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Background: Interpretation as Abduction

The framework adopted in this chapter is that of ``Interpretation as Abduction'' (henceforth, IA) (Hobbs et al., 1993). In this framework, the interpretation of a sentence is the least-cost abductive proof of the logical form of the sentence. That is, to interpret a sentence one tries to prove the logical form by using the most salient axioms and other information, exploiting the natural redundancy of discourse to minimize the size of the proof, and allowing the minimal number of consistent and plausible assumptions necessary to make the proof go through. Anaphora are resolved and predications are pragmatically strengthened as a by-product of this process.

More generally in the IA framework, the job of an agent is to interpret the environment by proving abductively, or explaining, the observables in the environment, thereby establishing that the agent is in a coherent situation. This perspective is expanded upon in Section 9 below.

The representational conventions used in this chapter are those of Hobbs (1985, 1983, and 1995). The chief features relevant to this chapter are the use of eventualities and the use of typical elements of sets to represent information about pluralities. The latter is described in Section 7. Here, eventualities will be explicated briefly.

Corresponding to every predication p(x) there is a predication p'(e,x) which says that e is the eventuality of p being true of x. Existential quantification in this notation is over a Platonic universe of possible individuals. The actual truth or existence of e in the real world is asserted by a separate predication Rexists(e). The relation between the primed and unprimed predicates is given by the following axiom schema:

$ (\forall \,x)[p(x) \, \equiv \,(\exists \,e)[p'(e,x) \& Rexists(e)]] $

That is, p is true of x if and only if there is an eventuality eof p being true of x and e exists or obtains in the real world.

Eventualities are posited not just for events, such as flying to someplace-- fly'(e,x,y)--and activities, such as reading-- read'(e,x,y)--but also for stable conditions, such as being a cigarette-- cigarette'(e,x)--and even having a particular proper name-- John'(e,x). For economy, in the examples below eventualities will only be introduced where they are material to what is being illustrated, primarily when they appear as arguments in other predications. Otherwise the unprimed predicates will be used.

Using these notational devices, the logical form of sentences is an existentially quantified conjunction of atomic predications. The translations of several specific grammatical constructions into this logical form are given in the examples in the rest of this paper. In general, instead of writing

$ (\exists \,\ldots ,x, \ldots)[\ldots \& p(x) \& \ldots] $

I will simply write

$ \ldots \& p(x) \& \ldots $

Knowledge in this framework is expressed in (generally defeasible) axioms of the form

$ (\forall \,x,y)[p(x,y) \, \supset \,(\exists \,z) q(x,z)] $

These will be abbreviated to expressions of the form

$ p(x,y) \, \supset \,q(x,z) $

The focus of the interpretation process is to make explicit the information conveyed by the text in context, rather than, for example, to determine its truth conditions.

In the IA framework, syntax, semantics, and pragmatics are thoroughly integrated, through the observation that the task of an agent in explaining an utterance by another agent is normally to show that it is a grammatical, interpretable sentence whose content is somehow involved in the goals of the speaker. It has already been said that a sentence is interpretable insofar as its logical form can be proved abductively. The linkage with goals is described in Section 9. A set of syntactic and lexical axioms characterize grammaticality and yield the logical form of sentences.

In Hobbs (1998) an extensive subset of English grammar is described in detail, largely following Pollard and Sag's (1994) Head-Driven Phrase Structure Grammar but cast into the uniform IA framework. In this treatment, the predicate Syn is used to express the relation between a string of words and the eventuality it conveys. Certain axioms involving Syn, the ``composition axioms'', describe how the eventuality conveyed emerges from the concatenation of strings. Other axioms, the ``lexical axioms'', link Syn predications about words with the corresponding logical-form fragments. There are also ``transformation axioms'' which alter the places in the string of words predicates find their arguments.

In this chapter, a simplified version of the predicate Syn will be used. We will take Syn to be a predicate of seven arguments.

Syn(w,e,f,x,a,y,b)

w is a string of words. e is the eventuality described by this string. f is the category of the head of the phrase w. If the string w contains the logical subject of the head, then the arguments x and a are the empty symbol ``-''. Otherwise, x is a variable refering to the logical subject and a is its category. Similarly, y is either the empty symbol or a variable refering to the logical object and b is either the empty symbol or the category of the logical object. For example,

Syn(``reads a novel'',e,v,x,n,-,-)

says that the string of words ``reads a novel'' is a phrase describing an eventuality e and has a head of category verb. Its logical object ``a novel'' is in the string itself, so the last two arguments are the empty symbol. Its logical subject is not part of the string, so the fourth argument is the variable x standing for the logical subject and the fifth argument specifies that the phrase describing it must have a noun as its head. In Hobbs (1998) the full Synpredicate contains argument positions for further complements and filler-gap information, and the category arguments can record syntactic features as well.

Two of the most important composition axioms are the following:

$ Syn(w_{1},x,a,-,-,-,-) \& Syn(w_{2},e,f,x,a,-,-) $
$
\, \supset \,Syn(w_{1} w_{2},e,f,-,-,-,-) $

$ Syn(w_{1},e,f,x,a,y,b) \& Syn(w_{2},y,b,-,-,-,-) $
$
\, \supset \,Syn(w_{1} w_{2},e,f,x,a,-,-) $

The first axiom corresponds to the traditional ``S $\rightarrow$ NP VP'' rule. It says that if w1 is a string describing an entity xand headed by a word of category a and w2 is a string describing eventuality e, headed by a word of category f, and lacking a logical subject x of category a, then the concatenation w1 w2is a string describing eventuality e and headed by a word of category f. The second axiom corresponds to the traditional ``VP $\rightarrow$ V NP'' rule. It says that if w1 is a string describing eventuality e, headed by a word of category f, and lacking a logical subject x of category a and a logical object yof category b and w2 is a string describing an entity y and headed by a word of category b, then the concatenation w1 w2 is a string describing eventuality e, headed by a word of category f, and lacking a logical subject x of category a, but not lacking a logical object.

A typical lexical axiom is the following:

$ read'(e,x,y) \& person(x) \& text(y)
\, \supset \,Syn($``read'',e,v,x,n,y,n)

That is, if e is the eventuality of a person x reading a text y, then the verb ``read'' can be used to describe e provided noun phrases describing x and y are found in the appropriate places, as specified by composition axioms. Lexical axioms thus encode the logical form fragment corresponding to a word ( read'(e,x,y)), selectional constraints (person(x) and text(y)), the spelling (or in a more detailed account, the phonology) of the word (``read''), its category (verb), and the syntactic constraints on its complements (that x and y must come from noun phrases). The lexical axioms constitute the interface between syntax and world knowledge; knowledge about reading is encoded in axioms involving the predicate read', whereas knowledge of syntax is encoded in axioms involving Syn, and these two are linked here. In the course of proving that a string of words is a grammatical, interpretable sentence, the interpretation process backchains through composition axioms to lexical axioms (the syntactic processing) and then is left with the logical form of the sentence to be proved. A proof of this logical form was the original IA characterization of the interpretation of a sentence.

The proof graph of the syntactic part of the interpretation of ``John read Ulysses'' is shown in Figure 1. Note that knowledge that John is a person and Ulysses is a text is used to establish the selectional constraints associated with ``read''.


  
Figure 1: Parse of ``John read Ulysses.''
\begin{figure}
\par\setlength{\unitlength}{0.0125in} %
\begin{picture}
(355,270)...
...[b]{\raisebox{0pt}[0pt][0pt]{\xipt\rm$text(y)$ }}}
\end{picture}\par\end{figure}

There are systematic alternations due to which the arguments of predicates are not found in the canonical locations specified by the lexical axioms. Transformation axioms accommodate these alternations. The (somewhat simplified) rule for passivation illustrates this.

Syn(w,e,v.en,x,a,y,n $)
\, \supset \,Syn(w,e,$pred,y,n,-,-)

If w is the past participle of a verb which takes a subject x of category a and an NP object y, then w can function as a predicate complement, taking an NP subject refering to y.

Metonymy can also be characterized by transformation axioms.


next up previous
Next: Axioms for Metonymy Up: Syntax and Metonymy Previous: Metonymy
Jerry Hobbs
2000-07-20