| { | |
| "paper_id": "Y18-1030", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T13:36:42.801892Z" | |
| }, | |
| "title": "Model-Theoretic Incremental Interpretation Based on Discourse Representation Theory", | |
| "authors": [ | |
| { | |
| "first": "Yoshihide", | |
| "middle": [], | |
| "last": "Kato", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "Nagoya University Furo-cho", | |
| "location": { | |
| "addrLine": "Chikusa-ku", | |
| "postCode": "464-8601", | |
| "settlement": "Nagoya", | |
| "country": "Japan" | |
| } | |
| }, | |
| "email": "yoshihide@icts.nagoya-u.ac.jp" | |
| }, | |
| { | |
| "first": "Shigeki", | |
| "middle": [], | |
| "last": "Matsubara", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "Nagoya University Furo-cho", | |
| "location": { | |
| "addrLine": "Chikusa-ku", | |
| "postCode": "464-8601", | |
| "settlement": "Nagoya", | |
| "country": "Japan" | |
| } | |
| }, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
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| "abstract": "This paper proposes a model-theoretic approach to incremental interpretation where all sentence prefixes have semantic values. The proposed semantics is based on Discourse Representation Theory (DRT), where semantic representations (called DRSs) are interpreted as assignment updates. In our semantics, a partial DRS of a sentence prefix is interpreted as two sets which stipulate the assignment updates. One denotes possible updates and the other denotes necessary updates. With the proposed semantics, we can assign truth values to sentence prefixes. 2 Discourse Representation Theory This section provides a brief introduction to Discourse Representation Theory (DRT).", | |
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| "paper_id": "Y18-1030", | |
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| "abstract": [ | |
| { | |
| "text": "This paper proposes a model-theoretic approach to incremental interpretation where all sentence prefixes have semantic values. The proposed semantics is based on Discourse Representation Theory (DRT), where semantic representations (called DRSs) are interpreted as assignment updates. In our semantics, a partial DRS of a sentence prefix is interpreted as two sets which stipulate the assignment updates. One denotes possible updates and the other denotes necessary updates. With the proposed semantics, we can assign truth values to sentence prefixes. 2 Discourse Representation Theory This section provides a brief introduction to Discourse Representation Theory (DRT).", | |
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| "section": "Abstract", | |
| "sec_num": null | |
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| "body_text": [ | |
| { | |
| "text": "Incremental semantic parsers construct semantic representations for each sentence prefix, and are useful for incremental dialogue systems (Allen et al., 2001; Aist et al., 2007) . While most research on incremental semantic parsers has focused on how to construct such representations incrementally (Pulman, 1985; Milward, 1995; Poesio and Rieser, 2010; Purver et al., 2011; Peldszus and Schlangen, 2012; Sayeed and Demberg, 2012; Kato and Matsubara, 2015) , there has been little work on how to formally interpret them.", | |
| "cite_spans": [ | |
| { | |
| "start": 138, | |
| "end": 158, | |
| "text": "(Allen et al., 2001;", | |
| "ref_id": "BIBREF1" | |
| }, | |
| { | |
| "start": 159, | |
| "end": 177, | |
| "text": "Aist et al., 2007)", | |
| "ref_id": "BIBREF0" | |
| }, | |
| { | |
| "start": 299, | |
| "end": 313, | |
| "text": "(Pulman, 1985;", | |
| "ref_id": "BIBREF12" | |
| }, | |
| { | |
| "start": 314, | |
| "end": 328, | |
| "text": "Milward, 1995;", | |
| "ref_id": "BIBREF8" | |
| }, | |
| { | |
| "start": 329, | |
| "end": 353, | |
| "text": "Poesio and Rieser, 2010;", | |
| "ref_id": "BIBREF11" | |
| }, | |
| { | |
| "start": 354, | |
| "end": 374, | |
| "text": "Purver et al., 2011;", | |
| "ref_id": "BIBREF13" | |
| }, | |
| { | |
| "start": 375, | |
| "end": 404, | |
| "text": "Peldszus and Schlangen, 2012;", | |
| "ref_id": "BIBREF10" | |
| }, | |
| { | |
| "start": 405, | |
| "end": 430, | |
| "text": "Sayeed and Demberg, 2012;", | |
| "ref_id": "BIBREF14" | |
| }, | |
| { | |
| "start": 431, | |
| "end": 456, | |
| "text": "Kato and Matsubara, 2015)", | |
| "ref_id": "BIBREF7" | |
| } | |
| ], | |
| "ref_spans": [], | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "An important issue with incremental interpretation, from a formal semantic viewpoint, is that sentence prefixes do not have propositional interpretations (Chater et al., 1995) . In other words, standard formal semantics cannot be applied to incremental interpretation directly.", | |
| "cite_spans": [ | |
| { | |
| "start": 154, | |
| "end": 175, | |
| "text": "(Chater et al., 1995)", | |
| "ref_id": "BIBREF4" | |
| } | |
| ], | |
| "ref_spans": [], | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "This paper proposes a model-theoretic approach to incremental interpretation where each sentence prefix has semantic values. The proposed semantics is an extension of Discourse Representation Theory (DRT) (Kamp and Reyle, 1993) . In DRT, semantic representations are called discourse representation structures (DRSs) and are interpreted in terms of (non-deterministic) assignment updates (An assignment is a function that maps discourse referents to entities.) In this paper, we define two types of interpretations of partial DRSs. One denotes possible assignment updates and the other denotes necessary updates. The proposed semantics monotonically specifies the semantic values of a sentence on a word-by-word basis, and finally assigns the same value to the sentence in terms of DRT's semantics. In addition, it can assign truth values to sentence prefixes that are not sentential clauses. To the best of our knowledge, this is the first attempt to interpret underspecified semantic representations of sentence prefixes.", | |
| "cite_spans": [ | |
| { | |
| "start": 205, | |
| "end": 227, | |
| "text": "(Kamp and Reyle, 1993)", | |
| "ref_id": "BIBREF6" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "This paper is organized as follows. Section 2 introduces DRT. Section 3 gives an overview of incremental semantic parsers that construct a partial DRS for each sentence prefix. Then, Section 4 proposes our incremental interpretation method based on DRT. Finally, Section 5 compares our work with previous studies, and Section 6 presents our conclusions.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "In DRT, semantic content is represented as a discourse representation structure (DRS). A DRS consists of a set of discourse referents and a set of conditions. A discourse referent denotes an entity, which is introduced by a sentence. A condition denotes a constraint imposed on discourse referents. DRSs are written as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Discourse Representation Structure", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "[x 1 , . . . x n | c 1 , \u2022 \u2022 \u2022 c m ]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Discourse Representation Structure", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Here, x 1 , . . ., x n are discourse referents, and c 1 , . . ., c m are conditions. For example, the following DRS intuitively represents a situation, where there is a student x 1 and a laptop x 2 , and x 1 uses x 2 :", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Discourse Representation Structure", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "[x 1 , x 2 | stu(x 1 ), laptop(x 2 ), use(x 1 , x 2 )]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Discourse Representation Structure", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "Below, we define a DRT language based on that of Bos (2009) , which adopts type theory to define expressions. The basic types are e (entities) and t (propositions). If \u03b1 and \u03b2 are types, then \u03b1\u03b2 is the type of a function from \u03b1 to \u03b2. The language is defined as follows.", | |
| "cite_spans": [ | |
| { | |
| "start": 49, | |
| "end": 59, | |
| "text": "Bos (2009)", | |
| "ref_id": "BIBREF3" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Discourse Representation Structure", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "1. A variable of type \u03b1 is an expression of type \u03b1.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Discourse Representation Structure", | |
| "sec_num": "2.1" | |
| }, | |
| { | |
| "text": "3. If P is an n-place predicate symbol and", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A discourse referent is an expression of type e.", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "x 1 , . . ., x n are expressions of type e, then P (x 1 , . . . , x n ) is a basic condition.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A discourse referent is an expression of type e.", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "4. If x 1 and x 2 are expressions of type e, then x 1 = x 2 is a basic condition.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A discourse referent is an expression of type e.", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "6. If X is a set of discourse referents and C is a set of conditions, then [X | C] is an expression of type t.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A basic condition is a condition.", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": ". If E 1 and E 2 are expressions of type t, then (E 1 ; E 2 ) is an expression of type t.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "8. If E is an expression of type t, then \u00acE is a condition.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "9. If E 1 and E 2 are expressions of type t, then E 1 \u2228 E 2 and E 1 \u21d2 E 2 are conditions.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "10. If X is a variable of type \u03b1 and E is an expression of type \u03b2, then (\u03bbX.E) is an expression of type \u03b1\u03b2 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "11. If E 1 is an expression of type \u03b1\u03b2 and E 2 is an expression of type \u03b1, then (E 1 E 2 ) is an expression of type \u03b2.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Below, we impose the following constraints on all expressions.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 All discourse referents are declared at most once. Here, we say that x is declared in E if E takes the form [. . . , x, . .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ". | C].", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 For all function types \u03b1\u03b2 , \u03b2 = e.", | |
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| "ref_spans": [], | |
| "eq_spans": [], | |
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| { | |
| "text": "Let E be an expression in \u03b2-normal form. 1 We say that E is complete if E does not include any variables. Otherwise, E is partial. If E is of type t, we call E a DRS. We can compositionally build up a DRS for a sentence in bottom-up fashion. In this paper, we adopt the approach of Bos (2008) , which is based on Combinatory Categorial Grammar (Steedman, 2000) . As an example, let us consider constructing a semantic expression for the noun phrase \"a student\" using the lexicon shown in Table 1 . The categories of \"a\" and \"student\" are NP/N and N, respectively. We can combine these, since NP/N means that it receives an expression of category N from the right and returns one of category NP. The corresponding semantic expression can be obtained by function application as follows 2 :", | |
| "cite_spans": [ | |
| { | |
| "start": 41, | |
| "end": 42, | |
| "text": "1", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 282, | |
| "end": 292, | |
| "text": "Bos (2008)", | |
| "ref_id": "BIBREF2" | |
| }, | |
| { | |
| "start": 344, | |
| "end": 360, | |
| "text": "(Steedman, 2000)", | |
| "ref_id": "BIBREF17" | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 488, | |
| "end": 495, | |
| "text": "Table 1", | |
| "ref_id": "TABREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u03bbP Q. ([x 1 | ]; P x 1 ); Qx 1 (\u03bbX.[ | stu(X)]) \u03b2 \u03bbQ.([x 1 | stu(x 1 )]; Qx 1 )", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Here, \u03b2 is the reflexive transitive closure of \u03b2reduction. Figure 1 shows the derivation of the DRS for the following example sentence:", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 59, | |
| "end": 67, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A student uses every red laptop.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "( ", | |
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| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "P x); Qx) p pt NP/N every \u03bbP Q.[ | ([x | ]; P x) \u21d2 Qx] p pt NP/N student \u03bbX.[ | stu(X)] p N laptop \u03bbX.[ | laptop(X)] p N blue \u03bbP X.([ | blue(X)]; P X) pp N/N red \u03bbP X.([ | red(X)]; P X) pp N/N use \u03bbP Q.Q(\u03bbX.P (\u03bbY.[ | use(X, Y )])) pt pt t (S\\NP)/NP don't \u03bbP Q.[ | \u00acP Q]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "pt t pt t (S/NP)/(S/NP) Here, we abbreviate the type et to p. ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "7", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "This section explains how DRSs are interpreted based on Muskens (1996) . A DRS is interpreted with respect to a model M , which is defined as a pair (D, I). Here, D is a (non-empty) set of entities called a domain and I is a function that maps n-place predicate symbols to sets of n-tuples of entities. d 1 , . . . , d n \u2208 I(P ) means that the entities d 1 , . . . , d n stand in the relation P . The interpretation of a DRS is a function that takes an assignment and returns a set of (updated) assignments. An assignment a is a partial function that maps discourse referents to entities, and we write an assignment a such that a", | |
| "cite_spans": [ | |
| { | |
| "start": 56, | |
| "end": 70, | |
| "text": "Muskens (1996)", | |
| "ref_id": "BIBREF9" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of DRSs", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "(x) = d as [x \u2192 d, . . .].", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of DRSs", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "The interpretation functions [[\u2022] ] for DRSs and conditions 3 are defined with respect to a model M as follows:", | |
| "cite_spans": [ | |
| { | |
| "start": 29, | |
| "end": 33, | |
| "text": "[[\u2022]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of DRSs", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "1. If E \u2261 P (x 1 , . . . , x n ), then [[E]] M = {a | a(x 1 ), . . . , a(x n ) \u2208 I(P )}. 2. If E \u2261 x 1 = x 2 , then [[E]] M = {a | a(x 1 ) = a(x 2 )}. 3. If E \u2261 [X | C], then [[E]] M (a) = {a | a \u2286 X a \u2227 a \u2208 c\u2208C [[c]] M }.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of DRSs", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "Here, a \u2286 X a means that Dom(a ) = Dom(a) \u222a X (Dom(a) is the domain of a. In addition, Dom(a) \u2229 X = \u2205) and a(x) = a (x) for all x \u2208 Dom(a). If there is a set X such that a \u2286 X a , we call a an extension of a and write a \u2286 a . 3 The interpretation of a condition is the set of assignments that satisfy the condition.", | |
| "cite_spans": [ | |
| { | |
| "start": 226, | |
| "end": 227, | |
| "text": "3", | |
| "ref_id": null | |
| } | |
| ], | |
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| "section": "Interpretation of DRSs", | |
| "sec_num": "2.2" | |
| }, | |
| { | |
| "text": "E \u2261 (E 1 ; E 2 ), then [[E]] M (a) = {a | \u2203a \u2208 [[E 1 ]] M (a) a \u2208 [[E 2 ]] M (a ) }. 5. If E \u2261 \u00acE 1 , then [[E]] M = {a | [[E 1 ]] M (a) = \u2205}. 6. If E \u2261 E 1 \u2228 E 2 , then [[E]] M = {a | [[E 1 ]] M (a) \u222a [[E 2 ]] M (a) = \u2205}. 7. If E \u2261 E 1 \u21d2 E 2 , then [[E]] M = {a | \u2200a \u2208 [[E 1 ]] M (a) [[E 2 ]] M (a ) = \u2205 }.", | |
| "cite_spans": [], | |
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| "section": "If", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "A DRS E is defined to be true in a model M if and only if [[E] ] M (\u03c6) = \u2205, where \u03c6 is the empty assignment (Dom(\u03c6) = \u2205). Otherwise, E is defined to be false in M .", | |
| "cite_spans": [ | |
| { | |
| "start": 58, | |
| "end": 62, | |
| "text": "[[E]", | |
| "ref_id": null | |
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| "section": "If", | |
| "sec_num": "4." | |
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| { | |
| "text": "As an example, consider the interpretation of the DRS S 6 for sentence (1), shown in Figure 1 . S 6 is true in M ex shown in Figure 2 , since the following holds:", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 85, | |
| "end": 93, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 125, | |
| "end": 133, | |
| "text": "Figure 2", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "If", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "[[S 6 ]] Mex (\u03c6) = {[x 1 \u2192 d 1 ]}.", | |
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| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "If", | |
| "sec_num": "4." | |
| }, | |
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| "text": "Intuitively, this interpretation means that there is a situation which satisfies the conditions of S 6 , and that the discourse referent x 1 introduced by the word \"a\" denotes the entity d 1 in the situation. 32nd Pacific Asia Conference on Language, Information and Computation Hong Kong, 1-3 December 2018", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "If", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "Copyright 2018 by the authors ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "If", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "Word Partial DRS A S1 \u2261 U (S\\NP) 2 (\u03bbQ.(([x1 | ]; U (N) 1 x1); Qx1)) student S2 \u2261 U (S\\NP) 2 (\u03bbQ.([x1 | stu(x1)]; Qx1)) uses S3 \u2261 [x1 | stu(x1)]; (U (NP) 3 (\u03bbY.[ | use(x1, Y)])) every S4 \u2261 [x1 | stu(x1), ([x4 | ]; U (N) 4 x4) \u21d2 [ | use(x1, x4)]] red S5 \u2261 [x1 | stu(x1), ([x4 | ]; ([ | red(x4)]; U (N) 5 x4)) \u21d2 [ | use(x1, x4)]] laptop S6 \u2261 [x1 | stu(x1), [x4 | red(x4), laptop(x4)] \u21d2 [ | use(x1, x4)]]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "If", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "We can construct a partial DRS for any sentence prefix by assigning variables to the underspecified parts of the sentence. Here, we will not discuss how to assign these variables, and instead refer the interested reader to the literature (Peldszus and Schlangen, 2012; Kato and Matsubara, 2015) . As an example, Table 2 shows the incremental construction of partial DRSs for the sentence (1). Here, the bracketed superscripts indicate the categories to which the variables correspond. To see how partial DRSs are constructed, let us consider the following sentence prefix, which will be followed by a category N:", | |
| "cite_spans": [ | |
| { | |
| "start": 238, | |
| "end": 268, | |
| "text": "(Peldszus and Schlangen, 2012;", | |
| "ref_id": "BIBREF10" | |
| }, | |
| { | |
| "start": 269, | |
| "end": 294, | |
| "text": "Kato and Matsubara, 2015)", | |
| "ref_id": "BIBREF7" | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 312, | |
| "end": 319, | |
| "text": "Table 2", | |
| "ref_id": "TABREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "A student uses every . . .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "We can obtain the partial DRS S 4 by assigning a variable U", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "4 of type et to the underspecified part of the sentence. Figure 3 shows the derivations of the partial DRSs.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 57, | |
| "end": 65, | |
| "text": "Figure 3", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "We treat incremental semantic construction as the process of substituting concrete expressions for variables in semantic representations, and formalize it as follows.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "\u2022 Let E 1 and E 2 be expressions in \u03b2-normal form, and let U be a free variable that occurs in E 1 . If there exists an E such that E 1 [U :", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "= E ] \u03b2 E 2 , then we write E 1 \u00a3 E 2 . Here, E[U := E ] is the capture-avoiding substitu- tion of E for U in E. When E 1 \u00a3 * E 2 , we say that E 2 is derived from E 1 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "The relation \u00a3 represents the incremental semantic construction process. For example,", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "S 2 \u00a3S 3 , because S 2 [U (S\\NP) 2 := E (S\\NP) ] \u03b2 S 3 where E (S\\NP)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "is obtained by combining the expression for \"use\" and a variable U", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "(NP) 3", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "of type et t :", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "E (S\\NP) \u2261 \u03bbR.R(\u03bbX.U (NP) 3 (\u03bbY.[ | use(X, Y)]))", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Incremental Construction of DRSs", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "In this section, we propose a method of semantically interpreting partial DRSs. Since sentence prefixes may not have any propositional content (Chater et al., 1995) , we give alternative semantic values to partial DRSs instead. The essential idea is to consider two types of interpretation: one stipulates which updates will necessarily be included by the complete DRS derived from the partial DRS (", | |
| "cite_spans": [ | |
| { | |
| "start": 143, | |
| "end": 164, | |
| "text": "(Chater et al., 1995)", | |
| "ref_id": "BIBREF4" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "[[\u2022]] 2 M )", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": ", and the other stipulates which updates may be included", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "([[\u2022]] 3 M )", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": ". We call these P-interpretations and Q-interpretations, respectively. At the end of this section, we will show that these interpretations can assign truth values to sentence prefixes in a consistent manner.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "The interpretations are defined as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "1. If E is a basic condition, then", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "[[E]] 2 M = [[E]] 3 M = [[E]] M .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "2. If E is a variable of type t or an expression of the form (E 1 E 2 ) of type t, then 4. If E \u2261 (E 1 ; E 2 ), then Theorem 1 (upper and lower bounds). For any partial DRS E, model M , and assignment a, the following statements hold:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "[[E]] 2 M (a) = \u2205, [[E]] 3 M (a) = {a}.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "3. If E \u2261 [X | C], then [[E]] 2 M (a) = {a | a \u2286 X a \u2227 a \u2208 c\u2208C [[c]] 2 M }. 32nd", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "[[E]] 2 M (a) = {a | \u2203a \u2208 [[E 1 ]] 2 M (a) a \u2208 [[E 2 ]] 2 M (a ) }. 5. If E \u2261 \u00acE 1 , then [[E]] 2 M = {a | [[E 1 ]] 3 M (a) = \u2205}. 6. If E \u2261 E 1 \u2228 E 2 , then [[E]] 2 M = {a | [[E 1 ]] 2 M (a) \u222a [[E 2 ]] 2 M (a) = \u2205}. 7. If E \u2261 E 1 \u21d2 E 2 , then [[E]] 2 M = {a | \u2200a \u2208 [[E 1 ]] 3 M (a) [[E 2 ]] 2 M (a ) = \u2205 }.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "[[E]] 2 M (a) \u2286 {a | a \u2286 DR(E) a } \u2229 E \u2208Comp(E) {a | a [[E ]] M (a)}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "(3)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "[[E]] 3 M (a) \u2287 {a | a \u2286 DR(E) a } \u2229 E \u2208Comp(E) {a | a [[E ]] M (a)}", | |
| "eq_num": "(4)" | |
| } | |
| ], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Here, Comp(E) is the set of complete expressions derived from E, and a A means that a \u2286 a holds for some assignment a \u2208 A. DR(E) is a set of discourse referents and defined as follows.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "DR(E) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 X (E \u2261 [X | C]) DR(E 1 ) \u222a DR(E 2 ) (E \u2261 (E 1 ; E 2 )) \u2205 (otherwise)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "The right-hand sides of the set inclusion relations (3) and (4) represent necessary and possible updates, respectively. We have equality in (3) and (4) when all free variables in E occur at most once. 5 We can obtain Theorem 2, which concerns the truth values of sentence prefixes by the following lemmas. Lemma 1 (monotonicity). Let E 1 and E 2 be expressions of type t such that E 1 \u00a3 * E 2 . For any model M and assignment a, the following statements hold: 32nd Pacific Asia Conference on Language, Information and Computation Hong Kong, 1-3 December 2018", | |
| "cite_spans": [ | |
| { | |
| "start": 201, | |
| "end": 202, | |
| "text": "5", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Copyright 2018 by the authors ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "use(X, Y )])) NP/N : \u03bbP Q.[| ([x4 | ];P x4) \u21d2 Qx4] N/N : \u03bbP X.([ | red(X)];P X) N : \u03bbX.[ | laptop(X)] N : \u03bbX.[ | red(X), laptop(X)] NP : \u03bbQ.[ | [x4 | red(x4), laptop(x4)] \u21d2 Qx4] S\\NP : \u03bbQ.Q(\u03bbX.[ | [x4 | red(x4), laptop(x4)] \u21d2 [ | use(X, x4)]]) S : S 6 \u2261 [x1 | stu(x1), [x4 | red(x4), laptop(x4)] \u21d2 [ | use(x1, x4)]]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "D = {d1, d2, d3, d4, d5, d6, d7} I(stu) = { d1 , d2 } I(laptop) = { d3 , d4 } I(tablet) = { d5 , d6 , d7 } I(red) = { d3 , d5 , d6 } I(blue) = { d4 , d7 } I(use) = { d1, d3 , d1, d4 , d1, d5 , d1, d6 , d2, d7 } Figure 2: Model M ex . \u2022 For any assignment a 1 \u2208 [[E 1 ]] 2 M (a), there is an assignment a 2 \u2208 [[E 2 ]] 2 M (a) such that a 1 \u2286 a 2 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "\u2022 For any assignment", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "a 2 \u2208 [[E 2 ]] 3 M (a), there is an assignment a 1 \u2208 [[E 1 ]] 3 M (a) such that a 1 \u2286 a 2 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Lemma 2 (consistency). For any complete DRS E, model M , and assignment a, the following holds:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "[[E]] 2 M (a) = [[E]] 3 M (a) = [[E]] M (a)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Theorem 2 (truth values of partial DRSs). Let E be a partial DRS. For any model M and DRS E \u2208 Comp(E), the following statements hold.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "\u2022 E is true in M if [[E]] 2 M (\u03c6) = \u2205. \u2022 E is false in M if [[E]] 3 M (\u03c6) = \u2205.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Below, we say that a partial DRS E is true in Table 3 shows interpretations of partial DRSs for example sentence (1). Below, we clarify our proposed semantics using this example.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 46, | |
| "end": 53, | |
| "text": "Table 3", | |
| "ref_id": "TABREF6" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "M if [[E]] 2 M (\u03c6) = \u2205, and false in M if [[E]] 3 M (\u03c6) = \u2205.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Interpretation of Partial DRSs", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "As an example, let us consider the interpretation of partial DRS S 4 (Table 2) in the model M ex (Figure 2) . For any entity d i (1 \u2264 i \u2264 7) , we have the following:", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 97, | |
| "end": 107, | |
| "text": "(Figure 2)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 129, | |
| "end": 140, | |
| "text": "(1 \u2264 i \u2264 7)", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "[[[x 4 | ]]] 3 Mex ([x 1 \u2192 d i ]) = {[x 1 \u2192 d i , x 4 \u2192 d j ] | 1 \u2264 j \u2264 7} (5)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "For any assignment a, Clause 2 gives us the following:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "[[U (N) 4 x 4 ]] 3", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "Mex (a) = {a} (6) From equations (5) and (6), and Clause 4, we have", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "[[\u2022]] 2 [[\u2022]] 3 A [[S1]] 2 Mex (\u03c6) = \u2205 [[S1]] 3 Mex (\u03c6) = {\u03c6} student [[S2]] 2 Mex (\u03c6) = \u2205 [[S2]] 3 Mex (\u03c6) = {\u03c6} uses [[S3]] 2 Mex (\u03c6) = \u2205 [[S3]] 3 Mex (\u03c6) = {[x1 \u2192 d1], [x1 \u2192 d2]} every [[S4]] 2 Mex (\u03c6) = \u2205 [[S4]] 3 Mex (\u03c6) = {[x1 \u2192 d1], [x1 \u2192 d2]} red [[S5]] 2 Mex (\u03c6) = {[x1 \u2192 d1]} [[S5]] 3 Mex (\u03c6) = {[x1 \u2192 d1], [x1 \u2192 d2]} laptop [[S6]] 2 Mex (\u03c6) = {[x1 \u2192 d1]} [[S6]] 3 Mex (\u03c6) = {[x1 \u2192 d1]}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "[[([x 4 | ]; U (N) 4 x 4 )]] 3 Mex ([x 1 \u2192 d i ]) = {[x 1 \u2192 d i , x 4 \u2192 d j ] | 1 \u2264 j \u2264 7} (7)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "On the other hand, d i , d 1 \u2208 I(use) in the model M ex gives us the following equation:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "[[[ | use(x 1 , x 4 )]]] 2 Mex ([x 1 \u2192 d i , x 4 \u2192 d 1 ]) = \u2205", | |
| "eq_num": "(8)" | |
| } | |
| ], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "By equations 7and 8, and Clause 7, we obtain the following:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "[x 1 \u2192 d i ] \u2208 [[([x 4 | ]; U (N) 4 x 4 ) \u21d2 [ | use(x 1 , x 4 )]]] 2", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Truth value of a partial DRS", | |
| "sec_num": "4.1.1" | |
| }, | |
| { | |
| "text": "This means any assignment a such that \u03c6 \u2286 {x1} a does not belong to the P-interpretaton of the second condition of S 4 , so we find the following:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "[[S 4 ]] 2 Mex (\u03c6) = \u2205", | |
| "eq_num": "(9)" | |
| } | |
| ], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In other words, there are no assignments that satisfy the conditions of any DRS derived from S 4 . Next, let us consider the Q-interpretation of", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "S 4 . From [[U (N) 4 x 4 ]] 2", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Mex = \u2205, we obtain the following:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "[[([x 4 | ]; U (N) 4 x 4 )]] 2 Mex ([x 1 \u2192 d i ]) = \u2205", | |
| "eq_num": "(10)" | |
| } | |
| ], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Therefore, for any assignment a, we have", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "a \u2208 [[([x 4 | ]; U (N) 4 x 4 ) \u21d2 [ | use(x 1 , x 4 )]]] 3 Mex", | |
| "eq_num": "(11)" | |
| } | |
| ], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Since the Q-interpretation of the first condition of S 4 has members", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "[x 1 \u2192 d 1 ] and [x 1 \u2192 d 2 ]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ", and that of the second condition also has the same members, we obtain the following:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "[[S 4 ]] 3 Mex (\u03c6) = {[x 1 \u2192 d 1 ], [x 1 \u2192 d 2 ]}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "By Theorem 2, we can therefore conclude that the partial DRS S 4 is neither true nor false in the model M ex .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Let us now consider another example, where the word \"red\" follows (2):", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A student uses every red . . .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The partial DRS of the prefix (12) is S 5 , and its interpretations are as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "[[S 5 ]] 2 Mex (\u03c6) = {[x 1 \u2192 d 1 ]}, [[S 5 ]] 3 Mex (\u03c6) = {[x 1 \u2192 d 1 ], [x 1 \u2192 d 2 ]}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The important point here is that any DRS derived from S 5 is guaranteed to be true, independently of any input that follows (12), due to Theorem 2. These examples demonstrate that our framework can discuss the propositional contents of sentence prefixes, even if they are not sentential clauses.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Mex", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Here, let us consider another sentence prefix, namely the following:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "A student doesn't use every . . .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "The partial DRS of this sentence prefix is as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "[ | \u00acS 4 ]", | |
| "eq_num": "(14)" | |
| } | |
| ], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "Since Next, we consider the following sentence prefix:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "(\u03c6) A U (S\\NP) 2 (\u03bbQ.(([x1 | ]; U (N) 1 x1); Qx1)) (([x1 | ]; U (N) 1 x1); Qx1) \u222ai=1,...,7{[x1 \u2192 di]} red U (S\\NP) 2 (\u03bbQ.(([x1 | ]; ([ | red(x1)];U (N) 3 )); Qx1)) (([x1 | ]; ([ | red(x1)];U (N) 3 )); Qx1) \u222ai=3,5,6{[x1 \u2192 di]} laptop U (S\\NP) 2 (\u03bbQ.([x1 | red(x1), laptop(x1)]; Qx1)) ([x1 | red(x1), laptop(x1)]; Qx1) {[x1 \u2192 d3]}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "A student doesn't use every red . . .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "Now, its partial DRS is", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "[ | \u00acS 5 ]. Since [[S 5 ]] 2 Mex (\u03c6) = \u2205, i.e., \u03c6 \u2208 [[\u00acS 5 ]] 3 Mex , we have [[[ | \u00acS 5 ]]] 3", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "Mex (\u03c6) = \u2205. By Theorem 2, we can therefore conclude that [ | \u00acS 5 ] is false in M ex .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Negation", | |
| "sec_num": "4.1.2" | |
| }, | |
| { | |
| "text": "By applying the Q-interpretation function to DRS's sub-expressions that include discourse referents, we can identify the entities to which the discourse referents refer. Here, let us consider the sentence prefix A red laptop . . . Table 4 shows the partial DRSs, the sub-expressions, and their Q-interpretations. In Q-interpretations, the entities to which discourse referents can refer are incrementally specified. This example demonstrates that our semantics has a potential to be useful for incremental reference resolution (Schlangen et al., 2009) .", | |
| "cite_spans": [ | |
| { | |
| "start": 527, | |
| "end": 551, | |
| "text": "(Schlangen et al., 2009)", | |
| "ref_id": "BIBREF15" | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 231, | |
| "end": 238, | |
| "text": "Table 4", | |
| "ref_id": "TABREF8" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Referential interpretation", | |
| "sec_num": "4.1.3" | |
| }, | |
| { | |
| "text": "Unlike incremental semantic construction, there has been little work on how to interpret partial semantic representations incrementally, with two exceptions: (Schuler et al., 2009) and (Hough and Purver, 2014) . These papers proposed an incremental referential interpretation where noun phrase prefixes are interpreted as entities to which the noun phrase derived from the prefix can refer. In their interpretation process, such entities are incrementally specified. Our Q-interpretaion provides a similar mechanism, as shown in Section 4.1.3. In addition, our approach provides a method of determining the truth values of sentence prefixes (Theorem 2), whereas that of the previous studies has no way to deal with truth values, and thus cannot offer sentential interpretations. Furthermore, their semantics cannot treat quantifiers, while ours provides interpretations of both existential and universal quantifiers. Chater et al. (1995) adopted another approach to incremental interpretation called two-level incremental interpretation. One level carries out incremental semantic construction, while the other serves as interpretation. At the first level, a first-order formula with \u03bb-operators is constructed incrementally. At the second level, this formula is then converted into a first-order formula without \u03bb-operators by an existential closure-like mechanism (replacing the \u03bboperators with existential quantifiers). Since the second level representation is a proper first-order formula, it can be interpreted by standard semantics. However, this approach has the drawback that the truth value of the formula may be inconsistent with that of the final formula obtained from the whole sentence, even when there is no misanalysis at the first level. This issue is inevitable as long as the principle of bivalence is adopted, because there are cases where truth value of sentence prefix cannot be determined. In contrast, this is not the case for our proposed incremental interpretation, because it allows sentence prefixes to have truth-value gaps, i.e., to be neither true nor false. This is achieved by using the two types of interpretations proposed above.", | |
| "cite_spans": [ | |
| { | |
| "start": 158, | |
| "end": 180, | |
| "text": "(Schuler et al., 2009)", | |
| "ref_id": "BIBREF16" | |
| }, | |
| { | |
| "start": 185, | |
| "end": 209, | |
| "text": "(Hough and Purver, 2014)", | |
| "ref_id": "BIBREF5" | |
| }, | |
| { | |
| "start": 917, | |
| "end": 937, | |
| "text": "Chater et al. (1995)", | |
| "ref_id": "BIBREF4" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Comparisons with Previous Work", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "This paper has proposed a model-theoretic incremental interpretation framework that can treat the propositional contents of sentence prefixes, including phenomena such as negation and quantification. We believe that this framework will help to clarify the roles semantic representations can play in incremental processing.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Conclusion", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "If an expression E does not contain a \u03b2-redex, namely an expression of the form (\u03bbX.E1)E2, we say that E is in \u03b2normal form.2 To simplify the notation, we allow merge operations that substitute [X1 \u222a X2 | C1 \u222a C2] for ([X1 | C1]; [X2 | C2]) in any expression. This does not affect the following discussion.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "If E is a variable of type \u03b1, any expression E of type \u03b1 can be derived from E, because E[E := E ] \u03b2 E . If E is of the form E1E2, E1 is a variable or an expression of the form E3E4 or \u03bbP.E5. If E1 is a variable, then any expression E of type \u03b1 can be derived from E1E2, since (E1E2)[E1 := \u03bbP.E ] \u03b2 E (where P is a fresh variable). If E1 is of the form E3E4, we can prove that \u03bbP.E can be derived from E3E4 by mathematical induction. Any expression E of type \u03b1 therefore can be derived from E1E2. The third case \u03bbP.E5 is not allowed, since E \u2261 (\u03bbP.E5)E2 is a \u03b2-redex, i.e., E is not in \u03b2-normal form.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "When a free variable occurs more than once, our interpretation functions treat each occurrence of the variable independently, which is why the equality does not always hold. We plan to solve this problem in future work.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [ | |
| { | |
| "text": "This research was partially supported by the Grantin-Aid for Scientific Research (C) (17K00303) of JSPS. ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Acknowledgements", | |
| "sec_num": null | |
| } | |
| ], | |
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| "FIGREF0": { | |
| "num": null, | |
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| "text": "Pacific Asia Conference on Language, Information and Computation Hong Kong, 1-3 December 2018 Copyright 2018 by the authors NP/N : \u03bbP Q.(([x1 | ];P x1): \u03bbP Q.(([x1 | ];P x1);Qx1) N : \u03bbX.[ | stu(X)] NP : \u03bbQ.([x1 | stu(x1)];Qx1) (S\\NP)/NP : \u03bbP Q.Q(\u03bbX.P (\u03bbY.[ | use(X, Y )])) NP/N : \u03bbP Q.[ | ([x4 | ];P x4) \u21d2 Qx4] \u21d2 [ | use(X, x4)]]) S : S 4 \u2261 [x1 | stu(x1), ([x4 |];U (N) 4 x4) \u21d2 [ | use(x1, x4)]] Derivations of partial DRSs." | |
| }, | |
| "FIGREF1": { | |
| "num": null, | |
| "uris": null, | |
| "type_str": "figure", | |
| "text": "NP/N : \u03bbP Q.(([x1 | ];P x1);Qx1) N : \u03bbX.[ | stu(X)] NP : \u03bbQ.([x1 | stu(x1)];Qx1) (S\\NP)/NP : \u03bbP Q.Q(\u03bbX.P (\u03bbY.[ |" | |
| }, | |
| "FIGREF2": { | |
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| "uris": null, | |
| "type_str": "figure", | |
| "text": "Derivation for sentence (1)." | |
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| "num": null, | |
| "text": "Semantic expressions for words.", | |
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| "num": null, | |
| "text": "Incremental semantic constructions for \"A student uses every red laptop.\"", | |
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| "text": "Incremental interpretations of \"A student uses every red laptop.\"", | |
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| "text": "Incremental reference resolution of \"A red laptop. . .\"", | |
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