<?xml version="1.0" standalone="yes"?>
<Paper uid="C90-3007">
  <Title>Partial Descriptions and Systemic Grammar</Title>
  <Section position="3" start_page="36" end_page="36" type="metho">
    <SectionTitle>
2 What's in a net?
</SectionTitle>
    <Paragraph position="0"> Our first task is to provide a precise charact.erisa-tion of the information expressed by ~ systemic network. We begin by defining a way of labelling systemic networks, then provide a translation which maps labelled networks into collections of axioms expressed in the tbrm of' propositional logic. This work is a slight retinement of a very similar approach used by Mellish.</Paragraph>
    <Paragraph position="1"> Figure 1 contains examples of each of the tbur types of syste.m which we need to consider, linked together in such a way as to produce a description of the possible \['(3rills of English pronouns. The leftmost system is a choice system expressing the opposition between qucslion~ persoual and dcmonslral, ive pronotlns. Within these broad classification further distinctions operate. For example queslion pronouns iila~y be a'lt{71?ralC or inani'maiG and m,lst also make a choice between various case'-;. The system which expresses t~he necessity of making i;wo sinmltane-otis choices is indicated wit.h a left. curly bracket, and is i,l,e and system. Note that ~here are two routes ~,o tile choice of case, one t?om q'ucslion and the other fro,, personal. The system which ties these two routes together is called the disjunctive system. Finally, the rightmost system is a choice between various grammatica! genders.</Paragraph>
    <Paragraph position="2"> This system can only be reached it' a pronoun is both third and singular. The system involving the right-hand curly bracket which expresses this is called the conjunctive system.</Paragraph>
  </Section>
  <Section position="4" start_page="36" end_page="40" type="metho">
    <SectionTitle>
3 Labelllngs for networks
</SectionTitle>
    <Paragraph position="0"> We now establish technical definitions of two types of labelling for systemic networks.</Paragraph>
    <Section position="1" start_page="36" end_page="36" type="sub_section">
      <SectionTitle>
3.1 Basic Labeltings
</SectionTitle>
      <Paragraph position="0"> A basic labelling is defined to be a partial function from lines to names such that C/~ A line receives a name if and only if there is a choice system to whose right hand side i~ is directly attached.</Paragraph>
      <Paragraph position="1"> No two lines carry the same name.</Paragraph>
      <Paragraph position="2"> Figure l ,~dlows a ba,~dc labelling</Paragraph>
    </Section>
    <Section position="2" start_page="36" end_page="38" type="sub_section">
      <SectionTitle>
3.2 Exhaustive labellings
</SectionTitle>
      <Paragraph position="0"> Let ~&amp;quot; be a 1-1 function from lines to names for  these lines. 5 c is an exhaustive labelling of a network if and only if the following conditions hold :1. That part of ~&amp;quot; which provides names for lines attached directly to the right of choice systems must be a basic labelling.</Paragraph>
      <Paragraph position="1"> . If 5 c assigns a line name lzh~ to the line which is directly attached to the left hand side of an and system, then it must also assign that name to the lines which are directly attached to the right hand side of that system.</Paragraph>
      <Paragraph position="2"> . If )c assigns line names 11,12,... lm to the lines entering a conjunctive system (where ll is the label for the line appearing at the top of the system and lm that at the bof tom), then it must assign the label ll A 12 A * .. lm to the line which leaves that system 4. If ~&amp;quot; assigns line narnes ll, 12,... l,~ to the lines entering a disjunctive system (adopting the same ordering convention as above), then it must assign the label 11 V l~ V ...Ira to the line which leaves that system.</Paragraph>
      <Paragraph position="3">  We use a translation scheme given by Mellish \[12\] to produce a set of logical axioms equivalent in meaning to the original network. Mellish's scheme can be applied to all four types of system and the correct results will be produced, but with our labelling scheme only choice systems contribute to the meaning of the network. Each choice system translates into two axioms: an accessibility axiom, expressing tile constraint that none of the labels to the right of the system can be selected unless the entry point of the system has been reached; and an exclusivity axiom expressing the fact that these labels are mutually exclusive.</Paragraph>
      <Paragraph position="4">  In this section we show that the problem of systemic cl~sification is at least as hard as problerns known to be NP-hard. This is done by constructing a polynomial time mapping 1~ from instances of the NP-hard problem called 3SAT to networks which can be tricked into solving this problem for us. For an introduction to similar linguistic applications of complexity theory see Barton et al \[1\].</Paragraph>
      <Paragraph position="5"> If there were a polynomial time algorithm tbr checking arbitrary systemic networks, it would follow that 3SAT could be solved by the composition of the mapping that constructs the network with the algorithm that cheeks the network. Since this composition is itself a polyno-. mial time algorithm we would then have a polynomial time solution fc)r 3SAT, and hence for all other problems of the complexity class AfT ) . Thus the successflfl construction of 1I implies that systemic classification is itself NP-hard.  bility of a boolean forrnnla, stated ill con imictive normal form, in which exactly three va~:iables occur in each clause, of the conj unctioa. These variables may either be positive or negated, and may be repeated fl:om clause to clause. It, is known that 3SAT is just as hard as the proble~rl of sat-isfiability for general boolean formulae (Barl, on at al provide a demonst, ration of this fact on pp 52-~35 of \[(I).</Paragraph>
    </Section>
    <Section position="3" start_page="38" end_page="40" type="sub_section">
      <SectionTitle>
5.2 The mapping from 3SAT in-
</SectionTitle>
      <Paragraph position="0"> stances to networks The mapping II takes a 3SAT instance and produces a network. Let tile name of the 3SAT instance be E and its length NI,;.</Paragraph>
      <Paragraph position="1"> * Make a list of the variable names used in E, counting positive and negative occurrences of a wtriable as tile same. This can certainly he done in time polynomial in Nt~ using a standard sorting algorithm such as merge sort. Let the name of tile list of variable names be V and its length Nv. We use the example of the very simple expression</Paragraph>
      <Paragraph position="3"> and system feeding Nv parallel binary choice systems. Each choice system carries two labels, one corresponding to a variahle name in V and the other formed by negating the label on the other branch of the system. The choice of prefix should be such that all labels on tile resulting network are unique. '\]'his part of the process is polynomial in the length of V.</Paragraph>
      <Paragraph position="4"> e For every clause in E, add a teraary disdeg junctive system linking the lines of the network having the label,; corresponding to the three symbols of the clause. This part of the process involves scanning down the Nv systems of the network once for each clause of E, and is thereR)re also polynomial in NE.</Paragraph>
      <Paragraph position="5"> Finally, binary eholee systems are attached to the outputs of all the disjunctive systems introduced in tile last stage. These systems are labelled with generated labels distinct from those already used in the network. 'l'his step is clearly also polynomia, l in Nt~', requiring the crealion of a nmnber of choice systems equal t,o the number of' clauses in P;.</Paragraph>
      <Paragraph position="6"> The network giveu i~ figure 3 is the Oile which would be produced lY=om t5'. In order to use the construcLed network to solve the satisfiability problem for /';, we check an expression correspouding t/) I;he conjnnction of all the three member (-lause,~; in t~:. This is lmili by choosing ~:,l arbitrary label from each of tim rightmost choice systems. 'fhe coa itmct, io~ of the:;e labels is a consisi:,ent descr~l ,,~on whenever all the&amp;quot;  clauses of E can be satisfied by the same value assignment. The choice systems to the left of the disjunction express the facts that no variable can be simultaneously true and false. A correct checking algorithm will succeed in just those circumstances where there is at least one value assignment for the variables of E which makes E come out true. Systemic classification is therefore at least as hard as the other problems in Alp,and we should be very surprised to find that it can in general be solved in polynomial time.</Paragraph>
      <Paragraph position="7"> 6 Checking systemic descriptions null Although accurate checking of systemic descriptions is an NP-hard problem, it is still possible to devise algorithms which carry out part of the process of checking without incurring the cost of complete correctness. Our algorithm depends on a pre-processing step in which the original network is split into two components, each of which embodies some but not all of the information that was present at the outset.</Paragraph>
      <Paragraph position="8"> The first component is a simplified version of the original network, in which no disjunctive systems are present. This is achieved by removing all disjunctive systems, then re-attaching any dangling systems to a suitable point to the left of the position of the disjunction. For convenience in book-keeping we introduce special generated features which take the place of the disjunctive expressions that appear in the labelling of the original network. Figure 5 shows the result of peeling awaythe disjunction in figure 4.</Paragraph>
      <Paragraph position="9"> The second component of the network consists of a collection of statements indicating ways in which the generated features may be discharged.</Paragraph>
      <Paragraph position="10"> For the example network we would have had to note that gen f eat ~_ cl V c2 Taken together the simplified version of the network and the statements about generated fentures contain all the information needed. The simplified network is now amenable to deterrninistic and efficient checking procedures, including reductions to term unification as proposed by Mellish. The efficiency of these techniques hinges on the removal of disjunctive systems.</Paragraph>
      <Paragraph position="11"> The second stage of checking involves the search for a consistent way of discharging all the  generated features introduced by the first stage. This is the potentially costly part of the checking process, since separate disjunctions may conspire to produce exponentially many different alternatives which have to be checl(ed. It was to be expected that the process of systemic checking would involve an exponential cost somewhere, so this is no surprise.</Paragraph>
      <Paragraph position="12"> Even the second stage of checking is cheap unless two separate conditions hoh:t 1.. The description produced by the first stage of checking must involve many generated features.</Paragraph>
      <Paragraph position="13"> 2. The generated features must be interdependent, in that the way in which one feature is discharged colmtrains the way in which olher fiea.tures can be discharged.</Paragraph>
      <Paragraph position="14"> We can't be sure whether the first condition is going to hold until we see the OUtl)ut of the first stage, but we ca.n estimate the extent to which features interact, by inspect, lag the checking rules which arise when (he net, work is pardt.ioned. Thus, while we can't promise that the use of systemic networks will ensure tractability !'or arbitrary grammars, v,'e can help linguists to catch potential t)roblerns in the formulation of their feature system,,~ during grammar development, and avoid the risk of unexpected combinatorial explosions during the exploitation of the grammars in question.</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>