Claude. Make me an app to track recipes. Make no mistakes.

We can model the problem of multi-agent synthesis formally as follows:

Given a prompt \(P :=\) "\(\textit{An app to track recipes}\)", we can define the formula \(\Phi(P)\) as the set of software consistent with the prompt:

\[ \Phi(P) := \{ \phi ~|~ \phi ~ \text{program} ∧ \phi ~\text{is consistent with the prompt}~P \} \]

The key point here is that a natural language prompt, by its very nature, is underspecified — i.e there may be multiple programs that are consistent with the prompt. When we use LLMs to build and write software systems, we're effectively asking the LLMs to select one element amongst many from this set.

Conversely, when we do multi-agentic software development, i.e we spin up several agents, \(A_1, ⋯, A_n\), and ask them to build a piece of software, we're essentially asking them each to produce software components \(\phi_1, ⋯, \phi_n\) such that they all refine one single consistent interpretation of the prompt:

\[ C(\phi_1, \cdots, \phi_n) := \exists \phi \in \Phi(P), \forall i, \phi_i ~\text{refines}~ \phi \]

This in other words, is nothing other than one big distributed consensus problem.

In other words, the user's prompt \(P\) first gets sent split, via a plan, into tasks for several agents \(a_1, \cdots, a_n\). Then, these agents work in parallel to implement their respective coding tasks \(\phi_1, \cdots, \phi_n\), and by the end, if the synthesis was successful, we're hoping that the final generated software system \(\phi\) composed of each of the individual constructions \(\phi := \phi_1 || \cdots || \phi_n\), satisfies the user's request.

This is inherently a consensus problem as the agents \(a_1,\cdots,a_n\) must work concurrently to produce their software artefacts \(\phi_1,\cdots,\phi_n\), but communicate and agree enough that the final piece of software \(\phi_1 || \cdots || \phi_n\) is well formed and satisfies the request. Design decisions or choices in one \(\phi_i\) will result in constraints that affect and influence the possible choices of \(\phi_j\) for other agents. For example, if the agent in charge of implementing network connections \(a_{\text{network}}\) chooses a library with a callback-style async API for requests, then whichever agent is responsible for the overall integration \(a_{\text{integration}}\) must organise the infrastructure around that choice and so on and so forth. Similar choices in other modules will influence the design spaces of other agents and overall the process proceeds as a joint synthesis problem.