Towards a Model-Theoretic View of Narratives (2021)

Abstract: In this paper, we propose the beginnings of a formal framework for modeling narrative qua narrative. Our framework affords the ability to discuss key qualities of stories and their communication, including the flow of information from a Narrator to a Reader, the evolution of a Reader’s story model over time, and Reader uncertainty. We demonstrate its applicability to computational narratology by giving explicit algorithms for measuring the accuracy with which information was conveyed to the Reader, along with two novel measurements of story coherence.

## introduction story understanding is both (1) the process through which a cognitive agent (human or artificial) mentally constructs a plot through the perception of a narrated discourse, and (2) the outcome of that process: i.e., the agent's mental representation of the plot. the best way to computationally model story understanding is contextual to the aims of a given research program, and today we enjoy a plethora of artificial intelligence (ai)-based capabilities. data-driven approaches-including statistical, neural, and neuro-symbolic ones-look to narrative as a benchmark task for demonstrating human-level competency on inferencing, question-answering, and storytelling. that is, they draw associations between event @xcite , causal @xcite , and purposive @xcite information extracted from textual or visual narrative corpora to answer questions or generate meaningful stories that depend on information implied and not necessarily expressed by stories (e.g. @xcite @xcite . symbolic approaches seek to understand narrative, its communication, and its effect by using ai techniques as computational modeling tools, including logic, constraint satisfaction, and automated planning. these include efforts to model creative storytelling as a search process @xcite , generating stories with predictable effects on their comprehension by audiences @xcite , and modeling story understanding through humanconstrained techniques @xcite . despite recent advances, few works have offered a thorough conceptual account of narrative in a way that affords reconciling how different research programs might relate to each other. without a foundation for shared progress, our community might strain to determine how individual results may build upon each other to make progress on story understanding ai that performs as robustly and flexibly as humans do @xcite . in this paper, we take steps toward such a foundation. we posit that such a foundation must acknowledge the diverse factors that contribute to an artifact being treated as a narrative. key among these factors is a narrative's communicative status: unlike more-general natural language generation (cf. @xcite , an audience's belief dynamics-the trajectory of belief expansions, contractions, and revisions @xcite -is core to what gives a narrative experience its quality @xcite . failure to engage with narratives on these grounds risks losing an essential aspect of what makes narrative storytelling a vibrant and unique form of literature. to that end, we define a preliminary theoretical framework of narrative centered on information entropy. our framework is built atop model theory, the set-theoretic study of language interpretation. model theory is a field of formal logic that has been used extensively by epistomologists, linguists, and other theorists as a framework for building logical semantics. contributions. in this paper, we propose the beginnings of a formal framework for modeling narrative qua narrative. our framework affords discussing the flow of information from a narrator to a reader, the evolution of a reader's story model over time, and reader uncertainty. our work is grounded in the long history of narratology, drawing on the rich linguistic and philosophical history of the field to justify our notions. we use our framework to make experimentally verifiable conjectures about how story readers respond to under-specification of the story world and how to use entropy to identify plot points. we additionally demonstrate its applicability to computational narratology by giving explicit algorithms for measuring the accuracy with which information was conveyed to the reader. we also propose two novel measurements of story coherence. ## pre-rigorous notions of narrative before we can begin defining narrative in a formal sense, we must examine the intuitive notions of what narrative is supposed to mean. while we cannot address all of the complexity of narratology in this work, we cover some key perspectives. ## a model-theoretic view of narrative the core of our framework for modeling narrative come from a field of mathematical logic known as model theory. model theory is a powerful yet flexible framework that has heavily influenced computer scientists, literary theorists, linguists, and philosophers @xcite . despite the centrality of model theory in our framework, a deep understanding of the topic is not necessary to work with it on an applied level. our goal in this section is thus to give an intuitive picture of model theory that is sufficient to understand how we will use it to talk about narratives. we refer an interested reader to @xcite ; chang and @xcite for a more complete presentation of the subject. ## conveying story information an important aspect of stories is that they are a way to convey information. in this section, we will discuss how to formalize this process and what we can learn about it. although stories can be constructed and conveyed in many different ways, we will speak of a narrator who tells the story and a reader who receives it for simplicity. the core of how we model storytelling as an act of communication can be seen in figure 1 . this diagram represents the transmission of information from the narrator's story-world (s n ) to the reader's (s r ), with each arrow representing the transmission from one representation to another. in an idealized world, stories would be conveyed by d : straight from the story world of the narrator (s n ) to the story world of the reader (s r ). in actuality, narrators must convey their ideas through media foot_0 . to do this, the narrator compresses their mental story world (via φ) into a fabula (f n ) which is then conveyed to the reader via speech, writing, etc. the conveyance of the fabula as understood by the narrator (f n ) to the fabula as understood by the reader (f r ) is denoted in our diagram by d. d is in many ways the real-world replacement for the function d the narrator is unable to carry out. once the discourse has been consumed by the reader, the reader then takes their reconstructed fabula (f r ) and uses the received information to update their story world model (s r , via ψ). ## a detailed look at temporal evolution, with applications to plot a commonly accepted notion in narratology is that at any given moment, a reader contains a potentially infinite set of possible worlds. determining which of these worlds agree with each other is a required attribute for consuming discourse. how do we discuss the notion of collapsing possible worlds upon acquiring new knowledge? assume that we have a narrator, n , and reader r with fabulas f n and f r respectively. given our definition of a story world model, s, we define s(t) as the set of all world models that satisfy f r (t). let ρ t+1 refer to the set of formulae that are contained in f r (t + 1)\f r (t). and similarly s r (t + 1) = sr (t + 1) ∩ sr (t) refer to the shared world models between the two adjacent time steps. note that it must follow ∀ρ ∈ p t+1 , ∀@xmath0, ρ ∈ s. that is to say, the story worlds that remain between the two time steps are the ones that agree on the propositions added by consuming f n (t + 1). since this can be repeated inductively, we can assume that for any such t we have that all such models agree on all such provided propositions. something to note that for ρ ∈ p t+1 , ρ will always be either true or false in sr (t)regardless if it is expressed in the fabula or not since sr (t) is the logical closure of s r (t). ## possible worlds and reader uncertainty so far we have spoken about the reader's storyworld model as if there is only one, but in light of the discussion in section 3 it is unclear it truly makes sense to do so. in actuality, the reader never learns to "true story-world model" (insofar as one can even be said to exist). rather, the reader has an evolving set of "plausible story-world models" that are extrapolated based on the incomplete information conveyed in the story. the purpose of this section is to detail how these "plausibilities" interact with each other and with plausibilities at other time steps. it likely seems natural to model the reader's uncertainty with a probabilistic model. unfortunately, the topological structure of first-order logic makes that impossible as there is no way to define a probability distribution over the set of models that are consistent with a set of sentences. instead, we are forced to appeal to filters, a weaker notion of size that captures the difference between "large" and "small" sets. again we develop the theory of ultrafil-ters only to the extent that we require, and refer an interested reader to a graduate text in mathematical logic for a thorough discussion. definition 6.1. let q be a set of sentences that make claims about a narrative. a non-empty collection f w ⊆ p(q) is a weak filter iff 1. ∀x, @xmath1, @xmath2. ∀@xmath3, @xmath4 we say that f w is a weak ultrafilter and denote it uf w if the second requirement is replaced by ∀@xmath5, @xmath6. a reader's beliefs at time t defines a weak filter over the set of possible story-world models {s i r }. call this filter f w , dropping the t when it is clear from context. each element @xmath7. this plausibility describes a set of propositions about the story that the reader thinks paints a coherent and plausible picture. formally, a plausibility identified with the largest set of sentences that is true for every model in u , or ∩ @xmath8. that is, the set of plausible facts. the intuition for the formal definition of a weak filter is that 1. means that adding worlds to an element of the filter (which decreases the number of elements in ∩ @xmath9; and that 2. means that it is not the case that both p and ¬p are plausible. it's important to remember that membership in f w is a binary property, and so a statement is either plausible or is not plausible. we do not have shades of plausibility due to the aforementioned lack of a probability distribution. as a framework for modeling the reader's uncertainty, weak filters underspecify the space of plausible story world as a whole in favor of capturing what the reader "has actively in mind" when reading. this is precisely because the ultrafilter axiom is not required, and so for some propositions neither p nor ¬p are judged to be plausible. when asked to stop and consider the truth of a specific proposition, the reader is confronted with the fact that there are many ways that they can precisify their world models. how a reader responds to this confrontation is an experimental question that we leave to future work, but we conjecture that with sufficient time and motivation a reader will build a weak ultrafilter uf w that extends f w and takes a position on the plausibility of all statements in the logical closure of their knowledge. once the reader has fleshed out the space of plausibilities, we can use uf w to build the ultraproduct of the reader's story-world models. an ultraproduct @xcite ) is a way of using an ultrafilter to engage in reconciliation and build a single consistent story world-model out of a space of plausibilities. intuitively, an ultraproduct can be thought of as a vote between the various models about the truth of individual propositions. a proposition is considered to be true in the ultraproduct if and only if the set of models in which it is true is an element of the ultrafilter. we conjecture that real-world rational agents with uncertain beliefs find the ultraproduct of their world models to be a reasonable reconciliation of their beliefs and that idealized perfectly rational agents will provably gravitate towards the ultraproduct as the correct reconciliation. ## applications to computational narratology finally, we demonstrate that our highly abstract framework is of practical use by using it to derive explicit computational tools that can benefit computational narratology.