File size: 44,676 Bytes
6fa4bc9 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 | {
"paper_id": "P96-1013",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T09:02:40.182778Z"
},
"title": "Parsing for Semidirectional Lambek Grammar is NP-Complete",
"authors": [
{
"first": "Jochen",
"middle": [],
"last": "Dfrre",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "Institut ffir maschinelle Sprachverarbeitung University of Stuttgart",
"location": {}
},
"email": ""
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "We study the computational complexity of the parsing problem of a variant of Lambek Categorial Grammar that we call semidirectional. In semidirectional Lambek calculus SD[ there is an additional nondirectional abstraction rule allowing the formula abstracted over to appear anywhere in the premise sequent's left-hand side, thus permitting non-peripheral extraction. SD[ grammars are able to generate each context-free language and more than that. We show that the parsing problem for semidireetional Lambek Grammar is NP-complete by a reduction of the 3-Partition problem.",
"pdf_parse": {
"paper_id": "P96-1013",
"_pdf_hash": "",
"abstract": [
{
"text": "We study the computational complexity of the parsing problem of a variant of Lambek Categorial Grammar that we call semidirectional. In semidirectional Lambek calculus SD[ there is an additional nondirectional abstraction rule allowing the formula abstracted over to appear anywhere in the premise sequent's left-hand side, thus permitting non-peripheral extraction. SD[ grammars are able to generate each context-free language and more than that. We show that the parsing problem for semidireetional Lambek Grammar is NP-complete by a reduction of the 3-Partition problem.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "Categorial Grammar (CG) and in particular Lambek Categorial Grammar (LCG) have their well-known benefits for the formal treatment of natural language syntax and semantics. The most outstanding of these benefits is probably the fact that the specific way, how the complete grammar is encoded, namely in terms of 'combinatory potentials' of its words, gives us at the same time recipes for the construction of meanings, once the words have been combined with others to form larger linguistic entities. Although both frameworks are equivalent in weak generative capacity --both derive exactly the context-free languages --, LCG is superior to CG in that it can cope in a natural way with extraction and unbounded dependency phenomena. For instance, no special category assignments need to be stipulated to handle a relative clause containing a trace, because it is analyzed, via hypothetical reasoning, like a traceless clause with the trace being the hypothesis to be discharged when combined with the relative pronoun. Figure 1 illustrates this proof-logical behaviour. Notice that this natural-deduction-style proof in the type logic corresponds very closely to the phrasestructure tree one would like to adopt in an analysis with traces. We thus can derive Bill misses ~ as an s from the hypothesis that there is a \"phantom\" np in the place of the trace. Discharging the hypothesis, indicated by index 1, results in Bill misses being analyzed as an s/np from zero hypotheses. Observe, however, that such a bottom-up synthesis of a new unsaturated type is only required, if that type is to be consumed (as the antecedent of an implication) by another type. Otherwise there would be a simpler proof without this abstraction. In our example the relative pronoun has such a complex type triggering an extraction.",
"cite_spans": [
{
"start": 31,
"end": 73,
"text": "particular Lambek Categorial Grammar (LCG)",
"ref_id": null
}
],
"ref_spans": [
{
"start": 1018,
"end": 1026,
"text": "Figure 1",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "A drawback of the pure Lambek Calculus !_ is that it only allows for so-called 'peripheral extraction', i.e., in our example the trace should better be initial or final in the relative clause. This inflexibility of Lambek Calculus is one of the reasons why many researchers study richer systems today. For instance, the recent work by Moortgat (Moortgat 94) gives a systematic in-depth study of mixed Lambek systems, which integrate the systems L, NL, NLP, and LP. These ingredient systems are obtained by varying the Lambek calculus along two dimensions: adding the permutation rule (P) and/or dropping the assumption that the type combinator (which forms the sequences the systems talk about) is associative (N for non-associative).",
"cite_spans": [
{
"start": 344,
"end": 357,
"text": "(Moortgat 94)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Taken for themselves these variants of I_ are of little use in linguistic descriptions. But in Moortgat's mixed system all the different resource management modes of the different systems are left intact in the combination and can be exploited in different parts of the grammar. The relative pronoun which would, for instance, receive category (np\\np)/ (np --o s) with --o being implication in LP, 1 i.e., it requires 1The Lambek calculus with permutation I_P is also called the \"nondirectional Lambek calculus\" (Benthem 88) . In it the leftward and rightward implication The present paper studies the computational complexity of a variant of the Lambek Calculus that lies between / and tP, the Semidirectional Lambek Calculus SDk. 3 Since tP derivability is known to be NPcomplete, it is interesting to study restrictions on the use of the I_P operator -o. A restriction that leaves its proposed linguistic applications intact is to admit a type B -o A only as the argument type in functional applications, but never as the functor. Stated prove-theoretically for Gentzen-style systems, this amounts to disallowing the left rule for -o. Surprisingly, the resulting system SD[. can be stated without the need for structural rules, i.e., as a monolithic system with just one structural connective, because the ability of the abstracted-over formula to permute can be directly encoded in the right rule for --o. 4",
"cite_spans": [
{
"start": 353,
"end": 363,
"text": "(np --o s)",
"ref_id": null
},
{
"start": 495,
"end": 524,
"text": "Lambek calculus\" (Benthem 88)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Note that our purpose for studying SDI_ is not that it might be in any sense better suited for a theory of grammar (except perhaps, because of its simplicity), but rather, because it exhibits a core of logical behaviour that any richer system also needs to include, at least if it should allow for non-peripheral extraction. The sources of complexity uncovered here are thus a forteriori present in all these richer systems as well.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "collapse. 2Morrill (Morrill 94) achieves the same effect with a permutation modality /k apphed to the np gap: (s/Anp)",
"cite_spans": [
{
"start": 19,
"end": 31,
"text": "(Morrill 94)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "SThis name was coined by Esther K6nig-Baumer, who employs a variant of this calculus in her LexGram system (KSnig 95) for practical grammar development.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "4It should be pointed out that the resource management in this calculus is very closely related to the handhng and interaction of local valency and unbounded dependencies in HPSG. The latter being handled with set-valued features SLASH, QUE and KEL essentially emulates the permutation potential of abstracted categories in semidirectional Lambek Grammar. A more detailed analysis of the relation between HPSG and SD[ is given in (KSnig 95).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "The semidirectional Lambek calculus (henceforth SDL) is a variant of J. Lambek's original (Lambek 58) calculus of syntactic types. We start by defining the Lambek calculus and extend it to obtain SDL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lambek calculus",
"sec_num": "2.1"
},
{
"text": "Formulae (also called \"syntactic types\") are built from a set of propositional variables (or \"primitive types\") B = {bl, b2,...} and the three binary connectives \u2022 , \\,/, called product, left implication, and right implication. We use generally capital letters A, B, C,... to denote formulae and capitals towards the end of the alphabet T, U, V, ... to denote sequences of formulae. The concatenation of sequences U and V is denoted by (U, V).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lambek calculus",
"sec_num": "2.1"
},
{
"text": "The (usual) formal framework of these logics is a Gentzen-style sequent calculus. Sequents are pairs (U, A), written as U =~ A, where A is a type and U is a sequence of types. 5 The claim embodied by sequent U =~ A can be read as \"formula A is derivable from the structured database U\". First of all, since we don't need products to obtain our results and since they only complicate matters, we eliminate products from consideration in the sequel.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lambek calculus",
"sec_num": "2.1"
},
{
"text": "In Semidirectional Lambek Calculus we add as additional connective the [_P implication --% but equip it only with a right rule.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lambek calculus",
"sec_num": "2.1"
},
{
"text": "U, B, V :=~ A (-o R) if T = (U, Y) nonempty.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lambek calculus",
"sec_num": "2.1"
},
{
"text": "5In contrast to Linear Logic (Girard 87) the order of types in U is essential, since the structural rule of permutation is not assumed to hold. Moreover, the fact that only a single formula may appear on the right of ~, make the Lambek calculus an intuitionistic fragment of the multiplicative fragment of non-commutative propositional Linear Logic.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "T :~ B --o A",
"sec_num": null
},
{
"text": "(Ax) T~B U,A,V=~C U, A/B, T, V =~ C (/L) U,B ~A U ::~ A/B (/1~) if U nonempty T ::v B U,A, V =v C U, T, B\\A, V =~ C (\\L) B,U~A U =~ B\\A (\\R) if U nonempty U,A,B, V =~ C (.L) U, AoB, V =~ C UsA V~B (.R) U,V =~ A.B T~A U,A,V=\u00a2,C (Cut) U, T, V =~ U Figure 2: Lambek calculus L",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "T :~ B --o A",
"sec_num": null
},
{
"text": "Let us define the polarity of a subformula of a sequent A1, \u2022 \u2022., Am ::~ A as follows: A has positive polarity, each of Ai have negative polarity and if B/C or C\\B has polarity p, then B also has polarity p and C has the opposite polarity of p in the sequent.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "T :~ B --o A",
"sec_num": null
},
{
"text": "A consequence of only allowing the (-o R) rule, which is easily proved by induction, is that in any derivable sequent --o may only appear in positive polarity. Hence, -o may not occur in the (cut) formula A of a (Cut) application and any subformula B -o A which occurs somewhere in the prove must also occur in the final sequent. When we assume the final sequent's RHS to be primitive (or --o-less), then the (-o R) rule will be used exactly once for each (positively) occuring -o-subformula. In other words, (-o R) may only do what it is supposed to do: extraction, and we can directly read off the category assignment which extractions there will be.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "T :~ B --o A",
"sec_num": null
},
{
"text": "We can show Cut Elimination for this calculus by a straight-forward adaptation of the Cut elimination proof for L. We omit the proof for reasons of space.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "T :~ B --o A",
"sec_num": null
},
{
"text": "The cut-free system enjoys, as usual for Lambek-like logics, the Subformula Property: in any proof only subformulae of the goal sequent may appear.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "SDL-derivable sequent has a cut-free proof.",
"sec_num": null
},
{
"text": "In our considerations below we will make heavy use of the well-known count invariant for The invariant now states that for any primitive b, the b-count of the RHS and the LHS of any derivable sequent are the same. By noticing that this invariant is true for (Ax) and is preserved by the rules, we immediately can state:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "SDL-derivable sequent has a cut-free proof.",
"sec_num": null
},
{
"text": "Proposition 2 (Count Invariant) If I-sb L U ==~",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "SDL-derivable sequent has a cut-free proof.",
"sec_num": null
},
{
"text": "Let us in parallel to SDL consider the fragment of it in which (/R) and (\\R) are disallowed. We call this fragment SDL-. Remarkable about this fragment is that any positive occurrence of an implication must be --o and any negative one must be / or \\. . This excludes many contextfree or even regular languages, but includes some context-sensitive ones, e.g., the permutation closure of a n b n c n .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "(]L) 2x(--on) (",
"eq_num": "/L"
}
],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "Concerning SD[, it is straightforward to show that all context-free languages can be generated by SDLgrammars\u2022 Proposition 4 Every context-free language is generated by some SDL-grammar.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "Proof. We can use a the standard transformation of an arbitrary cfr. grammar G = (N, T, P, S) to a categorial grammar G'. Since -o does not appear in G' each SDl_-proof of a lexical assignment must be also an I_-proof, i.e. exactly the same strings are judged grammatical by SDL as are judged by L. D Note that since the {(Ax), (/L), (\\L)} subset of I_ already accounts for the cfr. languages, this observation extends to SDL-.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "Moreover, some languages which are not context-free can also be generated. ----CI = C2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "The distinguished primitive type is x\u2022 To simplify the argumentation, we abbreviate types as indicated above\u2022 Now, observe that a sequent U =~ x, where U is the image of some string over E, only then may have balanced primitive counts, if U contains exactly one occurrence of each of A2, B2 and C2 (accounting for the one supernumerary x and balanced y and z counts) and for some number n >_ 0, n occurrences of each of A1, B1, and C1 (because, resource-oriented speaking, each Bi and Ci \"consume\" a b and c, resp., and each Ai \"provides\" a pair b, c). Hence, only strings containing the same number of a's, b's and c's may be produced. Furthermore, due to the Subformula Property we know that in a cut-free proof of U ~ x, the mMn formula in abstractions (right rules) may only be either c -o (b --o X) or b -o X, where X E {x,y}, since all other implication types have primitive antecedents. Hence, the LHS of any sequent in the proof must be a subsequence of U, with some additional b types and c types interspersed. But then it is easy to show that U can only be of the form Anl, A2, B~, B2, C~, C2, since any / connective in U needs to be introduced via (/L).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "It remains to be shown, that there is actually a proof for such a sequent\u2022 It is given in Figure 3 .",
"cite_spans": [],
"ref_spans": [
{
"start": 90,
"end": 98,
"text": "Figure 3",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "The sequent marked with * is easily seen to be derivable without abstractions.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "A remarkable point about SDL's ability to cover this language is that neither L nor LP can generate it. Hence, this example substantiates the claim made in (Moortgat 94 ) that the inferential capacity of mixed Lambek systems may be greater than the sum of its component parts. Moreover, the attentive reader will have noticed that our encoding also extends to languages having more groups of n symbols, i.e., to languages of the form n n n al a2 ... a k \u2022 Finally, we note in passing that for this grammar the rules (/R) and (\\R) are irrelevant, i.e. that it is at the same time an SOL-grammar.",
"cite_spans": [
{
"start": 156,
"end": 168,
"text": "(Moortgat 94",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "A, then #b(U) = #b(A) fo~ any b ~ t~.",
"sec_num": null
},
{
"text": "We show that the Parsing Problem for SDLgrammars is NP-complete by a reduction of the 3-Partition Problem to it. 6 This well-known NPcomplete problem is cited in (GareyJohnson 79) as follows.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Instance: Set ,4 of 3m elements, a bound N E Z +, and a size s(a) E Z + for each a E `4 such that ~ < s(a) < ~-and ~o~ s(a) = mN.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Question: Can `4 be partitioned into m disjoint sets `41,`42,...,Am such that, for 1 < i < m, ~ae.a s(a) = N (note that each `4i must 'therefore contain exactly 3 elements from `4)? Comment: NP-complete in the strong sense.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Here is our reduction. Let F = (`4, m,N,s) 6A similar reduction has been used in (LincolnWinkler 94) to show that derivability in the multiplicative fragment of propositional Linear Logic with only the connectives --o and @ (equivalently Lambek calculus with permutation LP) is NP-complete.",
"cite_spans": [],
"ref_spans": [
{
"start": 23,
"end": 42,
"text": "Let F = (`4, m,N,s)",
"ref_id": null
}
],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "The word we are interested in is v wl w2...w3m. We do not care about other words that might be generated by Gr. Our claim now is that a given 3-Partition problem F is solvable if and only if v wl ... w3m is in L(Gr). We consider each direction in turn.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Lemma 5 (Soundness) If a 3-Partition problem F = (A,m,N,s) has a solution, then vwl...w3m is in/(Gr).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Proof. We have to show, when given a solution to F, how to choose a type sequence U ~ l (vwl...wzm) and construct an SDL proof for U ==~ a. Suppose `4 = {al,a2,...,a3m}. From a given solution (set of triples) A1,`4~,... ,-Am we can compute in polynomial time a mapping k that sends the index of an element to the index of its solution triple, i.e., k(i) = j iff ai e `4j. To obtain the required sequence U, we simply choose for the wi terminals the type",
"cite_spans": [
{
"start": 88,
"end": 99,
"text": "(vwl...wzm)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "\u2022 cS(a3\"~) \u2022 c ~(\"~) (resp. d/bk(3m) k(3m) for W3m). did \u2022 bk(i) k(i) Hence the complete sequent to solve is:",
"cite_spans": [
{
"start": 21,
"end": 36,
"text": "(resp. d/bk(3m)",
"ref_id": null
},
{
"start": 57,
"end": 64,
"text": "\u2022 bk(i)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "N d) a/(b 3 \u2022b 3 \u2022...\u2022b3m ac N \u2022c N \u2022...\u2022c m -o did \u2022 bko) \u2022 %(1) cS(a3,.-1) (*) did \u2022 bk(3m-1) \u2022 k(am-1) dlb \u2022 cS(a3\") / k(3m) k(zm)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Let a/Bo, B1,...B3m ~ a be a shorthand for (*), and let X stand for the sequence of primitive types c~ (,,~,.) c~(,~.,,-~) c~(,~,)",
"cite_spans": [
{
"start": 103,
"end": 110,
"text": "(,,~,.)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "bk(3m), k(3m),bk(3m-l), k(3,~_l),...bko), k(1)\"",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Using rule (/L) only, we can obviously prove B1, . . . B3m , X ::~ d. Now, F = (.4, m, N, s 3m, Bi E l(wi) . Now, since the counts of this sequent must be balanced, the sequence B1,...B3m must contain for each 1 _< j < m exactly 3 bj and exactly N cj as subformulae. Therefore we can read off the solution to F from this sequent by including in Aj (for 1 < j < m) those three ai for which Bi has an occurrence of bj, say these are aj(1), aj(2) and aj(3). We verify, again via balancedness of the primitive counts, that s(aj(1)) \u00f7 s(aj(2)) + s(aj(3)) = N holds, because these are the numbers of positive and negative occurrences of cj in the sequent. This completes the proof.",
"cite_spans": [
{
"start": 55,
"end": 60,
"text": "B3m ,",
"ref_id": null
},
{
"start": 61,
"end": 74,
"text": "X ::~ d. Now,",
"ref_id": null
},
{
"start": 92,
"end": 95,
"text": "3m,",
"ref_id": null
},
{
"start": 96,
"end": 106,
"text": "Bi E l(wi)",
"ref_id": null
}
],
"ref_spans": [
{
"start": 75,
"end": 91,
"text": "F = (.4, m, N, s",
"ref_id": null
}
],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "[] The reduction above proves NP-hardness of the parsing problem. We need strong NP-completeness of 3-Partition here, since our reduction uses a unary encoding. Moreover, the parsing problem also lies within NP, since for a given grammar G proofs are linearly bound by the length of the string and hence, we can simply guess a proof and check it in polynomial time. Therefore we can state the following:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Theorem 7 The parsing problem for SDI_ is NPcomplete.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "Finally, we observe that for this reduction the rules (/R) and (\\R) are again irrelevant and that we can extend this result to SDI_-.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "NP-Completeness of the Parsing Problem",
"sec_num": "3"
},
{
"text": "We have defined a variant of Lambek's original calculus of types that allows abstracted-over categories to freely permute. Grammars based on SOl-can generate any context-free language and more than that. The parsing problem for SD[, however, we have shown to be NP-complete. This result indicates that efficient parsing for grammars that allow for large numbers of unbounded dependencies from within one node may be problematic, even in the categorial framework. Note that the fact, that this problematic case doesn't show up in the correct analysis of normal NL sentences, doesn't mean that a parser wouldn't have to try it, unless some arbitrary bound to that number is assumed. For practical grammar engineering one can devise the motto avoid accumulation of unbounded dependencies by whatever means.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "4"
},
{
"text": "On the theoretical side we think that this result for S01 is also of some importance, since SDI_ exhibits a core of logical behaviour that any (Lambek-based) logic must have which accounts for non-peripheral extraction by some form of permutation. And hence, this result increases our understanding of the necessary computational properties of such richer systems. To our knowledge the question, whether the Lambek calculus itself or its associated parsing problem are NP-hard, are still open.",
"cite_spans": [
{
"start": 143,
"end": 157,
"text": "(Lambek-based)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "4"
}
],
"back_matter": [],
"bib_entries": {
"BIBREF0": {
"ref_id": "b0",
"title": "Categorial Grammars and Natural Language Structures",
"authors": [
{
"first": "J",
"middle": [],
"last": "Van Benthem",
"suffix": ""
}
],
"year": 1988,
"venue": "",
"volume": "",
"issue": "",
"pages": "35--68",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "J. van Benthem. The Lambek Calculus. In R. T. O. et al. (Ed.), Categorial Grammars and Natural Lan- guage Structures, pp. 35-68. Reidel, 1988.",
"links": null
},
"BIBREF1": {
"ref_id": "b1",
"title": "Computers and Intractability--A Guide to the Theory of NP-Completeness",
"authors": [
{
"first": "M",
"middle": [
"R"
],
"last": "Garey",
"suffix": ""
},
{
"first": "D",
"middle": [
"S"
],
"last": "Johnson",
"suffix": ""
}
],
"year": 1979,
"venue": "J.-Y. Girard. Linear Logic. Theoretical Computer Science",
"volume": "50",
"issue": "1",
"pages": "1--102",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "M. R. Garey and D. S. Johnson. Computers and Intractability--A Guide to the Theory of NP- Completeness. Freeman, San Francisco, Cal., 1979. J.-Y. Girard. Linear Logic. Theoretical Computer Science, 50(1):1-102, 1987.",
"links": null
},
"BIBREF2": {
"ref_id": "b2",
"title": "LexGram -a practical categorial grammar formalism",
"authors": [
{
"first": "E",
"middle": [],
"last": "Khnig",
"suffix": ""
}
],
"year": 1995,
"venue": "Proceedings of the Workshop on Computational Logic for Natural Language Processing. A Joint COMPULOGNET/ELSNET/EAGLES Workshop",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "E. Khnig. LexGram -a practical categorial gram- mar formalism. In Proceedings of the Workshop on Computational Logic for Natural Language Process- ing. A Joint COMPULOGNET/ELSNET/EAGLES Workshop, Edinburgh, Scotland, April 1995.",
"links": null
},
"BIBREF3": {
"ref_id": "b3",
"title": "The Mathematics of Sentence Structure",
"authors": [
{
"first": "J",
"middle": [],
"last": "Lambek",
"suffix": ""
}
],
"year": 1958,
"venue": "American Mathematical Monthly",
"volume": "65",
"issue": "3",
"pages": "154--170",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "J. Lambek. The Mathematics of Sentence Struc- ture. American Mathematical Monthly, 65(3):154- 170, 1958.",
"links": null
},
"BIBREF4": {
"ref_id": "b4",
"title": "Constant-Only Multiplicative Linear Logic is NP-Complete",
"authors": [
{
"first": "P",
"middle": [],
"last": "Lincoln",
"suffix": ""
},
{
"first": "T",
"middle": [],
"last": "Winkler",
"suffix": ""
}
],
"year": 1994,
"venue": "Theoretical Computer Science",
"volume": "135",
"issue": "1",
"pages": "155--169",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "P. Lincoln and T. Winkler. Constant-Only Multi- plicative Linear Logic is NP-Complete. Theoretical Computer Science, 135(1):155-169, Dec. 1994.",
"links": null
},
"BIBREF5": {
"ref_id": "b5",
"title": "Lambek Calculus. Multimodal and Polymorphic Extensions, DYANA-2 deliverable RI.I.B. ESPRIT, Basic Research Project 6852",
"authors": [
{
"first": "M",
"middle": [],
"last": "Moortgat",
"suffix": ""
}
],
"year": 1994,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "M. Moortgat. Residuation in Mixed Lambek Sys- tems. In M. Moortgat (Ed.), Lambek Calculus. Mul- timodal and Polymorphic Extensions, DYANA-2 de- liverable RI.I.B. ESPRIT, Basic Research Project 6852, Sept. 1994.",
"links": null
},
"BIBREF6": {
"ref_id": "b6",
"title": "Type Logical Grammar: Categorial Logic of Signs",
"authors": [
{
"first": "G",
"middle": [],
"last": "Morrill",
"suffix": ""
}
],
"year": 1994,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "G. Morrill. Type Logical Grammar: Categorial Logic of Signs. Kluwer, 1994.",
"links": null
},
"BIBREF7": {
"ref_id": "b7",
"title": "Lambek grammars are context free",
"authors": [
{
"first": "M",
"middle": [],
"last": "Pentus",
"suffix": ""
}
],
"year": 1993,
"venue": "Proceedings of Logic in Computer Science",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "M. Pentus. Lambek grammars are context free. In Proceedings of Logic in Computer Science, Montreal, 1993.",
"links": null
}
},
"ref_entries": {
"FIGREF0": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "Extraction as resource-conscious hypothetical reasoning as an argument \"an s lacking an np somewhere\" .2."
},
"FIGREF1": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "Figure 2shows Lambek's original calculus t."
},
"FIGREF2": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "Lambek systems (Benthem 88), which is an expression of the resource-consciousness of these logics. Define #b(A) (the b-count of A), a function counting positive and negative occurrences of primitive type b in an arbi-97 trary type A, to be if A= b if A primitive and A ~ b #b(A)= #b(B)-#b(C)ifA=B/CorA=V\\B or A=C-o B [.#b(B) + #b(C) ifA = B. C"
},
"FIGREF3": {
"type_str": "figure",
"num": null,
"uris": null,
"text": ", C~, C2, c n+l, b n+l => y (*) B~, B2, C~, C2, c n, b n ~ c --o (b --o y) A2, B[, B2, C~, C2, c n, b n =* x n--1 A 1 , A2, B~, B2, C~, C2, c, b =v x A~ -1, A2, B~', B2, C~, C2 =~ c -0 (b -0 x) A?, A2, B~, B2, C{ ~, C2 ==> x Proof of A~, A2, B~, B2, C~, C2 =~ z 2x(-on)"
},
"FIGREF4": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "Example. Consider the following grammar G for the language anbnc n. We use primitive types B = {b, c, x, y, z} and define the lexical map for E = 98 {a, b, c} as follows: l(a) := { x/(c ---o (b -o x)), xl(c ---o (b -o y)) } = )41 = A2"
},
"FIGREF5": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "be a given 3-Partition instance. For notational convenience we abbreviate (...((A/BI)/B~)/...)/Bn by A/B~ \u2022...\u2022 B2 \u2022 B1 and similarly B, -o (... (B1 --o A)...) by Bn \u2022... \u2022 B2 \u2022 B1 --o A, but note that this is just an abbreviation in the product-free fragment. Moreover the notation A k stands for AoAo ...oA k t~mes We then define the SDL-grammar Gr = (~, ~, bs, l) as follows: p, := {v, wl,..., warn} 5 t\" := all formulae over primitive types m b ) := UJ.<./<m d/d \u2022 bj \u2022 c: (~')"
}
}
}
} |