A coinductive semantics of the Unlimited Register Machine

Abstract

We exploit (co)inductive specifications and proofs to approach the evaluation of low-level programs for the Unlimited Register Machine (URM) within the Coq system, a proof assistant based on the Calculus of (Co)Inductive Constructions type theory. Our formalization allows us to certify the implementation of partial functions, thus it can be regarded as a first step towards the development of a workbench for the formal analysis and verification of both converging and diverging computations

A Definitional Implementation of the Lax Logical Framework LLFP in Coq, for Supporting Fast and Loose Reasoning

“…On the other hand, due to the looping back via the J instruction, there are non-terminating computations. In our case study we consider only terminating computations (the interested reader may refer to [4] for a coinductive approach to diverging computations). The functions introduced in order to formalize evaluation are defined as follows: To introduce an LLF P signature, for the URM machine, we need first to encode infinite stores and non-structured programs.…”

“…On the other hand, due to the looping back via the J instruction, there are non-terminating computations. In our case study we consider only terminating computations (the interested reader may refer to [4] for a coinductive approach to diverging computations).…”

A supposedly fun thing i may have to do again

“…In other terms, once you go to the "dark side", there is no turning back. The connection between the two levels is "one way" only: the specification-level can never look outside of itself, although the meta-level may look at the specification-level, since the latter is embedded in the former 5 . Thus, we have not much choice in formalizing the candidate relation at the ML.…”

“…Those very detailed and non-trivial proofs should be compared to the more sketchy ones done in a Coq setting (e.g. [5]), which seem to appeal to a dubious "structural corecursion", that is to a notion of circular proofs in the mathematical informal world. It should be possible, albeit not without considerable effort, to reformulate this following [26].…”