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<Paper uid="P98-1057">
  <Title>Group Theory and Linguistic Processing*</Title>
  <Section position="2" start_page="0" end_page="348" type="metho">
    <SectionTitle>
2 Group Computation
</SectionTitle>
    <Paragraph position="0"> A MONOID AI is a set M together with a product M x  A GROUP is a monoid in which every element a has an inverse a -1 such that a- l a = aa -1 -- l. A PREORDER on a set is a reflexive and transitive relation on this set. When the relation is also symmetrical, that is, R(x, Y) ~ R(y, x), then the preorder is called an EQUIVALENCE RELATION. When it is antisymmetrical, that is that is, R(x, Y) A R(y, x) ~ x = Y, it is called a PARTIAL ORDER.</Paragraph>
    <Paragraph position="1"> A preorder R on a group G will be said to be COM-PATIBLE with the group product iff, whenever R(x, Y) and R( x', y'), then R( xx', yy').</Paragraph>
    <Paragraph position="2"> Normal submonoids of a group. We consider a compatible preorder notated x -4 y on a group G. The following properties, for any x, y E G, are immediate:</Paragraph>
    <Paragraph position="4"> Two elements x, x' in a group G are said to be CONJU-GATE if there exists y 6 G such that x' = yxy -1. The fourth property above says that the set A,/ of elements x 6 G such that x -41 is a set which contains along with an element all its conjugates, that is, a NORMAL subset of G. As M is clearly a submonoid of G, it will be called a NORMAL SUBMONOID of G.</Paragraph>
    <Paragraph position="5"> Conversely, it is easy to show that with any normal submonoid M of G one can associate a pre-order compatible with G. Indeed let's define x-+ y as xy -1 6 M. The relation --~ is clearly reflexive and transitive, hence is a preorder. It is also compatible with G, for if xl --)- yl and x2 -4 y_~, then xly1-1, x2yg. -1 and yl(x~y2-1)y1-1 are in M; hence XlX2y~.-ly1-1 : xlyl-lylx~.y2-1y1-1 is in M, implying that XlX2 -4 yly:, that is, that the preorder is compatible.</Paragraph>
    <Paragraph position="6"> If S is a subset of G, the intersection of all normal submonoids of G containing S (resp. of all subgroups of G containing S) is a normal submonoid of G (resp. a J ln general M is not a subgroup of G. It is iff x ~ y implies Y --+ x, that is, if the compatible preorder --~ is an equivalence relation (and, therefore, a CONGRUENCE) on G. When this is the case, M is a NORMAL SUBGROUPof G. This notion plays a pivotal role in classical algebra. Its generalization to submonoids of G is basic for the algebraic theory of computation presented here.</Paragraph>
    <Paragraph position="7">  normal subgroup of G) and is called the NORMAL SUB-</Paragraph>
  </Section>
  <Section position="3" start_page="348" end_page="348" type="metho">
    <SectionTitle>
MONOID CLOSURE NM(S) of S in G (resp. the NOR-
MAL SUBGROUP CLOSURE NG(S) of S in G).
</SectionTitle>
    <Paragraph position="0"> The free group over %'. We now consider an arbitrary set V, called the VOCABULARY, and we form the so-called SET OF ATOMS ON W, which is notated V t_J V -1 and is obtained by taking both elements v in V and the formal inverses v-1 of these elements.</Paragraph>
    <Paragraph position="1"> We now consider the set F(V) consisting of the empty string, notated 1, and of strings of the form zxx~....:e,, where zi is an atom on V. It is assumed that such a string is REDUCED, that is, never contains two consecutive atoms which are inverse of each other: no substring vv-1 or v-1 v is allowed to appear in a reduced string.</Paragraph>
    <Paragraph position="2"> When a and fl are two reduced strings, their concatenation c~fl can be reduced by eliminating all substrings of the form v v- 1 or v- 1 v. It can be proven that the reduced string 7 obtained in this way is independent of the order of such eliminations. In this way, a product on F(V) is defined, and it is easily shown that F(V) becomes a (non-commutative) group, called the FREE GROUP over V (Hungerford, 1974).</Paragraph>
    <Paragraph position="3"> Group computation. We will say that an ordered pair  GCS = (~, R) is a GROUP COMPUTATION STRUCTURE if: 1. V is a set, called the VOCABULARY, or the set of GENERATORS 2. R is a subset of F(V), called the LEXICON, or the set of RELATORS. 2  The submonoid closure NM(R) of R in F(V) is called the RESULT MONOID of the group computation structure GCS. The elements of NM(R) will be called COMPUTATION RESULTS, or simply RESULTS.</Paragraph>
    <Paragraph position="4"> If r is a relator, and if ct is an arbitrary element of F(V), then ct, rc~ -1 will be called a QUASI-RELATOR of the group computation structure. It is easily seen that the set RN of quasi-relators is equal to the normal sub-set closure of R in F(V), and that NM(RN) is equal to NM(R).</Paragraph>
    <Paragraph position="5"> A COMPUTATION relative to GCS is a finite sequence c = (rl .... , rn) of quasi-relators. The product rx * * * r,, in F(V) is evidently a result, and is called the RESULT OF THE COMPUTATION c. It can be shown that the result monoid is entirely covered in this way: each result is the result of some computation. A computation can thus be seen as a &amp;quot;witness&amp;quot;, or as a &amp;quot;proof&amp;quot;, of the fact that a given element of F(V) is a result of the computation structure. 3 For specific computation tasks, one focusses on results of a certain sort, for instance results which express a relationship of input-output, where input and output are 2 For readers familiar with group theory, this terminology will evoke the classical notion of group PRESENTATION through generators and relators. The main difference with our definition is that, in the classical case, the set of relators is taken to be symmetrical, that is, to contain r -1 if it contains r. When this additional assumption is made, our preorder becomes an equivalence relation.</Paragraph>
    <Paragraph position="6"> 3The analogy with the view in constructive logics is clear. There what we call a result is called a formula or a tbpe, and what we call a computation is called aprot~</Paragraph>
    <Paragraph position="8"> ev(N,X,P\[X\]) p\[x\]-1 ~-i X N -I ever)' -a sm(N,X,P\[X\]) p\[x\]-1 ~-i X N -1 some -x N -I tt(N,X,P\[X\]) p\[X\] -I a -I X ~ that -I Figure 1 : A G-grammar for a fragment of English assumed to belong to certain object types. For example, in computational linguistics, one is often interested in results which express a relationship between a fixed semantic input and a possible textual output (generation mode) or conversely in results which express a relationship between a fixed textual input and a possible semantic output (parsing mode).</Paragraph>
    <Paragraph position="9"> If GCS = (V, R) is a group computation structure, and if A is a given subset of F(V), then we will call the pair GCSA = (GCS, A) a GROUP COMPUTATION STRUCTURE WITH ACCEPTORS. We will say that A is the set of acceptors, or the PUBLIC INTERFACE, of GCSA. A result of GCS which belongs to the public interface will be called a PUBLIC RESULT of GCSA.</Paragraph>
  </Section>
  <Section position="4" start_page="348" end_page="349" type="metho">
    <SectionTitle>
3 G-Grammars
</SectionTitle>
    <Paragraph position="0"> We will now show how the formal concepts introduced above can be applied to the problems of grammatical description and computation. We start by introducing a grammar, which we will call a G-GRAMMAR (for &amp;quot;Group Grammar&amp;quot;), for a fragment of English (see Fig. 1).</Paragraph>
    <Paragraph position="1"> A G-grammar is a group computation structure with acceptors over a vocabulary V = Vlog U ~/pho~ consisting of a set of logical forms l/~og and a disjoint set of phonological elements (in the example, words) l/~ho,,. Examples of phonological elements are john, saw, ever).,, examples of logical forms j, s (j, 1), ev (re,x, sra(w,y, s (x,y)) ); these logical forms can be glossed respectively as &amp;quot;john&amp;quot;, &amp;quot;john saw louise&amp;quot; and &amp;quot;for every man x, for some woman y, x saw y&amp;quot;.</Paragraph>
    <Paragraph position="2"> The grammar lexicon, or set of relators, R is given as a list of&amp;quot;lexical schemes&amp;quot;. An example is given in Fig. 1. Each line is a lexical scheme and represents a set of relators in F(V). The first line is a ground scheme, which corresponds to the single relator j john-1, and so are the next four lines. The fifth line is a non-ground scheme, which corresponds to an infinite set of relators, obtained by instanciating the term meta-variable A (notated in uppercase) to a logical form. So are the remaining lines.</Paragraph>
    <Paragraph position="3"> We use Greek letters for expression meta-variables such as a, which can be replaced by an arbitrary expression of F(V); thus, whereas the term meta-variables A, B .....</Paragraph>
    <Paragraph position="4"> range over logical forms, the expression meta-variables ,~, fl ..... range over products of logical forms and phono- null logical elements (or their inverses) in F(V). 4 The notation p \[x\] is employed to express the fact that a logical form containing an argument identifier x is equal to the application of the abstraction P to x. The meta-variable X in p \[X\] ranges over such identifiers (x, y, z .... ), which are notated in lower-case italics (and are always ground). The meta-variable p ranges over logical form abstractions missing one argument (for instance Az. s ( j, z) ). When matching meta-variables in logical forms, we will allow limited use of higher-order unification. For instance, one can match P \[X\] to -~ (j ,x) by takingP = Az.s(j, z) and X = x.</Paragraph>
    <Paragraph position="5"> The vocabulary and the set of relators that we have just specified define a group computation structure GCS = (I,, _R). We will now describe a set of acceptors A for this computation structure. We take A to be the set of elements of F(V) which are products of the following form: S lI/n-lWr~_1-1 ...IV1-1 where S is a logical form (S stands for &amp;quot;semantics&amp;quot;), and where each II';- is a phonological element (W stands for &amp;quot;'word&amp;quot;). The expression above is a way of encoding the ordered pair consisting of the logical form S and the phonological string 111 l,I) ... l.I;~ (that is, the inverse of the product l, Vn- 11Vn- 1 - I ... I.V1-1).</Paragraph>
    <Paragraph position="6"> A public result SWn-lWn_l-1...t'Iq -1 in the group computation structure with acceptors ((V, R), A) -- the G-grammar --will be interpreted as meaning that the logical form S can be expressed as the phonological string IV1 l'l:~ ' .. lYn.</Paragraph>
    <Paragraph position="7"> Let us give an example of a public result relative to the grammar of Fig. 1.</Paragraph>
    <Paragraph position="8"> We consider the relators (instanciations of relator schemes):</Paragraph>
    <Paragraph position="10"> which means that s ( j, 1 ) louise-I saw- l john- 1 is the result of a computation (r~ ', r2', r3 ' ) * This result is obviously a public one, which means that the logical form s ( j, 1 ) can be verbalized as the phonological string john saw louise.</Paragraph>
    <Paragraph position="11"> 4Expression meta-variables are employed in the grammar for forming the set of conjugates c~ e:cp ~-1 of certain expressions ezp (in our example, earp is ov{N,X,P\[X\] ) P\[X\] -1, sm(N,X,P\[X\] ) P \[X\] -1 or X). Conjugacy allows the enclosed material exp to move as a bh, ck in expressions of F(V), see sections 3. and 4.</Paragraph>
    <Paragraph position="13"> N that a -a X -1 c~ P\[X\]</Paragraph>
  </Section>
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