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<Paper uid="C90-2030">
  <Title>Normal Form Theorem Proving for the Lambek Calculus ~</Title>
  <Section position="5" start_page="177" end_page="178" type="concl">
    <SectionTitle>
4 Discussion
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    <Paragraph position="0"> It can be seen that the form of any CNF proof is strongly tied to the form of the lambda expression it assigns a.s its meaning. As we have seen, the lambda term corresponding to the meaning of any (cut free) proof in I, is always a fl-NF term of the form: ~vl..v~.(hUi..U~) (n, m &gt; 0) where h is a variable, and the main branch of a CNF proof is always of the following form (starting at the root): zero or more right inferences, followed by zero or more left inferences, terminating with an axiom inference. The correspondence between the two is as follows: the initial sequence of right inferences corresponds to the lambda abstractions of ~he variables vi..v~, and the ensuing left inferences are just those required to apply the variable h (the semantics of the head) to each of its arguments Ui..Um in turn, with e~ch of the subterms Ui being 'constructed' in the subproof for a minor premise.</Paragraph>
    <Paragraph position="1"> These observations provide the basis for relating this approach to that of Khnig (1989), mentioned earlier. Khnig uses a non-standard method for arriving at a notion of NF proof which involves firstly mapping proofs into objects called 'syntax trees', where proofs that yield the same syntax tree form an equivalence class, and then mapping from each syntax tree to a single NF proof. From the form of such NF proofs, Khnig derives a set of Cnesting constraints' which are used to limit the operation of a top-down theorem prover, and which are such that they will never prevent the construction of any NF proof. As Khnig points out, however, the ~nesting constraints' do not exclude the construction all non-NF proofs when used with a standard propositional formulation of the Lambek Calculus (though better results are obtained with a unification-based version of the Lambek Calculus that Khnig describes). Khnig's syntax trees can be seen to bear a strong correspondence, in terms of their structure, to the lambda term for the meaning assigned by a proof (although the former include sufficient information, of types etc, to allow (re)construction of a complete proof for the initial sequent), and the relation of Khnig's NFs to the syntax trees used to define them closely parallels the relation between CNF proofs in the present approach and their corresponding lambda terms.</Paragraph>
    <Paragraph position="2">  A further topic worthy of comment is the relation between the current approach and natural deduction approaches such as that of Prawitz (1965).</Paragraph>
    <Paragraph position="3"> As Prawitz observes, sequent calculi can be understood as meta-calculi for corresponding natural deduction systems. Introduction rules correspond to right rules and elimination rules to left rules. In Prawitz's NFs, an introduction rule may never apply to the major premise of an elimination rule (such a subproof being a redex) so that eliminations always appear above introductions on the main branch of a NF proof, li which seems to parallel the form of CNF sequent proofs. However, the relation is not so straightforward. For a natural deduction formulation of the (product-free) Lambek Calculus, i2 the occurrence of a relevant redex in a natural deduction proof (i.e. where an introduction rule applies to the major premise of an elimination) corresponds to the occurrence of a fl-redex in the corresponding proof term. For sequent proofs, however, the occurrence of a fl-redex corresponds to a use of the cut rule in the proof--the lambda terms for cut-free proofs are always in fl-NF. Unfortunately, limitations of space prevent due discussion of this topic here.</Paragraph>
    <Paragraph position="4"> liThe terms main branch, major premise, etc have been borrowed from Prawitz, and are defined analogously for his system. 12Note that a natural deduction formulation of the Lambek Calculus differs from standard natural deduction systems in that the linear order of assumptions within a proof is important, (roughly) corresponding to the linear order of words combined. See, for example, the 'ordered' natural deduction formulations outlined in Hepple (1990) and Morrill (1990).</Paragraph>
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