Verification of a Rust Implementation of Knuth's Dancing Links using ACL2

Dancing Links connotes an optimization to a circular doubly-linked list data structure implementation which provides for fast list element removal and restoration. The Dancing Links optimization is used primarily in fast algorithms to find exact covers, and has been popularized by Knuth in Volume 4B of his seminal series The Art of Computer Programming. We describe an implementation of the Dancing Links optimization in the Rust programming language, as well as its formal verification using the ACL2 theorem prover. Rust has garnered significant endorsement in the past few years as a modern, memory-safe successor to C/C++ at companies such as Amazon, Google, and Microsoft, and is being integrated into both the Linux and Windows operating system kernels. Our interest in Rust stems from its potential as a hardware/software co-assurance language, with application to critical systems. We have crafted a Rust subset, inspired by Russinoff's Restricted Algorithmic C (RAC), which we have imaginatively named Restricted Algorithmic Rust, or RAR. In previous work, we described our initial implementation of a RAR toolchain, wherein we simply transpile the RAR source into RAC. By so doing, we leverage a number of existing hardware/software co-assurance tools with a minimum investment of time and effort. In this paper, we describe the RAR Rust subset, describe our improved prototype RAR toolchain, and detail the design and verification of a circular doubly-linked list data structure employing the Dancing Links optimization in RAR, with full proofs of functional correctness accomplished using the ACL2 theorem prover.Comment: In Proceedings ACL2-2023, arXiv:2311.08373. arXiv admin note: substantial text overlap with arXiv:2205.1170

Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP

Stateman: Using Metafunctions to Manage Large Terms Representing Machine States

When ACL2 is used to model the operational semantics of computing machines, machine states are typically represented by terms recording the contents of the state components. When models are realistic and are stepped through thousands of machine cycles, these terms can grow quite large and the cost of simplifying them on each step grows. In this paper we describe an ACL2 book that uses HIDE and metafunctions to facilitate the management of large terms representing such states. Because the metafunctions for each state component updater are solely responsible for creating state expressions (i.e., "writing") and the metafunctions for each state component accessor are solely responsible for extracting values (i.e., "reading") from such state expressions, they can maintain their own normal form, use HIDE to prevent other parts of ACL2 from inspecting them, and use honsing to uniquely represent state expressions. The last feature makes it possible to memoize the metafunctions, which can improve proof performance in some machine models. This paper describes a general-purpose ACL2 book modeling a byte-addressed memory supporting "mixed" reads and writes. By "mixed" we mean that reads need not correspond (in address or number of bytes) with writes. Verified metafunctions simplify such "read-over-write" expressions while hiding the potentially large state expression. A key utility is a function that determines an upper bound on the value of a symbolic arithmetic expression, which plays a role in resolving writes to addresses given by symbolic expressions. We also report on a preliminary experiment with the book, which involves the production of states containing several million function calls.Comment: In Proceedings ACL2 2015, arXiv:1509.0552

Formalization of Real Analysis: A Survey of Proof Assistants and Libraries

International audienceIn the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis