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<Paper uid="E89-1028">
  <Title>COMPUTATIONAL SEMANTICS OF MASS TERMS</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
REPRESENTING QUANTIFIED
MASS NOUNS
</SectionTitle>
    <Paragraph position="0"> We will make use of a very simple formal language for representing sentences containing quantified mass nouns, called LM (Lcnning, 1987). We refer to the original paper for motivation of the particular chosen format and more examples and repeat only the key points here.</Paragraph>
    <Paragraph position="1">  1. A particular LM language consists of a (nonempty) set of basic terms, say: water, gold, - 205 blue, hot, boil, disappear .... and a (possibly empty) set of non-logical determiners, say&amp;quot; much, little, less..than two kilos_of..</Paragraph>
    <Paragraph position="2"> 2. Common to all LM languages are the unary operator: -, the binary operator:., the logical determiners: : all, some, and the propositional connectives: : --1, ^, v, --~.</Paragraph>
    <Paragraph position="3"> 3. A term is either basic or of one of the two forms (t.s) and (--t) if t and s are terms.</Paragraph>
    <Paragraph position="4"> 4. An atomic formula has the form D(t)(s) where D is a determiner and t and s are terms.</Paragraph>
    <Paragraph position="5"> 5. More complex formulas are built by the propositional connectives in the usual way.</Paragraph>
    <Paragraph position="6"> A model for the particular language is a pair consisting of a Boolean algebra A = &lt;A, +, *, -, 0, 1&gt; and an interpretation function \[ \], such that 1. \[t\] is a member of A for all basic terms t, 2. \[/9\] is a set of pairs of members of A for all determiners D.</Paragraph>
    <Paragraph position="7"> The interpretation of more complex expressions is then defined as an extension of \[ \]: 1. \[-t\]= -\[t\], the Boolean complement of \[t\], \[t.s\]=\[t\]*\[s\], the Boolean product (or meet) of It\] and Is\].</Paragraph>
    <Paragraph position="8"> 2. \[D(t)(s)\] is true provided (\[t\],\[s\])~ \[D\], in particular \[All(t)(s)\] is true provided \[t\]_&lt;\[s\], and \[Some(t)(s)\] is true provided \[t\]*\[s\] ~ 0. 3. The propositional part is classical.</Paragraph>
    <Paragraph position="9">  To get an intuition of how the semantics work one can think of \[water\] as the totality of water in the world or in the more limited situation one considers, of \[blue\] as the totality of stuff that is blue and of \[disappear\] as the totality of stuff that disappeared. However, one shall not take this picture too literally since the model works as well for abstract as for concrete mass nouns.</Paragraph>
    <Paragraph position="10"> In the formalism, a sentence like (la) is represented as (lb) and (2a) is represented as (2b) if the  negation is read with narrow scope and as (2c) if the negation is read with wide scope.</Paragraph>
    <Paragraph position="11"> (1) (a) All hot water is water.</Paragraph>
    <Paragraph position="12"> (b) All(hot, water)(water) (2) (a) Much of the water that disappeared was not polluted.</Paragraph>
    <Paragraph position="14"> valid English inferences that become valid if a mass term like water is interpreted as quantities of water and all water is read as all quantities of water are also valid under the LM interpretation.</Paragraph>
    <Paragraph position="15"> In addition, this approach can explain several phenomena that are not easily explained on other approaches. Roeper (1983) pointed out that paraphrasing water as quantities of water was problematic when propositional connectives were considered. If some water disappeared and some did not disappear, there will be many quantities that partly disappeared and partly did not disappear. If disappear denotes the set of all quantities that wholly disappeared and did not disappear denotes the complement set of this set, then all quantities that partly disappeared will be members of the denotation of did not disappear. The sum of quantities that are members of the denotation of d/d not disappear will equal all the water there is. Roeper solved this problem by letting the quantities be members of a Boolean algebra and used a non-standard interpretation of the negation. In LM it naturally comes out by the Boolean complement as in (2b) and water that did not disappear is represented by (water.(--disappear)).</Paragraph>
    <Paragraph position="16"> A main feature of the current proposal is that it introduces non-logical quantitiers co-occurring with mass nouns, like much, little, most .... in a straightforward way. A sentence like much water was blue does not say anything about the number of quantities of blue water, but says something about the totality of blue water, which is the way it is interpreted in LM.</Paragraph>
    <Paragraph position="17"> It might seem a little like cheating that the system only introduces interpretations of mass quantitiers with minimal scope with respect to other quanriflers, that it is not possible to interpret one quantitier with scope over another quantifier. In particular, since this is a main reason to why the logic in the sequel becomes simple. However, it is characteristic for mass quantifiers that they get narrow scope.</Paragraph>
    <Paragraph position="18">  (3) (a) A boy ate many pizzas.</Paragraph>
    <Paragraph position="19"> (b) A girl drank much water.</Paragraph>
    <Paragraph position="20">  While it might be possible to get a reading of (3a) which involves more than one boy, i.e., one boy for each pizza, it is not possible to get a reading of (3b) involving more than one girl.</Paragraph>
    <Paragraph position="21"> The only determiners that get a fixed, or logical interpretation in LM are : all and some. For the other determiners we can add various sorts of constraints on the interpretations that will make more of the inferences we make in English valid in the logic. For example, it has been claimed that all natural language determiners are conservative (or &amp;quot;live on&amp;quot; their fast argumen0, i.e: (a,b)C/ \[D\] if and only if (a,b*a)~ \[/9\] (Barwise and Cooper, 1981).</Paragraph>
    <Paragraph position="22"> Several determiners are monotone, either increasing or decreasing, in one or both arguments, e.g., much is monotone increasing in its - 206 second argument: if (a,b) ~ \[much\] and b &lt; c then (a,c) e \[much\], and less than_two_kilos_of is monotone decreasing in-its second argument.</Paragraph>
    <Paragraph position="23"> Whether an inference like Much water evaporated.</Paragraph>
    <Paragraph position="24"> All that evaporated disappeared.</Paragraph>
    <Paragraph position="25"> * &amp;quot; Hence much water disappeared, becomes valid in the logic, will depend on whether the denotation of much is constrained to be monotone increasing in its second argument or not.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
LOGICAL PROPERTIES
</SectionTitle>
    <Paragraph position="0"> We will repeat shortly several of the properties of the logic LM shown in (LCnning, 1987) as a background for the decision algorithm for validity.</Paragraph>
    <Paragraph position="1"> A Hilbert style axiomatization was given and it was shown that any set of LM sentences consistent in the logic has a model: the logic is complete and compact.</Paragraph>
    <Paragraph position="2"> It was implicitly shown, but not stated, that any model for LM must be a Boolean algebra: let a model be any set A with one unary operation \[-\], one binary operation \[-\], and a binary relation JAil\] then the model is a Boolean algebra with \[-\] the Boolean complement, \[.\] the Boolean product (meet) operation and JAil\] the ordering &lt; on A.</Paragraph>
    <Paragraph position="3"> It was also shown that the logic was complete and compact with respect to the smaller model class: the atomic Boolean algebras, i.e. any consistent set of sentences has a model which is an atomic algebra, and in fact a finite such.</Paragraph>
    <Paragraph position="4"> From this, it was shown that LM with no non-logical determiners, let us call it LA, is equivalent to a subset of monadic first-order logic, hence it is decidable. It was also shown that the full LM is decidable. The argument is based on the fact that the number of possible models for a sentence is finite and decidable. This number grows rapidly, however.</Paragraph>
    <Paragraph position="5"> Already a sentence in LA with n different basic terms have 2(2n) different models, so the argument does not establish a good procedure for checking validity in practice. We will establish procedures that are better (in most cases) in the next section.</Paragraph>
    <Paragraph position="6"> Several natural restrictions on determiners, like conservativity and monotonicity can be expressed completely in LM. It is not surprising that this can be done for a fixed LM language given the finiteness of its nature. The more important point is that the properties can be expressed in a uniform way independently of the atomic terms of the language, which in next section will give rise to uniform inference rules.</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
A DECISION PROCEDURE
</SectionTitle>
    <Paragraph position="0"> We shall establish a procedure for deciding the validity of LM sentences. The procedure is a combination of a normal form procedure and a tableau procedure (see e.g. Smullyan, 1968).</Paragraph>
    <Paragraph position="1"> We start with an LM formula 9 for which we want to decide whether it is valid, ~. 9. This is equivalent to deciding whether ~tp is satisfiable.</Paragraph>
    <Paragraph position="2"> One can think of the process as an attempt on building a model for ~9. If we succeed, we have shown that tp is not valid and we have found an explicit counterexample; if we fail, we have shown that tp is valid. We assume all propositional connectives in 9 to be basic: 9, ^, v.</Paragraph>
    <Paragraph position="3"> 1. First we introduce a new unary quantifier Null such that Null(t) is a formula when t is a term. The meaning is that \[Null(t)\] is true ff and only if \[t\]=0.</Paragraph>
    <Paragraph position="5"> This step is not necessary, but it gives a more convenient notation to work with and fewer instances to consider in the sequel. The substitutions correspond to the fact that Some and Every can be taken to be unary and the one can be defined from the other, as in the case with count nouns.</Paragraph>
    <Paragraph position="6">  2. Then transform $ to conjunctive normal form, that is, a conjunction of disjunctions of literals, where a literal is an atomic or negated atomic formula. null 3. Observe that (i) and (ii) are equivalent.</Paragraph>
    <Paragraph position="7"> (i) P ~l^V2^...^~n (ii) N~I, NV2...andNVn,  Hence we can proceed with each conjunct separately. 4. We shall now check P ~lVXlt2v...VVn, where each Vi has the form Null(t), --Wull(t), D(t)(s) or -d)(t)(s). Observe that the following two formulas are equivalent, where t+s is shorthand for</Paragraph>
    <Paragraph position="9"> (This corresponds to the equivalence between 3xtp v 3xV and 3x(9 v ~) in first order logic.) Hence contract all the literals of the form --Wull(t) to one.</Paragraph>
    <Paragraph position="10"> 5. We are then left with a formula of the form -.-aVull(t)vNull(sl)v... vNull(srOvaF l V... VVm, where each ~Fi has the form D(u)(v) or --4)(u)(v), for some non-logical D. First assume that there are no ~i's. Then observe that (i) and (ii) are equivalent.</Paragraph>
    <Paragraph position="12"> (If there are no si's proceed with h ~ Null(t).) This equivalence might need some argument. That (ii) entails (i) is propositional logic. For the other way around, there are two possibilites. Either h--31ull(t), which yields the equivalence. Otherwise, there exists a model A for the language of d~ where \[t\]=0 and for all other terms s in the language: Is\]=0 if and only if All(s)(t) is a valid formula. Let V be the formula ~Null(t) v Null(sl) v ... v Null(sn).</Paragraph>
    <Paragraph position="13"> Then ~ is valid if and only if ~ is true in A. To show this, it is sufficient to show that if there is a model B in which q/is not true then W is not true in A. If B is a model in which W is not true, then It\]=0 and each \[si\]~0 in B. Hence All(sO(t) cannot be valid and \[si\]*:O in A for each si. Since \[t\]=0 in A, ~ cannot be true in A. The same type of argu- null ment yields that --,Vull(t) v Null(sO is valid if and only if it is true in A. If we write A h 11 for ~1 is true in A, the following equivalence is propositional logic and yields the equivalence above.</Paragraph>
    <Paragraph position="14"> (i) A h ~ Null(t) v Null(sl) v...v Null(sn) (ii) A h --1 Null(t) v Null(sl) or... or A h --, Null(t) v Null(sn).</Paragraph>
    <Paragraph position="15"> 6. a. We shall describe two different ways for  checking h ---dVull(t) v Null(sO. The first one proceeds by a transformation to normal form and may be the easiest one to understand if one is not accustomed to tableau calculus. The second one which uses a tableau approach is more efficient. First observe that (i) and (ii) are equivalent.</Paragraph>
    <Paragraph position="17"> is a valid LM-formula. To see that (ii) entails (i) observe that if --dVull(t)vNull(si) is valid, it will in particular be true in the model A described in step 5, hence h All(sO(t).</Paragraph>
    <Paragraph position="18"> To check hNull ((-t).si) rewrite the term (-t)degsi in disjunctive normal form: allow the symbol + and write the term on the form Sl+...+Sm where each si has the form Uldeg....Uk and each uj is either an atomic term or on the form -v for an atomic term v.</Paragraph>
    <Paragraph position="19"> Then h Null(sl+...+Sm) if and only if h Null(sl) and ... and hNull(sm), and hNull(ul.....Uk) if and only if there is a v such that one uj equals v and another uj equals -v.</Paragraph>
    <Paragraph position="20"> b. The checking of h ~Null(t)vNull(si) will be faster using a tableau procedure instead of rewriting to normal form. Note that the following are equivalent: null</Paragraph>
    <Paragraph position="22"> There is a close connection between propositional logic and Boolean algebras. To each term in LM, t, there corresponds a formula Pt in pure propositional logic such that --~Vull(t) is valid in LM if and only if the corresponding formula Pt is a tautology: shift each basic LM term t with a corresponding propositional constant Pt, and exchange - with --i, * with ^, and + with v. In particular, the following are equivalent:</Paragraph>
    <Paragraph position="24"> h (Pt v(-~Psi)) (in propositional logic) (By the earlier mentioned connection between LM and first order logic this corresponds to the fact that a first order formula 3xq~ is valid if and only if C/p is a tautology whenever q~ is quantifier free.) Step (6a) above is equivalent to checking this latter formula for validity by transformation to a normal form. Instead it can be checked by a standard tableau procedure (see e.g. Smullyan 1968).</Paragraph>
    <Paragraph position="25"> We give a short description of the tableau approach to propositional logic. In order to verify a formula V, we try to build a model that falsifies it.</Paragraph>
    <Paragraph position="26"> To ease the description we assume that V is on negation normal form, that is, built up from literals by ^, v. The attempt to build a model is done by building a tree for V. We start with a root node depicted by V and use the following two rules: 1. For each node a depicted by a formula of the form y v TI attach to each leaf below a in the tree constructed so far one successor node b depicted by y and one successor node c to b depicted by rl.</Paragraph>
    <Paragraph position="27"> 2. For each node a depicted by a formula of the form T ^ rl attach to each leaf below a in the tree constructed so far two new leaf nodes one depicted by T and another one depicted by rl.</Paragraph>
    <Paragraph position="28"> The tree is complete when all formulas of the forms y v 11 and &amp;quot;/^ 11 are reduced according to the rules above. A branch in a tree is called closed if there is a formula T such that one node along the branch is depicted by T and another node along the branch is depicted by --,7. A branch in a complete tree for V which is not closed describes a valuation that falsifies V. Conversely, if all branches in a complete tree for V are closed, W is valid. We illustrate with an example:</Paragraph>
    <Paragraph position="30"> The sign # indicates that a branch is closed. We have not completed the rightmost branch since it is already closed. Since there is one open branch in the tree, the formula is not valid. The literals along the open branch: ~p, ~q, r shows that any valuation V such that V(p) = T, V(q) = T, V(r) = .L, falsifies V.</Paragraph>
    <Paragraph position="31"> The strategy in step (6a) above with transformation to normal form corresponds to construction of separate copies for each branch, hence duplicating parts of the tree, while the tableau procedure exploits the possibility of structure sharing.</Paragraph>
    <Paragraph position="32"> Returning to our main point, we can observe one additional gain by using the tableau approach. Our goal is to check whether t&amp;quot; --Wull(t)vNull(sl) or .... or P- ---~ull(t)vNull(sn), which is equivalent to check whether (tv(~sl)) or .... or (tv(~sn)) is a tautology.</Paragraph>
    <Paragraph position="33"> The part of the tableau tree that corresponds to t can be constructed (and if possible reduced by removing closed branches) once and for all, and then be reused with all the different si's.</Paragraph>
    <Paragraph position="34"> 7. We now return to step 5 and consider the case where one or more disjuncts have the form D(u)(v) or ---~(u)(v), for some non-logical D. Then the following are equivalent.</Paragraph>
    <Paragraph position="35"> (i) b --~Null(t)vNull(sl)v...vNull(sn)WlV...V~m (ii)~- -Wull(t)vNull(si) for some si, 1 &lt; i _&lt; n, or --~Vull(t)VVkWj for some k andj between 1 and m, where Vk lias the form D(a)(b) and Vj has the form ---D(u)(v) for the same determin6r D.</Paragraph>
    <Paragraph position="36"> That (ii) entails (i) is immediate. For the other way around, suppose that (ii) does not hold. We shall then construct a model which falsifies the original formula in (i). Let A be the model where only terms provably less than t denote 0 and where a pair (\[d\],\[e\]) is a member of \[/9\] if and only if --~D(d)(e) is one of the disjuncts Vi's. By the construction, A will falsify --,Null(t) and each disjunct of the form ---~(d)(e). As in step 5 above, A will falsify each Null(si). It remains to show that A falsifies each disjunct of the form D(a)(b). Let ~/be one such disjunct, let rlj, 1 &lt; j &lt; s, be all the disjuncts of the form --dg(d)(e) with the same determiner D as in ~/ and let ej be ---aVull(t)v'l, vTIj&amp;quot; Since (ii) does not hold, theie exists a model B\] where ej is false, for each e&amp;quot; Then there also exists a model A&amp;quot; which J&amp;quot; . .J equals A except possibly for the interpretaUon of D, and where D gets the same interpretation as in Bj.</Paragraph>
    <Paragraph position="37"> Hence ej is false in A'j. Since there, exists such an Aj for each ej, ~/cannot be true m A.</Paragraph>
    <Paragraph position="38"> 8. Whether b -WuU(t)vD(a)(b)v--43(u)(v) holds, depends on which restrictions are put on D. With no restrictions, any possible counterexample is one where \[t\]=0, (\[u\],\[v\])~ \[D\] while (\[a\],\[b\])~ \[D\] . The only reason we should not be able to construct such a model is that \[a\] =\[u\] and\[b\] =Iv\] whenever It\] --0. We can hence proceed to check</Paragraph>
    <Paragraph position="40"> according to the same procedures as in step 6 above.</Paragraph>
    <Paragraph position="41"> If we have the additional constraint that the determiner in question is conservative, the last rule is changed such that the last conjunct, which above stated that the symmetric difference between b and v was zero, now instead states that the symmetric difference between a.b and u.v is zero.</Paragraph>
    <Paragraph position="43"> Similarly, if we know that the determiner is upwards monotone in its second argument D(a)(b)v---D(u)(v) has to be true in any model where \[al=\[u\] and \[v\]__,\[b\], so the last conjunct will be Null((-b.v)) instead of Null((-b.v)+(b.-v)). If the determiner is restricted to be both conservative and monotone, the last conjunct shall be Null((-(a.b).u.v)). Similar modifications of the rule can be done for determiners with other forms of monotone behaviour.</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
GENERALIZED QUANTIFIERS
</SectionTitle>
    <Paragraph position="0"> One main feature of the decision procedure is that it incorporates generalized quantifiers (step 7 and 8).</Paragraph>
    <Paragraph position="1"> The rules for generalized quantifiers correspond to axioms one will use in an axiomatization of LM.</Paragraph>
    <Paragraph position="2"> For example, the rule for quantifiers with no additional constraints correspond to the extensionality schemata: For all terms a, b, u, v:</Paragraph>
    <Paragraph position="4"> One should remember that we do not try to develop a logic for the strong logical interpretation of determiners like most, but a logic for some minimal constraints that interpretations of the determiners should at least satisfy.</Paragraph>
    <Paragraph position="5"> - 209 Just like there is a meaning preserving translation from LA into first order logic, LM can be translated into first order logic extended with generalized quantiflers. A proof procedure for first order logic, like a tableau or a sequent calculus, can be extended with rules for generalized quantifiers similar to the rules introduced here. If Q is a binary quantifier with no additional constraints on its interpretation then the following are equivalent.</Paragraph>
    <Paragraph position="7"> So to show that (i) is valid one has to show that (ii) is valid. This can be incorporated into a tableau or sequent calculus for In'st order logic. If the first order logic is monadlc, as the logic we get after translating LM into first order logic is, one can use a similar procedure as the one described here. If the extended first order logic is not monadie, the procedure one gets when rules corresponding to the reduction from (i) to (ii) are included, becomes more complex.</Paragraph>
  </Section>
  <Section position="8" start_page="0" end_page="0" type="metho">
    <SectionTitle>
EFFICIENCY
</SectionTitle>
    <Paragraph position="0"> We chose to transform the formula being tested to normal form early in the procedure (step 2).</Paragraph>
    <Paragraph position="1"> Alternatively to the described algorithm one could think of using a tableau procedure all the way, and not first transform to conjunctive normal form in step 2. In general, transformation to normal form is slower than using a tableau procedure (el. step 6 above). The reason we made the transformation to normal form was that this was necessary to split the formula in step 5 and step 7. In the procedure one gets by translating LM into first order logic (with generalized quantifiers) and using a tableau procedure from the start, it is not possible to split the tree similarly. If we for simplicity considers a formula with no generalized quantifiers, the pure tableau calculus will not lead to a separate tree for each si together with t but to one big tree containing all the si's and roughly one copy of t for each si. This corresponds to the quantifier rules in a tableau calculus for first order logic: (i) for each formula of the form Vxqb introduce one new formula dd(a) where a is some new term, (ii) for each formula 3xw introduce one new formula V(a) for each term a introduced in the tree at a branch to which 3x~ belongs. The successful separation in the described algorithm here will also be possible in a proof procedure for monadic first order logic.</Paragraph>
    <Paragraph position="2"> The two different procedures will be of the same time complexity in worst cases. In the practical applications we have in mind, the procedure described here will be faster. Typically we want to check whether a formula 13 follows from al ..... a n. This is the same as deciding whether ~tlV...V~nVl3 is valid or not. The transformation to normal form will produce one additional copy for each v within an ai and each ^ within 13. If each ai and \[3 are LM formulas that represent English sentences, they can each be expected to be relatively short and in particular not contain many v's, so the number of copies made will be relatively small. On the other hand, the number of ai's may be large if they represent the dialogue so far or the agent's knowledge. It is therefore important that each disjunct can be split up as much as possible.</Paragraph>
  </Section>
  <Section position="9" start_page="0" end_page="0" type="metho">
    <SectionTitle>
IMPLEMENTATION
</SectionTitle>
    <Paragraph position="0"> The inference algorithm has been implemented in PROLOG. To test it out we have built a small (toy) natural language question-answering-system around it. The program reads in simple sentences and questions from the terminal and answers the questions.</Paragraph>
    <Paragraph position="1"> It can handle simple statements, like If some of the hot coffee that did not disappear was black then much gold is valuable (the fragment in Lenning 1987) and yes/no questions like Did much water evaporate? and Was the old gold that disappeared valuable? We have written the grammar and translation to LM in the built in DCG rules (Pereira and Warren, 1980).</Paragraph>
    <Paragraph position="2"> Statements typed on the terminal are interpreted as facts about the world and stored as simple sentences ~bl ..... qb n. When a question like Did much water evaporate? is asked, it is parsed and turned into a formula like V: Much(water)(evaporate). Then the program proceeds to check the validity of (Ol^...^On) --) V- If it is valid, the program answers yes, otherwise it checks (ddl^...^0n) ---) ~V. If this is valid, the answer is no, otherwise the program answers that it does not know. When a statement is made the program checks whether it is consistent with what the program already knows before it is added to the knowledgebase.</Paragraph>
    <Paragraph position="3"> The system is mainly made to test the inference algorithm and is not meant as an application by itself. But it illustrates some general points. It is a system where natural language inferences are made from natural language sentences and not from a fixed database. The system contains a complete treatment of propositional logic and illustrates a sound treatment of negation where failure is treated as does not know instead of negation. On the other hand, there is also a price to pay for incorporating full propositional logic. The system can only handle examples of a limited size in reasonable time.</Paragraph>
  </Section>
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