{ "paper_id": "P02-1007", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:31:15.768891Z" }, "title": "OT Syntax: Decidability of Generation-based Optimization", "authors": [ { "first": "Jonas", "middle": [], "last": "Kuhn", "suffix": "", "affiliation": { "laboratory": "", "institution": "Stanford University", "location": {} }, "email": "jonask@stanford.edu" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "In Optimality-Theoretic Syntax, optimization with unrestricted expressive power on the side of the OT constraints is undecidable. This paper provides a proof for the decidability of optimization based on constraints expressed with reference to local subtrees (which is in the spirit of OT theory). The proof builds on Kaplan and Wedekind's (2000) construction showing that LFG generation produces contextfree languages. 3 OT-LFG Following (Bresnan, 2000; Kuhn, 2000; Kuhn, 2001), we define a restricted OT system based on Lexical-Functional Grammar (LFG) representations: c(ategory) structure/f(unctional) structure 2 Most computational OT work so far focuses on candidates and constraints expressible as regular languages/rational relations, based on (", "pdf_parse": { "paper_id": "P02-1007", "_pdf_hash": "", "abstract": [ { "text": "In Optimality-Theoretic Syntax, optimization with unrestricted expressive power on the side of the OT constraints is undecidable. This paper provides a proof for the decidability of optimization based on constraints expressed with reference to local subtrees (which is in the spirit of OT theory). The proof builds on Kaplan and Wedekind's (2000) construction showing that LFG generation produces contextfree languages. 3 OT-LFG Following (Bresnan, 2000; Kuhn, 2000; Kuhn, 2001), we define a restricted OT system based on Lexical-Functional Grammar (LFG) representations: c(ategory) structure/f(unctional) structure 2 Most computational OT work so far focuses on candidates and constraints expressible as regular languages/rational relations, based on (", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Optimality-Theoretic (OT) grammar systems are an interesting alternative to classical formal grammars, as they construe the task of learning from data in a meaning-based way: a form is defined as grammatical if it is optimal (most harmonic) within a set of generation alternatives for an underlying logical form. The harmony of a candidate analysis depends on a language-specific ranking ( \u00a1 ) of violable constraints, thus the learning task amounts to adjusting the ranking over a given set of constraints. The comparison-based setup of OT learning is closely related to discriminative learning approaches in probabilistic parsing (Johnson et al., 1999; Riezler et al., 2000; Riezler et al., 2002) , 1 however the comparison of generation alternatives -rather than parsing alternatives -adds the possibility of systematically learning the basic language-specific grammatical principles (which in probabilistic parsing are typically fixed a priori, using either a treebankderived or a manually written grammar for the given This work was supported by a postdoctoral fellowship of the German Academic Exchange Service (DAAD).", "cite_spans": [ { "start": 632, "end": 654, "text": "(Johnson et al., 1999;", "ref_id": "BIBREF8" }, { "start": 655, "end": 676, "text": "Riezler et al., 2000;", "ref_id": "BIBREF15" }, { "start": 677, "end": 698, "text": "Riezler et al., 2002)", "ref_id": "BIBREF16" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "1 This is for instance pointed out by (Johnson, 1998) . language). The \"base grammar\" assumed as given can be highly unrestricted in the OT setup. Using a linguistically motivated set of constraints, learning proceeds with a bias for unmarked linguistic structures (cf. e.g., (Bresnan et al., 2001) ).", "cite_spans": [ { "start": 38, "end": 53, "text": "(Johnson, 1998)", "ref_id": "BIBREF9" }, { "start": 276, "end": 298, "text": "(Bresnan et al., 2001)", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "For computational OT syntax, an interleaving of candidate generation and constraint checking has been proposed (Kuhn, 2000) . But the decidability of the optimization task in OT syntax, i.e., the identification of the optimal candidate(s) in a potentially infinite candidate set, has not been proven yet. 2", "cite_spans": [ { "start": 111, "end": 123, "text": "(Kuhn, 2000)", "ref_id": "BIBREF12" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Assume that the candidate set is characterized by a context-free grammar (cfg) , plus one additional candidate 'yes'. There are two constraints ( \u00a1 ) :", "cite_spans": [ { "start": 73, "end": 78, "text": "(cfg)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "is violated if the candidate is neither 'yes' nor a structure generated by a cfg ; is violated only by 'yes'. Now, 'yes' is in the language defined by this system iff there are no structures in that are also in . But the emptiness problem for the intersection of two context-free languages is known to be undecidable, so the optimization task for unrestricted OT is undecidable too. 3 However, it is not in the spirit of OT to have extremely powerful individual constraints; the explanatory power should rather arise from interaction of simple constraints.", "cite_spans": [ { "start": 383, "end": 384, "text": "3", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "pairs \u00a2 \u00a1 \u00a4 \u00a3 \u00a6 \u00a5 \u00a7 like (4),(5) \u00a7", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": ". Each c-structure tree node is mapped to a node in the f-structure graph by the function \u00a9 . The mapping is specified by fannotations in the grammar rules (below category symbols, cf. (2)) and lexicon entries (3). 4 (2) ROOT , i.e., the f-structure corresponding to the present node's mother category.", "cite_spans": [ { "start": 215, "end": 216, "text": "4", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "FP VP \" ! FP # NP FP $ TOPIC% $ COMP* OBJ% (NP) F& $ SUBJ% ' ! F& ( F FP VP ! VP (NP) V& ( SUBJ)= = V& ) V NP $ OBJ% 0 FP $ COMP% \" ! (3) Mary NP ( PRED)='Mary' ( NUM)=SG that F had F ( TNS)=PAST seen V ( PRED)='see 1 ( SUBJ) ( OBJ) 2 ' ( ASP)=PERF thought V ( PRED)='think 1 ( SUBJ) ( COMP) 2 ' ( TNS)=PAST laughed V ( PRED)='laugh 1 ( SUBJ) 2 ' ( TNS)=PAST (4) c", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "The correct f-structure for a sentence is the minimal model satisfying all properly instantiated fannotations.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "In OT-LFG, the universe of possible candidates is defined by an LFG", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "Q P S R U T V P X W \u1ef2", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "(encoding inviolable principles, like an X-bar scheme). A particular candidate set is the set Gena", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "c b d V e \u00a2 b f h g p i \u00a5 P S R r q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "-i.e., the c-/fstructure pairs in", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "s P C R U T p P S W \u1ef2", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": ", which have the input", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "\u00a5 P C R", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "as their f-structure. Constraints are expressed as local configurations in the c-/f-structure pairs. They have one of the following implicational forms:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "5 (6) tu w v t & u & where t y x t y &", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "are descriptions of nonterminals of", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "b d p e b f h g ; u x u &", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "are standard LFG f-annotations of constraining equations with as the only f-structure metavariable.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "(7) t u v t y & & & u & r & where t y x t y & x x & are descriptions of nonterminals of b d V e \u00a2 b f h g ; t y x t &", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "refer to the mother in a local subtree configuration,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "x & refer to the same daughter category;", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "x", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "& x x r &", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "are regular expressions over nonterminals; Any of the descriptions can be maximally unspecific; (6) can for example be instantiated by the OPSPEC constraint ( OP)=+ (DF ) (an operator must be the value of a discourse function, (Bresnan, 2000) ) with the category information unspecified.", "cite_spans": [ { "start": 227, "end": 242, "text": "(Bresnan, 2000)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "An OT-LFG system is thus characterized by a base grammar and a set of constraints, with a language-specific ranking relation", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "\u00a1 : 1 b d V e \u00a2 b f h g x 1 x 2 2 .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "The evaluation function Evald C e g f h r i picks the most harmonic from a set of candidates, based on the constraints and ranking. The language (set of analyses) 6 generated by an OT system is defined as", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "j $ % l k 1 n m p o U x q o r 2 ' b d p e b f h g s t q b d v u 1 n m o x w q o 2 Evalx z y U { | } $ Gen r \u00a2 X X $ q b d % %", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Undecidability for unrestricted OT", "sec_num": "2" }, { "text": "Our decidability proof for generation-based optimization builds on the result of (Kaplan and Wedekind, 2000) (K&W00) that LFG generation produces context-free languages. . K&W00 present a constructive proof, folding all fstructural contributions of lexical entries and LFG rules into the c-structural rewrite rules (which is possible since we know in advance the range of fstructural objects that can instantiate the f-structure meta-variables in the rules). I illustrate the specialization steps with grammar (2) and lexicon (3) and for generation from f-structure (5).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "Initially, the generalized format of right-hand sides in LFG rules is converted to the standard context-free notation (resolving regular expressions by explicit disjunction or recursive rules). Fstructure (5) contains five substructures: the root fstructure, plus the embedded f-structures under the paths SUBJ, COMP, COMP SUBJ, and COMP OBJ. Any relevant metavariable ( , \u00a4 ) in the grammar must end up instantiated to one of these. So for each path from the root f-structure, a distinct variable is introduced: \u00a5 , subscripted with the (abbreviated and possibly empty) feature path:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "\u00a5 \u00a3 \u00a5 \u00a7 \u00a6 \u00a3 \u00a5 \u00a9 \u00a3 \u00a5 \u00a9 \u00a6 \u00a3 \u00a5 \u00a9 .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "Rule augmentation step 1 adds to each category name a concrete f-structure to which the category corresponds. So for FP, we get FP:\u00a5 , FP:\u00a5 \u00a6 , FP:\u00a5 , FP:\u00a5\u00a6 , and FP:\u00a5 . The rules are multiplied out to cover all combinations of augmented categories obeying the original f-annotations. 7 Step 2 adds a set of instantiated f-annotation schemes to each symbol, based on the instantiation of metavariables from step 1. One instance of the lexicon entry Mary look as follows: 9NP:", "cite_spans": [ { "start": 285, "end": 286, "text": "7", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": ": $ PRED)='Mary' $ N UM)=SG ! Mary", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "The rules are again multiplied out to cover all combinations for which the set of f-constraints on the mother is the union of all daughters' fconstraints, plus the appropriately instantiated rulespecific annotations. So, for the VP rule based on the categories NP:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": ": $ ! PRED)='Mary' $ NUM)=SG ! and V& : ! : # $ ! P RED)='laugh' $ T NS)=PAST \"", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": ", we get the rule 7 VP:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "NP: V & :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "i s allowed, while VP:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "! NP: ! V & :", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "#", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "is excluded, since the = annotation of V& in the VP rule (2) enforces that", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "E $ VP% E $ V& X % . VP: : $ % % % % & % % % % ' $ S UBJ% $ P RED)='Mary' $ NUM)=SG $ PRED)='laugh' $ ! T NS)=PAST ( % % % % ) % % % % 0 NP: ! : $ P RED)='Mary' $ NUM)=SG ! V& : : # $ ! P RED)='laugh' $ TNS)=PAST ! ! \"", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "With this bottom-up construction it is ensured that each new category ROOT:\u00a5 :1 . . .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation", "sec_num": "4" }, { "text": "(corresponding to the original root symbol) contains a complete possible collection of instantiated f-constraints. To exclude analyses whose f-structure is not \u00a5 (for which we are generating strings) a new start symbol is introduced \"above\" the original root symbol. Only for the sets of f-constraints that have \u00a5 as their minimal model, rules of the form ROOT \u00a7 3 ROOT:\u00a5 :1 . . .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "2", "sec_num": null }, { "text": "are introduced (this also excludes inconsistent fconstraint sets).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "2", "sec_num": null }, { "text": "\u00a1 \u00a3 \u00a2 i \u00a3 \u00a6 \u00a5 q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "With the cfg", "sec_num": null }, { "text": ", standard techniques for cfg's can be applied, e.g., if there are infinitely many possible analyses for a given f-structure, the smallest one(s) can be produced, based on the pumping lemma for context-free languages. Grammar 2does indeed produce infinitely many analyses for the input f-structure (5). It overgenerates in several respects: The functional projection FP can be stacked due to recursions like the following (with the augmented FP reoccuring in the F rules):", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "With the cfg", "sec_num": null }, { "text": "FP:4 6 5 : $ % % % & % % % ' 7 4 8 5 P RED)='see 6 . . . 8 ' 7 4 6 5 T NS)=PAST 7 45 S UBJ9 A @ B 45 D C 7 45 E C P RED)='Mary' 7 4 6 5 O BJ9 A @ B 4 6 5 D F 7 45 E F P RED)='Titanic' 45@ B 45 ( % % % ) % % % 0 H G FI :4 6 5 : $ % % % & % % % ' 7 4 6 5 P RED)='see 6 . . . 8 ' 7 4 8 5 T NS)=PAST 7 45 S UBJ9 @ B 45 E C 7 45 D C P RED)='Mary' 7 4 8 5 O BJ9 \u00a7 @ B 4 8 5 E F 7 45 D F P RED)='Titanic' 45@ B 45 ( % % % ) % % % 0 FI :4 8 5 : $ % % % & % % % ' 7 4 8 5 P RED)='see 6 . . . 8 ' 7 4 6 5 T NS)=PAST 7 45 S UBJ9 @ B 45 D C 7 45 E C P RED)='Mary' 7 4 6 5 O BJ9 A @ B 4 6 5 D F 7 45 E F P RED)='Titanic' 45@ B 45 ( % % % ) % % % 0 G F:4 8 5 : P F P:4 6 5 : $ % % % & % % % ' 7 4 8 5 P RED)='see 6 . . . 8 ' 7 4 6 5 T NS)=PAST 7 45 S UBJ9 \u00a7 @ B 45 D C 7 45 E C P RED)='Mary' 7 4 6 5 O BJ9 A @ B 4 6 5 D F 7 45 E F P RED)='Titanic' 45@ B 45 ( % % % ) % % % 0 F:\u00a5 \u00a9 : Q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "With the cfg", "sec_num": null }, { "text": "is one of the augmented categories we get for that in (3), so \u00a1 \u00a3 \u00a2 ((2),(5)) generates an arbitrary number of thats on top of any FP. A similar repetition effect will arise for the auxiliary had. 8 Other choices in generation arise from the freedom of generating the subject in the specifier of VP or FP and from the possibility of (unbounded) topicalization of the object (the first disjunction of the FP rule in (2) contains a functional-uncertainty equation):", "cite_spans": [ { "start": 197, "end": 198, "text": "8", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "With the cfg", "sec_num": null }, { "text": "(10) a.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "With the cfg", "sec_num": null }, { "text": "John thought that Titanic, Mary had seen. b.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "With the cfg", "sec_num": null }, { "text": "Titanic, John thought that Mary had seen.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "With the cfg", "sec_num": null }, { "text": "While grammar (2) would be considered defective as a classical LFG grammar, it constitutes a reasonable example of a candidate generation grammar (", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "P S R U T V P X W \u1ef2 ) in OT.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "Here, it is the OT constraints that enforce language-specific restrictions, so", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "P S R U T V P X W \u1ef2", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "has to ensure that all candidates are generated in the first place. For instance, expletive elements as do in Who do you know will arise by passing a recursion in the cfg constructed during generation. A candidate containing such a vacuous cycle can still become the winner of the OT competition if the Faithfulness constraint punishing expletives is outranked by some constraint favoring an aspect of the recursive structure. So the harmony is increased by going through the recursion a certain number of times. It is for this very reason, that Who do you know is predicted to be grammatical in English.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "So, in OT-LFG it is not sufficient to apply just the \u00a1 \u00a3 \u00a2 construction; I use an additional step: prior to application of \u00a1 \u00a3 \u00a2 , the LFG grammar", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "P S R U T V P X W \u1ef2 is converted to a different form e i P C R U T p P S W \u1ef2 q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "(depending on the constraint set \u00a1 ), which is still an LFG grammar but has category symbols which reflect local constraint violations. When the \u00a1 \u00a2 construction is applied to", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "e i P S R U T V P X W \u1ef2 q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": ", all \"pumping\" structures generated by the cfg", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "\u00a1 \u00a3 \u00a2 i e i P C R U T p P S W \u1ef2 q \u00a3 \u00a6 \u00a5 P C R q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "can indeed be ignored since all OT-relevant candidates are already contained in the finite set of nonrecursive structures. So, finally the ranking of the constraints is taken into consideration in order to determine the harmony of the candidates in this finite subset.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "LFG generation in OT-LFG", "sec_num": "5" }, { "text": "\u00a2 e \u00a4 \u00a3 \u00a6 \u00a5 P C R U T p P S W \u1ef2 \u00a7", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "The conversion", "sec_num": "6" }, { "text": "Preprocessing Like K&W00, I assume an initial conversion of the c-structure part of rules into standard context-free form, i.e., the right-hand side is a category string rather than a regular expression. This ensures that for a given local subtree, each constraint (of form (6) or (7)) can be applied only a finite number of times: if \u00a9 is the arity of the longest right-hand side of a rule, the maximal number of local violations is \u00a9 (since some constraints of type (7) can be instantiated to all daughters).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "The conversion", "sec_num": "6" }, { "text": "With the number of local violations bounded, we can encode all candidate distinctions with respect to constraint violations at the local-subtree level with finite means: The set of categories in the newly constructed LFG grammar", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "e i P C R U T p P S W Y q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "is the finite set", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "(11) h S X", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": ": the set of categories in The rules in", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "y $ b d V e \u00a2 b f h g % k t :1 ! \u00a3 x \" \u00a5 x \" $ # & % ' % ( % ) 1 0 \u00a6 2 s t a nonterminal symbol of b d V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "e i P C R U T p P S W Y q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "are constructed in such a way that for each rule Note that the constraint profile of the daughter categories does not play any role in the determination of constraint violations local to the subtree under consideration (only the sequences 9 Q A are restricted by the conditions (12) and (13)). So for each new rule type, all combinations of constraint profiles on the daughters are constructed (creating a large but finite number of rules). 9 This ensures that no sentence that can be parsed (or generated) by analysis by applying a projection function Cat to all c-structure categories:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "X R X\u00a3 . . . X8 m \u00a3 m 8 in Q P C R U T p P S W \u1ef2 and each sequence @ 9 A \u00a3 B 9 A D C E C E C 9 G F A \u00a7 , H P I 9 R Q A I \u00a9 , all rules of the form X R :1 ! \u00a3 R x \" \u00a5 R % ( % ' % \" 0 R 2 X\u00a3 :1 ! \u00a3 \u00a3 % ' % B % S 0 \u00a32 . . . X8 :1 ! \u00a3 8 % ' % ' % \" 0 8 2 ,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "(12) for \u00a7 \u00a9 of form (6) U tu v t & u & ) V : a. \u00a8R 3 ; m & o m p o (W 4 Y X 4 b a ) if X R does not match the condition t ; b. \u00a8R 3 ; m & \u00a3 m \u00a3 d c f e u ; m & o m o (g 4 Y X h 4 7 a ) if X R matches t ; c. \u00a8R 3 ; m & \u00a3 m \u00a3 c u c u & ; m & o m p o (g 4 Y X h 4 b a ) if X R matches both t and t & ; d. \u00a8R W ; m & \u00a3 m \u00a3 d c u ; m & o m o (g 4 Y X h 4 b a ) if X R matches t but not t & ; e. \u00a8R W ; m & \u00a3 m \u00a3 c u c i e u & ; m & o m p o (g 4 p X & 4 \u00e0 ) if X", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "Cat $ t :1 ! \u00a3 x \" \u00a5 % ' % ' % \" 0 2 % t for every category in $ \u00a2 X X", "eq_num": "(11)" } ], "section": "Grammar conversion", "sec_num": null }, { "text": "9 For one rule/constraint combination several new rules can result; e.g., if the right-hand side of a rule (X R ) matches both the antecedent (t ) and the consequent (t & ) category description of a constraint of form (6), three clauses apply: (12b), (12c), and (12d). So, we get two new rules with the count of 0 local violations of the constraint and two rules with count 1, with a difference in the f-annotations.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "10 Providing all possible combinations of augmented category symbols on the right-hand rule sides in y $ %", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "ensures that the newly constructed rules can be reached from the root symbol in a derivation. It is also guaranteed that whenever a rule \u00a2 in contributes to an analysis, at least one of the rules constructed from \u00a2 will contribute to the corresponding analysis in y $ % . This is ensured since the subclauses in (12) and (13) cover the full space of logical possibilities.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "We can overload the function name Cat with a function applying to the set of analyses produced by an LFG grammar by defining encodes the number of local violations for all constraints. Since all constraints are locally evaluable by assumption, all constraints violated by a candidate analysis have to be incurred local to some subtree. Hence the total number of constraint violations incurred by a candidate can be computed by simply summing over all category-encoded local violation profiles: ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "Cat $ % k 1 n m c x V q 2 s 1 n m & x w q 2 ,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "1 n m x q 2 y $ b d V e \u00a2 b f h g % is \u00a3 \u00a5 \u00a4 \u00a7 \u00a6$ m % q \u00a9 x \u00a6 ~ f \" ! $ # \u00a8 D efine Totaly $ m % 1 \u00a3 \u00a4 & % $ m % V x \u00a3 \u00a4 \u00a7 ' $ m % V x ( % ' % ' % \u00a3 \u00a4 & ( $ m % 2 7 Applying ) 1 0 on \u00a2 e \u00a4 \u00a3 s \u00a5 P C R U T p P S W Y \u00a7 Since e i Q P C R U T p P S W \u1ef2 q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "is a standard LFG grammar, we can apply the \u00a1 \u00a3 \u00a2 construction to it to get a cfg for a given f-structure Since the e construction (strongly) preserves the language generated, coverage preservation holds also after the application of", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\u00a1 \u00a3 \u00a2 to e i P C R U T p P S W Y q and P S R U T V P X W \u1ef2", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": ", respectively:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "(17) Cat $ @ 9 B A $ y $ b d p e b f h g % V x q b d % % Cat $ @ 9 B A $ b d p e b f h g x w q b d % %", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "But since the symbols in", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "e i P C R U T p P S W \u1ef2 q reflect local constraint violations, Cati \u00a1 \u00a3 \u00a2 i e i P C R U T p P S W \u1ef2 q \u00a3 \u00a6 \u00a5 P S R q q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "has the property that all instances of recursion in the resulting cfg create candidates that are at most as harmonic as their non-recursive counterparts. Assuming a projection function CatCount i 4 3", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": ":\u00a5 :5 :6 q \u00a5 7 3", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": ":\u00a5 , we can state more formally: ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\u00a3x \" \u00a5 $ W %% 2 % from 1 ! \u00a3 \u00a3 % ' % B % S \u00a3 % B % ' % \" 0 \u00a32 Totaly $ m \u00a3%", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": ", and", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "1 ! \u00a3 \u00a5 % ' % ' % \" \u00a5 % ( % ' % \" 0 \u00a5 2 Totaly $ m \u00a5 % .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "This fact follows from definition of Total (16): the violation counts in the additional nodes in \u00a1 will add to the total of constraint violations (and if none of the additional nodes contains any local constraint violation at all, the total will be the same as in \u00a1 ) . Intuitively, the effect of the augmentation of the category format is that certain recursions in the pure \u00a1 \u00a3 \u00a2 construction (which one may think of as a loop) are unfolded, leading to a longer loop. The new loop is sufficiently large to make all relevant distinctions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "This result can be directly exploited in processing: if all non-recursive analyses are generated (of which there are only finitely many) it is guaranteed that a subset of the optimal candidates is among them. If the grammar does not contain any violation-free recursion, we even know that we have generated all optimal candidates. Note that if there is an applicable violation-free recursion, the set of optimal candidates is infinite; so if the constraint set is set up properly in a linguistic analysis, one would assume that violation-free recursion should not arise. (Kuhn, 2000) excludes the application of such recursions by a similar condition as offline parsability (which excludes vacuous recursions over a string in parsing), but with the \u00a1 \u00a3 \u00a2 construction, this condition is not necessary for decidability of the generation-based optimization task. The cfg produced by \u00a1 \u00a2 can be transformed further to only generate the optimal candidates according to the constraint ranking \u00a1 of the OT system", "cite_spans": [ { "start": 571, "end": 583, "text": "(Kuhn, 2000)", "ref_id": "BIBREF12" } ], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "7 P S R U T V P X W \u1ef2 \u00a3 \u00a1 \u00a3 \u00a1 \u00a7 \u00a7", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": ", eliminating all but the violation-free recursions in the grammar:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "(20) Creating a cfg that produces all optimal candidates a. Define", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "To prove fact (21) we will show that the c-structure of an arbitrary candidate analysis generated from", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\u00a5 P S R with P S R U T V P X W \u1ef2 is contained in Cati \" ! # b d q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "iff all other candidates are equally or less harmonic.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "Take an arbitrary candidate c-structure", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\u00a1 gen- erated from \u00a5 P C R with P S R U T V P X W \u1ef2 such that \u00a1 Cati \" ! # b d q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": ". We have to show that all other candidates would also be excluded (for lack of the relevant rules in the non-recursive part). On the other hand, if it were the recursion in by construction step (20c,d) (only violation-free recursion is possible). So we get another contradiction to the assumption that does incur some violation, not using the recursion leads to an even more harmonic candidate, for which again cases (i) and (ii) will apply. All possible cases lead to a contradiction with the assumptions, so no candidate is more harmonic than our", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\u00a1 Cati \" ! # b d q .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "We still have to prove that if the c-structure \u00a1 of a candidate analysis generated from", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\u00a5 P C R with Q P C R U T p P S W \u1ef2", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "is equally or more harmonic than all other candidates, then it is contained in Cati \" ! # . Now, there has to be a homomorphism from the categories in \u00a1 to the categories of some analysis in", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\" ! # b d . \" ! # b d is also based on \u00a1 \u00a3 \u00a2 i P S R U T V P X W \u1ef2 \u00a3 \u00a6 \u00a5 P C R q", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "(with an additional index \u00a1 on each category and some categories and rules of", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "\u00a1 \u00a3 \u00a2 i P C R U T p P S W \u1ef2 \u00a3 \u00a6 \u00a5 P S R q having no counterpart in \" ! # b d", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "). Since we know that \u00a1 is equally or more harmonic than any other candidate generated from ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Grammar conversion", "sec_num": null }, { "text": "We showed that for OT-LFG systems in which all constraints can be expressed relative to a local subtree in c-structure, the generation task from (noncyclic 13 ) f-structures is solvable. The infinity of the conceptually underlying candidate set does not preclude a computational approach. It is obvious that the construction proposed here has the purpose of bringing out the principled computability, rather than suggesting a particular algorithm for implementation. However on this basis, an implementation can be easily devised. The locality condition on constraint-checking seems unproblematic for linguistically relevant constraints, since a GPSG-style slash mechanism permits reference to (finitely many) nonlocal configurations from any given category (cf. fn. 5). 14 Decidability of generation-based optimization (from a given input f-structure) alone does not imply that the recognition and parsing tasks for an OT grammar system defined as in sec. 3 are decidable: for these tasks, a string is given and it has to be shown that the string is optimal for some underlying input f-structure (cf. (Johnson, 1998) ). However, a similar construction as the one presented here can be devised for parsing-based optimization (even for an LFG-style grammar that does not obey the offline parsability condition). So, if the language generated by an OT system is defined based on (strong) bidirectional optimality (Kuhn, 2001 , ch. 5), decidability of both the general parsing and generation problem follows. 15 For the unidirectionally defined OT language (as in sec. 3), decidability of parsing can be guaranteed under the assumption of a contextual recoverability condition in parsing (Kuhn, in preparation) .", "cite_spans": [ { "start": 771, "end": 773, "text": "14", "ref_id": "BIBREF2" }, { "start": 1102, "end": 1117, "text": "(Johnson, 1998)", "ref_id": "BIBREF9" }, { "start": 1411, "end": 1422, "text": "(Kuhn, 2001", "ref_id": "BIBREF13" }, { "start": 1506, "end": 1508, "text": "15", "ref_id": "BIBREF3" }, { "start": 1685, "end": 1707, "text": "(Kuhn, in preparation)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "9" }, { "text": "Note that with GPSG-style category-level feature percolation it is possible to refer to (finitely many) nonlocal configurations at the local tree level.6 The string language is obtained by taking the terminal string of the c-structure part of the analyses.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "The FR entries do not contribute any PRED value, which would exclude doubling due to the instantiated symbol character of PRED values (cf. K&W00, fn. 2).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null }, { "text": "The non-cyclicity condition is inherited from K&W00; in linguistically motivated applications of the LFG formalism, cru-", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "", "sec_num": null } ], "back_matter": [ { "text": "are the indexed symbols of step c.; S \u00a4 is a new start symbol; the ruleswhich were used in the analyses in Evalx With the index introduced in step (20c), the original recursion in the cfg is eliminated in all but the violation-free cases. The grammar Catiproduces (the c-structure of) the set of optimal candidates for the input, i.e., the set of c-structures for the optimal candidates for input f-structure .11 The projection function Cat is again overloaded to also remove the index on the categories.12 Like K&W00, I make the assumption that the input fstructure in generation is fully specified (i.e., all the candidates have the form), but the result can be extended to allow for the addition of a finite amount of f-structure information in generation. Then, the specified routine is computed separately for each possible f-structural extension and the results are compared in the end.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "annex", "sec_num": null } ], "bib_entries": { "BIBREF0": { "ref_id": "b0", "title": "Soft constraints mirror hard constraints: Voice and person in English and Lummi", "authors": [ { "first": "Joan", "middle": [], "last": "Bresnan", "suffix": "" }, { "first": "Shipra", "middle": [], "last": "Dingare", "suffix": "" }, { "first": "Christopher", "middle": [], "last": "Manning", "suffix": "" } ], "year": 2001, "venue": "Proceedings of the LFG 2001 Conference", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "Joan Bresnan, Shipra Dingare, and Christopher Manning. 2001. Soft constraints mirror hard constraints: Voice and person in English and Lummi. In Proceedings of the LFG 2001 Conference. 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Likewise, if the recursion in e i \u00a1 q", "type_str": "figure", "num": null, "uris": null }, "FIGREF17": { "text": "recursion or only violation-free recursion. If it does contain such violation-free recursions we map all categories \u00a2 on the recursion paths to the indexed form \u00a2 : H , and furthermore consider the variant of \u00a1avoiding the recursion(s). For our (non-recursive) tree, there is guaranteed to be a counterpart in the finite set of non-recursive trees in \"", "type_str": "figure", "num": null, "uris": null } } } }