Disambiguation of pronoun reference has been an important issue for both theoretical and computational linguists. While linguistic theories on binding conditions eliminate impossible readings to a certain extent, many inter-sentential anaphora remain ambiguous. Nishiguchi (2011Nishiguchi ( , 2012aNishiguchi ( ,b, 2014Nishiguchi ( , 2016a,b) ,b) consider pronoun resolution as a social choice among discourse participants which obeys Arrow's Impossibility Theorem (Arrow 1963). This paper further discusses discourse update of Social Welfare Function which provides updated variable assignment.
In (1), she has multiple candidates for its antecedent-Emma, Lisa and Lisa's mom. Proximity and saliency of antecedents have been considered to be key factors to decide (Leass 1991). In (1), the most proximate antecedent her (Lisa)'s mom is identified to be the antecedent for she.
(1) Frances: ...Not while Emma's not here. You know Emma Billy: Mm. Frances: she's, she was walking with Lisa and I weren't there and her Mum sh-jus-, like she muc-, she mucks about a lot and she told Leigh that if he don't serve her he's gonna die, she's gonna punch him right! However, proximity does not always resolve referential ambiguity of pronouns. Him in (2a) unambiguously means someone other than the closest John-some discourse-salient entity. In (2b), the pronoun is ambiguous.
(2) a. John i likes him { * i/j =i} . b. John i said he {i/j} likes himself {i/j} .
In linguistic binding theory (Chomsky 1981, Reinhart 1983), antecedents are called binders, which bind bindees that are anaphoric pronouns, e.g., him or himself. Condition B is that pronouns must be free in their local domain, meaning that they are not bound by the antecedent by means of coindexing and c-commanding relation. C-command is roughly equivalent to precedence, with some restrictions.
However, (3) is ambiguous in four ways and can have either one of the following interpretations: i) John broke John's leg, ii) John broke Bill's leg, iii) Bill broke Bill's leg, or iv) Bill broke John's leg. He and his can be bound by either John or Bill. The binding theories have no way of disambiguating these pronouns since there is no way of knowing speaker intention. Proximity does not predict the different readings in (3) either.
(3) Anna: Bill j is a good goalkeeper.
Kim: John i said he i/j broke his i/j leg recently.
Although Social Choice Theory (Arrow 1963, Moulin 1988, Taylor 2005, Gaertner 2009) has only been briefly mentioned in van Rooij (2011) in relation with interadjective comparison, Arrow's Impossibility Theorem is obeyed in a social choice of pronominal reference. Typically, social choice theory explains collective decision making in case of voting and has solved the problems with majority decision. Preferences are ordering between alternatives and should satisfy the following axioms. When R stands for a knowledge of all pairs and x, y and z for alternatives, Axiom 1. For all x and y, either xRy or yRx.
Axiom 2. For all x, y, and z, xRy and yRz imply xRz.
Axiom 1 states that the relation R is connected-every candidate is related to each other. Relations that satisfy Axiom 2 are transitive. In (4), N, a finite set of individuals or voters, consists of five individuals and χ, a nonempty set of alternatives or candidates, has three members. Let L(χ) denote the set of all linear orders on χ. A profile R is a vector of linear orders, or preferences. R i is a vector of preferences of an individual i. N R x>y denotes the set of individuals that prefer the candidate x to y. Supposing R the profile given in this model, N R o>c is a set of people who prefers Obama to Clinton, that are, Anna, Heather and George (cf. Endriss 2016).
A social welfare function (SWF) F is a function which takes individual's preferences and returns collective preference. Arrow demonstrated that any SWF for three or more alternatives the following conditions must be a dictatorship. Condition 2 states that the relative ranking of two candidates remains unchanged regardless of other candidates.
Theorem 1 (General Possibility Theorem (Impossibility Theorem)). If there are at least three alternatives which the members of the society are free to order in any way, then every social welfare function satisfying Conditions 1 and 2 and yielding a social ordering satisfying Axioms 1 and 2 must be either imposed or dictatorial. Condition 3 (Nondictatorship). There is no individual i such that for every element in the domain of rule f, ∀x, y ∈ X: xP i y → xPy (Sen 1979) 2 Application to Pronoun Resolution When individuals I = {c, n}, candidates χ = {j, b}, Chris and Naomi's ordering is jR c b ∧ bR n j, denote the set of linear orders on χ by L(χ). Preferences (or ballots) are taken to be elements of L(χ). A profile R ∈ L(χ) I is a vector of preferences. SCF or voting rule is a function F : L(χ) I → 2 χ \∅ mapping a given profile to a nonempty set of winners; e.g., a singleton set {b} for (5). SWF is a function F : L(χ) I → L(χ) mapping any given profile to a (single) collective preference order. Although the preferences between the candidates vary between the individuals, SWF returns a single preference order and ambiguities are resolved during the conversation.
There are three possible antecedents for she in (1)-Emma (e), Lisa (l) and Lisa's mother (m). Let us say that Billy (b) prefers e to l, and also l to m to be the antecedent. On the other hand, the speaker Francis (f) prefers m to l, and l to e according to the proximity. All three candidates are ordered in accordance with Axiom 1, i.e., eR b l ∧ lR b m and mR f l ∧ lR f e. Transitivity also holds for pronoun antecedent preferences. Each of them implies eR b lR b m and mR f lR f e. SWF for pronoun resolution also meets Pareto condition. When the interpretation of the addressees agrees with the one of the speaker, the decision of the society follows. It is unlikely that pronouns refer to someone else other than speaker's intention and hearer's interpretation. A SWF F satisfies IIA if the relative social ranking of two alternatives only depends on their relative individual rankings. Let us say that the preference relations are denoted by R and R'. Assume that IIA does not hold and consider a dialogue in (7) where the relative rankings between Bob and John is affected by irrelevant candidate Victor's ranking. The social decision differs from the relative ranking between John and Bob of speaker and hearer, which does not happen, in (8). ( 6
Then, N R b>j = N R b>j implies bF(R)j ⇔ bF(R')j The speaker's decision on pronominal reference dictates the social preference. Even when there is disagreement or misunderstanding, the speaker corrects unifies interpretation in general, as in ( 9). Pronoun resolution is dominated, or dictated, by the speaker's meaning. Lemma 1. The social welfare function for pronoun resolution is IIA and Pareto but is dictatorial.
In linguistic literature, a variable assignment function g has been assumed to assign the referent to indices indexed to pronouns. For example, g may assign John to the variable x: g(x) = John. Now, g can be considered to be SCF which selects a referent for a pronoun socially. Let us define g and the space as in (11). The assignment function g is updated throughout the discourse as in ( 12).
(11) a. g = {<x, i>: x refers to i} b. Information state σ consists of Social Welfare Function F, Social Choice Function g for variable assignment, individual's preferences R, individuals in the discourse X, a set of indices such as i, a set of discourse participants V, and relation between decisions B. h, d, v, m, b, r} e. g5 = {<x10, l>, <x11, l>} S ={a, h, d, v, m, b, r, l, n} f. [[her y ]] g1 = a G is regarded as SCF. Also, the set of best elements S' can be called its choice set of the whole set of alternaties, and is denoted g(S', R) (cf. Sen 1979) R is a sequence of individual's preferences where R x is a preference ordering of x. F n BF n+1 BF n+2 ,...
Dynamic Predicate Logic (Groenendijk and Stokhof 1991) consider update semantics where two states differ with respect to variable assignment. When h[x]g, the state g is updated with respect to the assignment to x. The current paper consider an abstract function B between two SCFs. Parkes and Procaccia (2013) model dynamic decision making under constantly changing preferences using Markov decision processes, in which the states coincide with preference profiles and a policy corresponds to a social choice function.
In order to implement Dynamic Social Choice for pronoun disambiguation, speaker's intention needs to be detected from the text. The phrases such as "I mean" are used to resolve ambiguity of pronominal reference in the discourse. as in ( 17).
(17) '...And Sarah Morgan likes the idea of Angela marrying someone in the government.' McLeish considered this cold and rational assessment. 'When did you last see her? Miss Angela Morgan, I mean.' (BNC AB9)
Out of 18 instances of "I mean PNP" (PNP stands for proper name" ) found with the query "I mean N" in BNC, 7 instances had a preceding pronoun, the caraphor.