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<?xml version="1.0" standalone="yes"?> <Paper uid="J82-1003"> <Title>From English to Logic: Context-Free Computation of &quot;Conventional&quot; Logical Translation 1</Title> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 6 The exact function of the intension operator need not con- </SectionTitle> <Paragraph position="0"> cern us here. Roughly speaking, it is used to bring meanings within the domain of discourse; e.g,, while an NP t denotes a property set at each index, the corresponding ANp~ denotes the entire NP intension (mapping from indices to property sets) at each index.</Paragraph> <Paragraph position="1"> American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 29 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic tic category corresponding to the syntactic category NP. What is the set-theoretic type of that category? Since (every v boy t) cannot be interpreted as denoting an individual (at least not without making the rules of semantic valuation for formulas depend on the structure of the terms they contain), neither can John w.</Paragraph> <Paragraph position="2"> The solution is to regard NPs as denoting sets of properties, where a property determines a set of individuals at each index, and VPs as sets of such property sets (or in functional terms, as truth functions over truth functions over properties). Thus John v does not denote an individual, but rather a set of properties, namely those which John has; (every w boy t) denotes the set of properties shared by all boys, (a v boy w) the set of all properties possessed by at least one boy, and so on. It is not hard to see that the interpretation of VPs as sets of property sets then leads to the appropriate truth conditions for sentences. 7 With respect to our objective of building a comprehensible, expandable natural language understanding system, the simplicity of Gazdar's semantic rules and their one-to-one correspondence to phrase structure rules is extremely attractive; however, the semantics of the intensional logic translations, as sketched above, seems to us quite unnatural.</Paragraph> <Paragraph position="3"> Admittedly naturalness is partly a matter of familiarity, and we are not about to fault Montague grammar for having novel features (as some writers do, e.g., Harman 1975). But Montague's semantics is at variance with pretheoretical intuitions as well as philosophical tradition, as Montague himself acknowledged (1970c:268). Intuitively, names denote individuals (when they denote anything real), not sets of properties of individuals; extensional transitive verbs express relations between pairs of individuals, not between pairs of property sets, and so on; and intuitively, quantified terms such as &quot;everyone&quot; and &quot;no-one&quot; simply don't bear the same sort of relationship to objects in the world as names, even though the evidence for placing them in the same syntactic category is overwhelming. Such objections would carry no weight if the sole purpose of formal semantics were to provide an explication of intuitions about truth and logical consequence, for in that area intensional logic is remarkably successful. But formal semantics should also do justice to our intuitions about the relationship between word and object, where those intuitions are clear - and intensional logic seems at odds with some of the clearest of those intuitions. 8 There is also a computational objection to intensional logic translations. As indicated in our introductory remarks, a natural language understanding system must be able to make inferences that relate the natural language input to the system's stored knowledge and discourse model. A great deal of work in AI has focused on inference during language understanding and on the organization of the base of stored knowledge on which the comprehension process draws. Almost all of this work has employed more or less conventional logics for expressing the stored knowledge. (Even such idiosyncratic formalisms as Schank's conceptual depen~lency theory (Schank 1973) are much more akin to, say, first order modal logic than to any form of intensional logic - see Schubert 1976). How are intensional logic formulas to be connected up with stored knowledge of this conventional type? One possible answer is that the stored knowledge should not be of the conventional type at all, but should itself be expressed in intensional logic. However, the history of automatic deduction suggests that higher-order logics are significantly harder to mechanize than lower-order logics. Developing efficient inference rules and strategies for intensional logics, with their arbitrarily complex types and their intension, extension and lambda abstraction operators in addition to the usual modal operators, promises to be very difficult indeed.</Paragraph> <Paragraph position="4"> Another possible answer is that the intensional logic translations of input sentences should be post-processed to yield translations expressed in the lowerorder, more conventional logic of the system's knowledge base. A difficulty with this answer is that discourse inferences need to be computed 'on the fly' to guide syntactic choices. For example, in the sentences &quot;John saw the bird without binoculars&quot; and &quot;John saw the bird without tail feathers&quot; the syntactic roles of the prepositional phrases (i.e., whether they modify &quot;saw&quot; or &quot;the bird&quot;) can only be determined by inference. One could uncouple inference from parsing by computing all possible parses and choosing among the resultant translations, but this would be cumbersome and psychologically implausible at best.</Paragraph> <Paragraph position="5"> As a final remark on the disadvantages of intensional translations, we note that Montague grammar relies heavily on meaning postulates to deliver simple consequences, such as A boy smiles - There is a boy; 7 This was the approach in Montague (1970b) and is adopted in Gazdar (1981a). In another, less commonly adopted approach NPs are still interpreted as sets of properties but VPs are interpreted simply as properties, the truth condition for a sentence being that the property denoted by the VP be in the set of properties denoted by the NP (Montague 1970c, Cresswell 1973).In other words, the NP is thought of as predicating something about the VP, rather than the other way around.</Paragraph> <Paragraph position="6"> 8 Thomason reminds us that &quot;...we should not forget the firmest and most irrefragable kind of data with which a semantic theory must cope. The theory must harmonize with the actual denotations taken by the expressions of natural languages,...&quot;, but confines his further remarks to sentence denotations, i.e., truth values (Thomason, 1974b:54).</Paragraph> <Paragraph position="7"> 30 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic (in this instance an extensionalizing postulate is required for &quot;smiles&quot; - see Montague 1970c:263). A conventional approach dispensing with postulates of this type would be preferable.</Paragraph> <Paragraph position="8"> Having stated our misgivings about Montague grammar, we need to confront the evidence in its favour. Are there compelling reasons for regarding sentential constituents as more or less directly and uniformly interpretable? In support of the affirmative, one can point out the simplicity and elegance of this strategy from a logical point of view. More tellingly, one can cite its success record: it has made possible for the first time the formal characterization of non-trivial fragments of natural languages, with precisely defined syntactic-semantic mappings; and as one would hope, the formal semantics accounts for many cases of entailment, ambiguity, contradictoriness, and other semantic phenomena, including some of the subtlest arising from intensional locutions.</Paragraph> <Paragraph position="9"> Concerning the simplicity of the strategy, we note that the connection between language and the world could be just as simple as Montague grammar would have it, without being quite so direct. Suppose, for a moment, that people communicated in first-order logic.</Paragraph> <Paragraph position="10"> Then, to express that A1, Bill and Clyde were born and raised in New York, we would have to say, in effect, &quot;Al was born in New York. A1 was raised in New York. Bill was born in New York .... Clyde was raised in New York.&quot; The pressure to condense such redundant verbalizations would be great, and might well lead to 'overlay' verbalizations in which lists enumerating the non-repeated constituents were fitted into a common sentential matrix. In other words, it might lead to something like constituent coordination. But unlike simple constituents, coordinated constituents would not be meaningful in isolation; they would realize their meaning only upon expansion of the embedding overlay verbalization into a set of first-order formulas. Yet the connection between language and the world would remain simple, assuming that the syntactic relation between overlay verbalizations and their first-order translations were simple. It would be quite pointless to reeonstrue the semantics of the enhanced language so as to align the denotations of names with the denotations of coordinated names, for example, as is done in Montague grammar. While formally simplifying the semantic mapping function, such a move would lead to complex and counterintuitive semantic types.</Paragraph> <Paragraph position="11"> The success of Montague grammar in characterizing fragments of natural languages, with a proper account of logical relations such as entailment, is indeed strong evidence in its favour. The only way of challenging this success is to offer an equally simple, equally viable alternative. In part, this paper is intended as a move in that direction. While we do not explicitly discuss logical relations between the translations of sentences, the kinds of translations produced by the sample grammar in Section 4 should at least provide some basis for discussion. To the extent that the translations are of a conventional type (or easily converted to conventional form), the entailment relations should be more or less self-evident.</Paragraph> <Paragraph position="12"> There is one linguistic phenomenon, however, which deserves preliminary comment since it might be thought to provide conclusive evidence in favour of Montague grammar, or at least in favour of the intensional treatment of NPs. This concerns intensional verbs such as those in sentences (1) and (2), and per- null haps (3): (1) John looks for a unicorn, (2) John imagines a unicorn, (3) John worships a unicorn.</Paragraph> <Paragraph position="13"> These sentences admit non-referential readings with respect to the NP &quot;a unicorn&quot;, i.e., readings that do not entail the existence of a unicorn which is the referent of the NP. In intensional logic the nonreferential reading of the first sentence would simply be ((looks-for t ^(a v unicornt)) ^Johnt).</Paragraph> <Paragraph position="14"> The formal semantic analysis of this formula turns out just as required; that is, its value can be &quot;true&quot; or &quot;false&quot; (in a given possible world) irrespective of whether or not there are unicorns (in that world). The referential reading is a little more complicated, but presents no difficulties.</Paragraph> <Paragraph position="15"> It is the non-referential reading which is troublesome for conventional logics. For the first sentence, there seems to be only one conventional translation, viz., 3x\[\[John looks-for x\] & \[x unicorn\]\], and of course, this is the referential reading. There is no direct way of representing the non-referential reading, since the scope of a quantifier in conventional logics is always a sentence, never a term.</Paragraph> <Paragraph position="16"> The only possible escape from the difficulty lies in translating intensional verbs as complex(non-atomic) logical expressions involving opaque sentential operators. 9 The extant literature on this subject supports the view that a satisfactory decomposition cannot be supplied in all cases (Montague 1970c, Bennett 1974, Partee 1974, Dowty 1978, 1979, Dowty, Wall & Peters 1981). A review of this literature would be out of place here; but we would like to indicate that the case against decomposition (and hence against conventional translations) is not closed, by offering the fol9 With regard to our system-building objectives, such resort to lexical decomposition is no liability: the need for some use of lexical decomposition to obtain &quot;canonical&quot; representations that facilitate inference is widely acknowledged by AI researchers, and carried to extremes by some (e.g., Wilks 1974, Schank 1975). American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 31 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic lowing paraphrases of the three sample sentences.</Paragraph> <Paragraph position="17"> (Paraphrase (1)w is well-known, except perhaps for the particular form of adverbial (Quine 1960, Bennett 1974, Partee 1974), while (2)I-(3) '' are original).</Paragraph> <Paragraph position="18"> These could be formalized within a conventional logical framework allowing for non-truth-functional sen- null tential operators: (1)' John tries to find a unicorn (by looking around), (2) I John forms a mental description which could apply to a unicorn, (3) 1 John acts, thinks and feels as if he worshipped a unicorn.</Paragraph> <Paragraph position="19"> (3)&quot; John worships an entity which he believes to be a unicorn.</Paragraph> <Paragraph position="20"> In each case the operator that is the key to the translation is italicized. Note that the original ambiguity of (1) and (2) has been preserved, but can now be con- null strued as a quantifier scope ambiguity in the conventional fashion. In (3) 1 and (3)&quot; the embedded &quot;worships&quot; is to be taken in a veridical sense that entails the existence of the worshippee. It is important to understand that the translations corresponding to (3) 1 and (3)&quot; would not be obtained directly by applying the rules of the grammar to the original sentence; rather, they would be obtained by amending the direct translation, which is patently false for a hearer who interprets &quot;worships&quot; veridically and does not believe in unicorns. Thus we are presupposing a mechanism similar to that required to interpret metaphor on a Gricean account (Grice 1975). The notion of &quot;acting, thinking and feeling as if...&quot; may seem rather ad hoc, but appears to be applicable in a wide variety of cases where (arguably) non-intensional verbs of human action and attitude are used non-referentially, as perhaps in &quot;John is communing with a spirit&quot;, &quot;John is afraid of the boogie-man in the attic&quot;, or &quot;John is tracking down a sasquatch&quot;. Formulation (3)&quot; represents a more radical alternative, since it supplies an acceptable interpretation of (3) only if the entity actually worshipped by John may be an 'imaginary unicorn'.</Paragraph> <Paragraph position="21"> But we may need to add imaginary entities to our 'ontological stable' in any event, since entities may be explicitly described as imaginary (fictitious, hypothetical, supposed) and yet be freely referred to in ordinary discourse. Also, sentences such as &quot;John frequently dreams about a certain unicorn&quot; (based on an example in Dowty, Wall and Peters 1981) seem to be untranslatable into any logic without recourse to imaginary entities. Our paraphrases of (3) have the important advantage of entailing that John has a specific unicorn in mind, as intuitively required (in contrast with (1) and (2)). This is not the case for the intensional logic translation of (3) analogous to that of (1), a fact that led Bennett to regard &quot;worships&quot; - correctly, we think - as extensional (Bennett 1974).</Paragraph> <Paragraph position="22"> In the light of these considerations, the conventional approach to logical translation seems well worth pursuing. The simplicity of the semantic rules to which we are led encourages us in this pursuit.</Paragraph> </Section> <Section position="6" start_page="0" end_page="0" type="metho"> <SectionTitle> 3. Syntactic and Semantic Preliminaries </SectionTitle> <Paragraph position="0"> The logical-form syntax provides for the formation of simple terms such as John1, x, quantified terms such as <somel man2>, <thel (little2 boy3)>, simple predicate formulas such as man2, loves3, P4, compound predicate formulas such as and lambda abstracts such as ~x\[x shaves2 x\], Xy\[y expects2 \[y wins4\]\]. Note the use of sharp angle brackets for quantified terms, square brackets or blunt angle brackets for compound predicate formulas, and round brackets for modified predicate formulas. (We explain the use of square brackets and blunt angle brackets below.) We also permit sentences (i.e., compound predicate formulas with all arguments in place) as operands of sentential operators, as in For coordination of expressions of all types (quantifiers, terms, predicate formulas, modifiers, and sentential operators) we use sharp angle brackets and prefix form, as in</Paragraph> <Paragraph position="2"> The resemblance of coordinated expressions to quantified terms is intentional: in both cases the sharp angle brackets signal the presence of an unscoped operator (viz., the first element in brackets) to be scoped later on.</Paragraph> <Paragraph position="3"> Finally, we may want to admit second-order predicates with first-order predicate arguments, as in \[Fidol little-for3 dog5\], \[bluel colour4\], \[Xx\[x kissesl Mary2\] is-fun3\], though it remains to be seen whether such second-order predications adequately capture the meaning of English sentences involving implicit comparatives and nominalization.</Paragraph> <Paragraph position="4"> 32 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic Fuller explanations of several of the above features follow. In outline, we first delve a little further into the syntax and semantics of predicate formulas; then we discuss the sources and significance of ambiguities in the formulas.</Paragraph> <Paragraph position="5"> Atomic sentences are of the form \[t n P t 1 ... tn_l\], (equivalently, (P t 1 ... tn)), where t I .... , t n are terms and P is a predicate constant, and the square brackets and blunt angle brackets distinguish infix and prefix syntax respectively. We regard this sentential form as equivalent to \[t n (...((e t 1) t 2) ... tn_ 1 )1, i.e., as obtained by applying an n-ary predicate successively to n terms. For example, \[John loves Mary\] = Cloves Mary John) =\[John Cloves Mary)\] = ((loves Mary) John). 10 As in Montague grammar, this predicate application syntax helps to keep the rules of translation simple: in most cases the translation of a phrase is just the composition of the translations of its top-level constituents. However, we saw earlier that a functional interpretation of predicate application leads to the interpretation of predicates as telescoped function-valued functions, whereas we wish to interpret predicates as n-ary relations (in each possible world) in the conventional way.</Paragraph> <Paragraph position="6"> We can satisfy this requirement by interpreting predicate application not as function application, but rather as leftmost section of the associated relation at the value of the given argument. For example, let V denote the semantic valuation function (with a particular interpretation and possible world understood) and</Paragraph> <Paragraph position="8"> where P is a triadic predicate symbol, x, y, and z are individual constants or variables, and a, b ..... g are elements of the individual domain D. Then</Paragraph> <Paragraph position="10"> We use the convention {<>} = true, {} = false.</Paragraph> <Paragraph position="11"> Lambda abstraction can be defined compatibly by Vl(~,x~b) = {{d} X Vi(x:d) (~b) I d * D}, where I is an interpretation, I(x:d) is an interpretation identical to I except that x denotes d, and X denotes Cartesian product (and a particular possible world is 10 We provide the double syntax for purely cosmetic reasons. In our use of the notation, expressions delimited by square brackets will generally be complete open or closed sentences, while expressions delimited by blunt angle brackets will be 'incomplete sentences', i.e., predicates with one or more arguments missing (and denoting a relation with adicity = number of missing arguments). understood). It can be verified that the usual lambdaconversion identities hold, i.e.,</Paragraph> <Paragraph position="13"> where P is a predicate of any adicity (including null, if we use{<>}XA= A for any set A).</Paragraph> <Paragraph position="14"> As far as modified predicate formulas such as (bright3 red4) are concerned, we can interpret the modifiers as functions from n-ary relations to n-ary relations (perhaps with n restricted to 1).</Paragraph> <Paragraph position="15"> We now turn to a consideration of the potential sources of ambiguity in the formulas. One source of ambiguity noted in the Introduction lies in the primitive logical symbols themselves, which may correspond ambiguously to various proper logical symbols. The ambiguous symbols are obtained by the translator via the first stage of a two-stage lexicon (and with the aid of morphological analysis, not discussed here). This first stage merely distinguishes the formal logical roles of a lexeme, supplying a distinct (but in general still ambiguous) symbol or compound expression for each role, along with syntactic information. For example, the entry for &quot;recover&quot; might distinguish (i) a predicate role with preliminary translation &quot;recovers-from&quot; and the syntactic information that this is a V admissible in the rule that expands a VP as a V optionally followed by a (PP from); (this information is supplied via the appropriate rule number); and (ii) a predicate role with preliminary translation &quot;recovers&quot; and the syntactic information that this is a V admissible in the rule that expands a VP as a V followed by an NP.</Paragraph> <Paragraph position="16"> Having obtained a preliminary translation of a lexeme in keeping with its apparent syntactic role, the translator affixes an index to it which has not yet been used in the current sentence (or if the translation is a compound expression, it affixes the same index to all of its primitive symbols). In this way indexed preliminary translations such as Maryl, good2, and recovers3 are obtained. For example, the verb translation selected for &quot;recovers&quot; in the sentence context &quot;John recovers the sofa&quot; would be recovers2, recovers-from2 being ruled out by the presence of the NP complement. The second stage of the lexicon supplies alternative final translations of the first-stage symbols, which in the case of &quot;recovers&quot; might be RE-COVERS, REGAINS, and so on. Naturally, the processors that choose among these final symbols would have to draw on knowledge stored in the propositional data base and in the representation of the discourse context.</Paragraph> <Paragraph position="17"> A second source of ambiguity lies in quantified terms. The sentence Someone loves every man American Journal of Computational Linguistics, Volume 8, Number i, January-March 1982 33 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic illustrates a quantifier scope ambiguity arising from a syntactically unambiguous construction. Its logical-form translation is \[<somel one2> loves3 <every4 man5>\], wherein the relative scopes of the quantifiers somel and every4 are ambiguous. Quantified terms are intended to be 'extracted' in the postprocessing phase to positions left-adjacent to sentential formulas (which may already be prefixed with other quantifiers). A new variable is introduced into each extracted quantifier expression, the angle brackets are changed to round brackets, and the new variable is substituted for all occurrences of the extracted term. (Thus the level of extraction must be 'high' enough to encompass all of these occurrences.) In the above formula, quantifier extraction reveals the implicit ambiguity, yielding ei-</Paragraph> <Paragraph position="19"> depending on the order of extraction.</Paragraph> <Paragraph position="20"> Assuming that somel and every4 correspond to the standard existential and universal quantifiers, these translations could be further processed to yield ~x\[\[x one2\] & Vy\[\[y man5\] => \[x loves3 y\]\]\] and Vy\[\[y man5\] => 3x\[\[x one2\] & \[x loves3 y\]\]\].</Paragraph> <Paragraph position="21"> However, we may not implement this last conversion step, since it cannot be carried out for all quantifiers. For example, as Cresswell remarks, &quot;most A's are B's&quot; cannot be rendered as &quot;for most x, either x is not an A or x is a B&quot; (Cresswell 1973: 137). (Consider, for instance, A = dog and B = beagle; then the last statement is true merely because most things are not dogs irrespective of whether or not most dogs are in fact beagles.) It appears from recent work by Goebel (to appear) that standard mechanical inference methods can readily be extended to deal with formulas with restricted quantifiers.</Paragraph> <Paragraph position="22"> A third source of ambiguity lies in coordinated expressions. For example, the logical form of the sentence &quot;Every man loves Peggy or Sue&quot; is \[<everyl man2> loves3 <or5 Peggy4 Sue6>\], which is open to the readings</Paragraph> <Paragraph position="24"> The postprocessing steps required to scope coordinators are similar to those for quantifiers and are illustrated in Section 4.1 l An important constraint on the disambiguation of the basic symbols as well as quantified terms and coordinated expressions is that identical expressions (i.e., expressions with identical constituent structure, including indices) must be identically disambiguated. For example, &quot;John shaves himself&quot; and &quot;John shaves John&quot; translate respectively into \[Johnl hx\[x shaves2 x\]\] = \[Johnl shaves2 Johnl\], and \[Johnl shaves2 John3\].</Paragraph> <Paragraph position="25"> The stated constraint ensures that both occurrences of Johnl in the first formula will ultimately be replaced by the same unambiguous constant. Similarly &quot;Someone shaves himself&quot; and &quot;Someone shaves someone&quot; translate initially into \[<somel one2> shaves3 <somel one2>\] and \[<somel one2> shaves3 <some4 one5>\] respectively, and these translations become (somel x:\[x one2\])\[x shaves3 x\] and (somel x:\[x one2\])(some4 y:\[y one5\])\[x shaves3 y\] respectively after quantifier extraction. Note that the two occurrences of <somel one2> in the first formula are extracted in unison and replaced by a common variable. Indexing will be seen to play a similar role in the distribution of coordinators that coordinate non-sentential constituents.</Paragraph> <Paragraph position="26"> By allowing the above types of ambiguities in the logical form translations, we are able to separate the problem of disambiguation from the problems of parsing and translation. This is an important advantage, since disambiguation depends upon pragmatic factors.</Paragraph> <Paragraph position="27"> For example, &quot;John admires John&quot; may refer to two distinct individuals or just to one (perhaps whimsically), depending on such factors as whether more than one individual named John has been mentioned in the current context. Examples involving ambiguities in nouns, verbs, determiners, etc., are easily supplied.</Paragraph> <Paragraph position="28"> Similarly, the determination of relative quantifier scopes involves pragmatic considerations in addition to level of syntactic embedding and surface order. This is true both for explicit quantifier scope ambiguities such as in the sentence &quot;Someone loves every man&quot;, and for scope ambiguities introduced by decomposition, such as the decomposition of &quot;seeks&quot; into hyhx\[x tries \[x finds y\]\], as a result of which a sentence like John seeks a unicorn admits the alternative translations 3x\[\[x unicorn\] & \[John tries \[John finds x\]\]\], and \[John tries 3x\[\[x unicorn\] & \[John finds x\]\]\], neglecting indices. It is simpler to produce a single output which can then be subjected to pragmatic post11 If first-order predicates are to be allowed as arguments of second-order predicates, then quantifier and coordinator scoping of the following types must also be allowed: \[P...<Q R>...\] -~ ~kx(Q y:\[y Rl)\[x P...y...\], <C P R> ~ ~kx\[\[x PI C \[x R\]\]. 34 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic processing to determine likely quantifier scopes, than to generate all possible orderings and then to make a pragmatic choice among them. Much the same can be said about scoping of coordinators.</Paragraph> <Paragraph position="29"> We also note that a grammar designed to generate all possible unambiguous translations of English phrases and sentences would have to supply multiple semantic rules for certain syntactic rules. For example, no one semantic rule can translate a quantifier-noun combination (rule 3 in Section 4) so as to deliver both readings of &quot;Someone loves every man&quot; upon combination of the verb translation with the translations of the NPs. Our use of an ambiguous logical form preserves the rule-to-rule hypothesis.</Paragraph> </Section> <Section position="7" start_page="0" end_page="0" type="metho"> <SectionTitle> 4. Sample Grammar </SectionTitle> <Paragraph position="0"> Our syntactic rules do not depart significantly from Gazdar's. The semantic rules formally resemble Gazdar's as well, but of course produce conventionally interpretable translations of the type described in the preceding section. As in Gazdar's semantic rules, constituent translations are denoted by primed category symbols such as NP' and V'. The semantic rules show how to assemble such translations (along with the occasional variable and lambda operator) to form the translations of larger constituents. The translations of individual lexemes are obtained as described above.</Paragraph> <Paragraph position="1"> In operation, the translator generates the minimum number of brackets consistent with the notational equivalences stated earlier. For example, in assembling \[NP' VP'\], with NP' = Johnl and VP t = \[loves2 Mary3\], the result is \[Johnl loves2 Mary3\], rather than \[Johnl (loves2 Mary3)\].</Paragraph> <Paragraph position="2"> Also, in binding a variable with lambda, the translator replaces all occurrences of the variable with a previously unused variable, thus minimizing the need for later renaming. Finally, it performs lambda conversions on the fly. For example, the result of assembling \[NP' VP'\] with NP' = Johnl and</Paragraph> <Paragraph position="4"> The rules that follow have been adapted from Gazdar (1981a). Note that each rule that involves a lexical category such as PN, N or V is accompanied by a specification of the subset of lexical items of that category admissible in the rule. This feature is particularly important for verb subcategorization. In addition, each rule is followed by (a) a sample phrase accepted by the rule, (b) an indication of how the logical translation of the phrase is obtained, and possibly (c) some words of further explanation.</Paragraph> <Paragraph position="5"> tives, unlike genuine predicate modifiers such as &quot;consummate&quot;, actually combine with terms. For such adjectives we might employ the semantic rule ~.x\[\[x ADJP'\] & \[x N'\]\]; in the case of &quot;little&quot;, we would use ADJP' = (little-for P), where P is an indeterminate predicate to be replaced pragmatically by a comparison-class predicate. Thus the translation of &quot;little boy&quot; (neglecting indices) would be ~kx\[\[x little-for PI & \[x boyl\].</Paragraph> <Paragraph position="6"> to extract quantifiers, the result might be S' = (thel x5:\[x5 (little2 boy3)\]) \[x5 smiles4\].</Paragraph> <Paragraph position="7"> Further postprocessing to determine referents and disambiguate operators and predicates might then yield S' = \[INDIVI7 SMILESl\], where INDIV17 is a (possibly new) logical constant unambiguously denoting the referent of (the\] x5:\[x5 (little2 boy3)\]) and SMILESl is an unambiguous logical predicate. 13 If constant INDIV17 is new, i.e., if the context provided no referent for the definite description, a supplementary assertion like \[INDIV17 (LITTLE2 BOYI)\] would be added to the context representation.</Paragraph> <Paragraph position="8"> (a)' John wants to give Fido to Mary</Paragraph> <Paragraph position="10"> embedded (subordinate) clause.</Paragraph> <Paragraph position="11"> The reader will observe that we have more or less fully traced the derivation and translation of the sentences &quot;The little boy smiles&quot; and &quot;John wants to give Fido to Mary&quot; in the course of the above examples.</Paragraph> <Paragraph position="12"> The resultant phrase structure trees, with rule numbers and translations indicated at each node, are shown in Figs. 1 and 2.</Paragraph> <Paragraph position="13"> 13 Definite singular terms often serve as descriptions to be used for referent determination, and in such cases it is the name of the referent, rather than the description itself, which is ultimately wanted in the formula.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 36 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 </SectionTitle> <Paragraph position="0"/> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 38 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 </SectionTitle> <Paragraph position="0"> Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic All of the above rules, as well as our versions of the remaining rules in Gazdar (1981a), are as simple as the intensional logic versions or simpler. For example, our semantic rule 8, i.e., Ax\[x V' \[x VP'\]\], may be contrasted with the corresponding rule suggested by Gazdar: XP{P Xx\[(V' A(Vp' XP(P x))) XP(P x)\]}.</Paragraph> <Paragraph position="1"> Here the lambda variable x, as in our formula, is used to feed a common logical subject to V' (the translation of the main verb) and to VP' (the translation of the embedded infinitive); the variables P and P, on the other hand, serve to ensure that the arguments of the V' and VP' functions will be of the correct type. Our 'conventional' rule is simpler because it makes no such use of lambda abstraction for type-raising and dispenses with the intension operator.</Paragraph> <Paragraph position="2"> Gazdar's approach to unbounded dependencies carries over virtually unchanged and can be illustrated with the sentence To Mary John wants to give Fido.</Paragraph> <Paragraph position="3"> Here the PP &quot;to Mary&quot; has been topicalized by extraction from &quot;John wants to give Fido to Mary&quot;, leaving a PP 'gap' at the extraction site. This 'gap' is syntactically embedded within the infinitive VP &quot;to give Fido&quot;, within the main VP &quot;wants to give Fido&quot;, and at the highest level, within the sentence &quot;John wants to give Fido&quot;. In general, the analysis of unbounded dependencies requires derived rules for propagating 'gaps' from level to level and linking rules for creating and filling them. The linking rules are obtained from the correspondingly numbered basic rules by means of the metarule</Paragraph> </Section> </Section> <Section position="8" start_page="0" end_page="0" type="metho"> <SectionTitle> \[AXCY\] ==> \[A/BXC/BY\], </SectionTitle> <Paragraph position="0"> where A, B and C may be any basic (i.e., non-slash) syntactic categories such that C can dominate B, and X, Y may be any sequences (possibly empty) of basic categories. The linking rules for topicalization are obtained from the rule schemata</Paragraph> <Paragraph position="2"> where B ranges over all basic phrasal categories, and t is a dummy element (trace). The first of these schemata introduces the free variable h as the translation of the gap, while the second lambda-abstracts on h and then supplies B' as the value of the lambda variable, thus 'filling the gap' at the sentence level. At syntactic nodes intermediate between those admitted by schemata 11 and 12, the B-gap is transmitted by derived rules and h is still free.</Paragraph> <Paragraph position="3"> Of the following rules, 6, 8, and 10 are the particular derived rules required to propagate the PP-gap in our example and 11 and 12 the particular linking rules that create and fill it: <11, \[(PP to)/(PP to) (a) t (b) PP' -> h t\], h> <6, \[(VP)/(PP to) (V) (NP) (PP to)/(PP to)\], (V' PP' NP')> (a) give Fido (b) with V' : gives5, NP' : Fido6, PP' = h, VP' -> (gives5 h Fido6) (c) Note that the semantic rule is unchanged. <8, \[(VP)/(PP to) (V) (VP INF)/(PP to)\], Ax\[x V' (a) wants to give Fido (b) with V' = wants3, VP' = (gives5 h Fido6), This translation is logically indistinguishable from the translation of the untopicalized sentence. However, the fronting of &quot;to Mary&quot; has left a pragmatic trace: the American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 39 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic corresponding argument Maryl has the lowest index, lower than that of the subject translation John2 (assuming that symbols are indexed in the order of occurrence of the lexical items they translate). In subsequent pragmatic processing, this feature could be used to detect the special salience of Maryl, without re-examination of the superficial sentence form.</Paragraph> <Paragraph position="4"> Another example of a sentence that can be analyzed by such methods, using relative clause rules similar to those for topicalization, is Every dog Mary wants to buy is small.</Paragraph> <Paragraph position="5"> The rules analyze &quot;Mary wants to buy&quot; as an S/NP with translation \[Mary wants \[Mary buys h\]\], neglecting indices. A further rule reduces the S/NP to an R (relative clause), and its semantic part abstracts on h to yield the predicate R' = Xh\[Mary wants \[Mary buys h\]\] as the translation of the relative clause. The rules for NPs can be formulated in such a way that &quot;every dog&quot; will be translated as <every kx\[\[x dog\] & \[x R\]\]> where R is a free predicate variable that is replaced by the translation of the relative clause when the NP-R rule <13, \[(NP) (NP) (R)\], <XRNP' R'>> is applied (cf., Gazdar 1981b; we have ignored multiple relative clauses). The resulting NP translation is <every hx\[\[x dog\] & \[Mary wants \[Mary buys x\]\]\]>.</Paragraph> <Paragraph position="6"> The translation of the complete sentence, after extraction of the quantifier and conversion of the constraint on the universally quantified variable to an implicative antecedent, would be Y=y\[\[\[y dog\] & \[Mary wants \[Mary buys y\]\]\] => \[y (small P)\]\], where P is an undetermined predicate (= dog, in the absence of contrary contextual information).</Paragraph> <Paragraph position="7"> As a further illustration of Gazdar's approach and how easily it is adapted to our purposes, we consider his metarule for passives: <\[(VP)(V TRAN) (NP) X\], (St NP&quot;)> ==> <\[(VP PASS) (V) X {(PP by)}\], ~,p((~r p) pp&quot;)>; i.e., &quot;for every active VP rule that expands VP as a transitive verb followed by NP, there is to be a passive VP rule that expands VP as V followed by what, if anything, followed the NP in the active VP rule, followed optionally by a by-PP&quot; (Gazdar 1981a). In the original and resultant semantic rules, (~&quot; ...) represents the original rule matrix in which NP&quot; is embedded; thus (~r p) is the result of substituting the lambda variable P (which varies over NP intensions) for NP&quot; in the original rule. Intuitively, the lambda variable 'reserves' the NP&quot; argument position for later binding by the subject of the passive sentence. It can be seen that the metarule will generate a passive VP rule corresponding to our rule 6 which will account for sentences such as &quot;Fido was given to Mary by John&quot;. Moreover, if we introduce a ditransitive rule <14, \[ (VP) (V TRAN) (NP) (NP) \] , (V' NP' NP')>I4 to allow for sentences such as &quot;John gave Mary Fido&quot;, the metarule will generate a passive VP rule that accounts for &quot;Mary was given Fido by John&quot;, in which the indirect rather than direct object has been turned into the sentence subject.</Paragraph> <Paragraph position="8"> The only change needed for our purposes is the replacement of the property variable P introduced by the metarule by an individual variable x: ...(~r NP' ) ..... > ...hx((~ r x) PP' )...</Paragraph> <Paragraph position="9"> Once the subject NP of the sentence is supplied via rule 10, x is replaced by the translation of that NP upon lambda conversion.</Paragraph> <Paragraph position="10"> Finally in this section, we shall briefly consider coordination. Gazdar has supplied general coordination rule schemata along with a cross-categorical semantics that assigns appropriate formal meanings to coordinate structures of any category (Gazdar 1980b). Like Gazdar's rules, our rules generate logical-form translations of coordinated constituents such as <and John Bill>, <or many few>, <and (hugs Mary) (kisses Sue)>, echoing the surface forms. However, it should be clear from our discussion in Section 2 that direct interpretation of expressions translating, say, coordinated NPs or VPs is not compatible with our conventional conception of formal semantics. For example, no formal semantic value is assigned directly to the coordinated term in the formula \[<and John Bill> loves Mary\].</Paragraph> <Paragraph position="11"> Rather, interpretation is deferred until the pragmatic processor has extracted the coordinator from the embedding sentence (much as in the case of quantified 14 In the computational version of the semantic rules, primed symbols are actually represented as numbers giving the positions of the corresponding constituents, e.g., (1 2 3) in rule 14. Thus no ambiguity can arise.</Paragraph> <Paragraph position="12"> 40 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic terms) and distributed the coordinated terms over duplicate copies of that sentence, yielding \[\[John loves Mary\] and \[Bill loves Mary\]\].</Paragraph> <Paragraph position="13"> We adopt the following coordination schemata without change. The superscript denotes sequences of length > 1 of the superscripted element. The schemata are accompanied by examples of phrases they admit, along with (unindexed) translations. The bracketing in (a) and (a)' indicates syntactic structure.</Paragraph> <Paragraph position="14"> The order in which coordinators are extracted and distributed is a matter of pragmatic choice. However, a crucial constraint is that multiple occurrences of a particular coordinated expression (with particular indices) must be extracted and distributed in a single operation, at the level of a sentential formula whose scope encompasses all of those occurrences (much as in the case of quantifier extraction). The following examples illustrate this process.</Paragraph> <Paragraph position="15"> (c) Note that once the and3-conjunction has been chosen for initial extraction and distribution, the simultaneous extraction and distribution of both occurrences of the or6-disjunction at the highest sentential level is compulsory. The resultant formula expresses the sense of &quot;John loves and admires Fido or loves and admires Kim&quot;. Initial extraction of the or6-disjunction would have led to the (implausible) reading &quot;John loves Fido or Kim and admires Fido or Kim&quot; (which is true even if John loves only Fido and admires only Kim).</Paragraph> <Paragraph position="17"> have been raised to the second highest sentential level, yielding (alll x: \[x man2\]) \[Ix wants3 Ix marries4 Peggy5\]\] or6 \[x wants3 \[x marries4 Sue7\]\]\], or to the highest sentential level, yielding American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 41 Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic \[(allq x:\[x man2\]) \[x wants3 \[x marries4 Peggy5\]\] or6 (alll x:\[x man2\]) \[x wants3 \[x marries4 Sue7\]\]\].</Paragraph> <Paragraph position="18"> The three readings are logically distinct and all are quite plausible (in the absence of additional context). The reader can verify that the first and second readings, but not the third, could have been obtained by extracting the coordinator first and the quantifier second.</Paragraph> <Paragraph position="19"> Finally, we should remark that the distributive rules are not appropriate for the group reading of coordinate structures in sentences such as John and Mary carried the sofa (together).</Paragraph> <Paragraph position="20"> We envisage a mereological interpretation in which John and Mary together comprise a two-component entity. However, we refrain from introducing a logical syntax for such entities here (but see the treatment of plurals in Schubert, 1982).</Paragraph> </Section> <Section position="9" start_page="0" end_page="0" type="metho"> <SectionTitle> 5. Parsing </SectionTitle> <Paragraph position="0"> Phrase structure grammars are relatively easy to parse. The most advanced parser for Gazdar-style grammars that we are aware of is Thompson's chart-parser (Thompson 1981), which provides for slash categories and coordination, but does not (as of this writing) generate logical translations. We have implemented two small parser-translators for preliminary experimentation, one written in SNOBOL and the other in MACLISP. The former uses a recursive descent algorithm and generates intensional logic translations. The latter is a 'left corner' parser that uses our reformulated semantic rules to generate conventional translations. It begins by finding a sequence of left-most phrase-structure-rule branches that lead from the first word upward to the sentence node. (e.g., Mary -~ PN -~ NP -~ S). The remaining branches of the phrase structure rules thus selected form a &quot;frontier&quot; of expectations. Next a similar initial-unit sequence is found to connect the second word of the sentence to the lowest-level (most immediate) expectation, and so on. There is provision for the definition and use of systems of features, although we find that the parser needs to do very little feature checking to stay on the right syntactic track. Neither parser at present handles slash categories and coordination (although they could be handled inefficiently by resort to closure of the grammar under metarules and rule schemata). Extraction of quantifiers from the logical-form translations is at present based on the level of syntactic embedding and left-to-right order alone, and no other form of postprocessing is attempted.l 5 15 Since submission of this paper for publication, we have become aware of several additional papers on parser-translators similar to ours. One is by Rosenschein & Shieber (1982), another by Gawron et al. (1982); in conception these are based quite directly on the generalized phrase structure grammar of Gazdar and his collaborators, and use reeursive descent parsers. A related Prolog-based approach is described by McCord (1981, 1982).</Paragraph> <Paragraph position="1"> It has been gratifyingly easy to write these parsertranslators, confirming us in the conviction that Gazdar-style grammars hold great promise for the design of natural language understanding systems. It is particularly noteworthy that we found the design of the translator component an almost trivial task; no modification of this component will be required even when the parser is expanded to handle slash categories and coordination directly. Encouraged by these resuits, we have begun to build a full-scale left-corner parser. A morphological analyzer that can work with arbitrary sets of formal affix rules is partially implemented; this work, as well as some ideas on the conventional translation of negative adjective prefixes, plurals, and tense/aspect structure, is reported in Schubert (1982).</Paragraph> </Section> class="xml-element"></Paper>