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<Paper uid="C92-4183">
  <Title>Uniform Recognition tbr Acyclic Context-Sensitive Grammars is</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
NITION FOR FD(ED GRAMMAR G ...&amp;quot;. The
</SectionTitle>
    <Paragraph position="0"> difference between uniform recognition and recognition for all fixed G can be illustrated with an example from Barton Jr., Berwick and l~istad \[1987\]. They show that uniform recognition for unordered context-free grammar (UCFG) can be done in time O(21C;In3). It has not been shown that the mfiform recognition problem is in 3 deg. For every G, however, tile fixed recognition problem can be solved in time O(n 3) and all these problems are in 7 ~. Barton Jr., Berwick and Ristad \[1987\] show the problem to be polynomial for any fixed grammar by a compilation step. The UCFG is compiled into a big context-free grammar. They use this grammar and the Earley algorithm in order to prove a polynomial bound. Just forgetting about the grammar size (replacing IGI by a constaslt) gives a polynomial bound too. It is not clear why Barton Jr., Berwick and Ristad \[1987\] always associate the fixed grarnmar problem with compilation (cf. their pp. 27-30, 64-79 and 202206). null This article is about uniform recognition for one type of restricted context-sensitive grammars, the acyclie context sensitive grammars (ACSG's). We prove it to be NP-complete. This means they are as complex as the Agreement Grammars and the Unordered CFG's of Barton Jr., Berwick and Ristad \[1987\]. ACSG's are the pure rewrite grammars in this group. They fit in the Chomsky hi~ erarchy.</Paragraph>
    <Paragraph position="2"> One might ask when we can use acycfic context-sensitive grammars. One can use them everywhere where one wants to use context-sensitive granlmars. But one has to be careful: cycles are not allowed. This property of acyclicity can be checked easily 1. For most purposes one does not need cycles at all. One field where context-sensitive grammars can be used is e.g. morphology. Characters in a word are often changed when 1 It is much easier than checking whether a CSG is t~ linear time CSG as defined by Book \[1978\]. One has to reason about length of possible derivations. In ACSG, derivations a.t'e short as a result of their acyclicity.</Paragraph>
    <Paragraph position="3"> ACnT~S DE COIJNG-92, NAIgfES, 2.3-28 ao~r 1992 1 15 8 PRO(:. OF COLlNG-92, N^rC/ll~s, AuG. 23-28, 1992 some suffix is added. These changes in a word are context-sensitive aald can be described by a context-sensitivc grammar. Once a character is changed, we normally do not want to change it back, the grammax we use is an acychc one.</Paragraph>
    <Paragraph position="4"> The complexity of recognition for ACSG is lower thmt in the unrestricted case (CSG, with complexity PSPACE) because we restrict the amount of information that can be passed through the sentence. The number of messages that e~'ut be sent is limited (and we do not block the messages by barriers as in Baker \[1974\] !). In the unrestricted case we can send messages that leave no trace. E.g. after a message that changes 0~s into l~s we can send a message that does the reverse.</Paragraph>
    <Paragraph position="5"> In sending a message from one position in the sentence to another~ the intermediate symbols are not chazlged. In fact they are changed twice: back and forth. With acycllc context-sensitive grammars, this is not possible. Every messages leaves a trace aatd the amount of information that ca~t be sent, is restricted by the gr~munar.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
Definitions
</SectionTitle>
      <Paragraph position="0"> A grammar is a 4-tuple, G = (V, E, R, S), where V is a set of symbols, :E C V is the set of terminal symbols. R C V ~ x V* is a relation defined on strings. Elements of _R arc called rules. S E V \ is the startsymbol.</Paragraph>
      <Paragraph position="1"> A grammar is context-sensitive if each rule is of the form aZfl ---* C/~7fl where Z ff V \ E ; cC/,/~, 7 G V* ; 7 5 L e. A granLmar is context-free if each rule is of the form Z -~ 3' where Z C V \ ;TEV*.</Paragraph>
      <Paragraph position="2"> Derivability (-%) between strings is defined as follows: uc~v ~ uflv (u,v,c*,fl E V*) iff (~,fl) E R. The transitive closure of -% is denoted by =L~. The transitive rettcxive closure of =4- is denoted by :~. The language generated by G is defined ms L(G) = {w E E* I S ~ w}.</Paragraph>
      <Paragraph position="3"> A derivation of a string ~ is a sequence of strings zl,x2,...,x,~ with xa = S, for &amp;quot;all i (1 &lt; i &lt; n) Xi =2- Xi+l and X n = ~.</Paragraph>
      <Paragraph position="4"> A context-free grammar is acyclic if there is no Z E V\E such that Z ~+ Z. Thisimphes that there is no string a E V* such that cr ~ a.</Paragraph>
      <Paragraph position="5"> We can map a context-sensitive grammar G onto its associated context-free grammar G ~ as follows: If G is (V,E,R,S) then G' is (V,E,R',S) where for every rule aZfl -~ oeTfl E R there is a rule Z -~ 7 ff R r. There axe no other rules in R I.</Paragraph>
      <Paragraph position="6"> Note that the associated grammar does not contain empty productions.</Paragraph>
      <Paragraph position="7"> We cefll G aeyclic iff the associated context-flee grammar C is acycllc.</Paragraph>
      <Paragraph position="8"> The notation we use for context-sensitive rules is ms follows: the rule aZfl ---* ceTfl is written</Paragraph>
      <Paragraph position="10"> An example of a context-sensitive grammar with the corresponding context-flee rules is: context-sensitive rules context-free part</Paragraph>
      <Paragraph position="12"> This contextMsensitive grammaris cyclic. Iris able to permute (}'s and its, Recognition is NP-complete</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
UNIFORM RECOGNITION FOR
ACYCLIC CONTEXT-SENSITIVE
GRAMMAR
</SectionTitle>
    <Paragraph position="0"> INSTANCE: An acyehc context-sensitive grammar G = (V, Z,R,S) and a string w G E*.</Paragraph>
    <Paragraph position="1"> QUESTION: Is w in the language generated by G ? The proof can be found in Aarts \[1991b\]. To prove that it is in NP wc have to prove that derivations ill ACSG's aa'e short (have polynomial length). Tiffs follows from the fact that derivations in context-free grammars have polynomial length. Derivations in an acyclie CSG are identical with derivations in the associated context-free grammar. The proof of NP-hardness is more complicated. The known NP-hard problem 3-SAT can be reduced to UNIFORM RECOGNITION for ACSG. Any 3-SAT formula can be translated in a grammar and an input for ACSG-recogultion.</Paragraph>
    <Paragraph position="2"> AcrEs DE COLING-92, NANTES, 23-28 Aour 1992 1 1 5 9 PROc, O1: COLING-92. NAbrl'ES, AUG. 23-28, 1992</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
Recognizing Power
</SectionTitle>
      <Paragraph position="0"> Any context-free grammar can be transformed into ant acyclic context-free grammar without loss of recognizing power. A cycle can be removed by introduction of a new symbol. This symbol rewrites to any member of the cycle. Any context-free grammar with empty productions can bc changed into a context-free grammar without empty productions that recognizes the same language. There's one exception here: languages containing the empty string can not be generated.</Paragraph>
      <Paragraph position="1"> Any acyclic context-free grammar withont empty productions is an acyclic context-sensitive granlmar. Therefore, ACSG's recognize all context-free 10alguages that do not contain the empty word.</Paragraph>
      <Paragraph position="2"> Furthermore, acyclic context-sensitive granamars recognize languages that are not contextfree. One example is the language {anb2~c** In &gt; 1} This language is recognized by the grammar (&amp;quot;X&amp;quot; is a nouternfinM):  aabbbbcc.</Paragraph>
      <Paragraph position="3"> With the pumping lemma one caal prove that the l~tbmage is not context-free.</Paragraph>
      <Paragraph position="4">  We have proved that UNIFORM RECOGNI-</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
TION FOR. ACYCLIC CONTEXT-SENSITIVE
</SectionTitle>
    <Paragraph position="0"> Gf\[AMMAR is NP-complete. It turns out to be important for complexity of recognition with context-sensitive grammars whether sending information leaves a trace.</Paragraph>
    <Paragraph position="1"> We have reduced 3-SAT to the uniform recognition problem for acyclic context-sensitive grammars. Every 3-SAT formula results in a different grammar. Probably it is not possible to construct an acyclic context-sensitive grammar that recognizes all 3-SAT formulas. My conjecture is that ACSG-recognition is not NP-hard for any fixed grammar. If this is not true, there would exist a grammar that recognizes all 3-SAT formulas. For this grammar the recognition problem would be NP-hard. In such a grammar, not every 3-SAT variable is encoded in a different symbol in the grammar. The variables are numbered and their numbers are encoded in sequences of O's and l's e.g.. A grammar that recognizes all 3-SAT formuta's must be able to compare such sequences.</Paragraph>
    <Paragraph position="2"> It must e.g. be able to recognize tile language {ww I w * V*}. Ifwis anumber, two numbers are compared. Context-sensitive grammars can recognize ww. Some can even recognize all 3-SAT formula's.</Paragraph>
    <Paragraph position="3"> ACSG's are not that strong. They can not even recognize ww. Any ACSG can compare only a fixed number of characters (only fixed amounts of information cazt be sent). Therefore my conjecture is that the recognition problem for any fixed grammar is not so hard: it's polynomial. Chart parsers for ACSG have been designed and implemented \[Aarts, 1991\]. They recognize inputs for many hard grammars in polynomial time. It is hard to prove, however, that they run in polynomial time for every grammar. If it could be proved, complexity of ACSG-recognition is similar to complexity of UCFG-recognition: NP-complete for the uniform case and a known algorithm that runs in time something like O(21GIna)) (polynomial in n but not in G).</Paragraph>
    <Paragraph position="4"> The polynmnial bound (which has not been proved yet) would be an explanation of the fact that humans can process language efllcicntly. Humans have a fixed grammar in mind which does not change. The complexity of recognition with a fixed grammar should be compared with the speed of human language processing. The arguments of Barton Jr., Berwick and Ristad \[1987\] against this are based on two kinds of arguments. The first has to do with compilation or preprocessing. We have polynomial bounds without compilation or preprocessing (just fix IGD. These arguments do not seem to hold. The other ones have to do with language acq~fisition. When a child is learning a language, the grammar she uses is changing. At every sentence utterance or understanding the graramar seems to be fixed. The difference between uniform recognition and recognition for any fixed grammar is that small that we can not draw conclusions about what kind of processing children perform when learning a language.</Paragraph>
    <Paragraph position="5"> AcrEs DE COLING-92, NAbrrES, 23-28 ^O~q&amp;quot; 1992 1 1 6 0 PROC. OF COLING-92, NAh'rES, AUG. 23-28, 1992</Paragraph>
  </Section>
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