| { |
| "paper_id": "P93-1018", |
| "header": { |
| "generated_with": "S2ORC 1.0.0", |
| "date_generated": "2023-01-19T08:52:10.742022Z" |
| }, |
| "title": "PARALLEL MULTIPLE CONTEXT-FREE GRAMMARS, FINITE-STATE TRANSLATION SYSTEMS, AND POLYNOMIAL-TIME RECOGNIZABLE SUBCLASSES OF LEXICAL-FUNCTIONAL GRAMMARS", |
| "authors": [ |
| { |
| "first": "Hiroyuki", |
| "middle": [], |
| "last": "Seki", |
| "suffix": "", |
| "affiliation": {}, |
| "email": "seki@ics.es.osaka-u.ac.jp" |
| }, |
| { |
| "first": "Ryuichi", |
| "middle": [], |
| "last": "Nakanishi", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Osaka University", |
| "location": { |
| "addrLine": "1-1 Machikaneyama", |
| "postCode": "560", |
| "settlement": "Toyonaka", |
| "region": "Osaka", |
| "country": "Japan" |
| } |
| }, |
| "email": "" |
| }, |
| { |
| "first": "Yuichi", |
| "middle": [], |
| "last": "Kaji", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Osaka University", |
| "location": { |
| "addrLine": "1-1 Machikaneyama", |
| "postCode": "560", |
| "settlement": "Toyonaka", |
| "region": "Osaka", |
| "country": "Japan" |
| } |
| }, |
| "email": "" |
| }, |
| { |
| "first": "Sachiko", |
| "middle": [], |
| "last": "Ando", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Osaka University", |
| "location": { |
| "addrLine": "1-1 Machikaneyama", |
| "postCode": "560", |
| "settlement": "Toyonaka", |
| "region": "Osaka", |
| "country": "Japan" |
| } |
| }, |
| "email": "" |
| }, |
| { |
| "first": "Tadao", |
| "middle": [], |
| "last": "Kasami", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Osaka University", |
| "location": { |
| "addrLine": "1-1 Machikaneyama", |
| "postCode": "560", |
| "settlement": "Toyonaka", |
| "region": "Osaka", |
| "country": "Japan" |
| } |
| }, |
| "email": "" |
| } |
| ], |
| "year": "", |
| "venue": null, |
| "identifiers": {}, |
| "abstract": "A number of grammatical formalisms were introduced to define the syntax of natural languages. Among them are parallel multiple context-free grammars (pmcfg's) and lexical-functional grammars (lfg's). Pmcfg's and their subclass called multiple context-free grammars (mcfg's) are natural extensions of cfg's, and pmcfg's are known to be recognizable in polynomial time. Some subclasses of lfg's have been proposed, but they were shown to generate an AlP-complete language. Finite state translation systems (fts') were introduced as a computational model of transformational grammars. In this paper, three subclasses of lfg's called nc-lfg's, dc-lfg's and fc-lfg's are introduced and the generative capacities of the above mentioned grammatical formalisms are investigated. First, we show that the generative capacity of fts' is equal to that of nc-lfg's. As relations among subclasses of those formalisms, it is shown that the generative capacities of deterministic fts', dc-lfg's, and pmcfg's are equal to each other, and the generative capacity of fc-lfg's is equal to that of mcfg's. It is also shown that at least one Af79-complete language is generated by fts'. Consequently, deterministic fts', dc-lfg's and fc-lfg's can be recognized in polynomial time. However, fts' (and nc-lfg's) cannot, if P \u00a2 AfT 9.", |
| "pdf_parse": { |
| "paper_id": "P93-1018", |
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| "abstract": [ |
| { |
| "text": "A number of grammatical formalisms were introduced to define the syntax of natural languages. Among them are parallel multiple context-free grammars (pmcfg's) and lexical-functional grammars (lfg's). Pmcfg's and their subclass called multiple context-free grammars (mcfg's) are natural extensions of cfg's, and pmcfg's are known to be recognizable in polynomial time. Some subclasses of lfg's have been proposed, but they were shown to generate an AlP-complete language. Finite state translation systems (fts') were introduced as a computational model of transformational grammars. In this paper, three subclasses of lfg's called nc-lfg's, dc-lfg's and fc-lfg's are introduced and the generative capacities of the above mentioned grammatical formalisms are investigated. First, we show that the generative capacity of fts' is equal to that of nc-lfg's. As relations among subclasses of those formalisms, it is shown that the generative capacities of deterministic fts', dc-lfg's, and pmcfg's are equal to each other, and the generative capacity of fc-lfg's is equal to that of mcfg's. It is also shown that at least one Af79-complete language is generated by fts'. Consequently, deterministic fts', dc-lfg's and fc-lfg's can be recognized in polynomial time. However, fts' (and nc-lfg's) cannot, if P \u00a2 AfT 9.", |
| "cite_spans": [], |
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| "eq_spans": [], |
| "section": "Abstract", |
| "sec_num": null |
| } |
| ], |
| "body_text": [ |
| { |
| "text": "Introduction A number of grammatical formalisms such as lexical-functional grammars (Kaplan 1982) , head grammars (Pollard 1984) and tree adjoining grammars (Joshi 1975 )(Vijay-Shanker 1987 were introduced to define the syntax of natural languages.", |
| "cite_spans": [ |
| { |
| "start": 84, |
| "end": 97, |
| "text": "(Kaplan 1982)", |
| "ref_id": "BIBREF6" |
| }, |
| { |
| "start": 114, |
| "end": 128, |
| "text": "(Pollard 1984)", |
| "ref_id": "BIBREF14" |
| }, |
| { |
| "start": 157, |
| "end": 168, |
| "text": "(Joshi 1975", |
| "ref_id": "BIBREF3" |
| }, |
| { |
| "start": 169, |
| "end": 189, |
| "text": ")(Vijay-Shanker 1987", |
| "ref_id": "BIBREF18" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "On the other hand, there has been much effort to propose well-defined computational models of transformational grammars. One of these is the one to extend devices which operate on strings, such as generalized sequential machines (gsm's) to devices which operate on trees. It is fundamentally significant to clarify the generative capacities of such grammars and devices.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "Parallel multiple context-free grammars (pmcfg's) and multiple context-free grammars (mcfg's) were introduced in (Kasami 1988a) (Seki 1991) as natural extensions of cfg's. The subsystem of linear context-free rewriting systems (Icfrs') (Vijay-Shanker 1987) which deals with only strings is the same formalism as mcfg's. The class of cfl's is properly included in the class of languages generated by pmcfg's, which in turn is properly included in the one generated by mcfg's. The class of languages generated by pmcfg's is properly included in that of context-sensitive languages (Kasami 1988a ).", |
| "cite_spans": [ |
| { |
| "start": 40, |
| "end": 49, |
| "text": "(pmcfg's)", |
| "ref_id": null |
| }, |
| { |
| "start": 85, |
| "end": 93, |
| "text": "(mcfg's)", |
| "ref_id": null |
| }, |
| { |
| "start": 113, |
| "end": 127, |
| "text": "(Kasami 1988a)", |
| "ref_id": "BIBREF7" |
| }, |
| { |
| "start": 128, |
| "end": 139, |
| "text": "(Seki 1991)", |
| "ref_id": "BIBREF16" |
| }, |
| { |
| "start": 227, |
| "end": 235, |
| "text": "(Icfrs')", |
| "ref_id": null |
| }, |
| { |
| "start": 236, |
| "end": 256, |
| "text": "(Vijay-Shanker 1987)", |
| "ref_id": "BIBREF18" |
| }, |
| { |
| "start": 579, |
| "end": 592, |
| "text": "(Kasami 1988a", |
| "ref_id": "BIBREF7" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "Pmcfg's have been shown to be recognized in polynomial time (Kasami 1988b ) (Seki 1991) .", |
| "cite_spans": [ |
| { |
| "start": 60, |
| "end": 73, |
| "text": "(Kasami 1988b", |
| "ref_id": "BIBREF8" |
| }, |
| { |
| "start": 76, |
| "end": 87, |
| "text": "(Seki 1991)", |
| "ref_id": "BIBREF16" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "A tree transducer (Rounds 1969 ) takes a tree as an input, starts from the initial state with its head scanning the root node of an input. According to the current state and the label of the scanned node, it transforms an input tree into an output tree in a top-down way. A finite state translation system (fts) is a tree transducer with its input domain being the set of derivation trees of a cfg (Rounds 1969) (Thatcher 1967) . A number of equivalence relations between the classes of yield languages generated by fts' and other computational models have been established (Engelfriet 1991) (Engelfriet 1980) (Weir 1992) . Especially, it has been shown that the class of yield languages generated by finite-copying fts' equals to the class of languages generated by lcfrs' (Weir 1992) , hence by mcfg's.", |
| "cite_spans": [ |
| { |
| "start": 18, |
| "end": 30, |
| "text": "(Rounds 1969", |
| "ref_id": "BIBREF15" |
| }, |
| { |
| "start": 398, |
| "end": 411, |
| "text": "(Rounds 1969)", |
| "ref_id": "BIBREF15" |
| }, |
| { |
| "start": 412, |
| "end": 427, |
| "text": "(Thatcher 1967)", |
| "ref_id": "BIBREF17" |
| }, |
| { |
| "start": 574, |
| "end": 591, |
| "text": "(Engelfriet 1991)", |
| "ref_id": "BIBREF1" |
| }, |
| { |
| "start": 592, |
| "end": 609, |
| "text": "(Engelfriet 1980)", |
| "ref_id": "BIBREF2" |
| }, |
| { |
| "start": 610, |
| "end": 621, |
| "text": "(Weir 1992)", |
| "ref_id": "BIBREF21" |
| }, |
| { |
| "start": 774, |
| "end": 785, |
| "text": "(Weir 1992)", |
| "ref_id": "BIBREF21" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "In lexical-functional grammars (Ifg's) (Kaplan 1982) , associated with each node v of a derivation tree is a finite set F of pairs of attribute names and their values. F is called the fstructure of v. An lfg G consists of a cfg Go called the underlying cfg of G and a finite set Pfs of equations called functional schemata which specify constraints between the f-structures of nodes in a derivation tree. Functional schemata are attached to symbols in productions of Go. It has been shown in (Nakanishi 1992 ) that the class of languages generated by lfg's is equal to that of re-cursively enumerable languages even though the underlying cfg's are restricted to regular grammars. In (Gazdar 1985) (Kaplan 1982) (Nishino 1991) , subclasses of lfg's were proposed in order to guarantee the recursiveness (and/or the efficient recognition) of languages generated by lfg's. However, these classes were shown to generate an A/P-complete language (Nakanishi 1992) .", |
| "cite_spans": [ |
| { |
| "start": 31, |
| "end": 38, |
| "text": "(Ifg's)", |
| "ref_id": null |
| }, |
| { |
| "start": 39, |
| "end": 52, |
| "text": "(Kaplan 1982)", |
| "ref_id": "BIBREF6" |
| }, |
| { |
| "start": 492, |
| "end": 507, |
| "text": "(Nakanishi 1992", |
| "ref_id": "BIBREF10" |
| }, |
| { |
| "start": 683, |
| "end": 696, |
| "text": "(Gazdar 1985)", |
| "ref_id": "BIBREF4" |
| }, |
| { |
| "start": 697, |
| "end": 710, |
| "text": "(Kaplan 1982)", |
| "ref_id": "BIBREF6" |
| }, |
| { |
| "start": 711, |
| "end": 725, |
| "text": "(Nishino 1991)", |
| "ref_id": "BIBREF11" |
| }, |
| { |
| "start": 941, |
| "end": 957, |
| "text": "(Nakanishi 1992)", |
| "ref_id": "BIBREF10" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "In this paper, three subclasses of lfg's called nc-lfg's, dc-lfg's and fc-lfg's are proposed, two of which can be recognized in polynomial time. Moreover, this paper clarifies the relations among the generative capacities of pmcfg's, fts' and these subclasses of lfg's.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "In nc-lfg's, a functional schema either specifies the vMue of a specific attribute, say atr, immediately (Tart = val) or specifies that the value of a specific attribute of a node v is equal to the whole f-structure of a child node of v (Tatr =l).", |
| "cite_spans": [ |
| { |
| "start": 105, |
| "end": 117, |
| "text": "(Tart = val)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "An nc-lfg is called a dc-lfg if each pair of rules P] : A --~ aa and P2 : A --~ a2 whose left-hand sides are the same is inconsistent in the sense that there exists no f-structure that locally satisfies both of the functional schemata of Pl and those of p2. Intuitively, in a dc-lfg G, for each pair (tl, t2) of derivation trees in G, if the f-structure and nonterminal of the root of tl are the same as those of t2, then t] and t2 derive the same terminal string. Let G be an nc-lfg. A multiset M of nonterminals of G is called an SPN multiset in G if the following condition holds: v2,...,v,~} of t such that the label ofvi is Ai (1 < i < n) and the fstructures of vi's are the same with each other by functional schemata of G.", |
| "cite_spans": [ |
| { |
| "start": 584, |
| "end": 598, |
| "text": "v2,...,v,~} of", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "Let M = {{A1,A2,'..,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "V = {v],", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "If the number of SPN multisets in G is finite, then G is called an fc-lfg.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "Our main result is that the generative capacity of nc-lfg's is equal to that of fts'. As relations among proper subclasses of the above mentioned formalisms, it is shown that the generative capacities of dc-lfg's, deterministic fts' and pmcfg's are equal to each other, and the generative capacity of fc-lfg's is equal to that of mcfg's. It is also shown that a (nondeterministic) fts generates an Af:P-complete language.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1", |
| "sec_num": null |
| }, |
| { |
| "text": "A parallel multiple context-free grammar (pmcfg) is defined to be a 5-tuple G = ( N, T, F, P, S) which satisfies the following conditions (G1) through (Gh) (Kasami 1988a ) (Seki 1991 ", |
| "cite_spans": [ |
| { |
| "start": 80, |
| "end": 96, |
| "text": "( N, T, F, P, S)", |
| "ref_id": null |
| }, |
| { |
| "start": 156, |
| "end": 169, |
| "text": "(Kasami 1988a", |
| "ref_id": "BIBREF7" |
| }, |
| { |
| "start": 172, |
| "end": 182, |
| "text": "(Seki 1991", |
| "ref_id": "BIBREF16" |
| } |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Parallel Multiple Context-Free Grammars", |
| "sec_num": "2" |
| }, |
| { |
| "text": "(T*) dl(:) x (T*) d2(f) \u00d7... x (T*)da(f) (1) to I", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Parallel Multiple Context-Free Grammars", |
| "sec_num": "2" |
| }, |
| { |
| "text": "T*)'(:) which satisfies the following condition fl). Let", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Parallel Multiple Context-Free Grammars", |
| "sec_num": "2" |
| }, |
| { |
| "text": "\u2022 i = (zil, zi2,..", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Parallel Multiple Context-Free Grammars", |
| "sec_num": "2" |
| }, |
| { |
| "text": "denote the ith argument of f for 1 < i < a(f).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "(fl) For 1 < h < r(f), the hth component of f, denoted by f[h], is defined as; f[h] [Xl, f~2,-\" -, Xa(f)] = OCh,OX#(h,O)rl(h,o)Oth,1", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 . 'ah, nh_lXu(h, nh_l)n(h, nh_Dah, n~ (2.1) where", |
| "cite_spans": [ |
| { |
| "start": 4, |
| "end": 8, |
| "text": "'ah,", |
| "ref_id": null |
| }, |
| { |
| "start": 9, |
| "end": 18, |
| "text": "nh_lXu(h,", |
| "ref_id": null |
| }, |
| { |
| "start": 19, |
| "end": 28, |
| "text": "nh_l)n(h,", |
| "ref_id": null |
| }, |
| { |
| "start": 29, |
| "end": 36, |
| "text": "nh_Dah,", |
| "ref_id": null |
| }, |
| { |
| "start": 37, |
| "end": 45, |
| "text": "n~ (2.1)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "ah,k \u2022 T* for 0 < k <_ nh, 1 < #(h,j) <_ a(f) and 1 <_ ~(h,j) <_ dt~(h,j)(f) for O ~ j ~_ nh --1. (G4) P is a finite set of productions of the form A ---* f[A1,A2,...,Aa(y)] where A, Aa,A2,...,Aa(/) \u2022 N, f \u2022 F, r(f) = d(A) and di(f) = d(Ai) (1 < i < a(f)). Ifa(f) = 0,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "i.e., f \u2022 (T*) r(f), the production is called a terminating production, otherwise it is called a nonterminating production. If all the functions of a pmcfg G satisfy the following Right Linearity condition, then G is called a multiple context-free grammar (mcfg).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "[Right Linearity ] For each xlj, the total number of occurrences of xij in the right-hand sides of (2.1) from h = 1 through r(f) is at most one.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "The language generated by a pmcfg G = (N, T, F, P, S) is defined as follows. For A \u2022 N, let us define LG(A) as the smallest set satisfying the following two conditions:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "(L1) If a terminating production A --* & is in P, then ~ \u2022 LG(A). (L2) If A --~ f[A1,A2,...,Aa(y)] \u2022 P and (~i \u2022 LG(Ai) ~1 < i < a(f)), then f[~1,~2,''', O~a(f)] \u2022 LG(A) .- Define L(G) a=La(S). L(G)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "is called the parallel multiple context-free language (pmcfl) generated by G. If G is an mcfg, L(G) is called the multiple context-free language (mcfl) generated by G.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "Example 2.1 (Kasami 1988a) :", |
| "cite_spans": [ |
| { |
| "start": 12, |
| "end": 26, |
| "text": "(Kasami 1988a)", |
| "ref_id": "BIBREF7" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "Let GEX1 ~---(N, T,F,P,S),N = {S}, T = {a},F = {f~,f},P = {r] : S --~ fa, ro : S --* f[S]}, where f~ = a,f[(x)] =", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "xx. GExl is a pmcfg but is not an mcfg since the function f does not satisfy Right Linearity. The language generated by GEx~ is {a 2\" In > 0}, which cannot be generated by any mcfg (see Lemma 6 of (Kasami 1988a) ).", |
| "cite_spans": [ |
| { |
| "start": 197, |
| "end": 211, |
| "text": "(Kasami 1988a)", |
| "ref_id": "BIBREF7" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "The empty string is denoted by \u00a2. Lamina 2.1 (Kasami 1988b) (Seki 1991) : Let C be a pmcfg. For a given string w, it is decidable whether w E L (G) or not in time polynomial of I~1, where I~1 denotes the length of w.", |
| "cite_spans": [ |
| { |
| "start": 45, |
| "end": 59, |
| "text": "(Kasami 1988b)", |
| "ref_id": "BIBREF8" |
| }, |
| { |
| "start": 60, |
| "end": 71, |
| "text": "(Seki 1991)", |
| "ref_id": "BIBREF16" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "., zid,(S))", |
| "sec_num": null |
| }, |
| { |
| "text": "A set ~ of symbols is a ranked alphabet if, for each cr E ~, a unique non-negative number p(c~) is associated, p(cr) is the rank of ~. For a set X, we define free algebra T~.(X) as the smallest set such that; * T~: (X) includes X.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "\u2022 If p(~) = 0 for cr E ~, then ~ E T~(X).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "\u2022 If p(o') = n (> 1) for a E, ~ and tl,..., E 7-~.(X), then t-= or(t1,.., tn) E T~(X). t~ is called the root symbol, or shortly, the root of t.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "Hereafter, a term in 7\"~ (X) is also called a tree, and we use terminology of trees such as subtree, node and so on. Let G -(N, T, P, S) be a context-free grammar (cfg) where N, T, P and S are a set of nonterminal symbols, a set of terminal symbols, a set of productions and the initial symbol, respectively. A derivation tree in cfg G is a term defined as follows.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "(T1) Every a E T is a derivation tree in G.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "(T2) Assume that there are a production p : XI,...,Xn E NUT) in P and n derivation trees tl,...t,~ whose roots are labeled with Pl,..., pn, respectively, and", |
| "cite_spans": [ |
| { |
| "start": 44, |
| "end": 60, |
| "text": "XI,...,Xn E NUT)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "A ---* X1...X,~ (A E N,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "\u2022 ifXi E N, then pl is a production Xi --~ \" \", whose left-hand side is Xi, and", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "\u2022 ifXiET, thenpi=ti=Xi.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "Then p (tl,..., t,~) is a derivation tree in G.", |
| "cite_spans": [ |
| { |
| "start": 7, |
| "end": 20, |
| "text": "(tl,..., t,~)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "(T3) There are no other derivation trees.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "Let T~(G) be the set of derivation trees in G, and 7\u00a2s(G) C 7\u00a2(G) be the set of derivation trees whose root is labeled with a production of which left-hand side is the initial symbol S. Clearly, T~s(G) C_ T~(\u00a2) holds. Remark that 7\u00a2s(G) is a multi-sorted algebra, where the nonterminals are sorts, and the terminals and the labels of productions are operators.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "A tree transducer (Rounds 1969 ) defines a mapping from trees to trees. Since we are mainly interested in the string language generated by a tree transducer, a \"tree-to-string\" version of transducer defined in (Engelfriet 1980 ) is used in this paper. For sets Q and X, let", |
| "cite_spans": [ |
| { |
| "start": 18, |
| "end": 30, |
| "text": "(Rounds 1969", |
| "ref_id": "BIBREF15" |
| }, |
| { |
| "start": 210, |
| "end": 226, |
| "text": "(Engelfriet 1980", |
| "ref_id": "BIBREF2" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "Q[X]~{q[x] l q e Q,x", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Finite State Translation Systems", |
| "sec_num": "3" |
| }, |
| { |
| "text": "A tree-to-string transducer (yT-transducer or simply transducer) is defined to be a 5-tuple M = (Q, ~., A, q0, R) where (1) Q is a finite set of states, (2) ~ is an input ranked alphabet, (3) A is an output alphabet, (4) q0 E Q is the initial state, and (5) R is a finite set of rules of the form A tree-to-string finite state translation system (yT-fts or fts) is defined by a yT-transducer M and a cfg G, written as (M,G) (Rounds 1969) (Thatcher 1967) . Engelfriet introduced a subclass of fts' called finite-copying fts' as follows (Engelfriet 1980) : Let (M,G) be an fts with output alphabet A and initial state q0, t be a derivation tree in G and t ~ be a subtree of t. Assume that there is a derivation a : q0[t] =~ w. Now, delete from this derivation a all the derivation steps which operates on t t. This leads to the following new derivation which keeps t ~ untouched; (Engelfriet 1980) . We note that an fts (M, G) can be considered to be a model of a transformational grammar: A deep-structure of a sentence is represented by a derivation tree of G, and M can be considered to transform the deep-structure into a sentence (or its surface structure).", |
| "cite_spans": [ |
| { |
| "start": 424, |
| "end": 437, |
| "text": "(Rounds 1969)", |
| "ref_id": "BIBREF15" |
| }, |
| { |
| "start": 438, |
| "end": 453, |
| "text": "(Thatcher 1967)", |
| "ref_id": "BIBREF17" |
| }, |
| { |
| "start": 535, |
| "end": 552, |
| "text": "(Engelfriet 1980)", |
| "ref_id": "BIBREF2" |
| }, |
| { |
| "start": 878, |
| "end": 895, |
| "text": "(Engelfriet 1980)", |
| "ref_id": "BIBREF2" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "e X).", |
| "sec_num": null |
| }, |
| { |
| "text": "q[c~(xl,..., xn)] --* v", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "e X).", |
| "sec_num": null |
| }, |
| { |
| "text": "A simple subclass of lfg's, called r-lfg's, is introduced in (Nishino 1992) , which is shown to generate all the recursively enumerable languages (Nakanishi 1992) . Here, we define a nondeterministic copying Ifg (nc-lfg) as a proper subclass of r-lfg's. An nc-lfg is defined to be a 6-tuple G = (N, T, P, S, N~t~, A~tr~) where: (1) N is a finite set of nonterminal symbols, (2) T is a finite set of terminal symbols, and (3) P is a finite set of annotated productions. Sometimes, a nonterminal symbol, a terminal symbol and an annotated production are abbreviated as a nonterminal, a terminal and a production, respectively, i 4) S \u2022 N is the initial symbol, (5) Nat~ is a finite set of attributes, and (6) A~tm is a finite set of atoms.", |
| "cite_spans": [ |
| { |
| "start": 61, |
| "end": 75, |
| "text": "(Nishino 1992)", |
| "ref_id": "BIBREF12" |
| }, |
| { |
| "start": 146, |
| "end": 162, |
| "text": "(Nakanishi 1992)", |
| "ref_id": "BIBREF10" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "An equation of the form T atr =~ (atr \u2022 Nat,) is called an S (structure synthesizing) schema, and an equation of the form T atr . -= val (atr \u2022 Natr, val \u2022 A~tm) is called a V (immediate value) schema. A functional schema is either an S schema or a V schema.", |
| "cite_spans": [ |
| { |
| "start": 130, |
| "end": 161, |
| "text": "-= val (atr \u2022 Natr, val \u2022 A~tm)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "Each production p \u2022 P has the following form: An f-structure of G is recursively defined as a set F -=-{(atrl, call), (atr2, val2>,..., latrk, valk)} where atr], atr2,..., and atrk are distinct attributes, and each of vail, val2,.\" \", and valk is an atom or an f-structure. We say that vali (1 < i < k) is the value of atri in F and write F.atri -= vali. Suppose Go = i N, T, P0, S) is the underlying cfg of an nc-lfg G = (N, T, P, S, Nat,, Aa,m). Let t be a derivation tree in Go. (In 4.,7. and 8., the label of a leaf of a derivation tree is allowed to be a nonterminal.) Every internal node v in t has an f-structure, which is called the f-structure of v and written as Fv. If an underlying production P0 :A ~ BI\".Bq \u2022 P0 is applied at v, then v is labeled with either P0 itself, or p (\u2022 P) of which P0 is the underlying production, if necessary. Let vi be the ith child ofv (1 < i < q). We define the values of both sides of a functional schema attached to the symbol in p (on v) as follows:", |
| "cite_spans": [ |
| { |
| "start": 118, |
| "end": 149, |
| "text": "(atr2, val2>,..., latrk, valk)}", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "p :A -~ B1 B2 ... Bq,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "* the value of T atr (atr \u2022 Nat,) is Fv.atr,", |
| "cite_spans": [ |
| { |
| "start": 21, |
| "end": 33, |
| "text": "(atr \u2022 Nat,)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "\u2022 the value of + in an S schema is Fv~ if the S schema is attached to the i(1 _< i _< q)th symbol in the right-hand side of p, and", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "\u2022 the value of atom atm in a V schema is arm itself.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "We say that v satisfies functional schemata if for each functional schema lls = rib of p, the values of lls and r/s on v are defined and equals with each other. In this case, it is also said that Fv locally satisfies the functional schemata of p.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "NOTE : Because the meaning of a V schema is independent of the position where it is annotated, V schemata are attached to the left-hand side in this paper. For a nonterminal A E N and a sentential form a E iN t_J T)*, let t be a derivation tree of a derivation A =** Go a. If all internal nodes in t satisfy functional schemata, then a is said to be derived from A and written as A =~* . a a In this case, the tree t is called a derivation tree of A:=~* G a. We also call t a derivation tree (of a) in G simply.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "The language generated by an nc-lfg G, denoted by LIG), The language generated by GEX4 is L(GEx4) =", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "{a2\" ]n > 0}. = {the, woman, men, and, drinks, smoke, respec-tively}, N.t, = {hum, list}, A.tm = {sg,pl, nil}, and productions in P are; G~xs generates \"respectively\" sentences such as \"the woman and the men drinks and smoke respectively\".", |
| "cite_spans": [ |
| { |
| "start": 14, |
| "end": 110, |
| "text": "= {the, woman, men, and, drinks, smoke, respec-tively}, N.t, = {hum, list}, A.tm = {sg,pl, nil},", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "For a set X of functional schemata, X is consistent iff neither the following (1) nor (2) holds.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "(1) {T atr = Call, T atr = val2 } c X for some atr E Na,, and some vall,val2 E Aatm such that call # val2.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "(2) iT atr = val, T atr =~} _C X for some atr E Nat~ and some val E Aatm.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "Productions pl,''',Pn are consistent iff Ul<i<_n E (0 is consistent where E (/) is the set of functional schemata of Pl. If productions are not consistent then they are called inconsistent.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "An nc-lfg G is called a deterministically copying Ifg (dc-lfg), if any two productions A --+ al and A --+ a2 whoes left-hand sides are the same are inconsistent.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "Suppose G = (N,T, P, S, Nat,, Aatm) is an nc-lfg. Let {{el,e2,-'.,en}} denote the multiset which consists of elements el, e2,\" \u2022 \u2022, en that are not necessarily distinct. An SPN (SubPhrase Nonterminal) multiset in G is recursively defined as the following 1 through 3:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "1. {{S}} is an SPN multiset.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Subclasses of Lexical-functional grammars", |
| "sec_num": "4" |
| }, |
| { |
| "text": "Ah E N) is an SPN multiset. Let A1 --~ al, NOTE : L (GExs) is generated by a tree adjoining grammar. Suppose that a sentence has three or more phrases which have co-occurrence relation like the one between the subject phrase and the verb phrase in the \"respectively\" sentence. Tree adjoining grammars can not generate such syntax while fc-lfg's or dc-lfg's can, although the authors do not know a natural language which has such syntax so far. By Lemma 2.1 and Theorem 8.1, fc-lfg's are polynomial-time recognizable. Hence, it is desirable that whether a given lfg G is an fc-lfg or not is decidable. Fortunately, it is decidable by the following lemma.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Suppose that {{A1, A2,'\", Ah}} (A1, A2,'\" \",", |
| "sec_num": "2." |
| }, |
| { |
| "text": "Lemma 4.1: For a given nc-lfg G, it is decidable whether the number of SPN multisets in G is finite or infinite. Proof. The problem can be reduced to the boundedness problem of Petri nets, which is known to be decidable (Peterson 1981) .", |
| "cite_spans": [ |
| { |
| "start": 220, |
| "end": 235, |
| "text": "(Peterson 1981)", |
| "ref_id": "BIBREF13" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Suppose that {{A1, A2,'\", Ah}} (A1, A2,'\" \",", |
| "sec_num": "2." |
| }, |
| { |
| "text": "Let ~'nc-lfg, ~'dc-lfg and ~-'fc-lfg denote the classes of languages generated by nc-lfg's, dc-lfg's and fc-lfg's, respectively, and let y~#,, Y~.d-fts and YElc-#s denote the classes of yield languages generated by fts', deterministic fts' and finite-copying fts', respectively. Let l:vmcla and \u00a3:mcfg be the classes of languages generated by pmcfg's and mcfg's, respectively. Also let \u00a3:ta9 be the class of language generated by tree adjoining grammars.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Overview of the Results", |
| "sec_num": "5" |
| }, |
| { |
| "text": "Inclusion relations among these classes of languages are summarized in Figure 2 . An equivalence relation *1 is shown in (Weir 1992) . Relations *2 are new results which we prove in this paper. We also note that all the inclusion rela-tions are proper; indeed, 0 l", |
| "cite_spans": [ |
| { |
| "start": 121, |
| "end": 132, |
| "text": "(Weir 1992)", |
| "ref_id": "BIBREF21" |
| } |
| ], |
| "ref_spans": [ |
| { |
| "start": 71, |
| "end": 79, |
| "text": "Figure 2", |
| "ref_id": "FIGREF9" |
| } |
| ], |
| "eq_spans": [], |
| "section": "Overview of the Results", |
| "sec_num": "5" |
| }, |
| { |
| "text": "{ala2a3a41n >_ E D -E a a2 n n _ ..... a2m_la2m [ n > E C -D for m > 3,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Overview of the Results", |
| "sec_num": "5" |
| }, |
| { |
| "text": "(by (Vijay-Shanker 1987).) {a 2\" In > 0} e S -C, (by (Kasami 1988a) (Seki 1991) .)", |
| "cite_spans": [ |
| { |
| "start": 53, |
| "end": 67, |
| "text": "(Kasami 1988a)", |
| "ref_id": "BIBREF7" |
| }, |
| { |
| "start": 68, |
| "end": 79, |
| "text": "(Seki 1991)", |
| "ref_id": "BIBREF16" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Overview of the Results", |
| "sec_num": "5" |
| }, |
| { |
| "text": "A relation B~ A is shown in (Engelfriet 1980) . By Lemma 2.1, all languages in the region enclosed with the bold line are recognizable in polynomial time. On the other hand, it is shown in this paper that Unary-3SAT, which is known to be A/P-complete (Nakanishi 1992) , is in A. Hence, if ~ ~ A/~, then Unary-3SAT E A -B and the languages generated by fts' (or equivalently, nclfg's) are not recognizable in polynomial time in general.", |
| "cite_spans": [ |
| { |
| "start": 28, |
| "end": 45, |
| "text": "(Engelfriet 1980)", |
| "ref_id": "BIBREF2" |
| }, |
| { |
| "start": 251, |
| "end": 267, |
| "text": "(Nakanishi 1992)", |
| "ref_id": "BIBREF10" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Overview of the Results", |
| "sec_num": "5" |
| }, |
| { |
| "text": "6.1 Deterministic fts' Here, the proof of an inclusion relation yEd-#s C_ /:vmc/g is sketched.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Generative Capacity of fts'", |
| "sec_num": "6" |
| }, |
| { |
| "text": "Let (M, G) be a deterministic yT-fts where M = (Q,~,A,ql,R) and G = (N,T,P,S)., We assume that Q = {ql,---,ql}, T = {al,... an} and P = {Pl,...,Pm}. Since the input for M is the set of derivation trees of G, we assume that = {Pl,.-. ,Pro, al,..., an} without loss of generality.", |
| "cite_spans": [ |
| { |
| "start": 47, |
| "end": 59, |
| "text": "(Q,~,A,ql,R)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Generative Capacity of fts'", |
| "sec_num": "6" |
| }, |
| { |
| "text": "We will construct a pmcfg G I -=-( N ~, T ~, F', P', S') such that yL(M, G) ----L(G') N A*. Since /:pmc/g is closed under the intersection with a regular set (Kasami 1988a) (Seki 1991) , it follows that yL(U, G) E \u00a3'pmclg. (1 < j < u) no output is derived from q,~ [t] (1 _< j < v).", |
| "cite_spans": [ |
| { |
| "start": 158, |
| "end": 172, |
| "text": "(Kasami 1988a)", |
| "ref_id": "BIBREF7" |
| }, |
| { |
| "start": 173, |
| "end": 184, |
| "text": "(Seki 1991)", |
| "ref_id": "BIBREF16" |
| }, |
| { |
| "start": 265, |
| "end": 268, |
| "text": "[t]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Generative Capacity of fts'", |
| "sec_num": "6" |
| }, |
| { |
| "text": "The basic idea is to simulate the move of tree transducer M which is scanning a symbol Ph (resp. ah) with state ql by the ith component of the nonterminal Rh (resp. Ah) of pmcfg G I. During the move of M, it may happen that no rule is defined for a current configuration and hence no output will be derived\u2022 The symbol b is introduced to represent such an undefined move explicitly.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "We define RS(X) (X E N tO T) as follows.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "{Rh [the left-hand side of Ph is X} RS(X) = if X E N { Ah } if X = ah E T.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Productions and functions are defined as follows.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Step 1: For each production Ph : Iio --'* Y~ \" \" Yk ( Yo ~ N , Y= E NtoT for 1 <u< k) of cfg G, construct nonterminating productions Rh -+ [&,..., zk] for every Z~ E RS(Y~) (1 < u < k), where fph is defined as follows: For 1 < i < g,", |
| "cite_spans": [ |
| { |
| "start": 139, |
| "end": 150, |
| "text": "[&,..., zk]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 if the transducer M has no rule whose lefthand side is qi~ah (Xl,..., xk) ], then", |
| "cite_spans": [ |
| { |
| "start": 63, |
| "end": 75, |
| "text": "(Xl,..., xk)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "(6.a) h \u2022''", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 if M has a rule -+", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 \" ai,n,-lq~(i,,~,_D[x~4~,,,_D] a~,n,, then", |
| "cite_spans": [ |
| { |
| "start": 2, |
| "end": 32, |
| "text": "\" ai,n,-lq~(i,,~,_D[x~4~,,,_D]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "EQUATION", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [ |
| { |
| "start": 0, |
| "end": 8, |
| "text": "EQUATION", |
| "ref_id": "EQREF", |
| "raw_str": "fp[i][\u2022x .. , 5:k] __a ei,ox~,(i,o),7(i,o)ei,] (6.4) h ~ \u2022 \u2022 \" \" Ot -, ni --l gl z\", ni --l ' rl ' i , ni --l ' Ogi , n\u0129,", |
| "eq_num": "(1, ) [ ) where =" |
| } |
| ], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "(1 <, < k).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "(Since M is deterministic, there exists at most one rule whose left-hand side is qi~h('\" \")] and hence the above construction is well defined\u2022)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Step 2: For each ah E T, construct a terminating production Ah -\"+ fah where f~h is defined as follows: For 1 < i < i,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 if M has no rule whose left-hand side is", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "qi[ah], then ~a~[i] ~--b.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 ifM has a rule qi[ah] --+ hi, then f[~&ai.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Step 3: For each Rh E RS(S), construct S' --+ /fi~st [Rh] where /fi,st [(x], ..., xl) ]~x]. Intuitively, the right-hand side of this production corresponds to the initial configuration, that is, M is in the initial state ql and scanning the root symbol Ph of a derivation tree, where the left-hand side of Ph is the initial symbol S.", |
| "cite_spans": [ |
| { |
| "start": 53, |
| "end": 57, |
| "text": "[Rh]", |
| "ref_id": null |
| }, |
| { |
| "start": 71, |
| "end": 85, |
| "text": "[(x], ..., xl)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "The pmcfg G I constructed above satisfies Property 6.1. Its proof is found in (Kaji 1992) and omitted in this paper. By Property 6.1, we obtain the following lemma. every language in this class can be recognized in time polynomial of the length of an input string\u2022 Our result here is: there is a nondeterministic fts that generates an A/'~-complete language\u2022 In the following, a language called Unary-3SAT, which is ArT'-complete (Nakanishi 1992) , is considered, and then it is shown to belong to yL:/,a.", |
| "cite_spans": [ |
| { |
| "start": 78, |
| "end": 89, |
| "text": "(Kaji 1992)", |
| "ref_id": "BIBREF5" |
| }, |
| { |
| "start": 430, |
| "end": 446, |
| "text": "(Nakanishi 1992)", |
| "ref_id": "BIBREF10" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "A Unary-3CNF is a (nonempty) 3CNF in which the subscripts of variables are represented in unary. A positive literal xi in a 3CNF is represented by 1i$ in a Unary-3CNF. Similarly , a negative literal --xl is represented by 12#. For example, a 3CNF", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "(xi v x2 v ~xa) A (xa V ~x] v ~x~)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "is represented by a Unary-3CNF 15115111# A I1151#Ii#.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Unary-3SAT is the set of all satisfiable Unary-3CNF's.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Next, we construct a nondeterministic yT-fts (M, G) that generates Unary-3SAT. Define a cfg G = (N,T,P, S) where N = {S,T,F}, T = {e} and the productions in P are as follows: {qo,q~,qt, qa}, ~ {rSS,... ,rFe}, z~ = {L^,$,#}.", |
| "cite_spans": [ |
| { |
| "start": 175, |
| "end": 208, |
| "text": "{qo,q~,qt, qa}, ~ {rSS,... ,rFe},", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "q =", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Since there are many rules in R, we will use an abbreviated notation. Next, Y\u00a3~s C_ \u00a3~c-zf9 is shown. For a given fts (M, G), the following algorithm constructs an nc-lfg G' such that L(G') ---yL (M, G) . [TRANS 7.2] Suppose that a given fts (M, G) is G --(N, T, P, S) and M --(Q, E, A, ql, R) where Q = {ql,q2,'\",qm}. Let n be the maximum length of the right-hand side of a production in P.", |
| "cite_spans": [ |
| { |
| "start": 196, |
| "end": 202, |
| "text": "(M, G)", |
| "ref_id": null |
| }, |
| { |
| "start": 205, |
| "end": 216, |
| "text": "[TRANS 7.2]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Define an nc-lfg G I = ( N', A, P', S I, N~r, Aatm) as follows.", |
| "cite_spans": [ |
| { |
| "start": 17, |
| "end": 51, |
| "text": "G I = ( N', A, P', S I, N~r, Aatm)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Step 1: N'={C [J] Step 2: A move when M at state qj reads a symbol p which is the label of a production p : C --+ ..., can be simulated by a production in G ~ whose left-hand side is", |
| "cite_spans": [ |
| { |
| "start": 14, |
| "end": 17, |
| "text": "[J]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D", |
| "sec_num": null |
| }, |
| { |
| "text": "Formally, the set P~ of productions of G I is constructed as follows.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "(i) Let p : C --* X1 \"\" Xh be a production in P where CE N, Xi E NUT (1 <i < h), and let: [z~,,, ]aj,...q,7,zj [X~,,L ' ] O~jL, be a rule in R where ~k E A* (0 < k < Lj), q' Tj~ E Q, and xvj~ e tXl,'\",Xh}(1", |
| "cite_spans": [ |
| { |
| "start": 90, |
| "end": 121, |
| "text": "[z~,,, ]aj,...q,7,zj [X~,,L ' ]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "qj[p(x],..., Xh)] --~ ajoq,7,,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "< l < L j).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "Then, the following production belongs to P~: By TRANS 7.1 and TRANS7.2, the following theorem is obtained. A formal proof is found in (Nakanishi 1993 ).", |
| "cite_spans": [ |
| { |
| "start": 135, |
| "end": 150, |
| "text": "(Nakanishi 1993", |
| "ref_id": "BIBREF9" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "y[r/jl] V", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "Theorem 7.1: f~nc-lfg = Y~'fts.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "Corollary 7.2: ~'dc-lfg ----Y~.d-fts. Proof. In TRANS 7.1, if G is a dc-lfg, then no sp E SP contains distinct productions whose left-hand sides are the same and hence the constructed transducer M becomes deterministic by the construction. Conversely, in TRANS 7.2, if M is deterministic, then there exist no consistent productions p~ and p~ in P~ whose left-hand sides are the same and hence the constructed nc-lfg is a dc-lfg.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C[J] {T ute = p}\"", |
| "sec_num": null |
| }, |
| { |
| "text": "Equivalence of ~fc-lfg and \u00a3~mcfg To prove f~fc-lfg C Lmcfg, we give an algorithm which translates a given fc-lfg G = (N, T, P, S, Nat,, Aatm) into an mcfg G I such that L (G') = L (G).", |
| "cite_spans": [ |
| { |
| "start": 118, |
| "end": 142, |
| "text": "(N, T, P, S, Nat,, Aatm)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "[TRANS 8] We explain the algorithm by using the fc-lfg GEX3 in Example 4.1. An mcfg G' = (N', T, F, P', S) is constructed as follows.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "Step 1: N' = (the set of nonterminals which has a one-to-one correspondence with the set of SPN multisets in G)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "= {(S), (A,B)} (for GEx3 in Example 4.1) P' = \u00a2, and F =\u00a2.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "Step 2: For each SPN multiset M0 = {{A1,A2,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 \".,Ak}} of G, consider every tuple (pl,P2,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "\"\",Pk) of productions in P whose lefthand sides are A1, A2,'\", Ak respectively and which are consistent. (Suppose that, if we write an SPN multiset as {{A1, A2,. \", Ak}}, then Aj's are arranged according to a predefined total order < on N, that is, A1 < A2 <_ \"'\" <_ Ak hold.) For an SPN multiset {{A, B}} in GEX3, the following two pairs of productions have to be considered: For (Pl, P2, '\", Pk) , a production p' and a function f of G' are constructed and added to P' and F, respectively as follows.", |
| "cite_spans": [ |
| { |
| "start": 381, |
| "end": 385, |
| "text": "(Pl,", |
| "ref_id": null |
| }, |
| { |
| "start": 386, |
| "end": 389, |
| "text": "P2,", |
| "ref_id": null |
| }, |
| { |
| "start": 390, |
| "end": 393, |
| "text": "'\",", |
| "ref_id": null |
| }, |
| { |
| "start": 394, |
| "end": 397, |
| "text": "Pk)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "The multiset M of the nonterminals appearing in the right-hand side of some pj (1 < j < k) are partitioned into multisets M1, M2,\" -., Mh with respect to the S schemata attached to the nonterminals in pj's. That is, (11//1, M2,-\", Mh) are the coarsest partition of M such that for each M,, (1 < u < h), the following condition holds.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "Each nonterminal in M~, has the same S schema.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "By the definition, each M= (1 < u < h) is an SPN multiset in G. _Construct a production of mcfg p': hit0 --* f [M1, ffI2, '\", Mh] where M= is the nonterminal of G' which corresponds to M=(1 < u < h). Addp' to P' and f to F where f is defined as follows. Suppose pj : Aj ~ ajoBjlajl \"'' BjL~ajL~ (1 < j < k) where Aj E N, Bfl E N(1 < l < Lj) and ajz E T* (0 < l < Lj), and let--=", |
| "cite_spans": [ |
| { |
| "start": 111, |
| "end": 115, |
| "text": "[M1,", |
| "ref_id": null |
| }, |
| { |
| "start": 116, |
| "end": 121, |
| "text": "ffI2,", |
| "ref_id": null |
| }, |
| { |
| "start": 122, |
| "end": 125, |
| "text": "'\",", |
| "ref_id": null |
| }, |
| { |
| "start": 126, |
| "end": 129, |
| "text": "Mh]", |
| "ref_id": null |
| }, |
| { |
| "start": 295, |
| "end": 306, |
| "text": "(1 < j < k)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "(1 < < h)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "where Cu, E N(1 < v < su). Then, for 1 < j < k, the jth component f[J] of f is:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "_ A f[J] (X-l, x2", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": ",\" \"', Xh )=otjoYjl Otjl Yj2 \"\" \u2022 YjLj OtjLj where x-u = (xul,xu2,'\",xus.) (1 < u < h).", |
| "cite_spans": [ |
| { |
| "start": 57, |
| "end": 74, |
| "text": "(xul,xu2,'\",xus.)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| }, |
| { |
| "text": "For j (1 <_ j < k) and l (1 _< l _< nj), if z~ Bjl = C~,, then yfl-=x~,v. Note that, since Mu's are a partition of M, f satisfies Right Linearity (see 2.) and hence G' is an mcfg. For example, consider the above (P12,P13)-The nonterminals appearing in the right-hand", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "8", |
| "sec_num": null |
| } |
| ], |
| "back_matter": [ |
| { |
| "text": "sides are A and B, and their S schemata are the same. Thus, we construct the following mcfg production:where fl [(Xl, x2) Consider the following pair of productions as another example:The multiset of nonterminals in the righthand sides are partitioned into M1 = I{ A, B}} (for arT1) and M~ = {{C, D, D}} for atr2). For (p~,p~), the following mcfg production is constructed:xilx .lc 3).", |
| "cite_spans": [ |
| { |
| "start": 112, |
| "end": 117, |
| "text": "[(Xl,", |
| "ref_id": null |
| }, |
| { |
| "start": 118, |
| "end": 121, |
| "text": "x2)", |
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| "ref_spans": [], |
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| { |
| "text": "Example 8.1: TRANS 8 translates fc-lfg Proof: \u00a3yc-tfg C \u00a3mcf9 can be proved by TRANS 8. Conversely, for a given mcfg G, an fc-lfg G' such that L (G') = L (G) can be constructed in a similar way to TRANS 8. Details are found in (Ando 1992) .[1 9 Conclusion In this paper, we introduce three subclasses of lfg's, two of which can be recognized in polynomial time. Also this paper clarifies the relations between the generative capacities of those subclasses, pmcfg's and fts'.", |
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| "FIGREF0": { |
| "text": "Gh) S \u2022 N is the initial symbol, and d(S) = 1.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF1": { |
| "text": "Example 2.2: Let GEx2 = (N, T, F, P, S) be a pmcfg, where N = {S,A), T = {a,b}, F = {g[(Xl,X2) ] ----XlX2, fa[(Xl,X2)] -~ (xla, x2a), ], Pl : A ---* f~[A], Pz : A --* fb[A], P3 : A ---* f~}. Note that GEx2 is an mcfg. L (GEx2) = {ww I w E {a, b)*}.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF2": { |
| "text": "xn}])*. If any different rules in R have different left-hand sides, then M is called deterministic(Engelfriet 1980).A configuration of a yT-transducer is an element in (A U Q[T~.(\u00a2)])*. Derivation of M is defined as follows. Let t -----alq[a(tl,..., tn)]a2 be a configuration where al, a2 E (A U Q[T~.(\u00a2)])*, q E Q, ~ E ~, p(a) = n and Q,...,tn E T~.(\u00a2). Assume that there is a rule q[cr(xl,..., Xn)] --* V in R. Let t ~ be obtained from v by substituting t],..., tn for xl,..., xn, respectively, then we define t ~M ultra2 \u2022 Let ::~ be the reflexive and transitive closure of :=~. If t =\u00a2.~ t ~, then we say t ~ is derived from t. If there is no w E A* such that t ~ w, then we say no output is derived from t.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF3": { |
| "text": "We define yL(M,G), called the yield language generated by yT-fts(M, G), as yL(M,a)~{w e A* 13t e ~s(a),qo[t] ~*M w} where A is an output alphabet and q0 is the initial state of M. An fts is called deterministic (Engelfriet 1980) if the transducer M is deterministic.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF4": { |
| "text": "whereA \u2022 N, B1,B2,.\",Bq \u2022 NUT. Ev is a finite set of V schemata and Esj (1 _< j <_ q) is a singleton of an S schema. A --~ B1B2\".. Bq in (4.2) is called the underlying production of p. Let P0 be the set of all the underlying productions of P. Cfg Go = (N, T, P0, S) is called the underlying c/g o/ C.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF5": { |
| "text": "For a cfgG' = ( N ~, T', P~, S~), derivation relations in G ~, denoted by A ::~a' a and A =~* G ~ (A \u2022 N',a \u2022 (N' u T')*) are defined in the usual way.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF6": { |
| "text": "is defined as L(G) = {w e T*[S ~* w I. G NOTE : In the definition of nc-lfg, even if\"Esj (1 < j < q) is a'singleton of an S schema\" A derivation tree of aabbccdd \"Esj (1 < j < q) is either a singleton of an S schema or an empty set\", the generative capacity of nc-lfg is not changed.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF7": { |
| "text": "Example 4.1: Let G~xs =(N, T, P, S, Nat,, A~tm) be an nc-lfg where N = {S,A,B}, T = {a, b,c, d}, Nat~ = {count}, Aatrn = {e}, and productions in P are;The language generated by GExs is L(GExs) = {a'~bncnd n In > 0}.Figure 1shows a derivation tree of S ~* aabbccdd in GEXS. GEX3 Example 4.2: Let Gsx4 = (N, T, P, S, N,t,, A~tm) be an nc-lfg where g = {S}, T = Ca}, N,t, = {log}, A,tm = {e}, and productions in P are;", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF8": { |
| "text": "Example 4.3: Let GEX5 = (N, T, P, S, Na~,, Aatm) be an nc-lfg where N = {S,S',A,B}, T", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF9": { |
| "text": "Let T' = A td {b} where b is a newly introduced symbol and let N' = {S',RI,...,Rm, AI,...,An} where d(Ri) = d(Aj) = t for 1 < i <_ m and 1 < j <_ n. Productions and functions of G ~ will be constructed to have the following property. Inclusion relations between classes of languages. (1) : The class of language generated by lcfrs' is equal to C. (2) : The class of language generated by head grammars is equal to D. Property 6.1: There is (a~,... ,a~) e LG,(Rh) (resp. LG,(Ah)) such that each of a,,,...,as~ does not contain b, and every remaining at,,..., a,~ contains b if and only if there is a derivation tree t of G such that the root is Ph (resp. ah) and { qs, [t] ==>~ c~s~", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF10": { |
| "text": "The reverse inclusion relation l:p,~c/g C_ Y~.d-B, can be shown in a similar way, and the following theorem holds\u2022Theorem 6.2: yf-.d./,s : E-pmcfg\u2022 0 6.2 Nondeterministic fts' In this section, the generative capacity of nondeterministic yT-fts' is investigated, from the viewpoint of computational complexity\u2022 We have already shown that Y~.d-~s : ~.pmcfg, and hence", |
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| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF11": { |
| "text": "rss : S--+S rsT : S--+ T rsF : S-+F \"rTT : T--+ T rTF : T--+ F. ?'Te : T--+ e rFT : F-+T ?'FF : F--+ F rFe : F-+ eLet M = (Q, E, A, qo, R) where", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF12": { |
| "text": "q,[r**(x)] = q&.~(~)] -* lq,[~] or 1~ q,[rr,(~)] ~ 1, q,[r~,(~)] = q,[r~(~)] -* lq,[x] or 1# q,[rF~(x)]---+ 1# qo[rr~(~)] = qo[rr~(~)] = qoirF~(~)] = qa[rFF(X)] lqa[X] or 15 or 1# qo[r~c(x)] = q.[r~(~)] ++ 1, or 1#. The readers can easily verify that this yT-fts generates Unary-3SAT. 7 Equivalence of f-'nc-lfg and Y\u00a3fts First, we show \u00a3,~c-lfg C_ Y\u00a3qt~. For a given nclfg G = (N, T, P, S, Nat,, A~m), an equivalent fts (M, G I) is constructed in the following way. Let t be a derivation tree in lfg G and the f-structure of the root node of t be F = {(atrl,F1),..., (atr,~,Fn)}. F is represented by a derivation tree r = p,p(Tl,'-., rn) in G', where ri (1 < i < n) is a derivation tree in G' which represents Fi recursively. And sp is a set of productions such that F locally satisfies the functional schemata of all productions in sp. M transforms r into the yield of t, i.e., the terminal string obtalned by concatenating the labels of leaves, in a top-down way. [TRANS 7.11 Let N = {A1,'\",Am}, S = A1 and Nat, = {atrl,-.., atr,~}. Define SP as the set of all consistent subsets of P. Step 1: G' = (N',{d},P',S'), where N' = {S,plsp e SP} U {S'} and P' = {p',p : S,p ---* S'-.. S't u{p;=~... : s' --+ Ss, l,p e sP} u{p~,m :~s' -+ deC:_/}.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF13": { |
| "text": "For a derivation tree r in G' and a node v ' is applied, the snbtree rooted by the where p,p ith child of v represents the value of attribute atr i.Step 2: M = (Q,E,T, ql,R) is defined as follows.Define Q ={ql,..., qm}. A state qj (1 < j _< m) corresponds to nonterminal A t in N. Define E --{d} where p(p'.,) = p(p .... .~) = ' = and p(d) = O. And define R by the following (i) through (iii). (i) qj~ .... .,(x)] -~ qj[x] (1 _< j < m) belongs to R for each sp \u2022 SP. (ii) Let r be a derivation tree in G '. When plsp is the production applied at the root of r and a state of M is q,o, M chooses a production p whose left-hand side is Auo , if exists, in sp. NOTE : Since productions in sp are consistent, there is an f-structure, which locally satisfies the functional schemata of all productions in sp. For each production p E sp in SP p : A~o --* a0 A m al ... OtL-1 At, L aL Ev {~ atrv~ =~} ... {~ atrvL =~}where A~z E N and al E T*(0 < l < L), the following rule belongs to R: No other rule belongs to R.", |
| "uris": null, |
| "type_str": "figure", |
| "num": null |
| }, |
| "FIGREF14": { |
| "text": "[nJLj] C[J] 7...40tjo-~vjl Otjl \"'\" AI~jLj OtjLj\" {Trute = p) {Tatr , {Tatr j (ii) Let qj[a] ---* flj be a rule in R where a 6 T and flj 6 A*. Then the production a[J] --~ flj belongs to P'. (iii) No other production belongs to P'.", |
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| "type_str": "figure", |
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| "TABREF4": { |
| "type_str": "table", |
| "html": null, |
| "text": "\u2022 .', Ah ~ O:h be consistent productions. For each atr E Nat,, let MS~,~ be the multiset consisting of all the nonterminals which appear in al,''',ah and have an S schema T atr --l. If MSat~ is not empty, then MS~t~ is also an SPN multiset.3. There is no other SPN multiset.", |
| "content": "<table><tr><td>An nc-lfg such that the number of SPN multisets</td></tr><tr><td>in G is finite is called a finite-copying lfg (fc-lfg).</td></tr><tr><td>Example 4.4: Consider GEX s in Example 4.1.</td></tr><tr><td>Productions /912 and P14 are inconsistent with</td></tr><tr><td>each other and so are P13 and Ply. SPN multisets</td></tr><tr><td>in GEX3 are {{S}} and {{A,B)). Hence GEXS</td></tr><tr><td>is a dc-lfg and is an fc-lfg. GEX5 is also a dc-lfg</td></tr><tr><td>and is an fc-lfg by the similar reason. Similarly,</td></tr><tr><td>GEX4 in Example 4.2 is a dc-lfg. SPN multisets</td></tr><tr><td>in C~x~ are {{S}}, {{S, S}), {{S, S, S, S)}, ....</td></tr></table>", |
| "num": null |
| } |
| } |
| } |
| } |
