a na\u00efve agreement module, S 5 . Combined with S 1 of Figure 1 , S 1 \u22d2 S 5 = S 5 \u22d2 S 1 = S 6 , where S 6 is depicted in Figure 5 . All dashed arrows are labeled AGR, but these labels are suppressed for readability.", "cite_spans": [], "ref_spans": [ { "start": 256, "end": 264, "text": "Figure 4", "ref_id": null }, { "start": 326, "end": 334, "text": "Figure 1", "ref_id": "FIGREF0" }, { "start": 392, "end": 400, "text": "Figure 5", "ref_id": null } ], "eq_spans": [], "section": "Definition 7 Let", "sec_num": null }, { "text": "Example 4 Let S 7 and S 8 be the PSSs depicted in Figures 6 and 7 , respectively. Then S 7 \u22d2 S 8 = S 8 \u22d2S 7 = S 9 , where S 9 is depicted in Figure 8 . By standard convention, Ap arcs that can be inferred by upward closure are not depicted. ", "cite_spans": [], "ref_spans": [ { "start": 50, "end": 65, "text": "Figures 6 and 7", "ref_id": null }, { "start": 141, "end": 149, "text": "Figure 8", "ref_id": "FIGREF3" } ], "eq_spans": [], "section": "Example 3", "sec_num": null }, { "text": "S 6 = S 1 \u22d2 S 5", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example 3", "sec_num": null }, { "text": "Proposition 2 Given two mergeable PSSs S 1 , S 2 , S 1 \u22d2 S 2 is a well formed PSS.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example 3", "sec_num": null }, { "text": "Proposition 3 PSS merge is commutative: for any two PSSs, S 1 , S 2 , S 1 \u22d2S 2 = S 2 \u22d2S 1 . In particular, either both are defined or both are undefined.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example 3", "sec_num": null }, { "text": "Proposition 4 PSS merge is associative: for all", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example 3", "sec_num": null }, { "text": "S 1 , S 2 , S 3 , (S 1 \u22d2 S 2 ) \u22d2 S 3 = S 1 \u22d2 (S 2 \u22d2 S 3 ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example 3", "sec_num": null }, { "text": "When developing large scale grammars, the signature can be distributed among several modules. A PSS encodes only partial information and therefore is not required to conform with all the constraints imposed on ordinary signatures. After all the modules are combined, however, the PSS must be extended into a signature. This process is done in 4 stages, each dealing with one property: 1. Name resolution: assigning types to anonymous nodes (section 7.1); 2. Determinizing Ap, converting it from a relation to a function (section 7.2); 3. Extending ' ' to a BCPO. This is done using the algorithm of Penn (2000); 4. Extending Ap to a full appropriateness specification by enforcing the feature introduction condition: Again, we use the ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Extending PSSs to type signatures", "sec_num": "7" }, { "text": "By the definition of a well-formed PSS, each anonymous node is unique with respect to the information it encodes among the anonymous nodes, but there may exist a marked node encoding the same information. The goal of the name resolution procedure is to assign a type to every anonymous node, by coalescing it with a similar marked node, if one exists. If no such node exists, or if there is more than one such node, the anonymous node is given an arbitrary type.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Name resolution", "sec_num": "7.1" }, { "text": "The name resolution algorithm iterates as long as there are nodes to coalesce. In each iteration, for each anonymous node the set of its similar typed nodes is computed. Then, using this computation, anonymous nodes are coalesced with their paired similar typed node, if such a node uniquely exists. After coalescing all such pairs, the resulting PSS may be non well-formed and therefore the PSS is compacted. Compactness can trigger more pairs that need to be coalesced, and therefore the above procedure is repeated. When no pairs that need to be coalesced are left, the remaining anonymous nodes are assigned arbitrary names and the algorithm halts. The detailed algorithm is suppressed for lack of space.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Name resolution", "sec_num": "7.1" }, { "text": "Example 5 Let S 6 be the PSS depicted in Figure 5 . Executing the name resolution algorithm on this module results in the PSS of Figure 9 (AGR-labels are suppressed for readability.) The two anonymous nodes in S 6 are coalesced with the nodes marked nagr and vagr, as per their attributes. Cf. Figure 1 , in particular how two anonymous nodes in S 1 are assigned types from S 5 (Figure 4) . ", "cite_spans": [ { "start": 41, "end": 49, "text": "Figure 5", "ref_id": null } ], "ref_spans": [ { "start": 129, "end": 137, "text": "Figure 9", "ref_id": "FIGREF4" }, { "start": 294, "end": 302, "text": "Figure 1", "ref_id": "FIGREF0" }, { "start": 378, "end": 388, "text": "(Figure 4)", "ref_id": null } ], "eq_spans": [], "section": "Name resolution", "sec_num": "7.1" }, { "text": "For each node q, the set of outgoing appropriateness arcs with the same label F , {(q, F, q \u2032 )}, is replaced by the single arc (q, F, q l ), where q l is marked by the lub of the types of all q \u2032 . If no lub exists, a new node is added and is marked by the lub. The result is that the appropriateness relation is a function, and upward closure is preserved; feature introduction is dealt with separately. The input to the following procedure is a PSS whose typing function, T , is total; its output is a PSS whose typing function, T , is total and whose appropriateness relation is a function. Let S = Q, T, , Ap be a PSS. For each q \u2208 Q and F \u2208 FEAT, let 2. If target(q, F ) has a lub, p, then: (a) for all q \u2032 \u2208 target(q, F ), remove the arc (q, F, q \u2032 ) from Ap.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Appropriateness consolidation", "sec_num": "7.2" }, { "text": "target(q, F ) = {q \u2032 | (q, F, q \u2032 ) \u2208 Ap} sup(q) = {q \u2032 \u2208 Q | q \u2032 q} sub(q) = {q \u2032 \u2208 Q | q q \u2032 } out(q) = {(F, q \u2032 ) | (q, F, q \u2032 ) \u2208 Ap", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Appropriateness consolidation", "sec_num": "7.2" }, { "text": "(b) add the arc (q, F, p) to Ap.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Appropriateness consolidation", "sec_num": "7.2" }, { "text": "(c) for all q \u2032 \u2208 Q such that q * q \u2032 , if (q \u2032 , F, p) / \u2208 Ap then add (q \u2032 , F, p) to Ap. (d) go to (1).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Appropriateness consolidation", "sec_num": "7.2" }, { "text": "3. (a) Add a new node, p, to Q with:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Appropriateness consolidation", "sec_num": "7.2" }, { "text": "\u2022 sup(p) = target(q, F ) \u2022 sub(p) = (target(q, F )) u \u2022 out(p) = q \u2032 \u2208target(q,F ) out(q \u2032 ) (b) Mark p with a fresh type from NAMES. (c) For all q \u2032 \u2208 Q such that q * q \u2032 , add (q \u2032 , F, p) to Ap. (d) For all q \u2032 \u2208 target(q, F ), remove the arc (q, F, q \u2032 ) from Ap. (e) Add (q, F, p) to Ap. (f) go to (1).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Appropriateness consolidation", "sec_num": "7.2" }, { "text": "The order in which nodes are selected in step 1 of the algorithm is from supertypes to subtypes. This is done to preserve upward closure. In addition, when replacing a set of outgoing appropriateness arcs with the same label F , {(q, F, q \u2032 )}, by a single arc (q, F, q l ), q l is added as an appropriate value for F and all the subtypes of q. Again, this is done to preserve upward closure. If a new node is added (stage 3), then its appropriate features and values are inherited from its immediate supertypes. During the iterations of the algorithm, condition 3b (maximality) of the definition of a PSS may be violated but the resulting graph is guaranteed to be a PSS. Figure 9 . Executing the appropriateness consolidation algorithm on this module results in the module depicted in Figure 10 . AGR-labels are suppressed. ", "cite_spans": [], "ref_spans": [ { "start": 673, "end": 681, "text": "Figure 9", "ref_id": "FIGREF4" }, { "start": 787, "end": 796, "text": "Figure 10", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Appropriateness consolidation", "sec_num": "7.2" }, { "text": "We advocate the use of PSSs as the correct concept of signature modules, supporting interaction among grammar modules. Unlike existing approaches, our solution is formally defined, mathematically proven and can be easily and efficiently implemented. Module combination is a commutative and associative operation which meets all the desiderata listed in section 4. There is an obvious trade-off between flexibility and strong typedeness, and our definitions can be finely tuned to fit various points along this spectrum. In this paper we prefer flexibility, following Melnik (2005) , but future work will investigate other options.", "cite_spans": [ { "start": 567, "end": 580, "text": "Melnik (2005)", "ref_id": "BIBREF12" } ], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "8" }, { "text": "There are various other directions for future research. First, grammar rules can be distributed among modules in addition to the signature. The definition of modules can then be extended to include also parts of the grammar. Then, various combination operators can be defined for grammar modules (cf. Wintner (2002) ). We are actively pursuing this line of research.", "cite_spans": [ { "start": 301, "end": 315, "text": "Wintner (2002)", "ref_id": "BIBREF18" } ], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "8" }, { "text": "Finally, while this work is mainly theoretical, it has important practical implications. We would like to integrate our solutions in an existing environment for grammar development. An environment that supports modular construction of large scale grammars will greatly contribute to grammar development and will have a significant impact on practical implementations of grammatical formalisms.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "8" }, { "text": "The examples in the paper are inspired by actual grammars but are obviously much simplified.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null } ], "back_matter": [ { "text": "We are grateful to Gerald Penn and Nissim Francez for their comments on an earlier version of this paper. 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Grammars, 5(1):41-63.", "links": null } }, "ref_entries": { "FIGREF0": { "type_str": "figure", "uris": null, "num": null, "text": "A partially specified signature, S 1" }, "FIGREF1": { "type_str": "figure", "uris": null, "num": null, "text": "A partially specified signature with indistinguishable nodes, S 2 The compacted partially specified signature of S 2 Proposition 1 If S is a PSS then compact(S) is a well formed PSS." }, "FIGREF2": { "type_str": "figure", "uris": null, "num": null, "text": "Figure 4: Na\u00efve agreement module, S 5" }, "FIGREF3": { "type_str": "figure", "uris": null, "num": null, "text": "S 9 = S 7 \u22d2 S 8 algorithm of Penn (2000)." }, "FIGREF4": { "type_str": "figure", "uris": null, "num": null, "text": "Name resolution result for S 6" }, "FIGREF5": { "type_str": "figure", "uris": null, "num": null, "text": "a node q and a feature F for which |target(q, F )| > 1 and for all q \u2032 \u2208 Q such that q \u2032 * q, |target(q \u2032 , F )| \u2264 1. If no such pair exists, halt." }, "FIGREF6": { "type_str": "figure", "uris": null, "num": null, "text": "Figure 10: Appropriateness consolidation result" }, "TABREF0": { "type_str": "table", "text": "over TYPE and FEAT is a finite, directed labeled graph S = Q, T, , Ap , where:1. Q is a finite, nonempty set of nodes, disjoint from TYPE and FEAT.", "content": "
2. T : Q \u2192 TYPE is a partial function, marking
some of the nodes with types.
3. \u2286 Q \u00d7 Q is a relation specifying (immedi-
ate) subsumption.
4. Ap \u2286 Q \u00d7 FEAT \u00d7 Q is a relation specifying
appropriateness.
Definition 6 A partially specified signa-
ture (PSS) over TYPE and FEAT is a PLG
S = Q, T, , Ap , where:
1. T is one to one.
2. ' ' is antireflexive; its reflexive-transitive
*
closure, denoted '', is antisymmetric.
3. (a) (Relaxed Upward Closure) for all
", "html": null, "num": null }, "TABREF1": { "type_str": "table", "text": "solid arrows represent the ' ' (subsumption) relation and dashed arrows, labeled by features, the Ap relation. S 1 stipulates two subtypes of cat, n and v, with a common subtype, gerund. The feature AGR is appropriate for all three categories, with distinct (but anonymous) values for Approp(n, AGR) and Approp(v, AGR). Approp(gerund, AGR) will eventually be the lub of Approp(n, AGR) and Approp(v, AGR), hence the multiple outgoing AGR arcs from gerund.", "content": "
Observe that in S 1 , ' ' is not a BCPO, Ap is
not a function and the feature introduction condi-
tion does not hold.
", "html": null, "num": null } } } }