Towards a Model-Theoretic View of Narratives
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Towards a Model-Theoretic View of Narratives
Louis Castricato∗ Stella Biderman∗ Rogelio E. Cardona-Rivera David Thue
Georgia Tech Georgia Tech University of Utah Carleton University
EleutherAI EleutherAI rogelio@cs.utah.edu david.thue@carleton.ca
lcastric@gatech.edu stella@eleuther.ai
Abstract including logic, constraint satisfaction, and auto-
mated planning. These include efforts to model
In this paper, we propose the beginnings of a creative storytelling as a search process (Riedl and
formal framework for modeling narrative qua Young, 2006; Thue et al., 2016), generating sto-
arXiv:2103.12872v1 [cs.CL] 23 Mar 2021
narrative. Our framework affords the ability ries with predictable effects on their comprehen-
to discuss key qualities of stories and their
sion by audiences (Cardona-Rivera et al., 2016),
communication, including the flow of informa-
tion from a Narrator to a Reader, the evolu- and modeling story understanding through human-
tion of a Reader’s story model over time, and constrained techniques (Martens et al., 2020).
Reader uncertainty. We demonstrate its appli-
However, despite excellent advances, few works
cability to computational narratology by giv-
ing explicit algorithms for measuring the ac-
have offered a thorough conceptual account of nar-
curacy with which information was conveyed rative in a way that affords reconciling how differ-
to the Reader and two novel measurements of ent research programs might relate to each other.
story coherence. Without a foundation for shared progress, our com-
munity might strain to determine how individual
1 Introduction results may build upon each other to make progress
on story understanding AI that performs as robustly
Story understanding is both (1) the process through and flexibly as humans do (Cardona-Rivera and
which a cognitive agent (human or artificial) men- Young, 2019). In this paper, we take steps toward
tally constructs a plot through the perception of a such a foundation.
narrated discourse, and (2) the outcome of that pro-
cess: i.e., the agent’s mental representation of the We posit that such a foundation must acknowl-
plot. The best way to computationally model story edge the diverse factors that contribute to an artifact
understanding is contextual to the aims of a given being treated as a narrative. Key among these fac-
research program, and today we enjoy a plethora tors is a narrative’s communicative status: unlike
of artificial intelligence (AI)-based capabilities. more-general natural language generation (cf. Gatt
Data-driven approaches—including statistical, and Krahmer, 2018), an audience’s belief dynam-
neural, and neuro-symbolic ones—look to narrative ics—the trajectory of belief expansions, contrac-
as a benchmark task for demonstrating human-level tions, and revisions (Alchourrón et al., 1985)—is
competency on inferencing, question-answering, core to what gives a narrative experience its qual-
and storytelling. That is, they draw associations ity (Herman, 2013). Failure to engage with nar-
between event (Chambers and Jurafsky, 2008), ratives on these grounds risks losing an essential
causal (Li et al., 2012), and purposive (Jiang and aspect of what makes narrative storytelling a vi-
Riloff, 2018) information extracted from textual brant and unique form of literature.
or visual narrative corpora to answer questions or To that end, we define a preliminary theoretical
generate meaningful stories that depend on infor- framework of narrative centered on information
mation implied and not necessarily expressed by entropy. Our framework is built atop model theory,
stories (e.g. Roemmele et al., 2011; Mostafazadeh the set-theoretic study of language interpretation.
et al., 2016; Martin et al., 2018; Kim et al., 2019). Model theory is a field of formal logic that has been
Symbolic approaches seek to understand narra- used extensively by epistomologists, linguists, and
tive, its communication, and its effect by using other theorists as a framework for building logical
AI techniques as computational modeling tools, semantics.Contributions In this paper, we propose the be- 2.2 Narratives as Mental Artifacts
ginnings of a formal framework for modeling nar-
Story psychologists frame the narration as instruc-
rative qua narrative. Our framework includes the
tions that guide story understanding (Gernsbacher
ability to discuss the flow of information from a
et al., 1990). The fabula in the audience’s mind
Narrator to a Reader, the evolution of a Reader’s
is termed the situation model—a mental repre-
story model over time, and Reader uncertainty. Our
sentation of the virtual world and the events that
work is grounded in the long history of narratology,
have transpired within it, formed from informa-
drawing on the rich linguistic and philosophical
tion both explicitly-narrated and inferable-from a
history of the field to justify our notions.
narration (Zwaan and Radvansky, 1998). The situa-
We use our framework to make experimentally
tion model itself is the audience’s understanding; it
verifiable conjectures about how story readers re-
reflects a tacit belief about the fabula, and is manip-
spond to under-specification of the story world and
ulated via three (fabula-belief) update operations.
how to use entropy to identify plot points. We
These work across memory retrieval, inferencing,
additionally demonstrate its applicability to compu-
and question-answering cognition: (1) expansion,
tational narratology by giving explicit algorithms
when the audience begins to believe something,
for measuring the accuracy with which informa-
(2) contraction, when the audience ceases to be-
tion was conveyed to the Reader and two novel
lieve something, and (3) revision, when the au-
measurements of story coherence.
dience expands their belief and contracts newly
inconsistent beliefs.
2 Pre-Rigorous Notions of Narrative
Before we can begin to define narrative in a formal 2.3 Narratives as Received Artifacts
sense, we must examine the intuitive notions of To the post-structuralist, the emphasis that the psy-
what narrative is supposed to mean. While we chological account puts on the author is fundamen-
cannot address all of the complexity of narratology tally misplaced (Barthes, 1967). From this point
in this work, we cover key perspectives. of view, books are meant to be read, not written,
and how they influence and are interpreted by their
2.1 Narratives as Physical Artifacts
readers is as essential to their essence as the inten-
We begin with the structuralist account within nar- tion of the author. In “Death of the Author” Barthes
ratology; it frames a narrative (story) as a commu- (Barthes, 1967) reinforces this concept by persis-
nicative, designed artifact—the product of a narra- tently referring to the writer of a narrative not as its
tion, itself a realization (e.g. book, film) of a dis- creator or its author, but as its sculptor - one who
course (Hühn and Sommer, 2013). The discourse shapes and guides the work but does not dictate to
is the story’s information layer (Genette, 1980): an their audience its meaning.
author-structured, temporally-organized subset of
the fabula; a discourse projects a fabula’s infor- 3 A Model Theoretic View of Narrative
mation. The fabula is the story’s world, which in-
cludes its characters, or intention-driven agents; lo- The core of our framework for modeling narrative
cations, or spatial context; and events, the causally-, come from a field of mathematical logic known as
purposely-, and chronologically-related situation model theory. Model theory is a powerful yet flexi-
changes (Bal, 1997; Rimmon-Kenan, 2002). ble framework that has been a heavily influential on
As a designed artifact, a narrative reflects au- people working in computer science, literary theory,
thorial intent. Authors design the stories they tell linguistics, and philosophy (Sider, 2010). Despite
to affect audiences in specific ways; their designs the centrality of model theory in our framework,
ultimately target effecting change in the minds of a deep understanding of the topic is not necessary
audiences (Bordwell, 1989). This design stems to work with it on an applied level. Our goal in
from the authors’ understanding of their fabula and this section is thus to give an intuitive picture of
of the narration that conveys its discourse. When model theory that is sufficient to understand how
audiences encounter the designed artifact, they per- we will use it to talk about narratives. We refer
form story understanding: they attempt to mentally an interested reader to Sider (2010); Chang and
construct a fabula through the perception of the Keisler (1990) for a more complete presentation of
story’s narration. the subject.3.1 An Outline of Model Theory in a particular application by simply adding them
to the underlying logic.
The central object of study in model theory is a
“model.” Loosely speaking, a model is a world in 3.2 Story-World Models and the Fabula
which particular propositions are true. A model As detailed in section 2, the fabula and story-world
has two components: a domain, which is the set of (i.e. situation) model are two central components of
objects the model makes claims about, and a theory, how people talk about storytelling. In this section
which is a set of consistent sentences that make we introduce formal definitions of these concepts
claims about elements of the domain. Models in and some of their properties.
many ways resemble fabulas, in that they describe
Definition 3.1. A language, L, is a set of rules for
the relational properties of objects. Model theory,
forming syntactically valid propositions. In this
however, requires that the theory of a model be
work we will make very light assumptions about L
complete – every expressible proposition must be
and leave its design largely up to the application.
either true or false in a particular model.
Meanwhile, our notion of a fabula can be incom- A language describes syntactic validity, but
plete - it can leave the truth of some propositions doesn’t contain a notion of truth. For that, we need
undefined. This means that the descriptions we are a model.
interested in do not correspond to only one model, Definition 3.2. A story world model, S, over a
but rather that there is an infinite set of models that language L is comprised of two parts: a domain,
are consistent with the description. This may seem which is the set of things that exist in the story,
limiting, but we will show in Section 6 that it is and an interpretation function, which takes logical
actually amenable to analysis. formulae and maps them to corresponding objects
As an example, consider a simple world in which in the domain. In other words, the interpretation
people can play cards with one another and wear function is what connects the logical expression “A
clothes of various colours. The description “Jay causes B” to the signified fact in the world that the
wears blue. Ali plays cards with Jay.” is incomplete thing we refer to as A causes the thing we refer to
because it does not say what colours Ali wears nor as B.
what other colours Jay wears. This description is Definition 3.3. The theory of a story world model,
consistent with a world in which there are charac- S, is the set of all propositions that are true in S. It
ters other than Jay and Ali or colours other than is denoted S̃. When we say “P is true in the model
blue (varying the domain), as well as one where S” we mean that P ∈ S 0 .
additional propositions such as “Ali wears blue.” Formalizing the concept of a fabula is a bit trick-
hold (varying the theory). ier. Traditionally, fabulas are represented diagram-
Although we learn more about the domain and matically as directed graphs. However this rep-
the theory of the narrator’s model as the story goes resentation gives little insight into their core at-
on, we will never learn every single detail. Some tributes. We posit that, at their core, fabulas are
of these details may not even be known to the nar- relational objects. Specifically, they are a collec-
rator! For this reason, our framework puts a strong tion of elements of the domain of the story-world
emphasis on consistency between models, and on model together with claims about the relationships
the set of all models that are consistent with a par- between those objects. Additionally, there is a
ticular set of statements. sense in which the fabula is a “scratch pad” for
Another very important aspect of model theory the story-world model. While a reader may not
is that it is highly modular. Much of model theory even be able to hold an entire infinite story-world
is independent of the underlying logical semantics, model in their head, they can more easily grasp the
which allows us to paint a very general picture. If distillation of that story-world model into a fabula.
a particular application requires augmenting the Definition 3.4. A reasoner’s fabula for a story
storytelling semantics with additional logical oper- world model S, denoted F , is a set of propositions
ators or relations, that is entirely non-problematic. that makes claims about S. A proposition P is a
For example, it is common for fabulas to contain member of F if it is an explicit belief of the rea-
Cause(X, Y) := “X causes Y” and Aft(X, Y) := “Y soner about the narrative that the reasoner deems
occurs after X.” Although we don’t specifically de- important to constructing an accurate story-world
fine either of these relations, they can be included model.4 Conveying Story Information the Reader to induce experiences such as suspense,
fear, and anticipation - the ability to discuss the
An important aspect of stories is that they are a
accuracy and consistency of the telling of the story
way to convey information. In this section, we
is an essential part of analyzing a narrative.
will discuss how to formalize this process and what
The d0 arrow in our diagram suggests a reason-
we can learn about it. Although stories can be
able criteria for accurate conveyance: a story is ac-
constructed and conveyed in many different ways,
curately conveyed if the path SN → FN → FR →
we will speak of a Narrator who tells the story and
SR and the path SN 99K SR compute the same (or,
a Reader who receives it for simplicity.
in practice, similar) functions. In mathematics, this
The core of our model of storytelling as an act property of path-independence is known as commu-
of communication can be seen in Figure 1. tativity and the diagram is called a “commutative di-
agram” when it holds. For the purposes of narrative
SN d0 SR
work, the essential aspect is that the arrows “map
corresponding objects correspondingly.” That is, if
φ ψ
a story is accurately conveyed from N to R then for
each proposition P ∈ SN there should be a corre-
FN d FR sponding P 0 ∈ SR such that the interpretations of
P and P 0 (with respect to their respective models)
Figure 1: Commutative diagram outlining storytelling
have the same truth value and (φ ◦ d ◦ ψ)(P ) = P 0 .
In other words, P and P 0 make the same claims
This diagram represents the transmission of in-
about the same things.
formation from the Narrator’s story-world to the
Reader’s, with each arrow representing the trans- 4.2 Time-Evolution of Story-World Models
mission from one representation to another. In an The transference of information depicted in fig. 1
idealized world, stories would be conveyed by d0 : gives rise to a straightforward way to understand
straight from the story world of the narrator (SN ) how the Reader gains knowledge during the course
to the story world of the reader (SR ). In actuality, of the story and incorporates new information
narrators must convey their ideas through media1 . into their existing story-world model. One pass
To do this, the narrator compresses their mental through the diagram from SN to SR represents
story world (via φ) into a fabula (FN ) which is “one time step” of the evolution of the Reader’s
then conveyed to the reader via speech, writing, world model2 .
etc. The conveyance of the fabula as understood Iterating this process over the the entire work
by the Narrator (FN ) to the fabula as understood gives a time series of story-world models, SR (t),
by the Reader (FR ) is denoted in our diagram by with SR (i) representing the Reader’s story-world
d. d is in many ways the real-world replacement model at time t = i. We are also typically inter-
for the function d0 the Narrator is unable to carry ested in how the story-world model changes over
out. Once the discourse has been consumed by the time, as the Reader revises their understanding of
Reader, the Reader then takes their reconstructed the story-world through consuming the discourse.
fabula (FR ) and uses the received information to This will be the subject of the next section.
update their story world model (SR , via ψ).
5 A Detailed Look at Temporal
4.1 Accurately Conveying Information Evolution, with Applications to Plot
Often times, information conveyed from the Narra-
A common accepted notion in narratology literature
tor to the Reader is “conveyed correctly.” By this,
is that at any given moment a reader contains a po-
we mean that the essential character of the story
tentially infinite set of possible worlds. Determin-
was conveyed from the Narrator to the Reader in
ing which of these worlds agree with each other is
such a way that the Reader forms accurate beliefs
a required attribute for consuming discourse. How
about the story-world. While accuracy is not al-
do we discuss the notion of collapsing possible
ways a primary consideration - some stories fea-
worlds upon acquiring new knowledge?
ture unreliable narrators or deliberately mislead
2
For simplicity we will speak of this as a discrete time
1 0
Nevertheless, having a conception of d is very important series, though for some media such as film it may make sense
on a formal level as we will see later. to model it as a continuous phenomenon.Assume that we have a narrator, N , and reader This in turn brings us to the notion of com-
R with fabulas FN and FR respectively. Given our pression and expansion. Namely that ψ, if left
definition of a story world model, S, we define S(t) unchecked, will continuously expand the fabula. In
as the set of all world models that satisfy FR (t). Let turn ζR is given the goal of compressing the story
ρt+1 refer to the set of formulae that are contained worlds that ψ produces by looking at the resulting
in FR (t + 1)\FR (t). Let transition functions that best match the author’s
0
intent.3
SR (t + 1) = SR (t + 1) ∩ SR (t)
5.2 Plot Relevance
and similarly
Stories contain many different threads and facts,
0
S̃R (t + 1) = S̃R (t + 1) ∩ S̃R (t) and it would be nice to be able to identify the ones
that are relevant to the plot. We begin with the idea
refer to the shared world models between the two
of the relevance of one question to another.
adjacent time steps. Note that it must follow ∀ρ ∈
Pt+1 , ∀s ∈ S̃0R (t + 1), ρ ∈ s. That is to say, Definition 5.1. Consider a question about a story,
the story worlds that remain between the two time q, of the form “if A then B" with possible values for
steps are the ones that agree on the propositions A = {T, F } and possible values for B = {T, F }.
added by consuming FN (t + 1). Since this can be We say that the relevance of B to A given some
repeated inductively, we can assume that for any prior γ is
such t we have that all such models agree on all
such provided propositions. H(A = ai |γ) − H(B = bj |A = ai , γ) (1)
Something to note that for ρ ∈ Pt+1 , ρ will
always be either true or false in S̃R (t)- regardless where ai and bj are the true answers to A and B
if it is expressed in the fabula or not since S̃R (t) is and H refers to binary entropy.
the logical closure of SR (t).
Note that the relevance of B to A depends on
5.1 Collapse of Worlds over Time the true answers. This is perhaps surprising, but
after some consideration it should be clear that this
Something to note is that a set of story worlds
has to be true. After all, the causal relationship be-
S̃R (t) does not provide us a transition function
tween A and B could depend on the true answers!
to discuss how the world evolves over time. Fur-
Consider the case where A is “is Harry Potter the
thermore, there is no reasonable way to infer
prophesied Heir of Slytherin?” and B is “can Harry
S̃R (t) 7→ S̃R (t + 1), as S̃R (t) provides no informa-
Potter speak Parseltongue because he is a descen-
tion about the actions that could inhibit or allow for
dent of Slytherin?” If Harry is a blood descendant
this transition- it simply provides us information
of Slytherin and that’s why he can speak Parsel-
about if a proposition is true within our story world.
tongue, then B is highly relevant to A. However,
To rectify this, we need to expand our commutative
the actual truth of the matter is that Harry’s abili-
diagram to act cross-temporally. The full diagram
ties are completely independent of his heritage and
can be found in the appendix.
arose due to a childhood experience. Therefore B
Let ζN denote the transition function from FN (t)
does not in fact have relevance to A even though it
to FN (t + 1). Define ζR likewise. See Figure 2
could have had relevance to A.
on page 10. Note that there is no inherent general
form of ζN or ζR as they are significantly context Having defined a notion of the relevance of Ques-
dependent. One can think of them as performing tion A to Question B, our next step is connecting to
graph edits on FN and FR respectively, to add the existing narratological analysis. Consider Barthes’
new information expressed in SN (t + 1) for ζN notion of kernels and satellites.(Barthes and Duisit,
and (d ◦ φ)(SN (t + 1)) for ζR . 1975)
The objective of ζR in turn is to guide the fab- Definition 5.2. A kernel is a narrative event such
ula to reach goals. This imposes a duality of ψ that after its completion, the beliefs a reader
and ζR . ψ attempts to generate the best candidate
3
story worlds for the reader’s current understanding, There is no single best way to define an author’s intent.
For instance, we could have easily said that ψ denotes author
where as ζR eliminates them by the direction the intent while ζR determines which intents are best grounded in
author wants to go. reality. The choice, however, needs to be made.holds as they pertain to the story have drastically “small” sets. Again we develop the theory of ultrafil-
changed.4 ters only to the extent that we require, and refer an
Definition 5.3. A satellite is a narrative event that interested reader to a graduate text in mathematical
supports a kernel. They are the minor plot points logic for a thorough discussion.
that lead up to major plot points. They do not result Definition 6.1. Let Q be a set of sentences that
in massive shift in beliefs. make claims about a narrative. A non-empty col-
lection Fw ⊆ P(Q) is a weak filter iff
Of importance to note is that satellites imply
the existence of kernels, e.g. small plot points will 1. ∀X, Y ∈ P(Q), X ∈ Fw and X ⊆ Y ⊆
explain and lead up to a large plot point, but kernels P(Q) implies Y ∈ Fw
do not imply the existence of satellites- kernels do
not require satellites to exist. One can think of this 2. ∀X ∈ P(Q), X 6∈ Fw or P(Q)\X 6∈ Fw
as when satellites exist kernels must always exist We say that Fw is a weak ultrafilter and denote
on their boundary whether they are referred to in it UF w if the second requirement is replaced by
the text or not. ∀X ∈ P(Q), X ∈ Fw ⇐⇒ P(Q)\X 6∈ Fw
A set of satellites, s = {s1 , . . . , sn }, is said (Askounis et al., 2016).
to be relevant to a kernel, k, if after the kernel’s A reader’s beliefs at time t defines a weak filter
competition, the reader believes that the set of ques- over the set of possible story-world models {SR i }.
tions posed by k are relevant to their understanding Call this filter Fw , dropping the t when it is clear
of the story world given prior s. dh Take note from context. Each element U ∈ Fw is a set of
of the definition of relevance. Simply put, A de- story world models that define a plausibility. This
notes the questions that define some notion of story plausibility describes a set of propositions about the
world level coherency where as B denotes the set story that the reader thinks paints a coherent and
of questions that define some notion of transitional plausible picture. Formally, a plausibility identified
coherency. with the largest set of sentences that is true for every
model in U , or ∩S∈U T (S) where T (S) denotes
6 Possible Worlds and Reader the set of true statements in S. That is, the set of
Uncertainty plausible facts.
So far we have spoken about the Reader’s story- The intuition for the formal definition of a weak
world model as if there is only one, but in light filter is that 1. means that adding worlds to an
of the discussion in section 3 it is unclear it truly element of the filter (which decreases the number
makes sense to do so. In actuality, the Reader never of elements in ∩S∈U T (S)) doesn’t stop it from
learns to “true story-world model” (insofar as one describing a plausibility since it is specifying fewer
can even be said to exist). Rather, the Reader has facts; and that 2. means that it is not the case
an evolving set of “plausible story-world models” that both P and ¬P are plausible. It’s important
that are extrapolated based on the incomplete in- to remember that membership in Fw is a binary
formation conveyed in the story. The purpose of property, and so a statement is either plausible or is
this section is to detail how these “plausibilities” not plausible. We do not have shades of plausibility
interact with each other and with plausibilities at due to the aforementioned lack of a probability
other time steps. distribution.
It likely seems natural to model the Reader’s un- As a framework for modeling the Reader’s un-
certainty with a probabilistic model. Unfortunately, certainty, weak filters underspecify the space of
the topological structure of first-order logic makes plausible story world as a whole in favor of captur-
that impossible as there is no way to define a prob- ing what the reader “has actively in mind” when
ability distribution over the set of models that are reading. This is precisely because the ultrafilter
consistent with a set of sentences. Instead, we are axiom is not required, and so for some propositions
forced to appeal to filters, a weaker notion of size neither P nor ¬P are judged to be plausible. When
that captures the difference between “large” and asked to stop and consider the truth of a specific
proposition, the reader is confronted with the fact
4
The notion of "drastic" is equivalent to "majority." To rig- that there are many ways that they can precisify
oriously define Barthes’ Kernel, and hence Barthes’ Cardinal,
we would require ultraproducts- which is outside of the scope their world models. How a Reader responds to this
of this paper. confrontation is an experimental question that weleave to future work, but we conjecture that with express the entropy of this as
sufficient time and motivation a Reader will build a
weak ultrafilter UF w that extends Fw and takes a H(Ps0 (q)) = H(q|s0 )
position on the plausibility of all statements in the = H(A = T |s0 ) − H(B = bj |A = T, s0 )
logical closure of their knowledge.
Once the Reader has fleshed out the space of Therefore averaging over H(Ps0 (q)) for all q ∈ Q
plausibilities, we can use UF w to build the ultra- is equivalent to determining the relevance of our
product of the Reader’s story-world models. An implication to our hypothesis. This now brings us
ultraproduct (Chang and Keisler, 1990) is a way to EWC, or entropy of world coherence. These
of using an ultrafilter to engage in reconciliation implications are of the form “Given something in
and build a single consistent story world-model out the ground truth that all story worlds believe, then
of a space of plausibilities. Intuitively, an ultra- X" where X is a proposition held by the majority
product can be thought of as a vote between the of story worlds but not all. We define EWC as
various models on the truth of individual propo-
sitions. A proposition is considered to be true in 1 X
EWC(s0 , Q) = Ps0 (q)
the ultraproduct if and only if the set of models in |Q|
q∈Q
which it is true is an element of the ultrafilter. We
conjecture that real-world rational agents with un- 7.2 Entropy of Transitional Coherence
certain beliefs find the ultraproduct of their world Note our definition of plot relevance. It is partic-
models to be a reasonable reconciliation of their ularly of value to not only measure the coherency
beliefs and that idealized perfectly rational agents of the rules that govern our story world but also
will provably gravitate towards the ultraproduct as to measure the coherency of the transitions that
the correct reconciliation. govern it over time. We can define a similar notion
to EWC, called Entropy of Transitional Coherence,
7 Applications to Computational
which aims to measure the agreement of how be-
Narratology
liefs change over time. In doing so, we can accu-
Finally, demonstrate that our highly abstract frame- rately measure the reader’s understanding of the
work is of practical use by using it to derive explicit laws that govern the dynamics of the story world
computational tools of use to computational narra- rather than just the relationships that exist in a static
tologists. frame.
To understand ETC we must first delve into the
7.1 Entropy of World Coherence dynamics of modal logic. Note that for a proposi-
Firstly it is important to acknowledge that a reader tion to be “necessary” in one frame of a narrative,
can never reason over an infinite set of worlds. it must have been plausible in a prior frame. (Sider,
Therefore, it is often best to consider a finite sam- 2010) Things that are necessary, the reader knows;
ple of worlds. Given the (non-finite) set of story hence, the set of necessary propositions is a subset
worlds, S(t), there must exist a set s0 ⊂ UF w (t) of a prior frame’s possible propositions.
such that every element in s0 is one of the "more We must define a boolean lattice to continue
likely" interpretations of the story world. This no- Definition 7.1. A boolean lattice of a set of propo-
tion of more likely is out of scope of this paper; sitions, Q, is a graph whose vertices are elements
however, in practice more likely simply denotes of Q and for any two a, b ∈ Q if a =⇒ b then
probability conditioned from S̃(t − 1). there exists an edge (a, b) unless a = b
It is equally important to note that every ele-
ment of s0 , by definition, can be represented in the Something to note is that a boolean lattice is a
reader’s mind by the same fabula, say F (t). Let Q directed acyclic graph (DAG) and as such as source
be some set of implications that we would like to vertices with no parents. In the case of boolean lat-
determine the truth assignment of. Let Ps0 (q) refer tices, a source vertex refers to an axiom, as sources
to the proportion of story worlds in s0 such that q is are not provable by other sources.
true.5 Clearly, Ps0 (q) is conditioned on s0 . We can q is true in the majority of story worlds, as defined by our
ultrafilter. Similarly, let P (q) = 0 otherwise. For those with
5
An equivalent form of P (q) exists for when we do not prior model theory experience, P (q) = 1 if q holds in an
have a form of measure. Particularly, define P (q) = 1 when ultraproduct of story world models.We define one reader at two times, denoted Reader’s active beliefs about the story can update
UF w (t) and UF w (t0 ) where t0 < t. We define as they receive that information.
a filtration of possible worlds s0 (t0 ) similar to how Thanks to this precision, we were able to define
we did in the previous section. a rigorous and measurable notion of plot relevance,
Given W (t) ∈ UF w (t), a ground truth at time which we used to formalize Barthes’ notions of
t, we restrict our view of W (t) to the maximal PW kernels and satellites. We also give a novel formu-
of time t0 . This can be done by looking at lation and analysis of Reader uncertainty, and form
experimentally verifiable conjectures on the basis
W 0 = argmaxW (t)∩s0 |B(W (t)) ∩ (∩s∈s0i B(s))|
i of our theories. We further demonstrated the value
Reason being is that it does not make sense to of our framework by formalizing two new narrative-
query about propositions that are undefined in prior focused measures: Entropy of World Coherence
frames. This effectively can be viewed as a pull- and Entropy of Transitional Coherence, which mea-
back through the commutative diagram outlined sure the agreement of story world models frames
previously. See Figure 2 on page 10. Something to and faithfulness of ζR respectively.
note however is that this pullback is not necessary Our framework also opens up new avenues for
for ETC in the theoretical setting, as all world mod- future research in narratology and related fields.
els would agree on any proposition not contained in While we were unable to explore their conse-
their respective Boolean lattices- this is not the case quences within the scope of this paper, the formula-
when testing on human subjects. Human subjects tion of narratives via model theory opens the door
would be more likely to guess if they are presented to leveraging the extensive theoretical work that’s
with a query that has no relevance to their current been done on models to narratology. The analysis
understanding. (Trabasso et al., 1982; Mandler and of the temporal evolution of models in section 5
Johnson, 1977) suggests connections with reinforcement learning
We can however similarly define ETC by uti- for natural language understanding. In section 6
lizing W 0 as our ground truth with EWC. Since we make testable conjectures about the behavior
W 0 is not the minimal ground truth for a particu- of Reader agents and in section 7 we describe how
lar frame, it encodes information about the ground to convert our theoretical musings into practical
truth where the narrative will be going by frame t. metrics for measuring consistency and coherency
Therefore, define Q similarly over time t0 relative of stories.
to W 0 . We can also use this to define Ps0 (t0 ) (q)
∀q ∈ Q. We denote ETC as
References
1 X
ETC(s0 (t0 ), Q) = Ps0 (t0 ) (q) C. E. Alchourrón, P. Gärdenfors, and D. Makinson.
|Q|
q∈Q 1985. On the Logic of Theory Change: Partial
Meet Contraction and Revision Functions. Journal
ETC differs from EWC in the form of implica- of Symbolic Logic, pages 510–530.
tions that reside in Q. Particularly since ETC wants
to measure the coherency of a reader’s internal tran- Dimitris Askounis, Costas D. Koutras, and Yorgos
sition model, ∀q ∈ Q where q := A =⇒ B we Zikos. 2016. Knowledge means all, belief means
have that A is the belief a reader holds before a most. Journal of Applied Non-Classical Logics,
26(3):173–192.
kernel and that B is a belief the reader holds after
a kernel. Since the kernel is defined as a plot point Mieke Bal. 1997. Narratology: Introduction to the the-
which changes the majority of a reader’s beliefs, we ory of narrative. University of Toronto Press.
are in turn measuring some notion of faithfulness
of ζR . Roland Barthes. 1967. The death of the author.
Fontana.
Conclusions and Future Work
Roland Barthes and Lionel Duisit. 1975. An introduc-
In this paper, we defined a preliminary theoreti- tion to the structural analysis of narrative. New liter-
cal framework of narrative that affords new preci- ary history, 6(2):237–272.
sion to common narratological concepts, including David Bordwell. 1989. Making Meaning: Inference
fabulas, story worlds, the conveyance of informa- and Rhetoric in the Interpretation of Cinema. Cam-
tion from Narrator to Reader, and the way that the bridge: Harvard University Press.Rogelio E. Cardona-Rivera, Thomas W. Price, David R. Lara J. Martin, Prithviraj Ammanabrolu, Xinyu Wang,
Winer, and R. Michael Young. 2016. Question An- William Hancock, Shruti Singh, Brent Harrison, and
swering in the Context of Stories Generated by Com- Mark O. Riedl. 2018. Event representations for au-
puters. Advances in Cognitive Systems, 4:227–246. tomated story generation with deep neural nets. In
Proceedings of the 32nd AAAI Conference on Artifi-
Rogelio E. Cardona-Rivera and R. Michael Young. cial Intelligence.
2019. Desiderata for a Computational Model of Hu-
man Online Narrative Sensemaking. In AAAI Spring Nasrin Mostafazadeh, Nathanael Chambers, Xiaodong
Symposium on Story-enabled Intelligence. He, Devi Parikh, Dhruv Batra, Lucy Vanderwende,
Pushmeet Kohli, and James Allen. 2016. A cor-
Nathanael Chambers and Dan Jurafsky. 2008. Unsu- pus and cloze evaluation for deeper understanding of
pervised learning of narrative event chains. In Pro- commonsense stories. In Proceedings of the 2016
ceedings of Annual Meeting of the Association for Conference of the North American Chapter of the
Computational Linguistics, pages 789–797. Association for Computational Linguistics: Human
Language Technologies, pages 839–849.
Chen Chung Chang and H Jerome Keisler. 1990.
Model theory. Elsevier. Mark O. Riedl and R. Michael Young. 2006. Story
A. Gatt and E. Krahmer. 2018. Survey of the state of planning as exploratory creativity: Techniques for
the art in natural language generation: Core tasks, expanding the narrative search space. New Genera-
applications and evaluation. Journal of Artificial In- tion Computing, 24(3):303–323.
telligence Research, 61:65–170. Shlomith Rimmon-Kenan. 2002. Narrative Fiction:
Gérard Genette. 1980. Narrative Discourse: An Essay Contemporary Poetics. Routledge.
in Method. Cornell University Press.
Melissa Roemmele, Cosmin Adrian Bejan, and An-
Morton A. Gernsbacher, Kathleen R. Verner, and drew S. Gordon. 2011. Choice of plausible alterna-
Mark E. Faust. 1990. Investigating differences in tives: An evaluation of commonsense causal reason-
general comprehension skill. Journal of Experimen- ing. In 2011 AAAI Spring Symposium Series.
tal Psychology: Learning, Memory, and Cognition,
Theodore Sider. 2010. Logic for philosophy. Oxford
16(3):430–445.
University Press, USA.
David Herman. 2013. Storytelling and the Sciences of
Mind. MIT Press. David Thue, Stephan Schiffel, Ragnar Adolf Árnason,
Ingibergur Sindri Stefnisson, and Birgir Steinarsson.
Peter Hühn and Roy Sommer. 2013. Narration in po- 2016. Delayed roles with authorable continuity in
etry and drama. In Peter Hühn, John Pier, Wolf plan-based interactive storytelling. In Interactive
Schmid, and Jörg Schönert, editors, the living hand- Storytelling, pages 258–269, Cham. Springer Inter-
book of narratology. Hamburg U., Hamburg, Ger- national Publishing.
many.
Tom Trabasso et al. 1982. Causal cohesion and story
Tianyu Jiang and Ellen Riloff. 2018. Learning proto- coherence. ERIC.
typical goal activities for locations. In Proceedings
of the 56th Annual Meeting of the Association for Rolf A. Zwaan and Gabriel A. Radvansky. 1998. Situ-
Computational Linguistics, pages 1297–1307. ation models in language comprehension and mem-
ory. Psychological Bulletin, 123(2):162–85.
Junyeong Kim, Minuk Ma, Kyungsu Kim, Sungjin
Kim, and Chang D. Yoo. 2019. Progressive atten-
tion memory network for movie story question an-
swering. In Proceedings of the IEEE Conference
on Computer Vision and Pattern Recognition, pages
8337–8346.
Boyang Li, Stephen Lee-Urban, Darren Scott Appling,
and Mark O. Riedl. 2012. Crowdsourcing narrative
intelligence. Advances in Cognitive Systems, 1:1–
18.
Jean M Mandler and Nancy S Johnson. 1977. Remem-
brance of things parsed: Story structure and recall.
Cognitive psychology, 9(1):111–151.
Chris Martens, Rogelio E. Cardona-Rivera, and Neil
Cohn. 2020. The visual narrative engine: A compu-
tational model of the visual narrative parallel archi-
tecture. In Proceedings of the 8th Annual Confer-
ence on Advances in Cognitive Systems.FN (t + 1) SN (t + 1)
φ
ζN
FN (t) SN (t) d0
d FR (t + 1) SR (t + 1)
ζR
FR (t) SR (t)
ψ
Figure 2: Commutative diagram expressing ζR and ζN . Some edge labels were removed for clarity. Refer to figure
1 on page 4.You can also read
