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<Paper uid="C90-3016">
  <Title>Structured Meanings in Computational Linguistics</Title>
  <Section position="3" start_page="0" end_page="85" type="metho">
    <SectionTitle>
2 An Old Idea: Structured
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="85" type="sub_section">
      <SectionTitle>
Meanings
</SectionTitle>
      <Paragraph position="0"> It has been argued that no theory of meaning that is con:positional and truth conditional can deal with propoMtional attitudes. For, whenever two expressions with different conlpositional (syntactic) structures boil down -- via the semantic operations connetted with their respective structures -- to the same meaning, a person can fail to see the equivalence', he can carry out the operations in the wrong way, or t, oo slowly (\[Cresswell 19851). Cresswell and others have concluded that syntactic structure has to take part in meaning representations: t-indistinguishable expressions may still have different meanings, due to differences in syntactic structure. D.Lewis, for instance, used semantically interpreted phrase markers (roughly: syntax trees with logical formulas attached to the nodes) as meanings for natural language expressions (\[Lewis 1972\]). However, this leads to an extremely strict notion of synonymy: Perhaps we would cut thereby meanings too finely. For instance, we will be unuble to ~gree with someone who says that a double negation has the same meaning as the corresponding affirmative. (\[Lewis 19721) Also, no relation of logical consequence has seen the light for any notion of structured meaning. In the sequel we will deal with the notion of meaning inherent in the Rosetta automatic translation project (e.g. \[Landsbergen 19821, \[Landsbergen 1985\], \[Landsbergen 1987\], or \[de Jong and Appelo 1987\]).</Paragraph>
      <Paragraph position="1"> This notion of meaning -- essentially an elaboration of the one proposed by Lewis -- allows a suitably weaker notion of synonymy, and can also be provided with a notion of logical consequence. Thus, some of the weak sides of older &amp;quot;structured meanings&amp;quot; prodeg posals are compensated for.</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="85" end_page="85" type="metho">
    <SectionTitle>
3 Structured Meanings in
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="85" end_page="85" type="sub_section">
      <SectionTitle>
Rosetta
</SectionTitle>
      <Paragraph position="0"> Rosetta uses a variant of Montague grammar (\[Montague 1974\]), in which each syntax rule has a semantic counterpart. Each node in the syntactic derivation tree (D-tree) for a sentence is associated with a semantic rule. Thus, each D-tree is associated with a semantic tree (M-tree), whose nodes are semantic rules and whose leaves are non-logical constants. By applying the semantic rules to their arguments, a logical formula can be calculated for each node in the M-tree that stands for its truthconditional me~ning. We will call this fornmla the corresponding formula of the node. Now in Rosetta, a sentence meaning is not, as in \[Montague 1974\], identified with the formula that corresponds to the top node of an M-tree, but with the entire tree.</Paragraph>
      <Paragraph position="1"> Thus, syntactic structure in Rosetta becomes a part of meaning in much the same way as proposed by D.Lewis (see above). For instance, the English Noun Phrase 'ItMian giri' and its Spanish equivalent 'muchach~ Italiana' might, if we simplify, both be represented by the same M-tree:</Paragraph>
      <Paragraph position="3"> where M2 stands for the sets of italians, M3 stands for the set of girls and M1 stands for the operation of set intersection. M1 is expressed by different syntax rules in English and Spanish: REIn1: If a is an Adjective and fl is a Noun, then o~fl is a Nom.</Paragraph>
      <Paragraph position="4"> RjSV&amp;quot;: If c~ is an Adjective and fl is ~L Noun, then fla t is a Nora, where a ~ is the adjective a, adjusted to number and gender of the llOUn ft.</Paragraph>
      <Paragraph position="5"> By mapping both of these rules onto M1, the two NPs are designated as translations of each other. Now M-trees in Rosetta are used as vehicles for inter-lingual translation, but we will view them as &amp;quot;general purpose&amp;quot; representations for the meanings of natural language expressions. Viewed in this way, the following definition of synonymy (notation: '=') between D-trees (and, derivatively, for natural language expressions) is forthcoming: Synonymy (first version): D1 ~ D2 C/*ae/</Paragraph>
      <Paragraph position="7"> R2(bl, ..., b,,), where Rt and R~ snap onto the same meaning rule, und where it holds for alll&lt;i&lt;nthat ai ~bi, or - D1 and D2 are basic expressions which map onto the same basic meaning.</Paragraph>
      <Paragraph position="8"> (Definition of synonynly for M-trees, at this stage, comes down to simple equality of the trees.) This notion of meaning takes syntactic structure into account, but does not &amp;quot;cut meanings too finely&amp;quot;, since any two linguistic constructions can be designated as synonymous. For instunce, Lewis' &amp;quot;double negation&amp;quot; problem can be countered as follows: the syntax rules of double negation (Raou~ler~e~) and plain affirmation (R~Hi .... ) can be mapped onto one and the same meaning rule, if the grammar writer decides that they are semantically indistinguishable. Alternatively, the semantic relation between a D-tree of  and its constituent tree D nl~y be accounted for if both trees are snapped onto one and the same M-tree. Effectively, this would come down to an extension of Rosetta with &amp;quot;rules of synonynly&amp;quot; for entire trees, rather than for individual syntax rules.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="85" end_page="87" type="metho">
    <SectionTitle>
4 Inference with M-trees
</SectionTitle>
    <Paragraph position="0"> Arguably, our grip on the notion of nleaning is incomplete if only the limiting case of structural equivalence is dealt with, leaving aside the more general case of structural consequence (~v/). Under what conditions does, for instance, one belief follow from another? Rosetta's isomorphy-based notion of meaning seems ill-equipped to deal with inference, but we claim theft an extrapolation is possible.</Paragraph>
    <Paragraph position="1"> A natural boundary conditions on ~M is logical validity: no rule may lead from true premises to a false conclusion. Writing '~-' for the relation holding between M-trees if the formulas corresponding to their top-nodes stand in the relation of logical consequence, this gives: Validity: T,, ~M Tb only if T,, ~ Tt,.</Paragraph>
    <Paragraph position="2"> Given validity as an upper bound, we seek reasonable lower bounds on structural inference. It is not generally valid to allow that a tree has all its sub-trees as structural consequences. (For instance, the 86 2 negation of a tree T does not have the subtree T as a consequence.) However, a solution can be found if we take the dual nature of our concept of meaning into account: M-trees combine structural and logical information. Therefore, if one tree is a subtree of another tree, and also a purely logical consequence of the bigger tree, then the inference is indisputable; for the inference is logically correct and there can be no difference in syntactic structure: Subtree Principle (1 't version): If (i) T, T~, and (ii) T1, is a subtree of T,~ then T,~ ~M Tb.</Paragraph>
    <Paragraph position="3"> However, we have to exclude as &amp;quot;pathological&amp;quot; cases all those situations ill which it is not one and the same subtree Tt, that takes care of the logical and the structural side: we cannot allow inferences such as the following -- where S abbreviates a &amp;quot;paraphrase&amp;quot; of S, namely a sentence that is logically, but not structurally, equivalent to S (see below) --. even though they fulfil both conditions of the Subtree Principle: (2) -, a (s -, v ~M St v s2.</Paragraph>
    <Paragraph position="4"> These inferences are not structurally valid, given the structural differences between the conclusion and the relevant part of the premise. Let an atomic sentential fragment (asj) be a sentential M-tree no proper part of which is sentential itseff. To be on the safe side, we might forbid that &amp;quot;par&amp;phases&amp;quot; of asps from the conclusion occur in the premisse: Subtree Principle (2 nd version): If (i) T,~ Tt, and (ii) Tb is a subtree of T,~ and (iii) If TI is an asf that occurs essentially in T,~ and T2 is an asf that occurs essentially ill Tb, then T\]. is not a paraphrase of T2, then Ta t=M T~,, where a paraphrase is a logical equivalent that falls short of structural equivalence: Paraphrase (1 ~t version): T1 is a paraphrase of T2 C/~D,j Tl ~ T2 and T2 ~ Tl but none of the two is a subtree of the other.</Paragraph>
    <Paragraph position="5"> The resulting logic is quite uncommon unless stronger lower bounds are given. For instance, if (ii) is a necessary condition, there cannot be any tree T such that ~M T. Consequently, the Deduction Theorem will not hold. Also, if (iii) is a necessary condition, then Conjunction Elimination fails to hold. In fact, it holds for all Sl and $2 that SI&amp;S2 ~M $2. To remedy this defect, (iii) may be weakened to allow logically inessential occurrences of paraphrases: Inessential occurrence: An occurrence of T in the premisse (conclusion) of an ino ference is inessential if the inference goes through when T is replaced by an arbitrary T' everywhere in the premisse (conclusion).</Paragraph>
    <Paragraph position="6"> For instance, the occurrence of S in (1) and (2) is essential, but its occurrence in S&amp;S ~/~ S is inessential and therefore harmless. As a result of this change, a restricted version of Conjunction Elimination holds, to the effect that a conjunction will structurally imply ally of its eonjuncts, provided the conchslon conjunct does not contain two asf's that are paraphrases of eachother. This concludes our formalization of the &amp;quot;subtree&amp;quot; intuition. If we want to cover more ground, we need a more liberal concept than the structural no~ion of one tree being a subtree of another. First, a more subtle structural notion may be employed. For instance, an inference from Each dog barks loudly to Each black dog barks must be allowed, it seems, even though none of the two M-trees is a p,'u't of the other. Therefore, a relation of constituent-wise comparability (~,, definition follows) is called for. It is important to note that the &amp;quot;direction&amp;quot; of the comparison (which of the two subsumes which) is irrelevant, since the logical requirement (i) determines the direction of the inference: Subtree Principle (3 r'~ version): If (i) T,, Wl, and (ii) Ta ~ T1, and (iii) (as above), then T,~ ~M Tb.</Paragraph>
    <Paragraph position="7"> If the notation ~-. stands for the symmetrical relation that holds between two trees if one of them is a sub-tree of the other, this is the definition of the relation Comparability: T,~ ~: TI, ~D~\]</Paragraph>
    <Paragraph position="9"> V Tai &amp;quot;-7 Tbi : Tai ~ Tbj or Y T~, s 3 T,,i : Tba' ~-- T,,.</Paragraph>
    <Paragraph position="10"> Here, T~ = &lt; TuI,...,T~,, &gt; means that T~ can be decomposed (at an arbitrary level of the tree) as the sequence W,,1 ,...,Tun.</Paragraph>
    <Paragraph position="11"> Example: The M-trees for Each black dog barks and Each dog barks stand in the relations ~c and ~, while the M-trees for Each dog barks loudly and Each black dog barks do not stand ill the relation ~, but they do stand in the relation &amp;quot;~c. They are constituent-wise comparable, so since the first logically implies (~) the second, the first must also have the second as a structural consequence (~M):  Here, T~ ~ Tb holds, for T,~ = &lt; B1, B2, T,~2 &gt;, and T~, = &lt; B1, T, n2, B3 &gt;, while B2 is a subtree of Tb12 and B3 is a subtree of T,~2. End of Example Note that, by replacing the subtree notion by the symmetrical notion ~.c, we now allow a conclusion to introduce asf's that do not occur in the premisse. For instance, under appropriate assumptions, it will hold that S ~-M S and SL, for logically true SL. This defect can be remedied simply if we add a clause that prevents a conclusion from contaiuing any novel asf's (see (iv), below) So far, the Subtree Principle still formalizes a strictly structural approach. But there ought to be more tllan that. In the ideolect of a given language user, two grammar rules, or two lexical items, may be semantically related without any strictly structural notion being involved. Within the bounds of Validity ~nd the Subtree Principle, the grammar writer is free to designate certain pairs of syntax rules or lexical items as semantically related. Since, again, the direction of the relation is irrelevant, this refinement can easily be built in into the definition of ~. If this is done, the relation ,~C/ will also hold between Each mammal barks loudly and Each black dog barks, assuming that 'mammal' and 'dog' are semantically related. Note, however, that these stipulations need not be the same for all language users: different stipulations of structural relatedness may reflect differences in linguistic competence (\[Partee 1982\]). In short, ore' proposal implements the hypothesis that structural relations hold for everyone, while linguistic relations allow individual variation.</Paragraph>
    <Paragraph position="12"> If all the suggested improvements on the Subtree Principle are taken into account, one might venture the followhlg definition of structural consequence: Subtree Principle(final version): T,, ~-'M Tb ~C/:~Def (i) We, ~ W~, and (ii) T,~ ~c Tb and (iii) If T1 is an asf that occurs essentially in T~ and T2 is an asf that occurs essentially ill Tb, theu T1 is not a paraphrase of T2, and (iv) all asf's of T~, occur in T~.</Paragraph>
    <Paragraph position="13"> Since the notion of a subtree has now been replaced by constituent-wise comparability, the notion of a paraphrase must be redefined: Paraphrase (final version): T1 is a paraphrase of T~. C/&gt;De.f T1 ~ T2 and T2 ~ T1 but T1 ~ T2.</Paragraph>
    <Paragraph position="14"> Assuming that a notion of inference has been established along these lines, synonymy between M-trees can now be defined as nmtual structural consequence (synonymy of D-trees is analogous): Synonymy (final version): T1 and T2 are synonymous C/&gt;D~f TI bM T~ and To. bM T1.</Paragraph>
    <Paragraph position="15"> If the clauses in the first or the second version of the Subtree Principle are taken as collectively sufficient and necessary, the defined notion of synonymy coincides with the original Rosetta notion of ~having the same M-tree&amp;quot;. (In this case, T,, ~M T~, and T~, ~M Z, C/~ T,~ = T~,.) This conveniently simple situation breaks down in later versions of the Sub-tree Principle, where the relation of constituent-wise comparability is used. A simple example'.</Paragraph>
    <Paragraph position="16">  the difference between their corresponding M-trees -- which would have made them nonsynonymous in RosettWs original notion of synonymy.</Paragraph>
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