I'm looking for a definition of the syntax for the Cypher query language. I tried the docs but they're very vague.
Ideally, I'd like a BNF (or any variant) definition, or one of those "graph" definitions like this or this. Really, anything resembling a formal definition.
What you are looking for will be available in openCypher. Several items will be released as part of the project, one of the first of which is the BNF grammar.
Take a look at the recently announced (Oct 2015) openCypher project. It involves releasing the language specification, among other things.
From the announcement:
1. Cypher reference documentation: A comprehensive user documentation describing use of the Cypher query language with examples and tutorials.
2. Technology compatibility kit (TCK): The TCK consists of a number of tests that a software supplier would run in order to self-certify support for a given version of Cypher.
3. Reference implementation: Distributed under the Apache 2.0 license, the reference implementation is a fully functional implementation of key parts of the stack needed to support Cypher inside a data platform or tool. The first planned deliverable is a parser that will take a Cypher statement and parse it into an AST (abstract syntax tree) representation. The reference implementation complements the documentation and tests by providing working implementations of Cypher – which are permissively licensed – and can be used as examples or as a foundation for one’s own implementation.
4. Cypher language specification: Licensed under a Creative Commons license, the Cypher language specification is a technical expression of the language syntax to enable parsers to auto-generate the query syntax. A full semantic specification is also planned as a part of the openCypher project.
The same announcement also says that the process is open and that it is possible to submit, review and comment on language proposals.