I highlight the three main achievements of the TypeSynth project below:
1. a new denotational semantics of higher-order store
2. an equational higher-order separation logic based on the above denotational semantics
3. the extension of synthetic Tait computability to support computational effects including guarded recursion and higher-order store
DENOTATIONAL SEMANTICS OF HIGHER-ORDER STORE. The TypeSynth plan to develop denotational semantics of higher-order store had two components: the first was to develop a practical semantic model of polymorphic higher-order store without garbage collection, and the second was to extend this model to support the equational theory of garbage collection. For the former, the results have greatly outstripped expectations: I have succeeded in constructing not only the indended model of store with polymorphism, but in fact a model of full dependent type theory with higher-order store — posing the prospect for being able to write and verify the correctness of programs in the same language. This result is the first of its kind for higher-order store, and is a significant advance. In the final months of the project, we have also gone further beyond this result in a different direction, and constructed a version of the model of higher-order store satisfying a new gamut of compelling representation independence equations which I refer to as the theory of "univalent reference types": in short, two programs can be considered equal even if they allocate memory cells of different types, so long as the two types are in bijection and programs' interaction with the memory cell respects this bijection.
I have begun to extend these results to include the equational theory of garbage collection, but as the TypeSynth project has been terminated nearly one year early (due to my recent appointment as Associate Professor at University of Cambridge), this further exploitation of the TypeSynth methodology will continue beyond the conclusion of the project.
EQUATIONAL HIGHER-ORDER SEPARATION LOGIC FOR HIGHER-ORDER STORE. In collaboration with my colleagues Frederik Ljerbjerg Aagaard and Professor Lars Birkedal, I have developed a guarded higher-order separation logic called TULIP over the TypeSynth denotational model of higher-order store. This logic represents a significant step forward in the march toward simple, abstract, and compositional reasoning about higher-order stateful programs. Prior program logics (such as Iris and the Verified Software Toolchain) interact with computer programs only indirectly, by verifying properties of a specific "transition function" that simulates the steps that a (highly idealized) computer would take when executing a program; this style is called "operational". Unfortunately, the important structural properties of programs are highly unnatural to express at the level of transition steps, and as such, existing operationally-based program logics impose a great deal of bureaucracy by forcing those verifying programs to manually mediate the mismatch between the viewpoint of the machine (which cannot see program structure), and the viewpoint of the programmer (which is entirely based on program structure). An equational, or "denotationally-based", program logic like TULIP instead treats programs directly without needing to pass through an encoding in terms of machine transitions. An immediate benefit of the equational approach is that program equivalences can be glued together directly in any context, a kind of practical compositionality that is unique to equational and denotationally-based program logics. Our development of the TULIP logic is a strong first step in this direction.
EFFECTFUL SYNTHETIC TAIT COMPUTABILITY. A third achievement of the Typesynth project was to extend the highly successful "Synthetic Tait Computability" method, developed in my doctoral thesis, to the case of realistic programming languages with computational effects. The purpose of this extension was to achieve strong representation independence results, by which we can show the computational indistinguishability of even two programs that allocate memory cells of different types that are linked not by a bijection but by a mere relation, in contrast to univalent reference types. We have gone quite a bit beyond our expectations, as our account of STC for higher-order store applies not only to the simple polymorphic store model, but also to the full dependent type theory. This allows many classic results from the literature that previously required very complicated reasoning with the semantic model to be reconstructed in a simpler and more direct fashion.
