Alex,

Those things were done and published years ago.   They are not research issues, and there is nothing controversial about them.   They were published in an official ISO standard.  The latest version was published in 2018, but it is more complex, and the subset that was defined in 2007 is the only version that has been implemented and used: ISO/IEC standard 24707 for Common Logic.  Even more important, it can be downloaded for free:  http://standards.iso.org/ittf/PubliclyAvailableStandards/c039175_ISO_IEC_24707_2007(E).zip 

The ISO standard for Common Logic specifies the core semantics in an abstract syntax that is independent of any readable notation of any kind.  Then it states that any concrete syntax (linear or diagrammatic) that has a formally defined mapping to the abstract syntax may be called a dialect of Common Logic.  Then three different concrete syntaxes are specified in the Appendices: (1)  Common Logic Interchange Format (CLIF), which has a LISP-like syntax: (2) Conceptual Graph Interchange Format (CGIF); and (3) an XML-based notation (XCL). 

In that standard, the core semantics is formally equivalent to Peirce's existential graphs.  The formal name for the notation is "core CGIF", but I use the name EGIF (Existential Graph Interchange Format) because the core can be mapped to and from the graphic notation for EGs.  Anything stated in the full CLIF or CGIF or XCL dialects can be mapped to CGIF and then to the core EGIF.  The mappings are defined in that standard.

For more details about the full graph notation plus extensions, see the peer-reviewed research publication in the International Journal of Applied Logics:  Sowa, John F. (2018) Reasoning with diagrams and images, http://www.collegepublications.co.uk/downloads/ifcolog00025.pdf  .   That issue of the journal contains several articles presented at a conference in Bogota, Columbia.  My article is the second one.  It defines an extension to EGs that also supports mappings to and from images.

But before reading all those formal publications, I recommend the slides from the talk that I presented at the European Sematic Web Conference in 2020:  https://jfsowa.com/talks/escw.pdf .  

These slides present a simpler overview, which may help smooth the way toward the more detailed formalism.  They also contain more links to other publications and presentations that can add useful background.  See the links at the bottom of most slides, and the suggested readings in the last slide.

John