Architecture specifications notionally define the fundamental interface between hardware and software: the envelope of allowed behaviour for processor implementations, and the basic assumptions for software development and verification. But in practice, they are typically prose and pseudocode documents, not rigorous or executable artifacts, leaving software and verification on shaky ground. In this paper, we present rigorous semantic models for the sequential behaviour of large parts of the mainstream ARMv8-A, RISC-V, and MIPS architectures, and the research CHERI-MIPS architecture, that are complete enough to boot operating systems, variously Linux, FreeBSD, or seL4. Our ARMv8-A models are automatically translated from authoritative ARM-internal definitions, and (in one variant) tested against the ARM Architecture Validation Suite. We do this using a custom language for ISA semantics, Sail, with a lightweight dependent type system, that supports automatic generation of emulator code in C and OCaml, and automatic generation of proof-assistant definitions for Isabelle, HOL4, and (currently only for MIPS) Coq. We use the former for validation, and to assess specification coverage. To demonstrate the usability of the latter, we prove (in Isabelle) correctness of a purely functional characterisation of ARMv8-A address translation. We moreover integrate the RISC-V model into the RMEM tool for (user-mode) relaxed-memory concurrency exploration. We prove (on paper) the soundness of the core Sail type system. We thereby take a big step towards making the architectural abstraction actually well-defined, establishing foundations for verification and reasoning. CCS Concepts: • General and reference → Verification; • Theory of computation → Semantics and reasoning; • Computer systems organization → Architectures; • Software and its engineering → Assembly languages;

Abstract. The inverse method, due to Maslov, is a forward theorem proving method for cut-free sequent calculi that relies on the subformula property. The Logic of Bunched Implications (BI), due to Pym and O'Hearn, is a logic which freely combines the familiar connectives of intuitionistic logic with multiplicative linear conjunction and its adjoint implication. We present the first formulation of an inverse method for propositional BI without units. We adapt the sequent calculus for BI into a forward calculus. The soundness and completeness of the calculus are proved, and a canonical form for bunches is given.

Interactive programs, such as GUIs or spreadsheets, often maintain dependency information over dynamically-created networks of objects. That is, each imperative object tracks not only the objects its own invariant depends on, but also all of the objects which depend upon it, in order to notify them when it changes.These bidirectional linkages pose a serious challenge to verification, because their correctness relies upon a global invariant over the object graph.We show how to modularly verify programs written using dynamically-generated bidirectional dependency information. The critical idea is to distinguish between the footprint of a command, and the state whose invariants depends upon the footprint. To do so, we define an application-specific semantics of updates, and introduce the concept of a ramification operator to explain how local changes can alter our knowledge of the rest of the heap. We illustrate the applicability of this style of proof with a case study from functional reactive programming, and formally justify reasoning about an extremely imperative implementation as if it were pure.

In this paper, we give a simple and efficient implementation of reverse-mode automatic differentiation, which both extends easily to higher-order functions, and has run time and memory consumption linear in the run time of the original program. In addition to a formal description of the translation, we also describe an implementation of this algorithm, and prove its correctness by means of a logical relations argument.

In this paper, we take a pervasively effectful (in the style of ML) typed lambda calculus, and show how to extend it to permit capturing pure expressions with types. Our key observation is that, just as the pure simplytyped lambda calculus can be extended to support effects with a monadic type discipline, an impure typed lambda calculus can be extended to support purity with a comonadic type discipline. We establish the correctness of our type system via a simple denotational model, which we call the capability space model. Our model formalises the intuition common to systems programmers that the ability to perform effects should be controlled via access to a permission or capability, and that a program is capability-safe if it performs no effects that it does not have a runtime capability for. We then identify the axiomatic categorical structure that the capability space model validates, and use these axioms to give a categorical semantics for our comonadic type system. We then give an equational theory (substitution and the call-by-value and laws) for the imperative lambda calculus, and show its soundness relative to this semantics. Finally, we give a translation of the pure simply-typed lambda calculus into our comonadic imperative calculus, and show that any two terms which are-equal in the STLC are equal in the equational theory of the comonadic calculus, establishing that pure programs can be mapped in an equation-preserving way into our imperative calculus.