13.3 Temporal reference and aspect in DRT

Discourse Representation Theory (DRT) and Temporal Reference

Discourse Representation Theory (DRT) provides a formal framework for tracking when things happen across stretches of discourse. It uses discourse referents and temporal variables to represent events, states, and the time relationships between them. This matters because natural language is full of implicit temporal connections: when someone says "John walked in. Mary smiled," you automatically understand the second event followed the first, even though no word explicitly says so. DRT gives us the tools to model that kind of reasoning.

This section covers how DRT encodes tense and aspect, how it resolves temporal anaphora, and what challenges arise with more complex temporal phenomena.

Temporal Representation in DRT

DRT distinguishes between two kinds of situations that get represented as discourse referents:

• Events are bounded occurrences with clear endpoints. "John ate dinner" introduces an event referent because the eating started and finished.
• States are ongoing situations without inherent boundaries. "Mary is happy" introduces a state referent because it describes a condition that holds over time without a built-in endpoint.

This event/state distinction matters because the two behave differently in discourse. Events typically advance the narrative timeline, while states provide background information.

Temporal variables and relations. DRT encodes time using temporal variables ($t$, $t_1$, $t_2$) that denote specific time points or intervals. It then uses temporal relations ($<$, $>$, $=$, $\subseteq$) to express how these variables relate to each other, capturing ordering and overlap.

Aspect is captured through the relationship between two key time intervals:

• Event time (ET): the time interval over which the event actually occurs
• Reference time (RT): the time interval from which the event is being viewed or evaluated

The two main aspectual categories work like this:

• Perfective aspect: The event is viewed as a complete whole. Formally, $ET \subseteq RT$, meaning the event time is contained within the reference time. Think of "John crossed the street" — you're looking at the entire crossing as a finished unit.
• Imperfective aspect: The event is viewed from the inside, as ongoing. Formally, $RT \subseteq ET$, meaning the reference time is contained within the event time. Think of "John was crossing the street" — you're zoomed in on a moment during the crossing, with no commitment about whether it was completed.

Discourse Analysis with DRT

DRT represents the temporal structure of a discourse using a series of discourse representation structures (DRSs). Each sentence or clause contributes a new DRS that introduces its own discourse referents, temporal variables, and conditions. These DRSs are then linked together by sharing temporal variables and applying temporal constraints.

Tense is represented by relating event time ($ET$) to speech time ($ST$) or a contextually determined reference time ($RT$):

• Past tense: $ET < ST$ or $ET < RT$ — the event precedes the speech or reference time
• Present tense: $ET = ST$ or $ET = RT$ — the event coincides with the speech or reference time
• Future tense: $ST < ET$ or $RT < ET$ — the event follows the speech or reference time

To see how this works in practice, consider a short discourse:

1. The first sentence introduces an event referent $e_1$ (the arriving) with a temporal variable $t_1$, and the past tense establishes $t_1 < ST$.
2. The second sentence introduces $e_2$ (the buying) with $t_2$, again with $t_2 < ST$.
3. Because both are perfective past-tense events in a narrative sequence, DRT applies a default temporal constraint: $t_1 < t_2$ (the arriving happened before the buying).

This is how DRT captures the intuition that narrative events are understood as occurring in the order they're mentioned, even without explicit temporal markers.

Temporal Anaphora Resolution

Temporal anaphora occurs when expressions like "then," "at that time," or "afterward" refer back to a previously mentioned time or event. Resolving these expressions requires identifying the right antecedent and establishing the correct temporal relation.

DRT resolves temporal anaphora using two key principles:

• Accessibility: The antecedent must be accessible from the current DRS or a superordinate (higher-level) DRS. A temporal referent buried inside a conditional or negation, for instance, may not be accessible from the main discourse.
• Specificity: When multiple antecedents are accessible, DRT selects the most specific one. In practice, this usually means the most recently introduced temporal referent.

Temporal relations between events can be established in two ways:

• Sharing temporal variables: If two events share a temporal variable, they're understood as simultaneous.
• Applying temporal constraints: Adding a condition like $e_1 < e_2$ specifies that one event precedes the other.

For example, in "John left the house. Then he caught the bus," the word "then" is a temporal anaphor. It picks up the reference time established by the first event and positions the second event after it: $e_1 < e_2$.

Challenges of Complex Temporal Phenomena

Several temporal phenomena push beyond what basic DRT can handle straightforwardly.

Progressive aspect is tricky because it presents an event as ongoing without committing to its completion. "John was building a house" doesn't tell you whether the house was ever finished. DRT needs additional machinery to represent this: the ongoing event is modeled as a non-completed part of a larger, potentially unrealized event. This often requires extra temporal variables and constraints to capture the "in-progress" reading.

Narrative progression involves the default temporal ordering of events in a story. Readers typically infer that narrated events happen in sequence, but this isn't always the case. A sentence like "John fell. Mary pushed him" reverses the expected order (the pushing caused the falling). DRT must rely on world knowledge and pragmatic principles to infer the appropriate temporal relations when they conflict with simple left-to-right ordering.

Habituals, iteratives, and generics require extensions to basic DRT:

• Habituals describe repeated patterns ("Mary jogs every morning") and require quantification over events or times.
• Iteratives involve repeated subevents within a single described event ("The light flickered") and need a way to represent internal repetition.
• Generics make general statements ("Dogs bark") that aren't tied to specific events at all, often requiring incorporation of aspectual operators or modal-like structures.

These phenomena show that while basic DRT handles simple tense, aspect, and temporal anaphora well, richer temporal reasoning demands formal extensions to the framework.