In formal specifications based on abstract algebraic types and equational theory you use formulas of equational theory to specify theory. System which will satisfy those constraints is called in formal logic a model.
Modeling is process of creating a model, which abstracts of some aspects, which are unnecessary details for a specific case. So concrete system has to adhere to created model in observed aspects.
Programming is a process of creating a program which will have specific behaviour – will perform specific algorithms – and programming languages through different paradigms enable us to think in a certain specific way, which abstracts of some details, usually machine specific ones.
So could we be doing all those things at the same time, because they are principially the same? Is declarative programming the nearest attempt to do that? Could we use some sort f programming languages which will be good for programming as well as for modeling and specification?
The scientist who has done the most to advance this point of view is Tony Hoare. Tony, along with his colleague Edsger Dijkstra, advocated nondeterministic programming languages so that there would be a smoother path from specification to implementation. Tony definitely wanted a single language for both specification and implementation. For more on this view, read his book on the Alegbra of Programming. Tony also did the seminal work on proving correctness of abstractions. All of this work was done in the context of simple, imperative languages with structured control flow and classic, side-effecting procedures. So there is not any connection with declarative programming of necessity. And historically, work on functional programming (the main branch of declarative programming) has followed more from Backus’s Turing lecture on “liberating programming from the von Neumann bottleneck”; functional programming has been about programming productivity as much as anything else.
What we discovered since Hoare is that formal specifications and formal modelsl are very expensive. The expense hasn’t been shown to be justified except in very special circumstances, like “if the software doesn’t work, the patient will die” or “if the software doesn’t work, the plane will crash.” Informal models and specifications are quite useful, and much cheaper to produce and work with. There is still interesting research going on around the fringes on modelling, model checking, and so on. One of my personal favorites is the Alloy language done by Daniel Jackson’s group at MIT. There’s also great stuff done at Microsoft Research and plenty of good stuff elsewhere. There’s some work in declarative programming as well, but it too is of the “cheap and cheerful” variety rather than a comprehensive, programmatic approach like Hoare’s. One of my favorites there is Claessen’s and Hughes’s QuickCheck, which provides a way to state formal properties and explore them by random testing. No proofs or theorems, but still jolly useful.
In summary, you describe an agenda of doing formal models, specifications, and programs, all within a single framework. There is still plenty of good work going on piecemeal, but the unified agenda has been abandoned.