Fix the Graph API example on the applications concepts page

[Excelius-Wang]

The Graph / Application split example on the applications page cannot be copy-pasted: it raises a SyntaxError before it gets anywhere.

Changes

Two lines in docs/concepts/state-machine.rst:

• with_state(chat_history=[]) was missing its leading dot, so the builder chain broke mid-expression.
• The import read from burr.core import ApplicationBuilder, default, expr, which brings in two names the snippet never uses while leaving GraphBuilder undefined. Changed to from burr.core import ApplicationBuilder, GraphBuilder.

How I tested this

Ran the corrected snippet with a human_input / ai_response action pair defined as in the surrounding docs. It builds the graph and the application without error. The original raises SyntaxError at the with_state line, and after fixing only that, NameError: name 'GraphBuilder' is not defined.

Notes

Docs only, and deliberately separate from #893 so this one can go in on its own. Both branches touch state-machine.rst but not the same lines, so there is no conflict either way round.

Checklist

• PR has an informative and human-readable title (this will be pulled into the release notes)
• Changes are limited to a single goal (no scope creep)
• Code passed the pre-commit check & code is left cleaner/nicer than when first encountered.
• Any change in functionality is tested
• New functions are documented (with a description, list of inputs, and expected output)
• Placeholder code is flagged / future TODOs are captured in comments
• Project documentation has been updated if adding/changing functionality.

Made with Cursor

[jernejfrank]

Thanks!