Cross-Situational Learning
3.1 Linking and Fracturing
Throughout much of part I of this thesis, I will represent meanings as terms, i.e. expressions com-posed of primitive constant and function symbols. For expository purposes, I will use primitives taken primarily from Jackendo's (1983) conceptual structure notation, though I will extend this
33
set arbitrarily as needed. Thus typical meaning expressions will include GO(
cup
;FROM(John
)) and SEE(John
;Mary
). None of the techniques in part I of this thesis attribute any interpretation to the primitives. In every way, the meaning expression GO(cup
;FROM(John
)) is treated the same as f(a;g(b)).2Variable-free meaning expressions such as those given above will denote the meanings of whole utter-ances. The meanings of utterance fragments in general, and words in particular, will be represented as meaning expression fragments that may contain variables as place holders for unlled portions of that fragment. Thus, the wordfrom might have the meaning FROM(x) while the wordJohn might have the meaning
John
.3 Crucial to many of the techniques discussed in part I of this thesis is a particular link-ing rule used to combine the meanings of words to form the meanings of phrases and whole utterances.This linking rule is adopted by numerous authors including Jackendo (1983, 1990), Pinker (1989), and Dorr (1990a, 1990b). Informally, the linking rule forms the meaning of the prepositional phrase from John by combining FROM(x) with
John
to form FROM(John
).This linking rule can be stated more formally as follows. Each node in a parse tree is assigned an expression to represent its meaning. The meaning of a terminal node is taken from the lexical entry for the word constituting that node. The meaning of a non-terminal node is derived from the meanings of its children. Every non-terminal node u has exactly one distinguished child called its head. The remaining children are called the complements of the head. The meaning of u is formed by substituting the meanings of each of the complements for all occurrences of some variable in the meaning of the head. To avoid the possibility of variable capture, without adding the complexity of a variable renaming process, we require that the meaning expression fragments associated with complements be variable-free. Notice that this rule does not stipulate which complements substitute for which variables. Thus if GO(x;TO(y)) is the meaning of the head of some phrase, and
John
is the meaning of its complement, the linking rule can produce either GO(x;TO(John
)) or GO(John
;TO(y)) as the meaning of the phrase.The only restriction on linking is that the head meaning must contain at least as many distinct variables as there are complements.
Some authors propose variants of the above linking rule that further species which variables are linked with which argument positions. For example, Pinker (1989) stipulates that the x in GO(x;y) is always linked to the direct internal argument. Irrespective of whether this is true, either for En-glish specically, or cross-linguistically in general, I refrain from adopting such restrictions here for two reasons. First, the algorithms presented in part I of this thesis apply generally to any expressions denoting meaning. They transcend a particular representation such as Jackendovian conceptual struc-tures. Linking restrictions such as those adopted by Pinker apply only to expressions constructed out of Jackendovian primitives. Since, for reasons to be discussed in part II of this thesis, the Jackendovian representation is inadequate, it does not make sense to base a learning theory on restrictions which are particular to that representation. Second, the learning algorithms presented here are capable of learning without making such restrictions. In fact, such restrictions could be learned if they were indeed true.
The standard motivation for assuming a faculty to be innate is the poverty of stimulus argument. This
1In part II of this thesis, I will discuss the inadequacies of both Jackendovian conceptual structure representations as well as substitution-based linking rules. Since much of the work in part I predates the work described in part II, it was formulated using the Jackendovian representation and associate linking rule as a matter of expedience, since that is what was prominent in the literature at the time. In more recent work, such as that described in section 4.3, I abandon the Jackendovian representation in favor of simple thematic role assignments, a much weaker form of semantic representation.
This also entails abandoning substitution-basedlinking in favor of-marking. In the future I hope to incorporate the more comprehensive semantic representations discussed in part II of this thesis into the techniques described in part I.2
This mitigates to some extent, the inadequacies of the Jackendovian representation. Nothing in part I of this thesis relies on the semantic content of a particular set of primitives. The techniques described here apply equally well to any representation provided that the representation adheres to a substitution-based linking rule. The inadequacies of such a linking rule still limit the applicability of these techniques, however.3
Throughout this chapter and the next I will use the somewhat pretentious phrase `the meaning ofx' to mean `the meaning expression associated withx'.
John slid the cup from Mary to Bill.
John
John slid the cup from Mary to BillCAUSE(x;GO(
cup
;[PathFROM(Mary
);TO(Mary
)]))
CAUSE(x;GO(y;[slid Pathu;v])) the cup
cup
Figure 3.1: A derivation of the meaning of the utteranceJohn slid the cup from Mary to Bill from the meanings of its constituent words using the linking rule proposed by Jackendo.
argument is falsied most strongly with a demonstration that something is learnable. It is important not to get carried away with our rationalist tendencies making unwarranted innateness assumptions in light of the rare observation of something that is indeed learnable by empiricist methods.
Some words, such as determiners and auxiliaries, appear not to have a meaning that can be easily characterized as meaning expressions to be combined by the above linking rule. To provide an escape hatch for semantic notions that fall outside the system described above, we provide the distinguished meaning symbol ?. Typically, words such as the will bear ? as their meaning. The linking rule is extended so that any complements that have?as their meaning are not substituted into the meaning of the head. This allows forming
cup
as the meaning ofthe cup whenthe has?andcup hascup
as their respective meanings. Using this linking rule, the meaning of phrases, and ultimately entire utterances can be derived from the meanings of their constituent words, given a parse tree annotated as to which children are heads and which are complements. A sample derivation is shown in gure 3.1. Note that the linking rule is ambiguous and can produce multiple meanings, even in the absence of lexical and structural ambiguity, since it does not specify which variables are linked to which complements. Also note that the aforementioned linking rule addresses only issues of argument structure. No attempt is made to support other aspects of compositional semantics such as quantication.Substitution-based linking rules are not new. They are widely discussed in the literature (cf. Jack-endo 1983, 1990, Pinker 1989, and Dorr 1990a, 1990b). The techniques in this thesis explore a novel application of such linking rules: the ability to use them in reverse. Traditionally, compositional seman-tics is viewed as a process for deriving utterance meanings from word meanings. This thesis will explore the opposite possibility: deriving the meanings of individual words from the meanings of utterances containing those words. I will refer to this inverse linking process as fracturing.
Fracturing is best described by way of an example. Assume some node in some parse tree has the meaning GO(
cup
;TO(John
)). Furthermore, assume that the node has two children. In this case there are four possibilities for assigning meanings to the children.Head Complement GO(x;TO(
John
))cup
GO(
cup
;x) TO(John
) GO(cup
;TO(x))John
GO(
cup
;TO(John
)) ?Note specically the last possibility of assigning?as the meaning of the complement. This option will always be present when fracturing any node. The above fracturing process can be applied recursively, starting at the root node of a tree, proceeding toward its leaves, to derive possible word meanings from the meaning of a whole utterance. More formally, fracturing a node u is accomplished by the following algorithm.
AlgorithmTo fracture the meaning expression associated with a node u into meaning expression fragments associated with the head of u and its complements:
Let e be the meaning of u. For each complement, either assign ?as the meaning of that complement or perform the following two steps.
1. Select some subexpression s of e and assign it as the meaning of that complement. The subexpression s must not contain any variables introduced in step 2.
2. Replace one or more occurrences of s in e with a new variable.
After all complements have been assigned meanings, assign e as the meaning of the head. 2 As stated above, the fracturing process is mediated by a parse tree annotated with head-child mark-ings. Given a meaningexpression e, one can enumerate all meaning expression fragments which can possi-bly link together to form e, irrespective of any parse tree for deriving e. Such a meaning fragment is called a submeaning of e. For example, the following are all of the submeanings of GO(
cup
;FROM(John
)).GO(
cup
;FROM(John
)) GO(x;FROM(cup John
))GO(x;FROM(y)) GO(x;y) GO(
cup
;FROM(x))GO(
cup
;x) FROM(John
)FROM(x)
John
?If an utterance has e as its meaning, then every word in that utterance must have a submeaning of e as its meaning. The set of submeanings for a meaning expression e can be derived by the following algorithm.
AlgorithmTo enumerate all submeanings of a meaning expression e:
Let s be some subexpression of e. Repeat the following two steps an arbitrary number of times.
1. Select some subexpression t of s not containing any variables introduced in step 2.
2. Replace one or more occurrences of t in s with a new variable.
Upon completion, s is a possible submeaning of e. Furthermore,?is a possible submeaning of every expression. 2
Both the fracturing algorithm, as well as the algorithm for enumerating all submeanings of a given
meaning expression, will play a prominent role throughout the remainder of part I of this thesis.