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<Paper uid="C96-1091">
  <Title>A Compilation-Chart Method for Linear Categorial Deduction</Title>
  <Section position="4" start_page="537" end_page="538" type="metho">
    <SectionTitle>
2 Implicational Linear Logic
</SectionTitle>
    <Paragraph position="0"> Linear logic is an example of a &amp;quot;resource-sensitive&amp;quot; logic, requiring that in any deduction, every assumption ('resource') is used precisely once. We consider only the implicational fragment of (intuitionistic) linear logic. 3 The set of formulae arises by closing a (nonempty) set of atomic types .4 under the linear implication operator o- (i.e.</Paragraph>
    <Paragraph position="1"> ~- ::= A I 5%--Y). Various alternative formulations are possible. We here use a natural deduction formulation, requiring the following rules (oelimination and introduction respectively):</Paragraph>
    <Paragraph position="3"> Eliminations and introductions correspond to steps of functional application and abstraction, respectively, as the lambda term labelling reveals.</Paragraph>
    <Paragraph position="4"> The introduction rule discharges precisely one assumption (B) within the proof to which it applies (ensuring linear use of resources, i.e. that each resource is used precisely once). Consider the following proof that Xo-Y, Yo-Z =~ Xo-Z</Paragraph>
    <Paragraph position="6"> xo-z: Following Prawitz (1965), a normal form for proofs can be defined using just tile fbllowing (meaning preserving) contraction rule (analogous to /4-conversion). This observation is of note in that it restricts the form of proofs that we must consider in seeking to prove some possible theorem. null  The normal form proofs of this system have a straightforward structural characterisation, that their main branch (the unique path fi'om an assumption to the proof's end-type that includes no 3It follows that tile parsing method to be developed applies only to categorial systems having only implicational connectives. It is standard in categorial calculi to include also a 'product' operator, enabliug matter like addition of substructures, e.g. L has a product (commonly notated as)., with the Lambek implicationals / and \ being its left and right residuals. Although it is appealing from a logical point of view to include such operators, their use is not motivated in grammar.</Paragraph>
    <Paragraph position="7"> minor premise of an elimination inference) consists of a sequence of (&gt;_ 0) eliminations followed by a sequence of (&gt; 0) introductions.</Paragraph>
    <Paragraph position="8"> The differential status of the left and right hand side formulae in a sequent may be addressed in terms of polarity, with left formulae being deemed to have positive polarity, and the right formula to have negative polarity. Polarity applies also to subformulae, i.e. in a formula Xo-Y with a given polarity p, the subformula X has the same polarity p, and Y has the opposite polarity. For example, a positively occuring higher-oi'der type might have the following pattern of positive and negative subformulae: (X + o- (Y- o- Z ~ )- )+ Consider the following proof involving this type:</Paragraph>
    <Paragraph position="10"> Observe that the involvement of 'hypothetical reasoning' in this proof (i.e. the use of an additional assumption that is later discharged) is driven by the presence of the higher-order formula, and that the additional assumption in fact corresponds to the positive subformula occurrence Z within that higher-order formula. In tile following proof that Xo-(yo-(Yo--Z)) ~ Xo-Z, hypothetical reasoning again arises in relation to positive subformulae, i.e. the subformula Yo-Z of the higher-order formula (X + o- (y- o- (Y+ o- Z- )4 )-)+, as well as tile subtbrmula Z of the (overall negative) goal formula (X- o- Z + )-.</Paragraph>
    <Paragraph position="12"> More specifically, additional assumptions link to maximal positive subformulae, i.e. a subformula Y+ in a context; of the form (X- o- Y+)-, but not in (Y+ o- Z-) ~.</Paragraph>
    <Paragraph position="13"> For an even more complex formula, e.g.</Paragraph>
    <Paragraph position="14"> (v+ o-(w- o-(x+ o-(Y o- z+ )- )+ )- )+ we might find that a proof would involve not only an additional assumption corresponding to the positive subformula Xo-(Yo-Z)), but that reasoning with that assumption would in turn involve a further additional assumption corresponding to its positive subformula Z.</Paragraph>
  </Section>
  <Section position="5" start_page="538" end_page="540" type="metho">
    <SectionTitle>
3 A Compilation-Chart Method
</SectionTitle>
    <Paragraph position="0"> Standard chart parsing for PSG has the adwmtage that a simple organising principle governs the storage of results and underpins search, namely span within a linear dimension, specified by limiting left, and right points. A fllrther crucial feature is that what we derive as all item for any span is purely a function of the results derived for substretches of that span, and ultimately of the lexical categories that it dominates (assuming a given grammar).</Paragraph>
    <Paragraph position="1"> l)eduction in implicational linear logic lacks both of these features, although, as we shall see shortly, some notion of 'span' can be specified. The crucial problem for developing a chart-like method is the fact that, in combining any two elements A,B ~ C, there is an infinite number of possible results C we could derive, and that what we in fact should derive depends not just on the formulae themselves, but upon other formulae that might combine with thai; result. More particularly, the reasoning needed to derive C is liable to involve hypothetical elements whose involvement is driven by the presence of some higher-order type elsewhere.</Paragraph>
    <Section position="1" start_page="538" end_page="538" type="sub_section">
      <SectionTitle>
First-Order Linear Deduction
</SectionTitle>
      <Paragraph position="0"> Let us t)egin by avoiding this latter l)roblem by considering the fl'agment involving only first-order fbrmulae, i.e. those defined by S ::= fl. t Yo--A, and furthermore allow only atomic goals (i.e. so A is atomic in any F ~ A). Consequently, tile \[o-I\] rule is not required, and hypothetical reasoning excluded. In combining types using just the remaining elimination rule, we must still ensure linear use of resources, i.e. that no resource may be used inore than once in any deduction, and that in any overall deduction, every resource has been used. These requirements carl be enforced using an indexation method, whereby each initial forinula in our dat, at)ase is marked with a unique index (or strictly a single(era set containing that index), and where a formula that results ti'om a combination is inarked with the union of the index sets of the two formulae combined. 4 We.</Paragraph>
      <Paragraph position="1"> can ensure that no initial assumption contributes more than once to any deduction by requiring that wherever two tbrmulae are combined, their index sets must be disjoint. Thus, we require the following modified \[o-El rule (where C/, '~/~, vr arc'. index sets, and t0 denotes union of sets that are required to be disjoint):  indexing in ensuring linear use of resources.</Paragraph>
      <Paragraph position="2"> A whose index set is the flfll set of indices assigned to the initial formulae in P. For' example, to prove Xo-X, Xo-X, Xo--Y, Y =&gt; X, we might start with a database containing entries as fbllows (the tmmbering of entries is purely for exposition):  I. i:Xo--X:v 2. j : Xo-X : w 3. k:Xo-Y:z 4. l:Y:y Use of the modified elimination rule gives additional fornmlae as follows: 5. {k,/}: X: zy \[3+4\] 6. {i, k, 1}: X: v(a:y) \[1-t-5\] 7. {j, k, l}: X: w(zy) \[2-1-5\] 8. {i,j,k,1}:X:v(w(xy)) \[1+7\] 9. {i,j,k,l}:X:w(v(a:y)) \[2+6\]  There are two successful analyses, numbered 8 and 9, which we recognise by the fact that they have the intended goal type (X), and are indexed with the full set of the indices assigned to the initial left hand side fornmlae. Note that the formula mnnbered 5 contributes to both of tile sucessflfl overall mtalyses, without needing to be recomtinted. Hence we can see that we have already gained the key benefit of a chart approach for PSG parsing, nanmly avoiding the need to recompute partial results. It can be seen that indexing in the above method plays a role sinfilar to that of 'spans' within standard (:hart parsing.</Paragraph>
      <Paragraph position="3"> An adequate algorithm for use with the above approach is easily stated. Given a possible theorem Br,... ,Bn =&gt; A, tire left hand side formulae are each assigned unique indices and semantic variables, and t)ul; on ail agenda. Then, a loop is followed in which a formula is .taken from the agenda and added to the database, and then the next formula is taken from the agenda and so on until the agenda is empty. Whenever a formula is added to the datahase, a check is made to see if it can combine with formulae ah'eady there, in which (:as(; new formulae are generated, which are added to tile agenda. When the agenda is empty, a check is made for any successful overall analsyses, identified as described above. Note that since the result of a combination always bears an index set larger than either of its parent formulae, and since the maximal index set that any fornmla c~n carry includes all and only the indices assigned to the original left hand side formulae, the above process nmst terminate.</Paragraph>
    </Section>
    <Section position="2" start_page="538" end_page="540" type="sub_section">
      <SectionTitle>
Higher-Order Linear Deduction
</SectionTitle>
      <Paragraph position="0"> I,et us turn now to the general case, where higher-order formulae are allowed. The method to be described involves compiling tile initial formulae (which may be higher-order) to give a new, possibly larger, set; of formulae which arc; all tirst order. We observed above how hypothetical reasoning in a proof is driven by the presence within higher-order fornuflae of positively occurring subforinu- null lae. The compilation inethod involves identifying and excising such subformulae (thereby simplifying the containing formulae) and including them as additional assumptions. For example, this method will simplify the higher-order formula Xo-(Yo-Z) to become Xo--Y, generating an additional assumption of Z. The two key challenges for such an approach are firstly ensuring that the additional assumptions are appropriately used (otherwise invalid reasoning will follow), and secondly ensuring that a proof term appropriate to the original type combination is returned.</Paragraph>
      <Paragraph position="1"> Consider an attempt to prove the (invalid) type combination: Xo-Zo-(Yo-Z), Y =&gt; X. Compilation of the tbrmula Xo-Zo-(yo-Z) yiehls two formulae Xo--Zo-Y and Z, so tile initial query becomes Xo-Zo-Y, Z, Y =&gt; X, which is provable.</Paragraph>
      <Paragraph position="2"> The problem arises due to inappropriate use of the additional formula Z, which should only be used to prove the argument Y (just as Z's role wouhl be to contribute to proving the argument Yo-Z in a standard proof involving the original formula Xo-Zo-(Yo--Z)). The solution to this problem relies upon the indexing method adopted above.</Paragraph>
      <Paragraph position="3"> The additional assumption generated in compiling a higher-order formula such as Xo--(yo-Z) will itself be marked with a unique index. By recording this index on tile argument position from which the additional assumption was generated, we can enforce the requirement that the assumption contributes to the derivation of that argument. Note that a single argument position inay give rise to inore dmn one addil;ional assumption, and so in fact all index set that should be recorded. For example, The (indexed) formula i: Xo-(yo-Zo-W) will compile to give three indexed formulae: i:Xo-(Y:{j,k}) j:Z k:W We, require a inodified elimination rule that will enforce appropriate usage: 5</Paragraph>
      <Paragraph position="5"> Note that the compilation process must also gencrate additional assumptions corresponding to the positive subformulae of the right hand side of a query, e.g. compilal;ion of Xo-Y, Yo-Z ~, Xo-Z simplifies the right hand side formula to atomic X, giving and additional assumption Z.</Paragraph>
      <Paragraph position="6"> The second challenge we noted for such an approach is ensuring that a proof term (loosely, the SNore the requirement that (t is a proper subset of ,/~, which will have the consequence that other assumptions must also contrihute to deriving the argunwnt B. This will block a derivation of the linear logically valid Xo-(yo-Y) =&gt; X. However, this move accords with general categorial practice, where it is standm'd to require that each deduction rests m, at least one assumption. The alternative regime is easily achieved, by making the condition c~ C ~/).</Paragraph>
      <Paragraph position="7"> 'serum:tic recipe' of the combination) ai)propriate to the original type, combination is returned. Let us illustrate how this can be achieved with a simple example. Consider the following proof:</Paragraph>
      <Paragraph position="9"> Deriving the argument Yc-Z of the higher-order fornmla involves a final introduction steI), whk:h, semanl;ically, corresponds l;o an abstraction step that binds the variable semantics of l;he additional assuinption Z. The possibility arises that, compilation inight insert tile absl;rael~ion into the semantics of the compiled tbrmula, so that it latex' binds the variable of the additional formula. For example, coinpilation of Xo- (Yo-Z) lnight yield Xo-Y with term Ay.z(Az.y) and Z with variable term z, so that combining the former with some formula derived from the latter (i.e. whose tern\] included z) would cause the free occurrence of z to become bound, giving a result such as x(iz.f(z)). In that case, we can see that all;hough C, olnpilation has eliminated the need tbr an explicit introduction step in the proof, the, sl;ep still occurs imtflicitly ill the semantics.</Paragraph>
      <Paragraph position="10"> Of course, anyone familiar with lambda calculus will immediately spot the flaw in the preceding proposal, namely that the substitution process that is used in ~-conversion is careflllly stated to avoid such 'accidental binding' of w~riables (by renaIning bound variables, wherever required). We will instead use a special variant of substition which specifically does not act to avoid accidental binding, notated __\[_//~\] (e.g. t,\[s//'v\] to indicate  substitution of s R)r v in t). Not(; that tim assignment of term variables in the apt)roach in general is such that other eases of 'accidental binding' (i.e. beyond those that we want) wilt not occur, incorporating this idea, we arrive at the fbllowing (final) version of tile elimination rule C/ : Ao-- (B:(~) : kv.a '~/J : B : b c,C ~ -- ~'~/~ re: A: a\[b//v\]  Note that the form of the rule requires the, implicational formula 1;hat; it, operates 111)Oll l;o t)e of a certain forin, i.e. involving an at)strael;ion (Av.a). This requirement is met by all implieationals, (as a side effect of the (:ompilation process.</Paragraph>
      <Paragraph position="11"> A precise statement of the compilation procedure (r) is given in Figure. 1. This takes a sequent F ~ A:x as input, where every left aml right hand side formula is labelled wil;h a Ulfique variable, and returns a strucl;ure (A, (C/ : G : u)), where A is a set, of indexed tirst order formulae, C/ is the flfll  T(Xl : 'd'l,..., Xn : ~1~,1, 0 X0 : a;0) :.: (~, ((/) : (~ : ?t)) where, i0,..., i~ \[resh i,t(li(:('.s neg(i0 :X0:x0) = (i0:G:u)WI' A : FUpos(il : X1 :x,) U. , .</Paragraph>
      <Paragraph position="12"> Upos(i,~ : X,~ : 'a:,,.) (/, :-indices(A).</Paragraph>
      <Paragraph position="13"> ,,o~(,: : x: t):-- (i: x :/,) whe~e X a,;omi(:.</Paragraph>
      <Paragraph position="14"> post,: : X~ o-y, :/,) (,i: x~(r,,, : (/,) : ~,,..~) LJF tJ A whe,'e neg('i : \]q : '.) = (i'}~ : '.)~Ul ~ (v a hesh variabh 0 pos(i : Xl : (t',))) : (i: X 2 : ,';)l~Jz~ (/~--indices(l').</Paragraph>
      <Paragraph position="15"> ,eg(i:X:v) =:(i: X:v) whereXa.tomic.</Paragraph>
      <Paragraph position="16"> neg(i:Xlo-}q :u)__(i:X2:w)Ol'UA where v. :-: Av.:r (v, :c fresh variables) neg(i : X, : z) = (i: X2 : w)Ul ~ pos(j : r, : v):= A (j a fr(;sh in,lex).</Paragraph>
      <Paragraph position="17">  set; of indices, (\] is an atx)mi(: Goal I;yp(',, and u a variable, l,et A* denote the result; of (:losiug A under (,he elimination rule. The, sequent ix proven iJ)&amp;quot; (() : G : u) (~ A* fi)r some assigmnent of a vahle 1;o 'a. Under t;hat assignment, the original right hand side va.riable x will return a (:omplete proof term for the imi)lMt I)roof of the original s(;qll(:IIIL Not, c {;hal l,he t)i'ot)f Lerllhq so t)ro(hlce(l have a form whi(:h (:orrest)onds, (m(h;r l;he Currylloward isomorphism, to itormal form dedu(:l;ions (as defin(xl earlier).</Paragraph>
      <Paragraph position="18"> A simt)le example. Compilation of the sequent:  Xo-(Yo-Z) :x, Yo--W:y, Wo-Z:w ~ X:v yields the goal Sl)ecification ({i, j, k, 1} :X:v) and fl)rmulae 11-4, with t'ormulae 5-7 m'ising under combinat;ion. Formula 7 meet;s t;he, goal spe(:ith:ation, so th(; inil;ial sequent ix proven, with l)roof term m()~Z.y(Wz) ) rel;urned.</Paragraph>
      <Paragraph position="19"> I. i:Xo~(y:{j}):Au.z(Az.,,,) 2. j:Z:z 3. k: Yo-W : A'u.y'u 4. 1 : Wo-Z : )vu.'w',, 5. {j,l}:W:wz \[2+4\] 6. {j,k,~}: v::q(~,,~) \[a+~\] 7. {,:, j, k, l}: x: :,;(A~.,(,,,,~)) \[7l+6\]  The indexed firsl;-ord(;r formulae generate(1 by the comI)ilation procedure can t)e processe, d using t)recisely the same algoril;hm as that des(:ribed above for handling formulae of the iirsl&gt;order fl'ag~ men% with precisely the same benctit, i.e. avoiding re(:ompul:ation of I)artiM results.</Paragraph>
      <Paragraph position="20"> Some efti(:iency questions tMse. Imagine a Prolog implementation of the method, with indexed fornmlae being stored as facts ('edges') in the Proh)g database. An imt)orl;ant, overhead will arise wh(;n adding an agenda item to the dal;al)ase fl'om lo(:~dng those, lbrmub~ Mrea(ly there that the curreid; t'orilltlla (:all combine with, i.e. if we ltlllSI; separat(Jy access every formula Mready stored to ewduate if in(l(,xation requiremelltS are satisii(d, a.d (:oml)iIladon possil)le. Note firstly dmt,, since (:omt)ih'd formulae are all tirst-order, if we are a&lt;tding an alomi(: f(&gt;rmula we nee,(t (&gt;nly h)ok to stored iml)li(:atiomfl formula.e for possible (:oral)i-. nations, and vice versa. 'Fhis is easily a&lt;:hiev(xl. 'l'he prol&gt;h'm (&gt;f (',valuating in&lt;lexation require,ments can be (~ase(11)y using at bit-vector e,n(:o(ling of in(h',x sets. The, (:Oml)iladon t)rocess will return a full set 1 of l:he mfi(lUe iudices assigmxl to any \[brnntlae. If we impose an arbitrary ord(n ()vet dm elements of this sol;, we (:&amp;n then (m(:o(h~ l;he exl;eltsiolt Of ally ill(lex set; We edl(;Ollill;er ttsillg aii 'n-1)la(:e bitove(:tor, where n is the c}udinality of l, i.e. if some, index set (:ontains the it;h ('\]e,m('m, of (ordered) 1, then the ii;h eh'ment of its bil&gt;v(w,l;()r is 1, otherwis(~ 0. \[t is uscl'ul to store fiflly sp(',cilie(l bit-vectors with al;omi(: formula(;, specifying l;heir imh'x set. For iml)li(:adonal fornmla, how-. ever, it is usehfl to store a. 1)it-ve,(:tor (m(:oding its 'requireme, nt.s for an appropriately indexed argu menl;, i.e. with 0s instantiate(l for tim (;\]em(!nts of the impli(:ational's own index sel; (to enfor(:(! disjoin(hess of index s(;ts), and with \] s a, pl)ea.ring; for those indi(:e,s that it requires hay(! l)e(m inv()lved in de, riving the argument. Other 1)ositi()ns will 1)e tilled with anonymous variM)les. The bil:-vet:tors for an imt)li(:ational and an at()mi(', formula will m;~t(:h just in case I:hcy ~r('~ permil;tcd tx) (:oml)inc, a(:(:or(ting to in(lexal;ion requir(;ments. (Tim one shortfM1 here is thai; tim the(hod allows the impli(:al,ioiml (;o spe(:ify t;ha.t (:crtain indices are a subsel; of those of l;he argumellt, but not that tlmy are a proper subset l;hereof.) l ly storing su(:h vet;tots with formulae in the datalm.se, indexation requirements cam l)e, (:he(:ked by the process of mat(:hing; 1;o the d~tabase, so dial; only at)prot)riate, entries ;~re brought out for further examination,</Paragraph>
    </Section>
  </Section>
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