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<?xml version="1.0" standalone="yes"?> <Paper uid="P86-1006"> <Title>COMPUTATIONAL COMPLEXITY OF CURRENT GPSG THEORY</Title> <Section position="4" start_page="0" end_page="33" type="metho"> <SectionTitle> 2 Complexity of GPSG Components </SectionTitle> <Paragraph position="0"> A generalized phrase structure grammar contains five language-particular components -- immediate dominance (ID) rules, metarules, linear precedence (LP) statements, feature co-occurrence restrictions (FCRs), and feature specification defaults (FSDs) -- and four universal components -- a theory of syntactic features, principles of universal feature instantiation, principles of semantic interpretation, and formal relationships among various components of the grammar, s Syntactic categories are partial functions from features to atomic feature values and syntactic categories. They encode subcategorization, agreement, unbounded dependency, and other significant syntactic information. The set K of syntactic categories is inductively specified by listing the set F of features, the set A of atomic feature values, the function po that defines the range of each atomic-valued feature, and a set R of restrictive predicates on categories (FCRs).</Paragraph> <Paragraph position="1"> The set of ID rules obtained by taking the finite closure of the metarules on the ID rules is mapped into local phrase structure trees, subject to principles of universal feature instantiation, FSDs, FCRs, and LP statements. Finally, local trees are 2We use the universal problem to more accurately explore the power of a grammatical formalism (see section 3.1 below for support). Ristad(1985) has previously proven that the universal recognition problem for the GPSG's of Gazdar(1981) is NP-hard and likely to be intractable, even under severe metarule restrictions.</Paragraph> <Paragraph position="2"> 3This work is based on current GPSG theory as presented in Gazdar et. al. (1985), hereafter GKPS. The reader is urged to consult that work for a formal presentation and thorough exposition of current GPSG theory.</Paragraph> <Paragraph position="3"> assembled to form phrase structure trees, which are terminated by lexical elements.</Paragraph> <Paragraph position="4"> To identify sources of complexity in GPSG theory, we consider the isolated complexity of the finite metarule closure Ol>station and the rule to tree mapping, using the finite closure membership and category membership problems, respectively.</Paragraph> <Paragraph position="5"> Informally, the finite closure membership problem is to determine if an ID rule is in the finite closure of a set of metarules M on a set of ID rules R. The category membership problem is to determine if a category or C or a legal extension of C is in the set K of all categories based the function p and the sets A, F and R. Note that both problems must be solved by any GPSG-based parsing system when computing the ID rule to local tree mapping.</Paragraph> <Paragraph position="6"> The major results are that finite closure membership is NP-hard and category membership is PSPACE-hard. Barton(1985) has previously shown that the recognition problem for ID/LP grammars is NP-hard. The components of GPSG theory are computationally complex, as is the theory as a whole.</Paragraph> <Paragraph position="7"> Assumptions. In the following problem definitions, we allow syntactic categories to be based on arbitrary sets of features and feature values. In actuality, GPSG syntactic categories are based on fixed sets and a fixed function p. As such, the set K of permissible categories is finite, and a large table containing K could, in princip}e, be given. 4 We (uncontroversially) generalize to arbitrary sets and an arbitrary function p to prevent such a solution while preserving GPSG's theory of syntactic features, s No other modifications to the theory are made.</Paragraph> <Paragraph position="8"> An ambiguity in GKPS is how the FCRs actually apply to embedded categories. 6 Following Ivan Sag (personal communication), I make the natural assumption here that FCRs apply top-level and to embedded categories equally.</Paragraph> <Paragraph position="9"> categories, given the 25 atomic features and 4 category-valued features, is:</Paragraph> <Paragraph position="11"> See page 10 for details. Many of these categories will be linguistically meaningless, but all GPSGs will generate all of them and then filter some out in consideration of FCRs, FSDs, universal feature instantiation, and the other admissible local trees and lexical entries in the GPSG. While the FCRs in some grammars may reduce the number of categories, FCRs are a language-particular component of the grammar. The vast number of categories cited above is inherent in the GPSG framework.</Paragraph> <Paragraph position="12"> SOur goal is to identify sources of complexity in GPSG theory. The generalization to arbitrary sets allows a fine-grained study of one component of GPSG theory (the theory of syntactic features) with the tools of computational complexity theory. Similarly, the chess board is uncontroverslally generalized to size n x a in order to study the computational complexity of chess.</Paragraph> <Paragraph position="13"> eA category C that is defined for a feature \], f E (F - Atom) n DON(C) (e.g. f = SLASH ), contains an embedded category C~, where C(f) --- C~.</Paragraph> <Paragraph position="14"> GKPS does not explain whether FCR's must be true of C~ as well as C.</Paragraph> <Section position="1" start_page="30" end_page="31" type="sub_section"> <SectionTitle> 2.1 Metarules </SectionTitle> <Paragraph position="0"> The complete set of ID rules in a GPSG is the maximal set that can be arrived at by taking each metarule and applying it to the set of rules that have not themselves arisen as a result of the application of that metarule. This maximal set is called the finite closure (FC) of a set R of lexical ID rules under a set M of metarules.</Paragraph> <Paragraph position="1"> The cleanest possible complexity proof for metarule finite closure would fix the GPSG (with the exception of metarules) for a given problem, and then construct metarules dependent on the problem instance that is being reduced. Unfortunately, metarules cannot be cleanly removed from the GPSG system.</Paragraph> <Paragraph position="2"> Metarules take ID rules as input, and produce other ID rules as their output. If we were to separate metarules from their inputs and outputs, there would be nothing left to study.</Paragraph> <Paragraph position="3"> The best complexity proof for metarules, then, would fix the GPSG modulo the metarules and their input. We ensure the input is not inadvertently performing some computation by requiring the one ID rule R allowed in the reduction to be fully specified, with only one 0-1evel category on the left-hand side and one unanalyzable terminal symbol on the right-hand side.</Paragraph> <Paragraph position="4"> Furthermore, no FCRs, FSDs, or principles of universal feature instantiation are allowed to apply. These are exceedingly severe constraints. The ID rules generated by this formal system will be the finite closure of the lone ID rule R under the set M of metarules.</Paragraph> <Paragraph position="5"> The (strict, resp.) finite closure membership problem for GPSG metarules is: Given an ID rule r and sets of metarules M and ID rules R, determine if 3r e such that r I ~ r (r I = r, resp.) and r I * FC(M, R).</Paragraph> <Paragraph position="6"> Theorem 1: Finite Closure Membership is NP-hard Proof: On input 3-CNF formula F of length n using the m variables zl... x,~, reduce 3-SAT, a known NP-complete problem, to Metarule-Membership in polynomial time.</Paragraph> <Paragraph position="7"> The set of ID rules consists of the one ID rule R, whose mother category represents the formula variables and clauses, and a set of metarules M s.t. an extension of the ID rule A is in the finite closure of M over R iff F is satisfiable. The metarules generate possible truth assignments for the formula variables, and then compute the truth value of F in the context of those truth assignments.</Paragraph> <Paragraph position="8"> Let w be the string of formula literals in F, and let wl denote the i th symbol in the string w.</Paragraph> <Paragraph position="9"> Vi, 1 < i < m {\[yi 0\],\[STAGE I\]} -* W (i) {\[Yi I\],\[STAGE 1\]} ~ W (b) one metarule to stop the assignment generation process null {\[STAGE 1\]) -~ W (2) {\[STAGE 2\]} --* W (c) I w\[ metarules to verify assignments Vi,j,k 1<i<1~ j, l <_ j <_ m, O < k < 2, if wsi-k : xj, then construct the metarule {\[yi 1\],\[ei 0\],\[STAGE 2\]) --+ W (3) {\[yj i\],\[ci 1\], \[STAGE 2\]} --' W Vi,j,k l<i<~ -1, l<_j<_m,O<k<_2, if wsi-k = ~, then construct the metarule {\[yj 0\], \[cl 0\], \[STAGE 2\]} -* W The reduction constructs O(I w l) metarules of size log(I w \[), and clearly may be performed in polynomial time: the reduction time is essentially the number of symbols needed to write the GPSG down. Note that the strict finite closure membership problem is also NP-hard. One need only add a polynomial number of metarules to &quot;change&quot; the feature values of the mother node C to some canonical value when C(STAGE ) = 3 -- all 0, for example, with the exception of STAGE . Let F = {\[Yi 0\] : l<i<m} U {\[c, O\]:l<i< ~}. Then A would be</Paragraph> <Paragraph position="11"> The major source of intractability is the finite closure operation itself. Informally, each metarule can more than double the number of ID rules, hence by chaining metarules (i.e. by applying the output of a metarule to the input of the next metarule) finite closure can increase the number of ID rules exponentiallyff</Paragraph> </Section> <Section position="2" start_page="31" end_page="33" type="sub_section"> <SectionTitle> 2.2 A Theory of Syntactic Features </SectionTitle> <Paragraph position="0"> Here we show that the complex feature system employed by GPSG leads to computational intractability. The underlying insight for the following complexity proof is the almost direct equivalence between Alternating Turing Machines (ATMs) and syntactic categories in GPSG. The nodes of an ATM computation correspond to 0-level syntactic categories, and the ATM computation tree corresponds to a full, n-level syntactic category. The finite feature closure restriction on categories, which limits the depth of category nesting, will limit the depth of the corresponding ATM computation tree. Finite feature closure constrains us to specifying (at most) a polynomially deep, polynomially branching tree in polynomial time. This is exactly equivalent to a polynomial time ATM computation, and by Chandra and Stockmeyer(1976), also equivalent to a deterministic polynomial space-bounded 'luring Machine computation. null As a consequence of the above insight, one would expect the GPSG Category-Membership problem to be PSPACE-hard.</Paragraph> <Paragraph position="1"> The actual proof is considerably simpler when framed as a reduction from the Quantified Boolean Formula (QBF) problem, a known PSPACE-complete problem.</Paragraph> <Paragraph position="2"> Let a specification of K be the arbitrary sets of features F, atomic features Atom, atomic feature values A, and feature co-occurrence restrictions R and let p be an arbitrary function, all equivalent to those defined in chapter 2 of GKPS.</Paragraph> <Paragraph position="3"> The category membership problem is: Given a category C and a specification of a set K of syntactic categories, determine if3C Is.t. C I~CandC IEK.</Paragraph> <Paragraph position="4"> The QBF problem is {QIF1Qzyz... Qmy,nF(yh YZ,..., y,n) I Qi 6 {V, 3}, where the yi are boolean variables, F is a boolean formula of length n in conjunctive normal form with exactly ~More precisely, the metarule finite closure operation can increase the size of a GPSG G worse than exponentially: from I Gi to O(\] G \[2~). Given a set of ID rules R of symbol size n, and a set M of m metarule, each of size p, the symbol size of FC(M,R) is O(n z~) = O(IGIZ~). Each met~ule can match the productions in R O(n) different ways, inducing O(n + p) new symbols per match: each metarule can therefore square the ID rule grammar size. There are m metarules, so finite closure can create an ID rule grammar with O(n 2~) symbols.</Paragraph> <Paragraph position="5"> three variables per clause (3-CNF), and the quantified formula is true}.</Paragraph> <Paragraph position="7"> we construct an instance P of the Category-Membership problem in polynomial time, such that f~ E QBF if and only if P is true.</Paragraph> <Paragraph position="8"> Consider the QBF as a strictly balanced binary tree, where the i th quantifier Qi represents pairs of subtrees < Tt, T! > such that (1) Tt and T! each immediately dominate pairs of subtrees representing the quantifiers Qi+l ... Qra, and (2) the i th variable yi is true in T~ and false in Tf. All nodes at level i in the whole tree correspond to the quantifier Qi. The leaves of the tree are different instantiations of the formula F, corresponding to the quantifier-determined truth assignments to the m variables. A leaf node is labeled true if the instantiated formula F that it represents is true. An internal node in the tree at level i is labeled true if 1. Qi = &quot;3&quot; and either daughter is labeled true, or 2. Qi -= &quot;V&quot; and both daughters are labeled true. Otherwise, the node is labeled false.</Paragraph> <Paragraph position="9"> Similarly, categories can be_understood as trees, where the features in the domain of a category constitute a node in the tree, and a category C immediately dominates all categories C ~ such that Sf e ((r - Atom) A DON(C))\[C(f) = C'\].</Paragraph> <Paragraph position="10"> In the QBF reduction, the atomic-valued features are used to represent the m variables, the clauses of F, the quantifier the category represents, and the truth label of the category. The category-valued features represent the quantifiers -- two category-valued features qk,qtk represent the subtree pairs < Tt, T I > for the quantifier Qk. FCRs maintain quantifier-imposed variable truth assignments &quot;down the tree&quot; and calculate the truth labeling of all leaves, according to F, and internal nodes, according to quantifier meaning.</Paragraph> <Paragraph position="11"> Details. Let w be the string of formula literals in F, and w~ denote the i th symbol in the string w. We specify a set K of permissible categories based on A, F, p,.and the set of FCRs R s.t. the category \[\[LABEL 1\]\] or an extension of it is an element of K iff ~ is true.</Paragraph> <Paragraph position="12"> First we define the set of possible 0-level categories, which encode the formula F and truth assignments to the formula variables. The feature wi represents the formula literal wi in w, yj represents the variable yj in f2, and ci represents the truth value of the i th clause in F.</Paragraph> <Paragraph position="14"> FCR's are included to constrain both the form and content of the guesses: 1. FCR's to create strictly balanced binary trees: Vk, l<k<m, \]LEVEL k\] = \[qk \[\[Yk 1\]\[LEVEL k + 1\]\]\]& \[ql \[\[Vk 0\]\[LEVEL k + 1\]\]\] 2. FCR's to ensure all 0-level categories are fully specified:</Paragraph> <Paragraph position="16"> Vk, 1 <k<m, \[LABEL\] --= \[yk\] 3. FCR's to label internal nodes with truth values determined by quantifier meaning: Vk, l<k<rn, if Qk = &quot;V&quot;, then include: \[LEVEL k\]&\[LABEL 1\] ------ \[qk \[\[LABEL ll\]\]&\[q~ \[\[LABEL 1\]\]1 \[LEVEL k\]&\[LABEL O\] ----- \[qk \[\[LABEL 0\]\]\] V \[q~ \[\[LABEL 0\]\]1 otherwise Qk = &quot;3&quot;, and include: \[LEVEL k\]&\[LABEL 1\] -- \[qk \[\[LABEL 11\]\] Y \[q~ \[\[LABEL I\]\]\] \[LEVEL k\]&\[LABEL O\] -- \[qk \[\[LABEL 0\]\]\]&Iq ~ \[\[LABEL 0\]\]\] The category-valued features qk and q~ represent the quantifier Qk. In the category value of qk, the formula variable yk = 1 everywhere, while in the category value of q~, Yk = 0 everywhere.</Paragraph> <Paragraph position="17"> 4. one FCR to guarantee that only satisfiable assignments are permitted: \[LEVEL 1\] ~ ILABEL 1\] 5. FCR's to ensure that quantifier assignments are preserved &quot;down the tree&quot;: Vi, k l<_i<k<m, \[Yi 1\] D \[qk \[\[Yi 1\]\]\]&\[q~ \[\[Yi 1\]\]\] \[~, O\] ~ \[q~ \[\[y~ o\]\]\]&\[q i \[\[y~ 0\]\]\] 6. FCR's to instantiate variable assignments into the formula</Paragraph> <Paragraph position="19"> 7. FCR's to verify the guessed variable assignments in leaf nodes:</Paragraph> <Paragraph position="21"> The reduction constructs O(1~1) features and O(m ~) FCRs of size O(log m) in a simple manner, and consequently may be seen to be polynomial time. 0.~.P The primary source of intractability in the theory of syntactic features is the large number of possible syntactic categories (arising from finite feature closure) in combination with the computational power of feature co-occurrence restrictions, s FCRs of the &quot;disjunctive consequence&quot; form \[f v\] D \[fl vl\] V ... V \[fn vn\] compute the direct analogue of Satisfiability: when used in conjunction with other FCRs, the GPSG effectively must try all n feature-value combinations.</Paragraph> </Section> </Section> <Section position="5" start_page="33" end_page="36" type="metho"> <SectionTitle> 3 Complexity of GPSG-Recognition </SectionTitle> <Paragraph position="0"> Two isolated membership problems for GPSG's component formal devices were considered above in an attempt to isolate sources of complexity in GPSG theory. In this section the recognition problem (RP) for GPSG theory as a whole is considered.</Paragraph> <Paragraph position="1"> I begin by arguing that the linguistically and computationally relevant recognition problem is the universal recognition problem, as opposed to the fixed language recognition problem. I then show that the former problem is exponential-polynomial (Exp-Poly) time-hard.</Paragraph> <Paragraph position="2"> SFinite feature closure admits a surprisingly large number of possible categories. Given a specification (F, Atom, A, R, p) of K, let a =lAteral and b =IF - Atom I. Assume that all atomic features are binary: a feature may be +,-, or undefined and there are 3 a 0-1evel categories. The b category-valued features may each assume O(3 ~) possible values in a 1-1evel category,</Paragraph> <Paragraph position="4"> where E~=o ~ converges toe ~ 2.7 very rapidly and a,b = O(IGI) ; a = 25, b = 4 in GKPS. The smallest category in K will be 1 symbol (null set), and the largest, maximally-specified, category wilt be of symbol-slze log I K I = oca. b!).</Paragraph> <Section position="1" start_page="33" end_page="34" type="sub_section"> <SectionTitle> 3.1 Defining the Recognition Problem </SectionTitle> <Paragraph position="0"> The universal recognition problem is: given a grammar G and input string x, is z C L(G)?. Alternately, the recognition problem for a class of grammars may be defined as the family of questions in one unkown. This fized language recognition problem is: given an input string x, is z E L for some fixed language L?. For the fixed language RP, it does not matter which grammar is chosen to generate L -- typically, the fastest grammar is picked.</Paragraph> <Paragraph position="1"> It seems reasonable clear that the universal RP is of greater linguistic and engineering interest than the fixed language RP.</Paragraph> <Paragraph position="2"> The grammars licensed by linguistic theory assign structural descriptions to utterances, which are used to query and update databases, be interpreted semantically, translated into other human languages, and so on. The universal recognition problem -- unlike the fixed language problem -- determines membership with respect to a grammar, and therefore more accurately models the parsing problem, which must use a grammar to assign structural descriptions.</Paragraph> <Paragraph position="3"> The universal RP also bears most directly on issues of natural language acquisition. The language learner evidently possesses a mechanism for selecting grammmars from the class of learnable natural language grammars/~a on the basis of linguistic inputs. The more fundamental question for linguistic theory, then, is &quot;what is the recognition complexity of the class /~c?&quot;. If this problem should prove computationally intractable, then the (potential) tractability of the problem for each language generated by a G in the class is only a partial answer to the linguistic questions raised.</Paragraph> <Paragraph position="4"> Finally, complexity considerations favor the universal RP.</Paragraph> <Paragraph position="5"> The goal of a complexity analysis is to characterize the amount of computational resources (e.g. time, space) needed to solve the problem in terms of all computationally relevent inputs on some standard machine model (typically, a multi-tape deterministic Turing machine). We know that both input string length and grammar size and structure affect the complexity of the recognition problem. Hence, excluding either input from complexity consideration would not advance our understanding. 9 Linguistics and computer science are primarily interested in the universal recognition problem because both disciplines are concerned with the formal power of a family of grammars. Linguistic competence and performance must be considered in the larger context of efficient language acquisition, while computational considerations demand that the recognition problem be characterized in terms of both input string and grammar size.</Paragraph> <Paragraph position="6"> Excluding grammar size from complexity consideration in order SThis ~consider all relevant inputs ~ methodology is universally assumed in the formal language and computational complexity literature. For example, Hopcraft and Ullman(1979:139) define the context-free grammar recognition problem as: &quot;Given a CFG G = (V,T,P, $) and a string z in Y', is x in L(G)?.&quot;. Garey and Johnson(1979) is a standard reference work in the field of computational complexity. All 10 automata and language recognition problems covered in the book (pp. 265-271) are universal, i.e. of the form &quot;Given an instance of a machine/grammar and an input, does the machine/grammar accept the input7 ~ The complexity of these recognition problems is alt#ays calculated in terms of grammar and input size. to argue that the recognition problem for a family of grammars is tractable is akin to fixing the size of the chess board in order to argue that winning the game of chess is tractable: neither claim advances our scientific understanding of chess or natural language.</Paragraph> </Section> <Section position="2" start_page="34" end_page="36" type="sub_section"> <SectionTitle> 3.2 GPSG-Recognition is Exp-Poly hard </SectionTitle> <Paragraph position="0"> alternating Turing Machine M on input w.</Paragraph> <Paragraph position="1"> Let S(n) be a polynomial in n. Then, on input M, a S(n) space-bounded one tape alternating Turing Machine (ATM), and string w, we construct a GPSG G in polynomial time such</Paragraph> <Paragraph position="3"> where ASPACE(S(n)) is the class of problems solvable in space Sin ) on an ATM, and DTIME(F(n)) is the class of problems solvable in time F(n) on a deterministic Turing Machine.</Paragraph> <Paragraph position="4"> As a consequence of this result and our following proof, we have the immediate result that GPSG-Recognition is DTIME(cS(n)) hard, for all constants c, or Exp-Poly time-hard.</Paragraph> <Paragraph position="5"> An alternating Turing Machine is like a nondeterministic TM, except that some subset of its states will be referred to as universal states, and the remainder as existential states. A nondeterministic TM is an alternating TM with no universal states. 10 The nodes of the ATM computation tree are represented by syntactic categories in K deg -- one feature for every tape square, plus three features to encode the ATM tape head positions and the current state. The reduction is limited to specifying a polynomial number of features in polynomial time; since these features are used to encode the ATM tape, the reduction may only specify polynomial space bounded ATM computations.</Paragraph> <Paragraph position="6"> The ID rules encode the ATM NextM() relation, i.e. C ---* NextM(C) for a universal configuration C. The reduction constructs an ID rule for every combination of possible head position, machine state, and symbol on the scanned tape square.</Paragraph> <Paragraph position="7"> Principles of universal feature instantiation transfer the rest of the instantaneous description (i.e. contents of the tape) from mother to daughters in ID rules.</Paragraph> <Paragraph position="8"> 1degOur ATM definition is taken from Chandra and Stockmeyer(1976), with the restriction that the work tapes are one-way infinite, instead of two-way infinite. Without loss of generality, we use a 1-tape ATM, so C (Q x r) x (Q x r k x (L,R} x (L,R)) Let NextM(C ) ---- {C0,Cl,... ,Ck}. If C is a universal configuration, then we construct an ID rule of the form</Paragraph> <Paragraph position="10"> A universal ATM configuration is labeled accepting if and only if it has halted and accepted, or if all of its daughters are labeled accepting. We reproduce this with the ID rules in 6 (or 8), which will be admissible only if all subtrees rooted by the RHS nodes are also admissible.</Paragraph> <Paragraph position="11"> An existential ATM configuration is labeled accepting if and only if it has halted and accepted, or if one of its daughters is labeled accepting. We reproduce this with the ID rules in 7 (or 9), which will be admissible only if one subtree rooted by a RHS node is admissible.</Paragraph> <Paragraph position="12"> All features that represent tape squares are declared to be in the HEAD feature set, and all daughter categories in the constructed ID rules are head daughters, thus ensuring that the Head Feature Convention (HFC) will transfer the tape contents of the mother to the daughter(s), modulo the tape writing activity specified by the next move relation.</Paragraph> <Paragraph position="14"> where a is the read-only (R/O) tape symbol currently being scanned b is the read-write (R/W) tape symbol currently being scanned dl is the R/O tape direction A category in K deg represents a node of an ATM computation tree, where the features in Atom encode the ATM configuration. Labeling is performed by ID rules. (a) definition of F, Atom, A</Paragraph> <Paragraph position="16"> {\[HEADO i\], \[i a\], \[HEAD1 j\], Jr; b\], \[STATE q\], \[A I\]} --* {ResultOM(i, a, dlk) U Result 1M(j, ck, Pk, d2k) : (Pk, ck, dlk, d2k) e TransM(q, a, b)} (s) where all categories on the RHS are heads.</Paragraph> <Paragraph position="17"> (b) otherwise q * Q - U (existential state) V(pk, ck, dlk, d2~) E TransM(q, a, b), {\[HEADO i\], \[i a\], \[HEAD1 j\], \[rj b\], \[STATE q\], \[A I\]} ---+ ResultOM({ , a, dlk ) U Result 1M(\], ck,pk , d2k ) (9) where all categories on the RHS are heads.</Paragraph> <Paragraph position="18"> (c) One ID rule to terminate accepting states, using nulltransitions. null {\[STATE h\], \[1 Y\]} --* ~ (10) (d) Two ID rules to read input strings and begin the ATM simulation. The A feature is used to separate functionally distinct components of the grammar. \[A 1\] categories participate in the direct ATM simulation, \[A 2\] categories are involved in reading the input string, and the \[A 3\] category connects the read input string with the ATM simulation start state. START---* {\[A 1\]},{\[A 21} (11) {\[a 2\]}--~ {\[A 2\]},{\[A 2\]} where all daughters are head daughters, and where</Paragraph> <Paragraph position="20"> The reduction plainly may be performed in polynomial time in the size of the simulated ATM, by inspection.</Paragraph> <Paragraph position="21"> No metarules or LP statements are needed, although rectarules could have been used instead of the Head Feature Convention. Both devices are capable of transferring the contents of the ATM tape from the mother to the daughter(s). One metarule would be needed for each tape square/tape symbol combination in the ATM.</Paragraph> <Paragraph position="22"> GKPS Definition 5.14 of Admissibility guarantees that admissible trees must be terminated, n By the construction above -- see especially the ID rule 10 -- an \[A 1\] node can be terminated only if it is an accepting configuration (i.e. it has halted and printed Y on its first square). This means the only admissible trees are accepting ones whose yield is the input string followed by a very long empty string. P.C.P **The admissibility of nonlocal trees is defined as follows (GKPS, p.104): Definition: Admissibility Let R be a set of ID rules. Then a tree t is admissible from R if and only if 1. t is terminated, and 2. every local subtree in. t is either terminated or locally admissible from some r 6 R.</Paragraph> </Section> <Section position="3" start_page="36" end_page="36" type="sub_section"> <SectionTitle> 3.3 Sources of Intractability </SectionTitle> <Paragraph position="0"> The two sources Of intractability in GPSG theory spotlighted by this reduction are null-transitions in ID rules (see the ID rule 10 above), and universal feature instantiation (in this case, the Head Feature Convention).</Paragraph> <Paragraph position="1"> Grammars with unrestricted null-transitions can assign elaborate phrase structure to the empty string, which is linguistically undesirable and computationally costly. The reduction must construct a GPSG G and input string x in polynomial time such that x E L(G) iff w E L(M), where M is a PSPACEbounded ATM with input w. The 'polynomial time' constraint prevents us from making either x or G too big. Null-transitions allow the grammar to simulate the PSPACE ATM computation (and an Exp-Poly TM computation indirectly) with an enormously long derivation string and then erase the string. If the GPSG G were unable to erase the derivation string, G would only accept strings which were exponentially larger than M and w, i.e. too big to write down in polynomial time.</Paragraph> <Paragraph position="2"> The Head Feature Condition transfers HEAD feature values from the mother to the head daughters just in case they don't conflict. In the reduction we use HEAD'features to encode the ATM tape, and thereby use the HFC to transfer the tape contents from one&quot; ATM configuration C (represented by the mother) to its immediate successors Co,... ,Cn (the head daughters}. The configurations C, C0,... ,Ca have identical tapes, with the critical exception of one tape square. If the HFC enforced absolute agreement between the HEAD features of the mother and head daughters, we would be unable to simulate the PSPACE ATM computation in this manner.</Paragraph> </Section> </Section> class="xml-element"></Paper>