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<Paper uid="P92-1045">
  <Title>INFORMATION STATES AS FIRST CLASS CITIZENS</Title>
  <Section position="4" start_page="303" end_page="303" type="metho">
    <SectionTitle>
AGENTS AND TEXTS
</SectionTitle>
    <Paragraph position="0"> Consider an agent processing the texts tl,..., tn-By processing we mean that the agent accepts the information conveyed by the texts. The texts are assumed to be declarative (purely informative) and unambiguous (uniquely informative).</Paragraph>
    <Paragraph position="1"> The texts are processed one by one (dynamically) -- not considered as a whole (statically). The dynamic interpretation of texts seems more realistic than the static interpretation.</Paragraph>
    <Paragraph position="2"> By a text we consider (complete) discourses -- although as examples we use only single (complete) sentences. We take the completeness to mean that the order of the texts is irrelevant. In general texts have expressions as parts whose order is important -- the completeness requirement only means that the (top level) texts are complete units.</Paragraph>
  </Section>
  <Section position="5" start_page="303" end_page="304" type="metho">
    <SectionTitle>
INFORMATION STATES
</SectionTitle>
    <Paragraph position="0"> We first consider an abstract notion of an information state (often called a knowledge state or a belief state). The initial information state I0 is assumed known (or assumed irrelevant). Changes are of the information states of the agent as follows: null</Paragraph>
    <Paragraph position="2"> where r/ is the change in the information state when the text t/is processed.</Paragraph>
    <Paragraph position="3"> An obvious approach is to identify information states with the set of texts already processed -hence nothing lost. Some improvements are possible (normalisation and the like). Since the texts are concrete objects they are easy to treat computationally. We call this approach the syntactical approach.</Paragraph>
    <Paragraph position="4"> An orthogonal approach (the semantical approach) identifies information states with sets of possibilities. This is the approach followed here.  Note that a possibility need not be a so-called &amp;quot;possible world&amp;quot; -- partiality and similar notions can be introduced, see Muskens (1989).</Paragraph>
    <Paragraph position="5"> A combination of the two approaches might be the optimal solution. Many of these aspects are discussed in Konolige (1986).</Paragraph>
    <Paragraph position="6"> Observe that the universal and empty sets are understood as opposites: the empty set of possibility and the universal set of texts represent the (absolute) inconsistent information state; and the universal set of possibility and the empty set of texts represent the (absolute) initial information state. Other notions of consistency and initiality can be defined.</Paragraph>
    <Paragraph position="7"> A partial order on information states (&amp;quot;getting better informed&amp;quot;) is easy obtained. For the syntactical approach this is trivial -- more texts make one better informed. For the semantical approach one could introduce previously eliminated possibilities in the information state, but we assume eliminative information state changes: r(I) C I for all I (this does not necessarily hold for non-monotonic logics / belief revision / anaphora(?) -- see Groenendijk and Stokhof (1991) for further details).</Paragraph>
    <Paragraph position="8"> Given the texts tl,...,t~ the agent is asked whether a text t can be inferred; i.e. whether processing t after processing tl,...,t~ would change the information state or not: Here r is the identity function.</Paragraph>
  </Section>
  <Section position="6" start_page="304" end_page="304" type="metho">
    <SectionTitle>
ELEMENTARY LOGIC
</SectionTitle>
    <Paragraph position="0"> When elementary logic is used as logical representation language for texts, information states are identified with sets of models.</Paragraph>
    <Paragraph position="1"> Let the formulas C/1,..., On, C/ be the translations of the texts tl,...,tn,t. The information state when tl .... ,tk has been processed is the set of all models in which C/1,..., C/n are all true.</Paragraph>
    <Paragraph position="2"> Q, * ..,tn entails t if the model set corresponding to the processing of Q,..., t,, does not change when t is processed. I.e. alternatively, consider a particular model M -- if C/1,-.., &amp;n are all true in M then C/ must be true in M as well (this is the usual formulation of entailment).</Paragraph>
    <Paragraph position="3"> Hence, although any proof theory for elementary logic matches the notion of entailment for &amp;quot;toy&amp;quot; example texts, the notion of information states is purely a notion of the model theory (hence in the meta-language; not available from the object language). This is problematic when texts have other texts as parts, like the embedded sentence in propositional attitudes, since a direct formalisation in elementary logic is ruled out.</Paragraph>
  </Section>
  <Section position="7" start_page="304" end_page="304" type="metho">
    <SectionTitle>
TRADITIONAL APPROACH
</SectionTitle>
    <Paragraph position="0"> When traditional intensional logics (e.g. modal logics) are used as logical representation languages for texts, information states are identified with sets of possible worlds relative to a model M = (W,...), where W is the considered set of possible worlds.</Paragraph>
    <Paragraph position="1"> The information state when tl,...,tk has been processed is, relative to a model, the set of possible worlds in which C/1,.--, ek are all true. The truth definition for a formula C/ allows for modal operators, say g), such that if C/ is (3C/ then is true in the possible worlds We C_ W if C/ is true in the possible worlds We _C W, where We -fv(WC/) for some function fC/~ : :P(W) --* :P(W) (hence U = (W, fv,...)).</Paragraph>
    <Paragraph position="2"> For the usual modal operator \[\] the function f:: reduces to a relation R:~ : W x W such that: WC/ -- fo(W,) - U {wC/ I Ro(w~,, wC/)} w~EWeb By introducing more modal operators the information states can be manipulated further (a small set of &amp;quot;permutational&amp;quot; and &amp;quot;quantificational&amp;quot; modal operators would suffice -- compare combinatory logic and variable-free formulations of predicate logic). However, the information states as well as the possible worlds are never directly accessible from the object language.</Paragraph>
    <Paragraph position="3"> Another complication is that the fv function cannot be specified in the object language directly (although equivalent object language formulas can often be found -- of. the correspondence theory for modal logic).</Paragraph>
    <Paragraph position="4"> Perhaps the most annoying complication is the possible interference with (extensional) notions like (standard) identity, where Leibniz's Law fails (for non-modally closed formulas) -- see Muskens (1989) for examples. If variables are present the inference rule of V-Introduction fails in a similar way.</Paragraph>
  </Section>
  <Section position="8" start_page="304" end_page="305" type="metho">
    <SectionTitle>
SIMPLE TYPE THEORY
</SectionTitle>
    <Paragraph position="0"> The above-mentioned complications becomes even more evident if elementary logic is replaced by a simple type theory while keeping the modal operators (cf. Montague's Intensional Logic). The ~calculus in the simple type theory allows for an elegant compositionality methodology (category to type correspondence over the two algebras). Often the higher-order logic (quantificational power) facilities of the simple type theory are not necessary -- or so-called general models are sufficient.</Paragraph>
    <Paragraph position="1"> The complication regarding variables mentioned above manifests itself in the way that /3reduction does not hold for the A-calculus (again,  see Muskens (1989) and references herein). Even more damaging: The (simply typed!) A-calculus is not Church-Rosser (due to the limited a-renaming capabilities of the modal operators).</Paragraph>
    <Paragraph position="2"> What seems needed is a logical representation language in which the information states are explicit manipulable, like the individuals in elementary logic. This point of view is forcefully defended by Cresswell (1990), where the possibilities of the information states are optimised using the well-known technique of indexing. Hence we obtain an ontology of entities and indices.</Paragraph>
    <Paragraph position="3"> In recent papers we have presented and discussed a categorial grammar formalism capable of (in a strict compositional way) parsing and translating natural language texts, see Villadsen (1991a,b,c). The resulting formulas are terms in a many-sorted simple type theory. An example of a translation (simplified): Mary believes that John lies. (5) )~i.believe(i, Mary, ()~j.lie(j, John))) (6) Adding partiality along the lines in Muskens (1989) is currently under investigation.</Paragraph>
  </Section>
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