The effectful forcing technique allows one to show that the denotation of a closed System T term of type (ι ⇒ ι) ⇒ ι in the set-theoretical model is a continuous function (N → N) → N. For this purpose, an alternative dialogue-tree semantics is defined and related to the set-theoretical semantics by a logical relation. In this paper, we apply effectful forcing to show that the dialogue tree of a System T term is itself System T-definable, using the Church encoding of trees.

Byzantine fault-tolerance is arguably an expensive characteristic for protocols to support, especially when considering its overhead on message complexity, number of communication phases, and number of nodes for resilience. Various works in the literature have addressed optimizing one or more of these dimensions through the use of algorithmic optimizations, trusted execution environments, and streamlined view changes.The best achievable message complexity, resilience, and latency to date are respectively linear complexity, by ⌊(N − 1)/2⌋, and two communication phases attained Damysus (EuroSys’22), a streamlined hybrid protocol. This paper strictly advances the aforementioned state of the art results by introducing OneShot, a streamlined hybrid protocol that uses one communication phase in the normal case, and one or two phases otherwise. OneShot exploits the information nodes receive about the system to dynamically modify and adapt views. We prove that OneShot is safe and live, and moreover demonstrate through experimental evaluation that it improves throughput and latency by respectively up to 150% and 59% compared to the state of the art.

Markov's principle (MP) is an axiom in some varieties of constructive mathematics, stating that Σ01 propositions (i.e. existential quantification over a decidable predicate on N) are stable under double negation. However, there are various non-equivalent definitions of decidable predicates and thus Σ01 in constructive foundations, leading to non-equivalent Markov's principles. While this fact is well-reported in the literature, it is often overlooked, leading to wrong claims in standard references and published papers.

Reliable communication is a fundamental distributed communication abstraction that allows any two nodes within a network to communicate with each other. It is necessary for more powerful communication primitives, such as broadcast and consensus. Using different authentication models, two classical protocols implement reliable communication in unknown and sufficiently connected networks. In the former, network links are authenticated, and processes rely on dissemination paths to authenticate messages. In the latter, processes generate digital signatures that are flooded throughout the network. This work considers the hybrid system model that combines authenticated links and authenticated processes. Additionally, we aim to leverage the possible presence of trusted nodes (e.g., network gateways) and trusted components (e.g., Intel SGX enclaves). We first extend the two classical reliable communication protocols to leverage trusted nodes. Then we propose DualRC, our most generic algorithm that considers the hybrid authentication model by manipulating dissemination paths and digital signatures, and leverages the possible presence of trusted nodes and trusted components. We describe and prove methods that establish whether our algorithms implement reliable communication on a given network.

$\text{TT}^{\Box}_{{\mathcal C}}$ is a generic family of effectful, extensional type theories with a forcing interpretation parameterized by modalities. This paper identifies a subclass of $\text{TT}^{\Box}_{{\mathcal C}}$ theories that internally realizes continuity principles through stateful computations, such as reference cells. The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. In this paper, the modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled internally in the theory.

Continuity is a key principle of intuitionistic logic that is generally accepted by constructivists but is inconsistent with classical logic. Most commonly, continuity states that a function from the Baire space to numbers, only needs approximations of the points in the Baire space to compute. More recently, another formulation of the continuity principle was put forward. It states that for any function F from the Baire space to numbers, there exists a (dialogue) tree that contains the values of F at its leaves and such that the modulus of F at each point of the Baire space is given by the length of the corresponding branch in the tree. In this paper we provide the first internalization of this "inductive" continuity principle within a computational setting. Concretely, we present a class of intuitionistic theories that validate this formulation of continuity thanks to computations that construct such dialogue trees internally to the theories using effectful computations. We further demonstrate that this inductive continuity principle implies other forms of continuity principles.

Current vehicular Intrusion Detection and Prevention Systems either incur high false-positive rates or do not capture zero-day vulnerabilities, leading to safety-critical risks. In addition, prevention is limited to few primitive options like dropping network packets or extreme options, e.g., ECU Bus-off state. To fill this gap, we introduce the concept of vehicular Intrusion Resilience Systems (IRS) that ensures the resilience of critical applications despite assumed faults or zero-day attacks, as long as threat assumptions are met. IRS enables running a vehicular application in a replicated way, i.e., as a Replicated State Machine, over several ECUs, and then requiring the replicated processes to reach a form of Byzantine agreement before changing their local state. Our study rides the mutation of modern vehicular environments, which are closing the gap between simple and resource-constrained "real-time and embedded systems", and complex and powerful "information technology" ones. It shows that current vehicle (e.g., Zonal) architectures and networks are becoming plausible for such modular fault and intrusion tolerance solutions—deemed too heavy in the past. Our evaluation on a simulated Automotive Ethernet network running two state-of-the-art agreement protocols (Damysus and Hotstuff) shows that the achieved latency and throughout are feasible for many Automotive applications.

Recently, streamlined Byzantine Fault Tolerant (BFT) consensus protocols, such as HotStuff, have been proposed as a means to circumvent the inefficient view-changes of traditional BFT protocols, such as PBFT. Several works have detailed trusted components, and BFT protocols that leverage them to tolerate a minority of faulty nodes and use a reduced number of communication rounds. Inspired by these works we identify two basic trusted services, respectively called the Checker and Accumulator services, which can be leveraged by streamlined protocols. Based on these services, we design Damysus, a streamlined protocol that improves upon HotStuff's resilience and uses less communication rounds. In addition, we show how the Checker and Accumulator services can be adapted to develop Chained-Damysus, a chained version of Damysus where operations are pipelined for efficiency. We prove the correctness of Damysus and Chained-Damysus, and evaluate their performance showcasing their superiority compared to previous protocols.

One of the differences between Brouwerian intuitionistic logic and classical logic is their treatment of time. In classical logic truth is atemporal, whereas in intuitionistic logic it is time-relative. Thus, in intuitionistic logic it is possible to acquire new knowledge as time progresses, whereas the classical Law of Excluded Middle (LEM) is essentially flattening the notion of time stating that it is possible to decide whether or not some knowledge will ever be acquired. This paper demonstrates that, nonetheless, the two approaches are not necessarily incompatible by introducing an intuitionistic type theory along with a Beth-like model for it that provide some middle ground. On one hand they incorporate a notion of progressing time and include evolving mathematical entities in the form of choice sequences, and on the other hand they are consistent with a variant of the classical LEM. Accordingly, this new type theory provides the basis for a more classically inclined Brouwerian intuitionistic type theory.

In this paper, we consider the Byzantine reliable broadcast problem on authenticated and partially connected networks. The state-of-the-art method to solve this problem consists in combining two algorithms from the literature. Handling asynchrony and faulty senders is typically done thanks to Gabriel Bracha's authenticated double-echo broadcast protocol, which assumes an asynchronous fully connected network. Danny Dolev's algorithm can then be used to provide reliable communications between processes in the global fault model, where up to f processes among N can be faulty in a communication network that is at least 2f+1-connected. Following recent works that showed how Dolev's protocol can be made more practical thanks to several optimizations, we show that the state-of-the-art methods to solve our problem can be optimized thanks to layer-specific and cross-layer optimizations. Our simulations with the Omnet++ network simulator show that these optimizations can be efficiently combined to decrease the total amount of information transmitted or the protocol's latency (e.g., respectively, −25% and −50% with a 16B payload, N=31 and f=4) compared to the state-of-the-art combination of Bracha's and Dolev's protocols.

The accelerated digitalisation of society along with technological evolution have extended the geographical span of cyber-physical systems. Two main threats have made the reliable and real-time control of these systems challenging: (i) uncertainty in the communication infrastructure induced by scale, and heterogeneity of the environment and devices; and (ii) targeted attacks maliciously worsening the impact of the above-mentioned communication uncertainties, disrupting the correctness of real-time applications. This article addresses those challenges by showing how to build distributed protocols that provide both real-time with practical performance, and scalability in the presence of network faults and attacks, in probabilistic synchronous environments. We provide a suite of real-time Byzantine protocols, which we prove correct, starting from a reliable broadcast protocol, called PISTIS, up to atomic broadcast and consensus. This suite simplifies the construction of powerful distributed and decentralized monitoring and control applications, including state-machine replication. Extensive empirical simulations showcase PISTIS's robustness, latency, and scalability. For example, PISTIS can withstand message loss (and delay) rates up to 50 percent in systems with 49 nodes and provides bounded delivery latencies in the order of a few milliseconds.

Powerful yet effective induction principles play an important role in computing, being a paramount component of programming languages, automated reasoning, and program verification systems. The Bar Induction (BI) principle is a fundamental concept of intuitionism, which is equivalent to the standard principle of transfinite induction. In this work, we investigate the compatibility of several variants of BI with Constructive Type Theory (CTT), a dependent type theory in the spirit of Martin-Löf’s extensional theory. We first show that CTT is compatible with a BI principle for sequences of numbers. Then, we establish the compatibility of CTT with a more general BI principle for sequences of name-free closed terms. The formalization of the latter principle within the theory involved enriching CTT’s term syntax with a limit constructor and showing that consistency is preserved. Furthermore, we provide novel insights regarding BI, such as the non-truncated version of BI on monotone bars being intuitionistically false. These enhancements are carried out formally using the Nuprl proof assistant that implements CTT and the formalization of CTT within the Coq proof assistant presented in previous works.

Byzantine fault-tolerant state-machine replication (BFT-SMR) is a technique for hardening systems to tolerate arbitrary faults. Although robust, BFT-SMR protocols are very costly in terms of the number of required replicas (3f+1 to tolerate f faults) and of exchanged messages. However, with "hybrid" architectures, where "normal" components trust some "special" components to provide properties in a trustworthy manner, the cost of using BFT can be dramatically reduced. Unfortunately, even though such hybridization techniques decrease the message/time/space complexity of BFT protocols, they also increase their structural complexity. Therefore, we introduce Asphalion, the first theorem prover-based framework for verifying implementations of hybrid systems and protocols. It relies on three novel languages: (1) HyLoE: a Hybrid Logic of Events to reason about hybrid fault models; (2) MoC: a Monadic Component language to implement systems as collections of interacting hybrid components; and (3) LoCK: a sound Logic of events-based Calculus of Knowledge to reason about both homogeneous and hybrid systems at a high-level of abstraction (thereby allowing reusing proofs, and capturing the high-level logic of distributed systems). In addition, Asphalion supports compositional reasoning, e.g., through mechanisms to lift properties about trusted-trustworthy components, to the level of the distributed systems they are integrated in. As a case study, we have verified crucial safety properties (e.g., agreement) of several implementations of hybrid protocols.

Our increasing dependence on complex and critical information infrastructures and the emerging threat of sophisticated attacks, ask for extended efforts to ensure the correctness and security of these systems. Byzantine fault-tolerant state-machine replication (BFT-SMR) provides a way to harden such systems. It ensures that they maintain correctness and availability in an application-agnostic way, provided that the replication protocol is correct and at least $$n-f$$ out of n replicas survive arbitrary faults. This paper presents Velisarios, a logic-of-events based framework implemented in Coq, which we developed to implement and reason about BFT-SMR protocols. As a case study, we present the first machine-checked proof of a crucial safety property of an implementation of the area’s reference protocol: PBFT.

Church-Turing computability was extended by Brouwer who considered non-lawlike computability in the form of free choice sequences. Those are essentially unbounded sequences whose elements are chosen freely, i.e. not subject to any law. In this work we develop a new type theory BITT, which is an extension of the type theory of the Nuprl proof assistant, that embeds the notion of choice sequences. Supporting the evolving, non-deterministic nature of these objects required major modifications to the underlying type theory. Even though the construction of a choice sequence is non-deterministic, once certain choices were made, they must remain consistent. To ensure this, BITT uses the underlying library as state and store choices as they are created. Another salient feature of BITT is that it uses a Beth-like semantics to account for the dynamic nature of choice sequences. We formally define BITT and use it to interpret and validate essential axioms governing choice sequences. These results provide a foundation for a fully intuitionistic version of Nuprl.

Building a verified proof assistant entails implementing and mechanizing the concept of a library, as well as adding support for standard manipulations on it. In this work we develop such mechanism for the Nuprl proof assistant, and integrate it into the formalization of Nuprl’s meta-theory in Coq. We formally verify that standard operations on the library preserve its validity. This is a key property for any interactive theorem prover, since it ensures consistency. Some unique features of Nuprl, such as the presence of undefined abstractions, make the proof of this property nontrivial. Thus, e.g., to achieve monotonicity the semantics of sequents had to be refined. On a broader view, this work provides a backend for a verified version of Nuprl. We use it, in turn, to develop a tool that converts proofs exported from the Nuprl proof assistant into proofs in the Coq formalization of Nuprl’s meta-theory, so as to be verified.

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) withnamed exceptionsandhandlers, as well as a nominalfreshoperator. Using these new features, we prove a version of Brouwer's continuity principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem, we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem, bar induction on monotone bars and the negation of the law of excluded middle.

We formalize the soundness theorem for differential dynamic logic, a logic for verifying hybrid systems. To increase confidence in the formalization, we present two versions: one in Isabelle/HOL and one in Coq. We extend the metatheory to include features used in practice, such as systems of differential equations and functions of multiple arguments. We demonstrate the viability of constructing a verified kernel for the hybrid systems theorem prover KeYmaera X by embedding proof checkers for differential dynamic logic in Coq and Isabelle. We discuss how different provers and libraries influence the design of the formalization.

We present an extension of the computation system and logic of the Nuprl proof assistant with intuitionistic principles, namely versions of Brouwer's bar induction principle, which is equivalent to transfinite induction. We have substantially extended the formalization of Nuprl's type theory within the Coq proof assistant to show that two such bar induction principles are valid w.r.t. Nuprl's semantics (the Good): one for sequences of numbers that involved only minor changes to the system, and a more general one for sequences of name-free (the Ugly) closed terms that involved adding a limit constructor to Nuprl's term syntax in our model of Nuprl's logic. We have proved that these additions preserve Nuprl's key metatheoretical properties such as consistency. Finally, we show some new insights regarding bar induction, such as the non-truncated version of bar induction on monotone bars is intuitionistically false (the Bad).

The world is becoming an immense critical information infrastructure, with the fast and increasing entanglement of utilities, telecommunications, Internet, cloud, and the emerging IoT tissue. This may create enormous opportunities, but also brings about similarly extreme security and dependability risks. We predict an increase in very sophisticated targeted attacks, or advanced persistent threats (APT), and claim that this calls for expanding the frontier of security and dependability methods and techniques used in our current CII. Extreme threats require extreme defenses: we propose resilience as a unifying paradigm to endow systems with the capability of dynamically and automatically handling extreme adversary power, and sustaining perpetual and unattended operation. In this position paper, we present this vision and describe our methodology, as well as the assurance arguments we make for the ultra-resilient components and protocols they enable, illustrated with case studies in progress.

This papers extends the Nuprl proof assistant (a system representative of the class of extensional type theories a la Martin-Lof) with named exceptions and handlers, as well as a nominal fresh operator. Using these new features, we prove a version of Brouwer's Continuity Principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's meta-theory using our formalization of Nuprl in Coq and show how we can reflect these meta-theoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key meta-theoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem and bar induction on monotone bars.

Fault-tolerant distributed systems often contain complex error handling code. Such code is hard to test or model-check because there are often too many possible failure scenarios to consider. As we will demonstrate in this paper, formal methods have evolved to a state in which it is possible to generate this code along with correctness guarantees. This paper describes our experience with building highly-available databases using replication protocols that were generated with the help of correct-by-construction formal methods. The goal of our project is to obtain databases with unsurpassed reliability while providing good performance. We report on our experience using a total order broadcast protocol based on Paxos and specified using a new formal language called Event ML. We compile Event ML specifications into a form that can be formally verified while simultaneously obtaining code that can be executed. We have developed two replicated databases based on this code and show that they have performance that is competitive with popular databases in one of the two considered benchmarks.

We present a fault-tolerant ordered broadcast service that is correct-by-construction. Our broadcast service allows for diversity in space, whereby the participants in the broadcast protocol run different code, as well as in time, whereby the protocol itself is changed periodically. We use the Nuprl proof assistant to specify the service, prove correctness, and synthesize the code. The paper includes initial performance results.

Expansion is a crucial operation for calculating principal typings in intersection type systems. Because the early definitions of expansion were complicated, E-variables were introduced in order to make the calculations easier to mechanise and reason

Reducibility, despite being quite mysterious and inflexible, has been used to prove a number of properties of the λ-calculus and is well known to offer general proofs which can be applied to a number of instantiations. In this paper, we look at two r