Chapter 0: Recap and Strategy for Binary Operations

In the previous units and sections, we have built a powerful set of tools. We can transform single timelines with map, create sequential, dependent chains with bind, and manage stateful transformations over time with scan.

However, we now face an entirely new challenge: combining multiple, independent timelines that run in parallel. Consider tasks like “adding the latest values of two separate counters” or “checking if two different form inputs are both valid.” These scenarios cannot be directly solved by our existing primitives alone.

What we need now is a general binary operation that takes two independent sources, Timeline<'a> and Timeline<'b>, and produces a new Timeline<'c>.

Re-evaluating the Classifications from Unit 4

To solve this, let’s revisit the classifications for Applicative Functors from Unit 4.

1. Cartesian: In the context of List, this approach generated all possible combinations of elements. If applied to Timeline, this would mean combining every past event from timelineA with every past event from timelineB. This is computationally explosive and does not align with our goal of determining the current combined state. Therefore, the Cartesian approach is unsuitable.

2. Pointwise: For List, this approach paired elements at the same index, like a zip operation. However, the concept of a synchronized “index” does not exist for Timeline. Timelines are asynchronous; an event on timelineA at time t1 has no guarantee of a corresponding event on timelineB at the exact same time t1. A strict, zip-like combination based on synchronized “points” in time would be practically useless. Therefore, the Pointwise classification, as defined for synchronous structures, is also unsuitable.

The Need for a New Concept: “Latest Value Combination”

Since the classifications from Unit 4 do not fit, we need a new concept tailored to the asynchronous nature of Timeline.

What we truly need is not to synchronize discrete events (points), but to combine continuously available states. This leads us to propose a new concept: “Latest Value Combination.”

This idea is defined as follows:

The moment any source timeline is updated, its new value is combined with the most recently known value (the latest state) of the other source(s) to produce a new result.

This powerful model fully embraces the asynchronous nature of the inputs while ensuring a consistent, combined state is always available. It is a pattern uniquely suited for reactive systems.

The Strategy for this Section

In this section, we will systematically build out the tools for this “Latest Value Combination” concept.

• First, we will introduce combineLatestWith, the core combinator that implements this idea (Chapter 1).
• Then, we will use this new tool to build concrete, Monoid-based logical operations like TL.And and TL.Or (Chapter 2).
• Finally, we will expand these patterns to aggregate entire lists of timelines and apply them to a practical case study.