In this section, we will give justifications for three Winograd Schema answers from the list compiled by Ernest Davis. For these examples, we will not exactly follow the procedure outlined in the previous section. Instead of constructing all possible DRSs, we will only prove the coherence of the DRS that happens to be the correct interpretation. Nevertheless, it is clear in each case that with the background infor- mation and commonsense axioms provided, the coherence of the incorrect answer cannot be proven. Unfortunately, in order to establish that concretely, we would need to establish broad claims about what should not be entailed by the knowledge base. This will be discussed briefly for each example.
Yelling at Kevin Schema 18 on Davis’s list reads:
(19) Jim [yelled at/comforted] Kevin because he was so upset. Who was so upset? We will justify the answer Jim for the first instance as follows. First, we construct the DRS (for the correct interpretation):
a b c e1 e2 Jim(a) Kevin(b) c = a e1: a yell at b e2: c is upset because(e1, e2) .
Then we extract the correlation formula:
yell-at (a, b) ⊕ upset(c). We then list the background information
1. a = Jim 2. b = Kevin 3. c = a
and the relevant commonsense knowledge: 4. yell-at (x, y) ⊕ upset (x).
The derivation continues as follows:
5. yell-at (a, b) ⊕ upset (a). by substitution from 4 6. upset (a) ↔ upset (c) entailed by 3
7. yell-at (a, b) ⊕ upset (c) by replacement from 5 and 6. Here we required the commonsense knowledge that x yelling at y is positively cor- related with x being upset. The opposite answer would be provable if and only if the
formula
yell-at (a, b) ⊕ upset (b)
were provable from the knowledge base and background knowledge. This is actually quite a reasonable axiom—which raises a problem. See Section 5.3 for details. Envying Martin Schema 20 on Davis’s list reads:
(20) Pete envies Martin [because/although] he is very successful. Who is very successful?
We will justify the answer Martin for the first instance. First, we construct the DRS: a b c e1 e2 Pete(a) Martin(b) c = b e1: a envy b e2: c is successful because(e1, e2) .
Then we extract the correlation formula:
envy(a, b) ⊕ successf ul(c). We then list the background information
1. a = Pete 2. b = Martin 3. c = b
and the relevant commonsense knowledge: 4. envy(x, y) ⊕ successful (y).
The derivation continues as follows:
5. envy(a, b) ⊕ successful (b) by substitution from 4 6. successful (b) ↔ successful (c) entailed by 3
7. envy(a, b) ⊕ successful (c) by replacement from 5 and 6. Here we required the commonsense knowledge that x envying y is positively corre- lated with y being successful. The incorrect answer Pete would be provable if
envy(a, b) ⊕ successful (a) (4.2) were provable from the knowledge base. One way to ensure that this is not the case is to ensure that
envy(a, b) successful (a)
is entailed by the knowledge base—this axiom seems reasonable, expressing that successful people tend to be less envious. Given that the formulas on either side are not entailed (which they definitely should not be, given that a and b are new object constants) this formula is contradictory (by the probabilistic semantics) to the correlation formula corresponding to the answer we would wish to avoid. Thus, because the calculus is sound, as long as the knowledge base is consistent, the undesirable formula (4.2) would not be derivable.
Ceding the Election Schema 139 on Davis’s list reads:
(21) Kirilov ceded the election to Shatov because he was [more/less] popular. Who was [more/less] popular?
We will justify the answer Shatov for the first instance of this schema. First, we construct the DRS:
e1 a b c e2 d f Kirilov (a) Shatov (b) election(c) d = b f = a e1: a cede c to b
e2: d is more popular than f because(e1, e2)
.
Here, we have assumed a slightly more complicated DRS that relies on the more complicated syntactic analysis that would arise from the comparative more popular. This is comparing against something, and whether the syntactic analysis resolves the issue as one of anaphora or of ellipsis, the result must involve a referent to denote that thing. We use f . Then this problem involves resolving what amounts to two anaphors: d and f . Of all of the possible choices, we will jointly justify the choice of d = b and f = a.
So the next step is to extract the correlation formula: cede-to(a, c, b) ⊕ more-popular (d, f ). We then list the background information
1. a = Kirilov 2. b = Shatov 3. election(c) 4. d = b 5. f = a
and the relevant commonsense knowledge:
The derivation continues as follows:
7. cede-to(a, c, b) ∧ election(c) ⊕ more-popular (b, a) by substitution from 6 8. (cede-to(a, c, b) ∧ election(c)) ↔ cede-to(a, c, b) entailed by 3
9. cede-to(a, c, b) ⊕ more-popular (b, a) by replacement from 7 and 8 10.more-popular (b, a) ↔ more-popular (d, f ) entailed by 4 and 5
11.cede-to(a, c, b) ⊕ more-popular (d, f ) by replacement from 9 and 10. Here we required the commonsense knowledge that x ceding an election z to y is positively correlated with y being more popular than z. The knowledge that would prevent the incorrect answer might as well be another axiom:
cede-to(x, y, z) ∧ election(y) more-popular (x, z). This in turn may even be justified by the simple facts
∀x∀y(more-popular (x, y) ↔ less-popular (y, x)) and
∀x∀y(more-popular (x, y) ↔ ¬less-popular (x, y))
(though the latter admittedly ignores the case that the two are equally popular). In this example, the Correlation Calculus has solved a slightly more complex decision than the WSC problem poses: it has resolved two anaphors instead of just the one. This shows how the approach may have explanatory power for discourse coherence more generally, rather than just in the Winograd Schema Challenge.
Nevertheless, the formulation of this last example (indeed, all of these exam- ples) might raise some questions. For instance, is it correct to include the election constraint on the z argument to cede-to? Or, should we instead say that cede is polysemic, and has one specific meaning (and thus, predicate in the first-order sig- nature) corresponding to ceding elections in particular (thus avoiding the need for the election constraint altogether)? Or perhaps worse, the more-popular predicate seems to miss out on the deeper common structure of gradable adjectives or com- paratives. These are important questions for the knowledge-representation side of a system to solve the Winograd Schema Challenge. But they are not as important to the Correlation Calculus: the fact that these proofs can be done in a relatively
simple system is a first step to showing that they may be doable in systems with richer ontologies. We will discuss some of these issues in Chapter 5.
Chapter 5
Discussion
5.1
Beyond “Because”
Though we developed the system with the word “because” in mind, not all of the Winograd Schema problems take this form. However, there’s a fairly straightforward way of generalizing the approach. In the first 100 Winograd Schemas of the collection compiled by Ernest Davis,1
• 96 put the pronoun in a separate independent clause,
• 71.5 of the answers seem to be justifiable by positive correlation,2 and
• of those justifiable by positive correlation, the clause containing the pronoun is connected to the discourse by one of the following: a full stop, a semicolon, because, so, if, since, when, but, now, and, and then, until.
So it seems that generalizing the approach simply to apply to more connectives could be a fruitful endeavor. The task would not be simple: for example but, which ap- pears above, might be expected to indicate negative correlation. But examples with but, 4.5 seemed justifiable by negative correlation, 1 seemed justifiable by positive correlation, and 6 did not seem to be related to correlation—so it is clearly a difficult case. The connectives though and although, though uncommon, seemed to indicate negative correlation in their 2.5 occurrences. All of the other connectives mentioned above appeared in no schemas that could be justified by negative correlation, and
1
https://www.cs.nyu.edu/davise/papers/WS.html
2
mostly schemas that could be justified by positive correlation. For details, the an- notations may be found at cs.utexas.edu/~julianjm/wsc-annotations.txt.
Even in cases where positive correlation seems to apply, the technique discussed in this thesis may be difficult to generalize to cases where the observed correlation is not between statements of eventuality, or the correlated statements appear far apart in the discourse. But the most difficult test of the ability of this method to generalize will be how it may be adapted to deal with richer and more general systems of commonsense knowledge, or the kind of commonsense knowledge that may be obtained from data. What these systems or this knowledge may look like is an open question for future work.
However, the Correlation Calculus is not just a system for justifying WSC an- swers. It is primarily a system for establishing discourse coherence on the basis of commonsense knowledge. Especially because of the state of the rest of the technol- ogy needed to actually apply the approach to the WSC, it is more useful to think of the Correlation Calculus approach as a contribution to our thinking on discourse coherence.