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<?xml version="1.0" standalone="yes"?> <Paper uid="E95-1001"> <Title>On Reasoning with Ambiguities</Title> <Section position="4" start_page="2" end_page="3" type="metho"> <SectionTitle> 3 A short introduction to UDRSs </SectionTitle> <Paragraph position="0"> The base for unscoped representations proposed in \[7\] is the separation of information about the structure of a particular semantic form and of the content of the information bits the semantic form combines.</Paragraph> <Paragraph position="1"> In case the semantic form is given by a DRS its structure is given by the hierarchy of subDRSs, that is determined by ==v, -% V and (>. We will represent this hierarchy explicitly by the subordination relation <.</Paragraph> <Paragraph position="2"> The semantic content of a DRS consists of the set of its discourse referents and its conditions. To be more precise, we express the structural information by a language with one predicate _< that relates individual constants l, called labels. The constants are names for DRS's. < corresponds to the subordination relation between them, i.e. the set of labels with < is a upper semilattice with one-element (denoted by/7-).</Paragraph> <Paragraph position="3"> Let us consider the DRSs (11) and (12) representing the two readings of (10).</Paragraph> <Paragraph position="4"> (10) Everybody didn't pay attention.</Paragraph> <Paragraph position="5"> (11) I hum:n(x) \] =~ \] .~\[x pay attention\] I I (12) -, hum:n(x) I =*z I x pay attention \] \] The following representations make the distinction between structure and content more explicit. The subordination relation <_ is read from bottom to top. (13) 1 hum:n(x) I=C/~J Ix pay attention\] Ix pay attention 1 Having achieved this separation we are able to represent the structure that is common to both, (11) and (12), by (14).</Paragraph> <Paragraph position="7"> (14) is already the UDRS that represents (10) with scope relationships left unresolved. We call the nodes of such graphs UDRS-components. Each UDRScomponent consists of a labelled DRS and two functions scope and res, which map labels of UDRS-components to the labels of their scope and restrictor, respectively. DRS-conditions are of the form (Q, l~1, l~2), with quantifier Q, restrictor//1 and scope li2, of the form lil~li2, or of the form li:-~lil. A UDRS is a set of UDRS-components together with a partial order ORD of its labels.</Paragraph> <Paragraph position="8"> If we make (some) labels explicit we may represent (14) as in (15).</Paragraph> <Paragraph position="9"> If ORD in (15) is given as {12 <_ scope(ll),13 <_ scope(12)} then (15) is equivalent to (11), and in case ORD is {11 _< scope(12), 13 <_ scope(ll)} we get a description of (12). If ORD is {13 _< scope(ll), 13 <_ scope(12)} then (15) represents (14), because it only contains the information common to both, (11) and (12).</Paragraph> <Paragraph position="10"> In any case ORD lists only the subordination relations that are neither implicitly contained in the partial order nor determined by complex UDRSconditions. This means that (15) implicitly contains the information that, e.g., res(/2) < lT, and also that res(/2) ~ 12, res(ll) ~_ lT and scope(ll) ~ lT.</Paragraph> <Paragraph position="11"> In this paper we consider the fragment of UDRSs without disjunction. For reason of space we cannot consider problems that arise when indefinites occurring in subordinate clauses are interpreted specifically. 2 We will, therefore assume that indefinites behave like generalized quantifers in that their scope is clause bounded too, i.e. require l<_l' for all i in clause (ii.c) of the following definition.</Paragraph> <Paragraph position="12"> Definition 1: (i) (I:<UK,C K U C~>,res(1), scope(l),ORDt) is a UDRS-component, if (UK, CSK) is a DRS containing standard DRS-conditions only, and C~: is one of the following sets of labelled DRS-conditions, where//1 and/(2 are standard DRSs, Qx is a generalized quantification over x, and l' is the upper bound of a (subordinate) UDRS-clause (l':(7o,...,Tn),ORD~) (defined below).</Paragraph> <Paragraph position="13"> (a) {}, or {sub(l')} (b) {l 1 ::~/2, ll :K1,/2:1(2}, or</Paragraph> <Paragraph position="15"> distinguished condition of K, referred to by l:7.</Paragraph> <Paragraph position="16"> res and scope are functions on the set of labels, and ORDt is a partial order of labels, res(l), scope(l), and ORDt are subject to the following restrictions: ~These problems axe discussed extensively in \[7\] and the solution given there can be taken over to the rules presented here.</Paragraph> <Paragraph position="17"> 3Whenever convenient we will simply use implicative conditions of the form ll =:~ /2, to represent universally quantified NPs (instead of their generalized quantifier representation (every, 11, /2) ).</Paragraph> <Paragraph position="19"> 11~12 C ORDt.</Paragraph> <Paragraph position="20"> (5') Otherwise res(1) -- scope(l) = l (b) If k:sub(l~)E C~, then l'<k E ORDz and ORD~, c ORD~.</Paragraph> <Paragraph position="21"> (ii) A UDRS-clause is a pair (l:(~0, ...,'Yn), ORDt), where 7~ -~ (li:Ki,res(li),scope(li),ORDl,), 0 <_ i _< n, are UDRS components, and ORDl contains all of the conditions in (a) to (c) and an arbitrary subset oif those in (d) and (e).</Paragraph> <Paragraph position="22"> (a) ORDI, C ORDI, for all i, 0 < i < n (b) IQ<_scope(li) E ORDt for all i, 1 < i < n (c) li<<_l e ORDI for all i, 1 < i < n.</Paragraph> <Paragraph position="23"> (d) l~<_scope(lj) E ORDt, for some i,j 1 <_ i,j <_ n such that ORD is a partial order.</Paragraph> <Paragraph position="24"> For each i, 1 < i < n, li is called a node. I is called upper bound and/0 lower bound of the UDRS-clause. Lower bounds neither have distinguished conditions nor is there an/I such that l ~<l.</Paragraph> <Paragraph position="25"> (iii) A UDRS-database is a set of UDRSs ((/iT:F, ORDl~))i. A UDRS-goal is a UDRS.</Paragraph> <Paragraph position="26"> For the fragment of this paper UDRS-components that contain distinguished conditions do not contain anything else, i.e. they consist of labelled DRSs K for which UK = C~ = {) if C~: ~ {). We assume that semantic values of verbs are associated with lower bounds of UDRS-clauses and NP-meanings with their other components. Then the definition of UDRSs ensures that 5 (i) the verb is in the scope of each of its arguments, (clause (ii.b)), (ii) the scope of proper quantifiers is clause bounded, (clause (ii.c)) For relative clauses the upper bound label l ~ is subordinated to the label I of its head noun (i.e. the restrictor of the NP containing the relative) by l'<l (see (ii)). In the case of conditionals the upper bound label of subordinate clauses is set equal to the label of the antecedent/consequent of the implicative condition. The ordering of the set of labels of a UDRS builds an upper-semilattice with one-element IT. We assume that databases are constructed out of sequences $1, ..., S~ of sentences. Having a unique one-element /t r associated with each UDRS representing a sentence Si is to prevent any quantifier of Si to have scope over (parts of) any other sentence. ons see \[2\], this volume.</Paragraph> </Section> <Section position="5" start_page="3" end_page="5" type="metho"> <SectionTitle> 4 Rules of Inference </SectionTitle> <Paragraph position="0"> The four inference rules needed for the fragment without generalized quantifiers 6 and disjunction are non-empty universe (NeU), detachment (DET), ambiguity introduction (AI), and ambiguity elimination (DIFF). NeU allows to add any finite collection of discourse referents to a DRS universe. It reflects the assumption that there is of necessity one thing, i.e. that we consider only models with non-empty universes. DET is a generalization of modus ponens.</Paragraph> <Paragraph position="1"> It allows to add (a variant of) the consequent of an implication (or the scope of a universally quantified condition) to the DRS in which the condition occurs if the antecedent (restrictor) can be mapped to this DRS. AI allows one to add an ambiguous representation to the data, if the data already contains all of its disambiguations. And an application of DIFF reduces the set of readings of an underspecified representation in the presence of negations of some of its readings. The formulations of NeU, DET and DIFF needed for the consequence relation (8) defined in Section 2 of this paper are just refinements of the formulations needed for the consequence relation (1). As the latter case isextensively discussed in \[7\] and a precise and complete formulation of the rules is also given there we will restrict ourselves to the refinements needed to adapt these rules to the new consequence relation.</Paragraph> <Paragraph position="2"> As there is nothing more to mention about NeU we start with DET. We first present a formulation of DET for DRSs. It is an extended formulation of standard DET as it allows for applications not only at the top level of a DRS but at levels of any depth.</Paragraph> <Paragraph position="3"> Correctness of this extension is shown in \[4\].</Paragraph> <Paragraph position="4"> DET Suppose a DRS K contains a condition of the form K1 ::~ K2 such that K1 may be embedded into K by a function f, where K is the merge of all the DRSs to which K is subordinate. Then we may add K~ to K, where K~ results from K2 by replacing all occurrences of discourse referents of UK2 by new ones and the discourse referents x declared in UK1 by f(x).</Paragraph> <Paragraph position="5"> We will generalize DET to UDRSs such that the structure that results from an application of DET to a UDRS is again a UDRS, i.e. directly represents some natural language sentence. We, therefore, incorporate the task of what is usually done by a rule of thinning into the formulation of DET itself and also into the following definition of embedding. We define an embedding f of a UDRS into a UDRS to be a function that maps labels to labels and discourse referents to discourse referents while preserving all conditions in which they occur. We assume that f is one-to-one when f is restricted to the set of discour6We will use implicative conditions of the form (=}, 11, 12), to represent universally quantified NPs (instead of their generalized quantifier representation (every, Zl, 12)).</Paragraph> <Paragraph position="6"> se referents occurring in proper sub-universes. Only discourse referents occurring in the universe associated with 1T may be identified by f. We do not assume that the restriction of f to the set of labels is one-to-one also. But f must preserve -~, :=> and V, i.e. respect the following restrictions.</Paragraph> <Paragraph position="7"> (i) if l:~(ll,12) occurs in K', then f(/)::=~(f(ll),f(12)), (ii) if l:-~ll occurs in K', then f(/):-~f(ll).</Paragraph> <Paragraph position="8"> For the formulation of the deduction rules it is convenient to introduce the following abbreviation. Let \]C be a UDRS and l some of its labels. Then \]Ct is the sub-UDRS of )~ dominated by l, i.e. Kz contains all conditions l':~ such that l'<_l and its ordering relation is the restriction of \]C's ordering relation. Suppose 7 = lo:ll==>12 is the distinguished condition of a UDRS component l:K occurring in a UDRS clause \]Ci of a UDRS K:. And suppose there is an embedding f of \]G1 into a set of conditions ?:5 of \]C such that l <: ?. Then the result of an application of DET to 7 is a clause \]~ that is obtained from \]Cl by (i) eliminating/C h from K:l (ii) replacing all occurrences of discourse referents in the remaining structure by new ones and the discourse referents x declared in the universe of/i, by f(x); (iii) substituting l' for l, /1, and /2 in ORDt; and (iv) replacing all other labels of K:l by new ones.</Paragraph> <Paragraph position="9"> But note that applications of DET are restricted to NPs that occur 'in the context of' implicative conditions, or monotone increasing quantifiers, as shown in (16). Suppose we know that John is a politician, then: (16)Few problems preoccupy every politician.</Paragraph> <Paragraph position="10"> t/Few problems preoccupy John.</Paragraph> <Paragraph position="11"> Every politician didn't sleep.</Paragraph> <Paragraph position="12"> ~/John didn't sleep.</Paragraph> <Paragraph position="13"> At least one problem preoccupies every pol.</Paragraph> <Paragraph position="14"> }- At least one problem preoccupies John.</Paragraph> <Paragraph position="15"> (16) shows that DET may only be applied to a con- null l':K I such that the distinguished condition l':7' of K' is either a monotone decreasing quantifier or a negation, and such that for some disambiguation of the clause in which 7 occurs we get l <_ scope(l').</Paragraph> <Paragraph position="16"> As the negation of a monotone decreasing quantifier is monotone increasing and two negations neutralize each other the easiest way to implement the restriction is to assign polarities to UDRS components and restrict applications of DET to components with positive polarity as follows.</Paragraph> <Paragraph position="17"> Suppose l:K occurs in a UDRS clause (/0:(7o,...,Tn),ORDzo), where l0 has positive polarity, written lo +. Then l has positive (negative) polarity if for each disambiguation the cardinality of the set of monotone decreasing components (i.e. monotone decreasing quantifiers or negations) that takes wide scope over l is even (odd). Negative polarity of l0 is induces the complementary distribution of polarity marking for l. If l is the label of a complex condition, then the polarity of l determines the polarity of the arguments of this condition according to the following patterns: l+:l-~, l-:~12-, /+ :-~, and l-:-~, l~ has positive polarity for every i. The polarity of the upper bound label of a UDRS-clause is inherited from the polarity of the label the UDRS-clause is attached to. Verbs, i.e. lower bounds of UDRS-clauses, always have definite polarities if the upper bound label of the same clause has.</Paragraph> <Paragraph position="18"> Two remarks are in order before we come to the formulation of DET. First, the polarity distribution can be done without explicitly calculating all disambiguations. The label l of a component l:K is positive (negative) in the clause in which occurs, if the set of components on the path to the upper bound label l + of this clause contains an even (odd) number of polarity changing elements, and all other components of the clause (i.e. those occurring on other paths) do not change polarity. Second, the fragment of UDRSs we are considering in this paper does not contain a treatment of n-ary quantifiers. Especially we do not deal with resumptive quantifiers, like <no boy, no girl> in No boy likes no girl. If we do not consider the fact that this sentence may be read as No boy likes any girl the polarity marking defined above will mark the label of the verb as positive. But if we take this reading into account, i.e. allow to construe the two quantified NPs as constituents of the resumptive quantifier, then one negation is cancelled and the label of the verb cannot get a definite value. 7 To represent DET schematically we write (IT:a(F:7),ORD) to indicate that i~:K is a component of the UDRS K:IT with polarity 7r and distinguished condition 7.</Paragraph> <Paragraph position="19"> A (lT:a(~:~ ~ ~),ORD) f:/Q,, ~-+ A exists The scheme for DET allows the arguments of the implicative condition to which it is applied still to be ambiguous. The discussion of example (6) in Section 2 focussed on the ambiguity of its antecedent only.</Paragraph> <Paragraph position="20"> (We ignored the ambiguity of the consequent there.) To discuss the case of ambiguous consequents we consider the the following argument.</Paragraph> <Paragraph position="21"> (17)If the chairman talks, everybody doesn't sleep.</Paragraph> <Paragraph position="22"> The chairman talks. ~- Everybody doesn't sleep.</Paragraph> <Paragraph position="23"> There is a crucial difference between (17) and (6): The truth of the conclusion in (17) depends on the fact that it is derived from the conditional. It, therefore, must be treated as correlated with the consequent of the conditional under any disambiguation. No non-correlated disambiguations are allowed. To ensure this we must have some means to represent 7A general treatment of n-ary quantification within the theory of UDRSs has still to be worked out. In \[6\] it is shown how cumulative quantification may be treated using identification of labels.</Paragraph> <Paragraph position="24"> the 'history' of the clauses that are added to a set of data. As (8) suggests this could be done by coindexing K:l,1 and/Cf(ln) in the representation of (17). In contrast to the obligatory coindexing in the case of (17) the consequence relation in (8) does allow for non-correlated interpretations in the case of (2). Such interpretations naturally occur if, e.g., the conditional and the minor premiss were introduced by very distinct parts of a text from which the database had been constructed. In such cases the interpreter may assume that the contexts in which the two sentences occurred are independent of each other.</Paragraph> <Paragraph position="25"> He, therefore, leaves leeway for the possibility that (later on) each context could be provided with more information in such a way that those interpretations trigger different disambiguations of the two occurrences. In such cases &quot;crossed interpretations&quot; must be allowed, and any application of DET must be refused by contraindexing - except the crossed interpretations can be shown to be equivalent. For the sake of readability we present the rule only for the propositional case.</Paragraph> <Paragraph position="26"> A oq =~ fl.i o~k i = k V (i # k A A F- c~i 4:~ c~k) at But the interpreter could also adopt the strategy to accept the argument also in case of non-correlated interpretations without checking the validity of aiC/* ak. In this case he will conclude that fit holds under the proviso that he might revise this inference if there will be additional information that forces him to disambiguate in a non-correlated way. If then ai 4:~ ak does not hold he must be able to give up the conclusion nit and every other argument that was based on it. To accomodate this strategy we need more than just coindexing. We need means to represent the structure of whole proofs. As we have labels available in our language we may do this by adopting the techniques of labelled deductive systems (\[3\]). For reasons of space we will not go into this in further detail.</Paragraph> <Paragraph position="27"> The next inference rule, AI, allows one to introduce ambiguities. It contrasts with the standard rule of disjunction introduction in that it allows for the introduction of a UDRS a that is underspecified with respect to the two readings al and a2 only if both, al and as, are contained in the data. This shows once more that ambiguities are not treated as disjunctions. null Ambiguitiy Introduction Let or1 and a2 be two UDRSs of A that differ only w.r.t, their ORDs.</Paragraph> <Paragraph position="28"> Then we may add a UDRS a3 to A that is like al but has the intersection of ORD and ORD ~ as ordering of its labels. The index of aa is new to A.</Paragraph> <Paragraph position="29"> We give an example to show how AI and DET interact in the case of non-correlated readings: Suppose the data A consists of a~, 0&quot;2 and a3 ~ % We want to derive 3'. We apply AI to al and 62 and add au to A. As the index of a3 is new we must check whether al ~=> a2 can be derived from A. Because A contains both of them the proof succeeds.</Paragraph> <Paragraph position="30"> The last rule of inference, DIFF, eliminates ambiguities on the basis of structural differences in the ordering relations. Suppose ~1 and c~2 are a under-specified representations with three scope bearing components 11, 12, and 13. Assume further that al has readings that correspond to the following orders of these components: (h, /2, 11), (h, h, ll), and (h, ll, /3), whereas a2 is ambiguous between (/2, /3, /1) and (/2, ll, /3). Suppose now that the data contains al and the negation of a2. Then this set of data is equivalentto the reading given by (/3, /2, 11). To see that this holds the structural difference between the structures ORD,~ and ORD~ has to be calculated. The structural difference between two structures ORD~ and ORDa2 is the partial order that satisfies ORD~ but not ORD~2, if there is any; and it is falsity if there is no such order. Thus the notion of structural difference generalizes the traditional notion of inconsistency. Again a precise formulation of DIFF is given in \[7\].</Paragraph> </Section> <Section position="6" start_page="5" end_page="6" type="metho"> <SectionTitle> 5 Rules of Proof </SectionTitle> <Paragraph position="0"> Rules of proof are deduction rules that allow us to reduce the complexity of the goal by accomplishing /~ subproof. We will consider COND(itionalization) and R(eductio)A(d)A(bsurdum) and show that they may not be applied in the case of ambiguous goals (i.e. goals in which no operator has widest scope).</Paragraph> <Paragraph position="1"> Suppose we want to derive everybody didn't snore from everybody didn't sleep and the fact that snoring implies sleeping. I.e. we want to carry out the proof in (18), where ORD = {13 <</Paragraph> <Paragraph position="3"> Let us try to apply rules of proof to reduce the complexity of the goal. We use the extensions of COND and RAA given in \[7\]. There use is quite simple.</Paragraph> <Paragraph position="4"> An application of COND to the goal in (18) results in adding <IT:\] a I, { }) to the data and leaves (/tc:(lT:q q ,ls:~ }, ORD&quot; ) to be shown, where ORIY' results from ORIY by replacing 16 and scope(16) with l~-. RAA is now applicable to the new goal in a standard way. It should be clear, however, that the order of application we have cho- null sen, i.e. COND before RAA, results in having given the universal quantifier wide scope over the negation. This means that after having applied COND we are not in the process of proving the original ambiguous goal any more. What we are going to prove instead is that reading of the goal with universal quantifier having wide scope over the negation. Beginning with RAA instead of COND assigns the negation wide scope over the quantifier, as we would add (l~r:(l~:\[~ ~ ~, Is:~),ORD&quot;)to the data in order to derive a contradiction, s Here ORlY' results from ORU by replacing 17 and scope(17) with l~-. If we tried to keep the reduction-of-the-goal strategy we would have to perform the disambiguation steps to formulas in the data that the order of application on COND and RAA triggers. And in addition we would have to check all possible orders, not only one. Hence we would perform exactly the same set of proofs that would be needed if we represented ambiguous sentences by sets of formulas. Nothing would have been gained with respect to any traditional approach. null We thus conclude that applications of COND and RAA are only possible if either =v or -, has wide scope in the goal. In this case standard formulations of COND and RAA may be applied even if the goal is ambiguous at some lower level of structure.</Paragraph> <Paragraph position="5"> In case the underspecification occurs with respect to the relative scope of immediate daughters of 1T, however, we must find some other means to relate non-identical UDRSs in goal and data. What we need are rules for UDRSs that generalize the success case for atoms within ordinary deduction systems.</Paragraph> </Section> <Section position="7" start_page="6" end_page="6" type="metho"> <SectionTitle> 6 Deduction rules for top-level </SectionTitle> <Paragraph position="0"> ambiguities The inference in (18) can be realised very easily if we allow components of UDRSs that are marked negative to be replaced by components with a smaller denotation. Likewise components of UDRSs that are marked positive may be replaced by components with a larger denotation. If the component to be replaced is the restrictor of a generalized quantifier, then in addition to the polarity marking the soundness of such substitutions depends on the persistence property of the quantifier. In the framework of UDRSs persistence of quantifiers has to be defined relative to the context in which they occur. Let NPi be a persistent (anti-persistent) NP. Then NPi is called persistent (anti-persistent) in clause S, if sIf we would treat ambiguous clauses as the disjunctions of their meanings, i.e. take the consequence relation in (1), then this disambiguation could be compensated for by applying RESTART (see \[7\] for details). But relative to the consequence relation under (8) RESTART is not sound! this property is preserved under each disambiguation of S. So everybody is anti-persistent in (19e), but not in (19a), because the wide scope reading for the negation blocks the inference in (19b). It is not persistent in (19c) nor in (19d).</Paragraph> <Paragraph position="1"> (19)a. Everybody didn't come.</Paragraph> <Paragraph position="2"> b. Everybody didn't come.</Paragraph> <Paragraph position="3"> Every woman didn't come.</Paragraph> <Paragraph position="4"> c. More than half the problems were solved by everybody.</Paragraph> <Paragraph position="5"> d. It is not true that everybody didn't come.</Paragraph> <Paragraph position="6"> e. Some problem was solved by everybody.</Paragraph> <Paragraph position="7"> The main rule of inference for UDRSs is the following R(eplacement)R(ule).</Paragraph> <Paragraph position="8"> RR Whenever some UDRS K:~- occurs in a UDRS-database A and A I-K:~- >>/C~ holds, then K:g may be added to A.</Paragraph> <Paragraph position="9"> RR is based on the following substitution rule. The >>-rules are given below.</Paragraph> <Paragraph position="10"> SUBST Let hK be a DRS component occurring in some UDRS )U, A a UDRS-database. Let K:' be the UDRS that results from K: by substituting K' for K.</Paragraph> <Paragraph position="11"> Then A KK: >>/C', if (i) or (ii) holds.</Paragraph> <Paragraph position="12"> (i) l has positive polarity and A K K >> K'.</Paragraph> <Paragraph position="13"> (ii) l has negative polarity and A K K' >> K.</Paragraph> <Paragraph position="14"> Schematically we represent the rule (for the case of positive polarity) as follows.</Paragraph> <Paragraph position="15"> 3- +' l+:K if A K l+:K >> l+:K I A, IC~- +--, l+:K ' For UDRS-components we have the following rule. >> DRS: A K K>>K' if there is a function f: UK--r UK, such that for all 7' E CK, there is a &quot;\[ E CK with A ~- f(7)>>7'. 9 Complex conditions are dealt with by the following set of rules. Except for persistence properties they are still independent of the meaning of any particular generalized quantifier. The success of the rules can be achieved in two ways. Either by recursively applying the >>-rules. Or, by proving the implicative condition which will guarantee soundness of SUBST. The following rules involve lexical meaning of words. We give some examples of determiner rules to indicate how we may deal with the logic of quantifiers in this rule set. Rules for nouns and verbs refer to a further inference relation, t -n. This relation takes the meaning postulates into account that a particular lexical theory associates with particular word meanings.</Paragraph> </Section> class="xml-element"></Paper>