Most attempts to replace/improve SQL derive from the fact SQL was a poorly conceived and designed interface, that originally was meant to be a very small DSL for end users, but unfortunately, was allowed to become a poor, complicated, confusing mess for app developers:
https://cacm.acm.org/research/50-years-of-queries/
SQL is not an orthogonal language... This is because, in the early days, Ray Boyce and I did not think we were designing a language for programmers. ..
As it turned out, Ray and I were wrong about the predominant usage of SQL...
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So the #1 thing any actual replacement or alternative to SQL is how actually become a good language for development, so it is actually composable, can be actually be used to reason about it, has minimal foot guns, etc.
There is a lot of misunderstanding and pushback, similar to how people in the past fight improvements over JS/C/C++ until typescript, rust comes.
But, oh boy, SQL need their typescript!
Full disclosure: I am one of the co-authors of this paper and an associated patch implenting it in PostgreSQL that we have proposed.
I am happy to see more people than us think this is useful.
(edit: in retrospect, this is just the "getting started" header link, despite the URL)
Unfortunately none that I've followed seems to do much to describe much of anything except the math foundations. They might cover pieces of syntax, but not how to use or think about them, and they seem to be rather excited about the data-generating features that I can't imagine anyone using outside tech demos (unless that's the only insert method? what about update?) :/
(edit 2: you can't even rely on the manual apparently, I'm fairly confident https://categoricaldata.net/help/Demo.html (manual -> examples -> demo) won't work because it has no schema definition for `Animal`. and it like all the others I've checked are little more than syntax and output, no real explanation)
SQL is like Java, CQL is like Haskell. SQL has been around and used in production. CQL is a research language, possibly cleaner foundation but YMMV.
The math fields you list are connected, but whether they are the same monster - again it's kinda like claiming all programming languages and implementations are the same (Turing-complete?) monster.
Most of SQL is not imperative, but it certainly also includes some imperative commands.
Inserting a new row into an existing table is an imperative command, which may be the most frequently used of the SQL features, in certain applications concerned with recording transactions.
Only the subset of SQL that is used for queries can be said to not be an imperative programming language.
Imperative SQL is the procedural elements that mostly do not exist at the standard level. Variables, control flow, and cursors.
Compilers are free to rearrange these accesses if the final result is same as if executed by these ordered microtransactions.
Consider loop fusion, loop splitting and/or loop skewing.
printf("Hello world!");
Anyone who claims that an SQL command like insert a row, create a table or destroy a table is not imperative, is just plainly wrong.
Even in real life, when humans communicate commands to each other, from where the term "imperative" comes, the commands normally do not tell anything to the recipient about how to accomplish the order, they just name the action, because it is supposed that whoever receives the command knows how to do it.
After programming languages were invented there was a clear partition between commands a.k.a. imperative statements, expressions and definitions a.k.a. declarations ("definition" and "declaration" were originally synonymous terms, "declaration" being the ALGOL 60 term and "definition" being the CPL term; the use of the 2 words with different meanings, like in C, where it is required by a defect of the compiler, which needs additional declarations besides a definition, is only recent.)
Later the meanings of these terms have become muddled by their careless use by many authors, which usually had political reasons, i.e. because they somehow considered the words "imperative" or "command" as shameful, they attempted to present their pet languages as purely functional or purely declarative, so they twisted the words in various ways, despite the fact that any practical program must contain imperative parts, not only functional or declarative parts.
I always thought this was splitting hairs.
C#,C++, Java seems imperative. You are in control? But you aren't really telling it how to move values between registers, the compiler is making a million decisions for you on how the computer will execute that 'imperative' code. Just like SQL isn't really telling the DB how to do it either..
Contrast that if I create a class/object/field structure/hierarchy in Java, or put a HashMap somewhere with a certain set of keys, I've written something in stone which requires significant refactoring if the data needs to be accessed from a different direction.
What you're really talking about is one of the things Codd wanted to emphasize: representational independence. Which actually was the primary thrust of his famous paper: the user should not need to know how the data is stored in order to use or manipulate it.
The other thing that people are talking about with "declarative" is probably another level up in abstraction. Talking about the business logic or problem in terms that are closer to logic than a sequence of instructions, and then letting the machine sort out what those steps are.
Consider in a Datalog users don't customarily perform DDL type operations; they declare data rules and the system decides the form of the underlying relations. That's a small step up the declarative ladder from SQL, even if it's somewhat analogous to "create view".
So I think there's a blurry definitional line between the two. But I don't think your very blunt "No." is doing much to help clear that up?