{ "paper_id": "C92-1020", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T12:34:43.349428Z" }, "title": "On the Satisfiability of Complex Constraints", "authors": [ { "first": "Luis", "middle": [], "last": "Damas", "suffix": "", "affiliation": { "laboratory": "", "institution": "Universidade do Porto", "location": { "addrLine": "Rua do Campo Alegre 823", "postCode": "4000", "settlement": "Porto", "country": "Portugal" } }, "email": "" }, { "first": "Giovanni", "middle": [ "B" ], "last": "Varile", "suffix": "", "affiliation": { "laboratory": "", "institution": "CEC", "location": { "postCode": "2920", "settlement": "Luxembourg", "country": "Luxembourg" } }, "email": "" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "The main problem arising from the use of complex constraints in Computational Linguists is due to the NP-hardness of checking whether a given set of constraints is satisfiable and, in the affirmative case, of generating the set of minimal models which satisfy the constraints. In this \"paper we show how the CLG approach to rewriting constraints while constructing a partial model can greatly reduce the size of the original constraints and thus contribute to reduce the computational problem.", "pdf_parse": { "paper_id": "C92-1020", "_pdf_hash": "", "abstract": [ { "text": "The main problem arising from the use of complex constraints in Computational Linguists is due to the NP-hardness of checking whether a given set of constraints is satisfiable and, in the affirmative case, of generating the set of minimal models which satisfy the constraints. In this \"paper we show how the CLG approach to rewriting constraints while constructing a partial model can greatly reduce the size of the original constraints and thus contribute to reduce the computational problem.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Most of the more recent formalisms in Computational Linguistics involve complex constraints, i.e. involving either disjunction or and negation of equality constraints [Kay85, KB82, PS87] .", "cite_spans": [ { "start": 175, "end": 180, "text": "KB82,", "ref_id": null }, { "start": 181, "end": 186, "text": "PS87]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "From a practical point of view, the main problem arising from the introduction of complex constraints is due to the NP-hardness of checking whether a given set of constraints is satisfiable and, in the affirmative case, of generating tile set of minimal models which satisfy thc constraints.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Since the NP-hardness of the problem tends to manifest in practical applications in a dra~ matic way there were several proposals of algorithms to minimize this problem [Kas87, ED88, MK91] .", "cite_spans": [ { "start": 169, "end": 176, "text": "[Kas87,", "ref_id": null }, { "start": 177, "end": 182, "text": "ED88,", "ref_id": null }, { "start": 183, "end": 188, "text": "MK91]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "However, in our opinion, any practical approach to the problem should rely on applying an exponential satisfaction algorithm as seldom as possible and also, when that cannot be avoided, to reduce as much as possible the size of tile input formulae to which the algorithm is applied.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "A classical way in which one can reduce the size of input formulae is by factoring, i.e. ex-pressing it as a conjunction of smaller formulae which do not have variables in common. As a matter of fact if instead of checking the satisfiability of a formula of length nk we check the satisfiability of n formulae of size k we are gaining an exponential reduction in time. This strategy is also explored in [MK91] .", "cite_spans": [ { "start": 403, "end": 409, "text": "[MK91]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Another way of reducing the size of input formulae which has been used in CLG implementations [DV89, BDMVg0, DMVglb] , and which can be combined with the previous approaches, is to build a partial model of the input formula and to apply the algorithm to the remaining constraints. Since in many cases this is able to do away with all the constraints this also achieves the goal of using the exponential satisfaction algorithm as seldom as possible. This method can also be seen as a generalization of steps 1 and 2 of the algorithm described in [Kas87], while avoiding some of its pitfalls.", "cite_spans": [ { "start": 94, "end": 100, "text": "[DV89,", "ref_id": null }, { "start": 101, "end": 108, "text": "BDMVg0,", "ref_id": null }, { "start": 109, "end": 116, "text": "DMVglb]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "In this paper we start by showing, by translating to the feature logics context the methods for first order terms introduced in [DMV91a, DMV91b] , how the CLG approach to rewriting constraints while constructing a partial model can greatly reduce the size of the original constraints and thus contribute to reduce the cornputational problem.", "cite_spans": [ { "start": 128, "end": 136, "text": "[DMV91a,", "ref_id": "BIBREF2" }, { "start": 137, "end": 144, "text": "DMV91b]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "After that will review the notion of factorization for feature logic formulae showing under which conditions it can be done.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "From an intuitive point of view, our approach attempts to explore the fact that complex constraints arising from combining descriptions of linguistic objects, tend to take the form of a large conjunction of smaller constraints.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "An example", "sec_num": "2" }, { "text": "To illnstrate this approach we will use the following example from ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "An example", "sec_num": "2" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "V [(f num) = pl ^ (f case) \u00a2 dat]]", "eq_num": "(1)" } ], "section": "An example", "sec_num": "2" }, { "text": "By looking at the top equality conjuncts in as it will be slmwn in the next section, is equivalent to the satisfiability of (1). However we should point out that the above process, wtfieh takes at most quadratic time since, by using adequate data structures, each rewrite can be done in linear time and the number of rewrites is limited by the size of the fornmla, has succeed in drastically reducing the size of the original formula.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "An example", "sec_num": "2" }, { "text": "(1) we can conclude that any model of (1) is In this section we iutroduee a rewriting system, subsumed by which is essentially an adaptation to feature", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "A rewriting system", "sec_num": "3" }, { "text": "gend = mase ] f = iv .... a j ('~)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "A rewriting system", "sec_num": "3" }, { "text": "Using this information and the standard axioms of Feature Logics we can rewrite 1 Since 11o atomic top conjunct remains we would now have to resort to the exponentiM algorithm to cheek the satisfiability of (6), which, structures of the one presented in [DMV91a] for constraints Oil first order terms.", "cite_spans": [ { "start": 254, "end": 262, "text": "[DMV91a]", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "A rewriting system", "sec_num": "3" }, { "text": "The purpose of this rewriting system is to fornlalize the rewriting process illustrated in thc previous section. The rules of the rewriting system, whidt are justified by theorems of the logic underlying features structures, have been dcsigncd with the aim of bringing out conjunctions to the top while avoiding more than a linear growth of the size of constraints.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "A rewriting system", "sec_num": "3" }, { "text": "Tc~ formalize the rewriting system we will use Feature Logics [Smo89] and its notation. Thus we will use a and b to denote atoms, f to denote a feature name, x and y to denote variables and and t to denote either an atom or a variable. We start by recalling the notion of solved feature clause of [Smo89] which is the feature logic w~rsion of the standard definition of solved form [lier30, Mah88] .", "cite_spans": [ { "start": 62, "end": 69, "text": "[Smo89]", "ref_id": "BIBREF9" }, { "start": 297, "end": 304, "text": "[Smo89]", "ref_id": "BIBREF9" }, { "start": 382, "end": 390, "text": "[lier30,", "ref_id": null }, { "start": 391, "end": 397, "text": "Mah88]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "A rewriting system", "sec_num": "3" }, { "text": "A set of formulae C is said to be a solved feature clause if it satisfies the following conditions:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "A rewriting system", "sec_num": "3" }, { "text": "1. every constraint in 6' has one of the follow- Smolka a]so shows that any satisfiable set C of, possibly negated, atomic formulae can be reduced to solved form by using the following simplification rules, which again are the feature logic version of llerbrand's rules for solving sets of equations of first order terms: We call a solved feature clause positive iff it includes only constraints of the form fx-s and x-s.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "A rewriting system", "sec_num": "3" }, { "text": "We can now make precise the notion of partial model used in the previous section as a positive solved feature clause .\u00a34.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "{x--s) LI C ---", "sec_num": "1." }, { "text": "Note that the form required for A~ is essentially the one produced by an unification algorithm for feature structures.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "{x--s) LI C ---", "sec_num": "1." }, { "text": "Given a set of constraints C we say that a feature clause C ~ is a minimal model of C if every model ofC r is a model of C and no proper subset of C ~ satisfies this condition.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "{x--s) LI C ---", "sec_num": "1." }, { "text": "From Theorem 5.6 of [Smo89] we can conclude that for any C there is a finite uumber of minimal models of C.", "cite_spans": [ { "start": 20, "end": 27, "text": "[Smo89]", "ref_id": "BIBREF9" } ], "ref_spans": [], "eq_spans": [], "section": "{x--s) LI C ---", "sec_num": "1." }, { "text": "\"]2he aim of the CLG rewriting system is to produce from a set of constraints Co a partial model .\u00a34 and a smaller set of constraints C such that any minimal model of Co can be obtained by conjoining (i.e. \"unifying\") a minimal model of C with .~ and moreover for any minimal model of C the reunion Ad U C is satisfiable.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "{x--s) LI C ---", "sec_num": "1." }, { "text": "We start by defining a set of rewriting rules ~ for terms and constraints:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "{x--s) LI C ---", "sec_num": "1." }, { "text": "z -*~ t if x='t E 34 b-x ---+~ x-b fa-t --~ false fx T~M false if fx-I C 3d a--b ---*~ false t-t ~ ~rue f x-s ~ t-s if f x-I E 34 ~true --*~4 false ~false ~ true ~C ~ C ~(C v C') ~.~ ~C A ~C' C A true-*~ C C A false -.2~ false C V true---+A~ true C V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "{x--s) LI C ---", "sec_num": "1." }, { "text": "We now define a rewriting system for pairs (34,17) by first closing C under ---*s~4 and then using the following rules", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "false -.~,~ C", "sec_num": null }, { "text": "(M, c u {false}) -~ ({rut,e}, ~) (M,Cu {true)) ~ (34,C) (34,c u {~-s}) ~ (34 v {~-s},C) (34,c u {/z-t}) --(34 u {fz-'-t}, c)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "false -.~,~ C", "sec_num": null }, { "text": "with the convention that after each application of one of the rewrite rules the new partial model is reduced to solved tbrm and the resulting set of constraints is closed nmler ~M.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "false -.~,~ C", "sec_num": null }, { "text": "As in [DMV91a] the rewrite process could be extended to use negated atomic constraints in C.", "cite_spans": [ { "start": 6, "end": 14, "text": "[DMV91a]", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "false -.~,~ C", "sec_num": null }, { "text": "We will now sketch the proof of the claims made above about the rcwriting system.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "false -.~,~ C", "sec_num": null }, { "text": "We will first argue that if given an initial set of constraints Co we apply the rewriting system to (~,Co) to obtain {34,C} then Co ( ....... precisely the conjunct of all the constraints in Co) is equivalent to 3.4 UC. As a matter of fact this follows from the fact that each rewrite rule is associated with a similar meta-theorem of l:irst Order Logic and/or the axioms of Feature Logits.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "false -.~,~ C", "sec_num": null }, { "text": "As for the other property of the rewriting system, namely that the minimal models of Co are obtained by conjoining NI with those of C, we will only sketch the argument of the proof which follows from the fact that if some minimal model C' of C was inconsistent witl~ .M then, after reducing C to disjunctive form, at least one of the disjunctions s|muld subsume C ~ and thus should be inconsistent with A4 while admiring itself a model. Now, since every atomic formula occurring ill the disjunction already occurcd in C it is possible to derive a contradiction with the hypothesis that C was closed under --~M. To see this we notice that each disjunct in the disjunctive form of C can be regarded as feature clause C'. Now if A4 U C\" was unsatisfiable then some sequence of simplification rules should lead to a clash. Now, noticing that any atomic formula in C\" umst already be present in C, it is easy to check that any sequence of simplification rules would involve only fornmlae from C\" and a clash with a formula in f14 would thus be impossible.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "false -.~,~ C", "sec_num": null }, { "text": "On the factorization of feature constraints Although tile rewriting system introduced in tile previous section can be seen as a factorization of tile original constraint it is also important, in accordance with tile discussion in the introduction, to further factorizing tile con-straint produced before an exponential satisfaction algorithm is api)licd. Now given a conjunction C A C' of feature constraints it is easy to provc, using argumcnts similar to the ones used in thc previuus section, that a sufficient condition for tim conjunction to be satisliablc iff each of the conjunets is independently satisfiable, can be expressed ms follows:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "4", "sec_num": null }, { "text": "1. any variable x which occurs in C in an atolnic formula x--s or s--a: does not occur in C' and vlce-versa 2. for every variable x and feature f such that fx occurs in C, /x does nnt occur in C'", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "4", "sec_num": null }, { "text": "and vice-versa.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "4", "sec_num": null }, { "text": "Note that given a conjunctiw~ wet of constraints it is possible to l)artition it into miidmal disjoint sets satisfying the. conditions above in quadratic time.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "4", "sec_num": null }, { "text": "Using the above criteria results in achievillg the objective described in [MK91] of separating not only the treatment of constraints dealing with completely indel)endcnt linguistic descriptions but also of independent phenomena for the same the [iaguistic descriptions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "4", "sec_num": null }, { "text": "[DMV91b]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "5" }, { "text": "[DVSg]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "5" }, { "text": "[El)aS]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "5" }, { "text": "[oaw86]", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "5" }, { "text": "Tim strategy for handling complex constraints described in this paper has I)een successfully put to use in ditl~rent CLG implementations.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "5" }, { "text": "One of the adwmtagcs of using a rewrite sys-[I[cr30] tcm is that it is easily extended to other atomic constraints like the ones in CLG(2) dealing with lists. Similarly it can bc adapted to cope with the extensions to feature constraints proposed [Job91] in [Joh9i] .", "cite_spans": [ { "start": 247, "end": 254, "text": "[Job91]", "ref_id": null }, { "start": 258, "end": 265, "text": "[Joh9i]", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Conclusions", "sec_num": "5" } ], "back_matter": [], "bib_entries": { "BIBREF0": { "ref_id": "b0", "title": "Nehna Moreira, and Giovanni I1. Varile. CLG: Constraint logic grammars", "authors": [ { "first": "Sergio", "middle": [], "last": "Balari", "suffix": "" }, { "first": "Luis", "middle": [], "last": "Damas", "suffix": "" } ], "year": null, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Sergio Balari, Luis Damas, Nehna Moreira, and Giovanni I1. Varile. CLG: Constraint logic grammars.", "links": null }, "BIBREF1": { "ref_id": "b1", "title": "P~vceediugs of the 13th htternutional Confercuce on Computa~ioual Linguistics [Kay85] (COLING)", "authors": [], "year": 1990, "venue": "", "volume": "3", "issue": "", "pages": "7--12", "other_ids": {}, "num": null, "urls": [], "raw_text": "In ll.Karlgrcn, editor, P~vceediugs of the 13th htternutional Confer- cuce on Computa~ioual Linguistics [Kay85] (COLING), volume 3, pages 7-12, llelsinki, 1990.", "links": null }, "BIBREF2": { "ref_id": "b2", "title": "The forlnal and processing models of CLG", "authors": [ { "first": "Nelma", "middle": [], "last": "Luis L)amas", "suffix": "" }, { "first": "Giovanni", "middle": [ "B" ], "last": "Moreira", "suffix": "" }, { "first": "", "middle": [], "last": "Varilc", "suffix": "" } ], "year": 1991, "venue": "Fiifth Coufereucc of the Emopeaa Chap= ter of the Association for Computational Linguistics", "volume": "", "issue": "", "pages": "173--178", "other_ids": {}, "num": null, "urls": [], "raw_text": "Luis l)amas, Nelma Moreira, and Giovanni B. Varilc. The forlnal and processing models of CLG. In Fiifth Coufereucc of the Emopeaa Chap= ter of the Association for Compu- tational Linguistics, pages 173-178, Berlin, 1991.", "links": null }, "BIBREF3": { "ref_id": "b3", "title": "CLG -a general logic grammar formalism, ht Proceediugs of the Workshop on Constraiut Propagation and Linguistic Description, forthcoming, I,ugano", "authors": [ { "first": "Luis", "middle": [], "last": "Damns", "suffix": "" }, { "first": "Nelma", "middle": [], "last": "Moreira", "suffix": "" }, { "first": "Giovamd", "middle": [ "B" ], "last": "Varile", "suffix": "" } ], "year": 1991, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Luis Damns, Nelma Moreira, and Giovamd B. Varile. CLG -a general logic grammar formalism, ht Pro- ceediugs of the Workshop on Con- straiut Propagation and Linguistic Description, forthcoming, I,ugano, Switzcrlan(l, 1991.", "links": null }, "BIBREF4": { "ref_id": "b4", "title": "Vartie. CLG: A grammar formalism based oll constraint resolution", "authors": [ { "first": "Giovanni", "middle": [ "B" ], "last": "Luis L)ama", "suffix": "" } ], "year": 1989, "venue": "Lecture Notes in Artificial Intelligence", "volume": "89", "issue": "", "pages": "175--186", "other_ids": {}, "num": null, "urls": [], "raw_text": "Luis l)ama, s and Giovanni B. Var- tie. CLG: A grammar formal- ism based oll constraint resolution. In l';.M.Morgado and J.P.Martitm, editors, EPIA 89, volume 390 of Lecture Notes in Artificial Intelli- gence, pages 175-186. Springer Ver- lag, 1989.", "links": null }, "BIBREF5": { "ref_id": "b5", "title": "Unification of disjunctive feature descriptions", "authors": [ { "first": "A", "middle": [], "last": "Eisete", "suffix": "" }, { "first": "J", "middle": [], "last": "D6rre", "suffix": "" } ], "year": 1930, "venue": "~6th Annual Meeting of the Association [or Computational Linguistics", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "A. Eisete and J. D6rre. Unification of disjunctive feature descriptions. In ~6th Annual Meeting of the As- sociation [or Computational Lin- guistics, Buffalo, New York, 1988. Barbara Grosz, Karen Sparck Jones, and Bomty ],ynn Webber, editors. Readings in Natural Lan- guage Processing. Morgan Ka/ff- mann, 1986. Jacques llerbrand. Recherches sur la theovie de la dgmoustration, 7'h~se de Doctorat. Univ. Paris, Paris, 1930.", "links": null }, "BIBREF6": { "ref_id": "b6", "title": "Features and formulae", "authors": [ { "first": "Mark", "middle": [], "last": "Johnson", "suffix": "" } ], "year": 1991, "venue": "Computational Linguistics", "volume": "17", "issue": "2", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Mark Johnson. Features and for- mulae. Computational Linguistics, 17(2), 1991.", "links": null }, "BIBREF7": { "ref_id": "b7", "title": "A unification metbod for disjm~etive t;eature deseril)tions", "authors": [ { "first": "Il", "middle": [ "T" ], "last": "Kasper", "suffix": "" } ], "year": 1987, "venue": "25th Annual Meeting of the Association for Computational Linguistics", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "IL T. Kasper. A unification metbod for disjm~etive t;eature deseril)tions. In 25th Annual Meeting of the As- sociation for Computational Lin- guistics, Stanford, CA, 1987.", "links": null }, "BIBREF8": { "ref_id": "b8", "title": "l>arsing in fimctional unification grammar", "authors": [ { "first": "Martin", "middle": [], "last": "Kay", "suffix": "" } ], "year": null, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Martin Kay. l>arsing in fimc- tional unification grammar. In l)avid It. Dowty, I,auri Karttunen, and Arnold M. Zwicky, editors, [KB82]", "links": null }, "BIBREF9": { "ref_id": "b9", "title": "Lexical functional grammar: A formal system for grammatical representation", "authors": [ { "first": "R", "middle": [], "last": "Kaplan", "suffix": "" }, { "first": "J", "middle": [], "last": "Bresban", "suffix": "" } ], "year": 1982, "venue": "", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "R. Kaplan and J. Bresban. Lex- ical functional grammar: A for- mal system for grammatical repre- sentation. In Joan Bresnan, edi- tor, The Mental Representation of Grammatical Relations. MIT Press, 1982.", "links": null }, "BIBREF10": { "ref_id": "b10", "title": "Complete a.xiomatizations of the algebras of finite, rational and infinite trees", "authors": [ { "first": "J", "middle": [], "last": "Michael", "suffix": "" }, { "first": "", "middle": [], "last": "Mailer", "suffix": "" } ], "year": 1988, "venue": "", "volume": "704", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Michael J. Mailer. Complete a.x- iomatizations of the algebras of fi- nite, rational and infinite trees. Technical report, IBM Thomas J. Watson Research Center, P.O. Box 704, Yorktown tleights, NY 10598, U.S.A., 1988.", "links": null }, "BIBREF11": { "ref_id": "b11", "title": "A method for disjunctive constraint satisfaction", "authors": [ { "first": "John", "middle": [ "T" ], "last": "Maxwell", "suffix": "" }, { "first": "Ronald", "middle": [ "M" ], "last": "Kaplan", "suffix": "" } ], "year": 1991, "venue": "Current Issues in Parsing Technology", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "John T. Maxwell and Ronald M. Kaplan. A method for disjunctive constraint satisfaction. In Masaru Tomita, editor, Current Issues in Parsing Technology. Kluwer Aca- demic Publishers, 1991.", "links": null }, "BIBREF12": { "ref_id": "b12", "title": "Center for the Study of Language and Information Stanford", "authors": [ { "first": "Carl", "middle": [], "last": "Pollard", "suffix": "" }, { "first": "Ivan", "middle": [], "last": "Sag", "suffix": "" } ], "year": 1987, "venue": "", "volume": "1", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Carl Pollard and Ivan Sag. [n- formation Based Syntax: and Se- mantics, Volume 1, Fundamentals, volume 13. Center for the Study of Language and Information Stan- ford, 1987.", "links": null }, "BIBREF13": { "ref_id": "b13", "title": "Feature constraint logics for unification grammars", "authors": [ { "first": "Gert", "middle": [], "last": "Smolka", "suffix": "" } ], "year": 1989, "venue": "IBM Wissenschafliches Zentrum", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Gert Smolka. Feature constraint logics for unification grammars. Technical report, IBM Wissenschafliches Zen- trum, Institut fiir Wissensbasierte Systeme, 1989. IWBS Report 93.", "links": null } }, "ref_entries": { "FIGREF0": { "num": null, "text": "[MK91] originated by conjoining the agreement features of the lexi-AcrEs DE COLING-92. NANTES, 23-28 AOr3T 1992 1 0 g PROC. OF COLING-92, NANTES. AUG. 23-28, 1992 eal information for tim German words die and Koffer. [(feas~) = ,~om v (/ease) = acc] ^ [[(f gend) = fern A (fnum)= sg]V (f sum) = pt] ^ (f aend) = mase ^ (f pers) = 3 ^ [[(f num) = sg ^ (lease) \u00a2 gen]", "type_str": "figure", "uris": null }, "FIGREF1": { "num": null, "text": "as follows [(f case) = nora V (f ease) = ace] A [[false A (f num)= sg] V (f n~,n) : pl] ^ true A true A [[(f .... ) = sg A (f .... ) 7 k Ben] V [(f,,ur.) = pl A (f e~se)\u00a2 ,tall] (a) which, by using the standard rules of Propositional Calculus, can be simplified to [(f case) =noru V (lease) = ace] A (f sum) = pl ^ [[(f.um) = sg ^ (f ease) \u00a2 ge.] V [(f nun,) = pl m (f ease) \u00a2 datl](4)Again, froin the atomic top conjunct in (4) we can refine (2) to obtain[ gend = mase ] f=/p~,.s--3", "type_str": "figure", "uris": null }, "FIGREF2": { "num": null, "text": "* {x \"-s} U [s/x]C if \u2022 ~k .... I x occurs in C 2, {a--'=}uc-, {='----a}UC 3. {fx--s, fx--t} UC ---* {fx--s,s--f} UC 4. {s-s}uC~C 5. {Sa T}uC ~C 7. {atb}uC-*Cifutb.", "type_str": "figure", "uris": null } } } }