{
    "paper_id": "W98-0136",
    "header": {
        "generated_with": "S2ORC 1.0.0",
        "date_generated": "2023-01-19T06:04:33.608503Z"
    },
    "title": "On Defining TALs with Logical Constraints",
    "authors": [
        {
            "first": "James",
            "middle": [],
            "last": "Rogers",
            "suffix": "",
            "affiliation": {
                "laboratory": "",
                "institution": "Univ. of Central Florida",
                "location": {}
            },
            "email": ""
        }
    ],
    "year": "",
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    "identifiers": {},
    "abstract": "",
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        "paper_id": "W98-0136",
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            {
                "text": "\u2022rn Rogers (1997b) we introduced a new dass of models, three-dimensional tree manifolds (3-TM), that can serve as both thc derived and clerivation structures for TAGs in the same way that trees serve as both clerived and derivation structures for CFGs. The1->e tree-manifolds are higher-dimensional analogs of trees; in a 3-Tl'v! the children of a node form an o~\u2022dinary (two-dimensional) tree just as in ordinary tree1-> the children of a node form a string. From t.his point of view the elementary struct.ures of a TAG can bc interpretcd as labeled local 3-Tl\\faa root node and it.s set of children (a pyramidal structure)-analogous to the interpretation of the rewrite rules of a CFG as local trees. Adjunction in TAGs and substitntion in CFGs both rcduce to a form of concatcnation, of local trees in CFGs, of local 3-TMs in TAGs. In Figure 1 , for examplc, the local 3-Tl\\'1s corresponding to the elementary trees o 1 aud \u00df 1 are concatenated to form the 3-TM corresponding to the result of adjoining \u00df 1 into 0:1. The two-dimcnsional yicld of this structure is the corresponding derivcd tree and its onc-dimensional yield is the derived string. This analogy can be extended downward to encompass the regular languagcs and upward generating thc control lnnguage hierarchy of Vijay-Shanker et al. (1987 ), \\Veir (1988 , Weir (1992) . And it turns out. to be quite deep . Thc ordinary finite-state aut.omata (over strings-the one-dimensional level) atcepting the regular languages become, at the twodimensional levcl, the tree-aut.omata accepting the rec:ognizable sct.s of trees. The corresponding automata ovcr 3-TM turn out to accept exactly the sets of t.rec manifolds that. are gcneratcd by TAGs (with adjoining constraints) modulo a relaxation of the usual requircment that the root. and foot of an aux-i1iary Lr~e be labeled identically to euch other and to the ~1ode at which it adjoins. {\\Ve rcfer to these sets as thc recognizable sets of three-dimensional tree manifolds.) l\\foreover, essentially all of the familiar ant.omat.a-t.heoret.ic proofs of properties of regular languages lift dire.ctly to automata O\\'er treemanifolds of arbitran-dimension-the dimensionalit.\u2022 of t.he st.ruct.urcs 0 is simply a paramct.er of the proof and plays no essential role.",
                "cite_spans": [
                    {
                        "start": 4,
                        "end": 18,
                        "text": "Rogers (1997b)",
                        "ref_id": "BIBREF6"
                    },
                    {
                        "start": 1280,
                        "end": 1306,
                        "text": "Vijay-Shanker et al. (1987",
                        "ref_id": "BIBREF10"
                    },
                    {
                        "start": 1307,
                        "end": 1321,
                        "text": "), \\Veir (1988",
                        "ref_id": null
                    },
                    {
                        "start": 1324,
                        "end": 1335,
                        "text": "Weir (1992)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [
                    {
                        "start": 838,
                        "end": 846,
                        "text": "Figure 1",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "In Rogers (1998) we exploit. this regularit.y to obtain results analogous t.o B\u00fcchi's characterization of the regular languages in terms of definability in wSlS (the weak monadic ser:ond-order t.hcory of the natural numbers with successor) (B\u00fcchi, 1960) and Doner's (1970) and Thatcher and Wright's (1968) characterizations of the recognizable sets (of trecs) in terms of definability in wSnS (the weak monadic second-order theory of 11 successor functions-the complete n-branching tree). Thc recognizable sets of 3-TM are cxactly t.he finite 3-TM definable in the weak monadic second-order t.heory of t.he complete n-branching three-dimensional tree manifold, which wc i:efer t.o as wSnT3. This raises t.he prospect of defining TALs through the medium of collcctions of logical constraints expresscd in the signature of wSnT3 rather than with explicit TAGs . In this paper, we introduce this approach and begin t.o cxplore some of its ramifications in t.he contcxt. of TAGs for natural languages.",
                "cite_spans": [
                    {
                        "start": 3,
                        "end": 16,
                        "text": "Rogers (1998)",
                        "ref_id": "BIBREF7"
                    },
                    {
                        "start": 240,
                        "end": 253,
                        "text": "(B\u00fcchi, 1960)",
                        "ref_id": null
                    },
                    {
                        "start": 258,
                        "end": 272,
                        "text": "Doner's (1970)",
                        "ref_id": null
                    },
                    {
                        "start": 277,
                        "end": 305,
                        "text": "Thatcher and Wright's (1968)",
                        "ref_id": "BIBREF8"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
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            {
                "text": "Rat.her than work in wSnT3 dircctly, we work with an equivalent class of struct.ures t.hat is linguistically more natural. A Labcled Headed Finite 3-TM is a structure:",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "where T is a rooted, connected, finite subset. of the complete n-branching 3-TM (for somc n); <1; is immediate domination, <i; is local proper domination (among siblings) and <IJ is global proper domination (inherited), all in thc i 1 \" <limension; 1 H 1 is the set of Hends (exactly one in cach st.ring of childrenthese are underlined in the figurcs) an<l Pu are the labels (each picking out thc sct. of nodes labelcd er , not necessarily mutually exclusive).",
                "cite_spans": [],
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                "section": "",
                "sec_num": null
            },
            {
                "text": "We begin by looking at a simple cxample: assignmcnt of case in XT\"4\\G rnain verb (a' 1 ) and auxiJiary verb (\u00df 1 ) trces. We int.erpret. node namcs as firstorder variables and tree namcs as mona<lic secondordcr variables with, e.g\" o 1 (x) sat.isficd iff x is s VP Figure 1 : Tree J\\fanifolds the (3rct -dimensional) root of the local 3-TM corresponding to a 1 :",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 265,
                        "end": 273,
                        "text": "Figure 1",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "n1(s)H (3sr, npo, vp, v, np1 )[ l \u2022\" <l:l Sr f, s <l~ npo /\\ s <13 vp /\\ s <13 v /\\ s <l, 3 np1 /\\ 1viin2(sr) /\\ Max2(np 0 ) /\\ rviax 2 (v) /\\ Max 2 (np 1 )/\\ Hr <l2 npo /\\Sr <J2 vp /\\ H1 (vp)/\\ lVIin1(npo) /\\npo <l 1 vp/\\ ;\\faxi(vp)/\\ vp <12 v /\\ vp <l:! np 1 /\\ Hi(v)/\\ Mini (v) /\\ v <l1 np 1 /\\ Max 1 (11p 1 )/\\ Initial(s) /\\ Anchor(v) /\\ Subst.(11p 0 ) /\\ Subst(np 1 )",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "Here Mini and Maxi pick out minimal (root) and maximal (leaf) nodes wrt the 1\u2022th dimension-these are defined predicates:",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "Min;(x) = \u2022(3y)[y <li x].",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "Initial(x) is true at the root of each local 3-TM encoding an initial tree, Anchor(x) is true at each anchor node ( we will ignore insertion of the lexical itmm;), and Subst.(x) is true at. each node marked for subst.itutiou--these are labels, in E. We require all Subst nodes to have children in the 3rd -dimension and require the set of Initial nodes to be exactly the Subst nodes plus the root of the entire 3-TM: Figure 2 shows the disttibution of foatures respon-Hible for case assignment in the XTAG grammar. Following the approach of Ro\u00b5;ers (1997a) we interpi\u2022ct. the pat.hs occurring in the feature struct.ures decorating the trees as monadic predicates: E inr.ludes each sequence of features that. is a prefix of a path occurring in a feature-structure derivable in tllf' grammar. 2 'Ve will refer to this set of sequences ~ ,\\i; is typical in FTJ\\G, we a.'iSume finite featuret ruct.urc.'i.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 417,
                        "end": 425,
                        "text": "Figure 2",
                        "ref_id": "FIGREF1"
                    }
                ],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "('v'x)[Subst(x) -t (3y)[:r: <l;~ y]J ('v'x)[Initial(x) H (Subst.(x) V Min 3 (x)))",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "as Feat. Each node is multiply labeled: t.hc featurestructure associated with it is the union of the paths labeling it. In order to capt.ure the distinct.ion between top and bottom featuri?-structures we will prefix their paths with 't' ail<:I 'b', respectivcly. 'Ve can then add to the definition of n 1 :",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "(t: case: acc)(np 1 ) /\\ (h: assign-case: nom)(v).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "This encoding of feature-struct.ures gives us a straightforward definition of predicates for path equations as well. Fm any sequences w, v E Feat: The Iabeling of the elementary t.rees can then be interpreted as a collection of c:onstraint.s on local 3-TM, with thc set of st.ruct.ures licensecl by t.he grammar being the set of 3-TM in whir.h every node satisfies one of these collect.ions of constraints. Note that for a 3-TM in which the /3 1 3-TM expands the VP node in an a 1 3-TM to hr: liccnsed, the VP node must satisfy both the constraints of the a 1 3-TM and the constraints on the root. of the /J 1 3-TM. Thus the top feature-structure of tlw VP is unified with the top feat.ure-structure of VP r and the bottom featurestructure with thc bot.tom fcature-structurc of the foot. VP by simple transitivity of equalit.y. There is no need for additionai pat.h cquat.ions anci no extralogical mcchanisms of any sort; licensing is simply a matter of ordinary modcl-theoretic satisfaction. To get the {default) unification of top and bott.om feat.ure struct.ures of nodcs t.hat are not. expandcd by adjunction we ade! a singlc universal principle: Taken literally, t.his approach yielcls little more than a fullv declarative rcstatement of thc original grnmmar. But. in fact, a large proportion of the feat.ures decorat.ing elementary trees arc there onl:r to facilitatc t.he transport of fcatures through the tree: thcre is no obvious linguistic motivation for posit.ing that \"assign-case\" is a feat.ure of VPs or of S. In t.he language of wSnT3 there is no need for these int.ermediatc \"functional\" feat.ures or even any need to clistinguish top and bottom foature structureswc can stat.c directlv tlrnt the value of the ca.~e featurc of the subject NP, for instance, must agree with the value of the assi,qn-case feature of the verb. Of course, what is int.eresting about this relationship is the effect. of adjoinecl auxiliaries. The TAG analysis iududes an assign-r:ase feature for the intermediate \\'P in order to allow auxiliary verbs adjoined at the \\-p t.o int.ercept t.his relat.ionship by interposing betwcen thc VP's top and bott.om feature structures. In wSnT3 WC' obtain the same result from the way iu which we identify the relevant verb. For instance, if we take it to be \u2022tl1e last adjoined verb 3 -the one most deeply embedded in the third dimension-we r:an add to t.lw definition of a 1 :",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "(w = v)(x,y) = ((w: u)(:z:) H (v: u)(y)).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "S [(O)J \u00dfi VP [assign-case:(l)J ,,. \" 1 \"\\-_-_ ~ _ _ _ _ _ _ / ~ -la~ign",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "(3.T, y)[v71 <J~ :1: /\\ l\\fax;i(.T) /\\ :c <i2 y/\\ {assign-case)(y) /\\ {assign-case = case)(npo,y)J.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "In somewhat man' linguistically natural terms 4 we mi\u00b5;ht say that a verbal head governs, for the purposes of case assignment, all arguments in it.s local t.ree manifold (i.e., the minimal associated struct.uw). Furthermore a yerbal head in an auxiliary tree p;<werns all nodes iu the st.ruct.ure it acljoins into, as well as all nodes governed b~\u2022 them-effectively each <'.aSP asiiigner governs ever\u00bb child of each node prop-<~rly above it up to the first Initial node: Then v assigns case to 11Jlo iff it p;m'erns it and is not, itself, governed by some other case assigner:",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "(Vx, y)((Governs(x, 11) /\\ \u2022(3z) [Governs(z, :1:)]) -t",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "{assign-case = casc)(:1:,y)J.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "Alternatively, we could adopt existing accounts based on the more familiar relationships in the twodimcnsional projections of thc 3-TMs such as traditional GB accounts or niz;d's (1990) Relativized Mir1imality. All of these are definable in wSnT3 and all, therefore, correspond to s01ne TAG account of case assignment to subjects . Thc ccntral question, perhaps, is which comcs closest to t.he intuitions informing the existing grammar. This fact.oring of a TAG grammar int.o component linguistic principles is not a ne\\\\' idea. Vija~'-Shanker and Schabes's (1992) hierarchical encoding of TAG lexicons using partial dC'.scriptions of trees hecomes, from this perspective, a matter of classifying the lexicon on the basis of shared properties-every verbal anchor is associated with a :mbject. and t.he associated structure (see Figure 3) : and so on. Note that\" since concatenation of 3-Tivis does not disturb relationships int.ernal to them, there is no non-monotonicit.y hcre (or,rather, the apparent. non-monotonicity is an artifact of the yield operation)-there is no need to distinguish top and hottom quasi-nodes, no need for partial trees. ... b.:a. !nore cbvlcus cannect.!on can be made to Frank's (1992) explorat.ion of universal grammatical principles as interactions of the TAG mcchanism with linguistically motivat.ed <:onst.rnints ou the elementary structures. From t.he current perspective, these c~nstraints are just. properties of tlw local 3-Tivls occurring in wcll-formed p;rammat.ical st.rwtures. Here, again, the const raints an~ not disturbed s s VP /\"\"-.",
                "cite_spans": [
                    {
                        "start": 1198,
                        "end": 1212,
                        "text": "Frank's (1992)",
                        "ref_id": "BIBREF2"
                    }
                ],
                "ref_spans": [
                    {
                        "start": 828,
                        "end": 837,
                        "text": "Figure 3)",
                        "ref_id": "FIGREF4"
                    }
                ],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "(Vv)[{Anchor(v) /\\ Verb{v))---+ (3s\", n]Jo, vp)[s 1 \u2022 <J2 11pn /\\ 8\" <iz vp /\\ vp <l2 v",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "/\"\"-. liy the process of building 3-Tl'vis from thcsc local st.ruct.nres-t.hese are propertics not just of the el-Pment.ary structurcs but. of every local 3-Ttvl in all well-formed st.ructurcs. Ivlore interestingly, not all of t.lwsc const.raints are simple properties of thc clemcntary trces, some dcpcnd on the deri\\'ations. The Specifier Licensing Condition (SLC), for instance, in its hasic form , can only be satisfied once an adjunct.ion has t.aken place. As it. t.urns out, t.he mechanism employed in capt.uring this as a condition on t.lw elementary trees is to encode it. as a requirenwnt. t.hat. r.crtain features of the sort we havc been calling \"functional\" are instantiatccl. 5 Again in this rnntext\" in abstracting away from such implementat.ion cletails, wSnT3 offcrs a more direct exprcssion of t.hc const.raint. The key feature of t.his approach is that. it isolat.i>s t.lw linguist.ic thcory being expressed from thc 11wchanical det.ails of the grammar formalism exprnssing it.-in t.his rcspect. there is a strong par-all<d to l\\fosier's r.at.egorr theoretic approach t.o HPSG (P.fosier. 1997)-without losing the restric:t.iom; t.hat t.lie formalism imposes. Tlms, while the linp;uist.ir. principles can usually be stat.ed dircctly, thc fact that t.he.r must bc expressible within the signature of wSnT3 limits t.hem to principles which c:an lw enforced by TAGs. In fact. the characterizat.ions of the rer.ognizablc sets of 3-TM by dcfinabil-it~\u2022 in wSnT3 ancl of TAG tree and st.ring languages as tlw .\"ields of rec:ognizablc sets of 3-Tl'vl are construct.i VI.' and when these constructions are carried out man~\u2022 \"functional\" fcatures of the sort t.hat the lo~ical approach eschews are instantiated in the result.ing TAG. This raises thc possibility of using the loi;ical definitions not just. as an abstract means of cliscussing t.l1e lingui~tic theory, bnt. also as a sort of higher-level languap;e which can be compiled into TAGs of t.he familiar sort. 0 '' Pm\u2022haps coinciclentall,v, t.hese attribute case-assignment to IP:< and Is in duse parallel to the XTAG examplc we started \\\\'itli .",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "NP VP",
                "sec_num": null
            },
            {
                "text": "\u2022 \"Then\u2022 are somc formidabh! ubstacles to realizing this i<lea, 1101 tlH' least of which is thr fact that t.he procC'ss uf compiling wS11T:l formttlf\u00fc\u2022 iuto 3-Tl\\I automata has, at least potentially, 11n11-Plm111~11tar.\\\u2022 complrxity. Nonr.theless, prior experience at t lil' 0111\u2022\u2022 and two-climensional levels suggcsts t.hat tlw process m;I\\' ht\u2022 foasible t !\\'t?Jt for rnlath\u2022ch\u2022 substantial thcories ai1<l h<';I' \\\\'\\' haw 1 lir knowledg<' t.hal ~ea~onably compacl grammarn for ~imilar\u2022 t lll'ories exist (as wit11C'sscd by the XTAG ).\\r;immar). Tims. iri ~ome ~ense, the potr.ntial intractability",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "NP VP",
                "sec_num": null
            },
            {
                "text": "Domination, in its familiar form iu trees, is domination in the se.cond dimensiou here. Domination in thc first dimension is U8ll:tlly known a8 linear precedenn\u2022. \\ Vt\u2022 will refor to domination in the third dimcnsion as nbo11c.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            }
        ],
        "back_matter": [],
        "bib_entries": {
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        "ref_entries": {
            "FIGREF0": {
                "uris": null,
                "type_str": "figure",
                "text": "can add the re-entrancy tags: (b : assign-case = t : assign-case)(11p, 11)/\\ (b : assign-case = t : assign-case)(s\", \u2022up)/\\ (b: assign-case = t.: casc)(8ri np 0 )/\\ (t. = t)(s,sr).",
                "num": null
            },
            "FIGREF1": {
                "uris": null,
                "type_str": "figure",
                "text": "Case assignment. in XTAG.",
                "num": null
            },
            "FIGREF2": {
                "uris": null,
                "type_str": "figure",
                "text": "Governs(:1:, y) = (a:-;sig11-rn:;~) (:1:)/\\ (3z )(z <lt .1: /\\ z <J;i y/\\ (\\!':;')[(.: <Jt z' /\\ z' <it :r.) ~ \u2022lnitial(z')Jl.' 1 Thii; is c.orrr'.ct 011!.1\u2022 if tlw font uode8 haw null-adjoiniug c\u2022o11Rlraints, as b nsual. \u2022IThis is nnt nwant tn b\" a proposal of au analysis of as-si~11nw111 of casc\u2022 iu XT:\\(;. only tn IH' au c\u2022xamplc of thc style ol' anal.1\u2022ses that can b1~ supporrc\u2022d by t.his approach.",
                "num": null
            },
            "FIGREF3": {
                "uris": null,
                "type_str": "figure",
                "text": "/\\ {case)(np 0 ) /\\ (assign-r.ase: nom)(v) t\\ \u2022 \u2022 \u2022JJ, transitive verbs, in addition, are associated with an object: (Vv)[(Anchor(v) /\\ Verb(11) /\\ Transitive(11))---+ (3npi)[v <J1 11.JJi /\\ (case:acc)(n7Ji) /\\ \u2022 \u2022 \u2022]],",
                "num": null
            },
            "FIGREF4": {
                "uris": null,
                "type_str": "figure",
                "text": "Sharccl structure as shared properties",
                "num": null
            }
        }
    }
}