Real root isolation is a fundamental subroutine in computer algebra, with applications ranging from algebraic number arithmetic to solving polynomial systems. Modern implementations typically employ subdivision methods based on root counting via Descartes' rule of signs. In contrast, most existing formally verified root isolation procedures rely on Sturm's theorem for root counting, leading to a noticeable gap between practical implementations and formally verified approaches. We take an initial step towards efficient verified real root isolation by formally verifying two simple algorithms based on Descartes' rule of signs: a classical bisection procedure and a Newton-accelerated variant. In this paper, we describe the algorithms, present formal proofs of termination, soundness, and completeness, and discuss our code-generation efforts. Brief experiments show promising performance improvements over existing formally verified algorithms in Isabelle/HOL.

We propose an alternative, explicit state only, approach to concurrent system synthesis. In particular, the focus of this work is on the synthesis of distributed protocols. Given a correctness specification and a protocol skeleton (i.e. incomplete with holes), the goal is to synthesize the holes. At the heart of our technique is a dynamic programming based algorithm that prunes inferred failure candidates. The algorithm exploits the fact that typically only a few transitions are needed to reach an erroneous state in a faulty distributed protocol. Therefore, it is unlikely that every hole to be synthesized is contributing towards the error; thus, faulty protocol candidates where only a subset of holes were used can be used to infer failures of later candidates with a superset of holes. We evaluate the tool using a cache coherence protocol synthesis case study. Specifically, we study a directory based MSI protocol, assuming an unordered interconnect which gives rise to numerous race conditions which must be resolved via introducing transient states - a common cause of complexity and bugs in such protocols. In the case study, we therefore focus on synthesizing the transient state actions (we consider up to 12 holes out of possible 35). With the proposed candidate pruning optimization, we report up to 43x improvement over a naïve candidate enumeration scheme. We make available the tool and C++ library, VerC3.

In this paper, we verify a modern lazy cache coherence protocol, TSO-CC, against the memory consistency model it was designed for, TSO. We achieve this by first showing a weak simulation relation between TSO-CC (with a fixed number of processors) and a novel finite-state operational model which exhibits the laziness of TSO-CC and satisfies TSO. We then extend this by an existing parameterisation technique, allowing verification for an unbounded number of processors. The approach is executed entirely within a model checker, no external tool is required and very little in-depth knowledge of formal verification methods is required of the verifier.