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.

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.

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 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).

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.