We report on a new course Verified Functional Data Structures and Algorithms taught at the Technical University of Munich. The course first introduces students to interactive theorem proving with the Isabelle proof assistant. Then it covers a range of standard data structures, in particular search trees and priority queues: it is shown how to express these data structures functionally and how to reason about their correctness and running time in Isabelle.
This talk will describe our work to establish and use mechanised and rather complete semantics for full-scale architectures: the mainstream Armv8-A architecture, the emerging RISC-V architecture, the CHERI-MIPS and CHERI-RISC-V research architectures that use hardware capabilities for improved security, and Arm’s prototype Morello architecture - an industrial demonstrator incorporating the CHERI ideas.
It brings together a variety of tools, especially our Sail ISA definition language and Isla symbolic evaluation engine, to build semantic definitions that are readable, executable as test oracles, support reasoning within the Coq, HOL4, and Isabelle proof assistants, support SMT-based symbolic evaluation, support model-based test generation, and can be integrated with operational and axiomatic concurrency models.
For Armv8-A, we have the complete sequential semantics of the instruction-set architecture (ISA) automatically translated from the Arm-internal model to Sail, and for RISC-V, we have hand-written what is now the offically adopted model. For their concurrent semantics, the "user" semantics, partly as a result of our collaborations with Arm and within the RISC-V concurrency task group, have become simplified and well-defined, with multiple models proved equivalent, and we are currently working on the "system" semantics. Recently, we have added a symbolic execution tool for Sail specifications, Isla, which supports axiomatic concurrency models but now for tests that span the full range of the instruction-set architecture, and including instruction cache maintenance instructions and self-modifying code.
For CHERI-MIPS and CHERI-RISC-V, we have used Sail models as the golden reference during the design process, working with our systems colleagues in the CHERI team to use lightweight formal specification routinely in documentation, testing, and test generation. We have stated and proved (in Isabelle) some of the fundamental intended security properties of the full CHERI-MIPS ISA.
Morello, supported by the UKRI Digital Security by Design programme, offers a path to hardware enforcement of fine-grained memory safety and/or secure encapsulation in the production Armv8-A architecture, potentially excluding or mitigating a large fraction of today’s security vulnerabilities for existing C/C++ code with little modification. We are currently working with Arm to prove fundamental security properties for the complete Morello ISA definition, and to generate tests which are being used during hardware development.
All these tools and models are (or will soon be) available under open-source licences for others to use and build on.
We present a formal proof in Lean of probably approximately correct (PAC) learnability of the concept class of decision stumps. This classic result in machine learning theory derives a bound on error probabilities for a simple type of classifier. Though such a proof appears simple on paper, analytic and measure-theoretic subtleties arise when carrying it out fully formally. Our proof is structured so as to separate reasoning about deterministic properties of a learning function from proofs of measurability and analysis of probabilities.
Reinforcement learning algorithms solve sequential decision-making problems in probabilistic environments by optimizing for long-term reward. The desire to use reinforcement learning in safety-critical settings inspires a recent line of work on formally constrained reinforcement learning; however, these methods place the implementation of the learning algorithm in their Trusted Computing Base. The crucial correctness property of these implementations is a guarantee that the learning algorithm converges to an optimal policy.
This paper begins the work of closing this gap by developing a Coq formalization of two canonical reinforcement learning algorithms: value and policy iteration for finite state Markov decision processes. The central results are a formalization of the Bellman optimality principle and its proof, which uses a contraction property of Bellman optimality operator to establish that a sequence converges in the infinite horizon limit. The CertRL development exemplifies how the Giry monad and mechanized metric coinduction streamline optimality proofs for reinforcement learning algorithms. The CertRL library provides a general framework for proving properties about Markov decision processes and reinforcement learning algorithms, paving the way for further work on formalization of reinforcement learning algorithms.
This paper shows how a small verified bootstrapped compiler can be developed inside an interactive theorem prover (ITP). Throughout, emphasis is put on clarity and minimalism.
We report on a new verified Verilog compiler called Lutsig. Lutsig currently targets (a class of) FPGAs and is capable of producing technology mapped netlists for FPGAs. We have connected Lutsig to existing Verilog development tools, and in this paper we show how Lutsig, as a consequence of this connection, fits into a hardware development methodology for verified circuits in the HOL4 theorem prover. One important step in the methodology is transporting properties proved at the behavioral Verilog level down to technology mapped netlists, and Lutsig is the component in the methodology that enables such transportation.
We present an Isabelle/HOL formalization of an alternative execution model---optimizing interpreters---and mechanically verify its correctness. Specifically, we formalize advanced speculative optimizations similar to those used in just-in-time compilers and prove semantics preservation. As a result, our formalization provides a path towards unifying vital performance requirements with desirable security guarantees.
The Michael-Scott queue (MS-queue) is a concurrent non-blocking queue. In an earlier pen-and-paper proof it was shown that a simplified variant of the MS-queue contextually refines a coarse-grained queue. Here we use the Iris and ReLoC logics to show, for the first time, that the original MS-queue contextually refines a coarse-grained queue. We make crucial use of the recently introduced prophecy variables of Iris and ReLoC. Our proof uses a fairly simple invariant that relies on encoding which nodes in the MS-queue can reach other nodes. To further simplify the proof, we extend separation logic with a generally applicable persistent points-to predicate for representing immutable pointers. This relies on a generalization of the well-known algebra of fractional permissions into one of discardable fractional permissions. We define the persistent points-to predicate entirely inside the base logic of Iris (thus getting soundness “for free”).
We use the same approach to prove refinement for a variant of the MS-queue resembling the one used in the java.util.concurrent library.
We have mechanized our proofs in Coq using the formalizations of ReLoC and Iris in Coq.
Reasoning about monotonicity is of key importance in concurrent separation logics. For instance, one needs to reason about monotonicity to show that the value of a concurrent counter with an increment operation only grows over time. Modern concurrent separation logics, such as VST, FCSL, and Iris, are based on resource models defined using partial commutative monoids. For any partial commutative monoid, there is a canonical ordering relation, the so-called extension order, and in a sense the logics are designed to reason about monotonicity wrt. the extension ordering.
Thus a natural question is: given an arbitrary preorder, can we construct a partial commutative monoid, where the extension order captures the given preorder.
In this paper, we answer this question in the affirmative and show that there is a canonical construction, which given any preorder produces a partial commutative monoid for which the extension order, restricted to the elements of the preorder, is exactly the given preorder. We prove that our construction is a free construction in the category-theoretic sense.
We demonstrate, using examples, that the general construction is useful. We have formalized the construction and its properties in Coq. Moreover, we have integrated it in the Iris program logic framework and used that to formalize our examples.
We implement extraction of Coq programs to functional languages based on MetaCoq's certified erasure. As part of this, we implement an optimisation pass removing unused arguments. We prove the pass correct wrt. a conventional call-by-value operational semantics of functional languages. We apply this to two functional smart contract languages, Liquidity and Midlang, and to the functional language Elm.
Our development is done in the context of the ConCert framework that enables smart contract verification. We contribute a verified boardroom voting smart contract featuring maximum voter privacy such that each vote is kept private except under collusion of all other parties.
We also integrate property-based testing into ConCert using QuickChick and our development is the first to support testing properties of interacting smart contracts. We test several complex contracts such as a DAO-like contract, an escrow contract, an implementation of a Decentralized Finance (DeFi) contract which includes a custom token standard (Tezos FA2), and more.
In total, this gives us a way to write dependent programs in Coq, test them semi-automatically, verify, and then extract to functional smart contract languages, while retaining a small trusted computing base of only MetaCoq and the pretty-printers into these languages.
Authenticated Append-Only Skiplists (AAOSLs) enable maintenance and querying of an authenticated log (such as a blockchain) without requiring any single party to store or verify the entire log, or to trust another party regarding its contents. AAOSLs can help to enable efficient dynamic participation (e.g., in consensus) and reduce storage overhead.
In this paper, we formalize an AAOSL originally described by Maniatis and Baker, and prove its key correctness proper- ties. Our model and proofs are machine checked in Agda. Our proofs apply to a generalization of the original construction and provide confidence that instances of this generalization can be used in practice. Our formalization effort has also yielded some simplifications and optimizations.
Hardware-assisted reference monitoring is receiving increasing attention as a way to improve the security of existing software. One example is the PIPE architecture extension, which attaches metadata tags to register and memory values and executes tag-based rules at each machine instruction to enforce a software-defined security policy. To use PIPE effectively, engineers should be able to write security policies in terms of source-level concepts like functions, local variables, and structured control operators, which are not visible at machine level. It is the job of the compiler to generate PIPE-aware machine code that enforces these source-level policies. The compiler thus becomes part of the monitored system’s trusted computing base---and hence a prime candidate for verification.
To formalize compiler correctness in this setting, we extend the source language semantics with its own form of user-specified tag-based monitoring, and show that the compiler preserves that monitoring behavior. The challenges of compilation include mapping source-level monitoring policies to instruction-level tag rules, preserving fail-stop behaviors, and satisfying the surprisingly complex preconditions for conventional optimizations. In this paper, we describe the design and verification of Tagine, a small prototype compiler that translates a simple tagged WHILE language to a tagged register transfer language and performs simple optimizations. Tagine is based on the RTLgen and Deadcode phases of the CompCert compiler, and hence is written and verified in Coq. This work is a first step toward verification of a full-scale compiler for a realistic tagged source language.
In multiple domains, large amounts of data are daily generated and combined to be analyzed. The interpretation of these analyses requires to track back the provenance of combined data with respect to initial, raw data. The correctness of the provenance is crucial in many critical domains, such as medicine to prescribe treatments. In this article, we propose the first provenance-aware extended relational algebra formalized in a proof assistant (Coq), for a non trivial subset of database queries: queries containing aggregates, null values, and correlated sub-queries. The formalization is validated by an adequacy proof with respect to standard evaluation of queries. This development is a first step towards a posteriori certification of provenance for data manipulation, with strong guaranties.
We introduce a static analysis and two program transformations for Datalog to circumvent performance ssues that arise with the implementation of primitive predicates, notably in the framework of a large scale telecommunication application. To this effect, we introduce a new trace semantics for Datalog with a verified mechanization. This work can be seen as both a first step and a proof of concept for the creation of a full-blown library of verified Datalog optimizations, on top of an existing Coq/MathComp formalization of Datalog towards the development of a realistic environment for certified data centric applications.
Session types—a family of type systems for message-passing concurrency—have been subject to many extensions, where each extension comes with a separate proof of type safety. These extensions cannot be readily combined, and their proofs of type safety are generally not machine checked, making their correctness less trustworthy. We overcome these shortcomings with a semantic approach to binary asynchronous affine session types, by developing a logical relations model in Coq using the Iris program logic. We demonstrate the power of our approach by combining various forms of polymorphism and recursion, asynchronous subtyping, references, and locks/mutexes. As an additional benefit of the semantic approach, we demonstrate how to manually prove typing judgements of racy, but safe, programs that cannot be type checked using only the rules of the type system.
In theorem provers based on dependent type theory such as Coq and Lean, induction is a fundamental proof method and induction tactics are omnipresent in proof scripts. Yet the ergonomics of existing induction tactics are not ideal: they do not reliably support inductive predicates and relations; they sometimes generate overly specific or unnecessarily complex induction hypotheses; and they occasionally choose confusing names for the hypotheses they introduce.
This paper describes a new induction tactic, implemented in Lean 3, which addresses these issues. The tactic is particularly suitable for educational use, but experts should also find it more convenient than existing induction tactics. In addition, the tactic serves as a moderately complex case study for the metaprogramming framework of Lean 3. The paper describes some difficulties encountered during the implementation and suggests improvements to the framework.
Proof engineering efforts using interactive theorem proving have yielded several impressive projects in software systems and mathematics. A key obstacle to such efforts is the requirement that the domain expert is also an expert in the low-level details in constructing the proof in a theorem prover. In particular, the user needs to select a sequence of tactics that lead to a successful proof, a task that in general requires knowledge of the exact names and use of a large set of tactics.
We present Lassie, a tactic framework for the HOL4 theorem prover that allows individual users to define their own tactic language by example and give frequently used tactics or tactic combinations easier-to-remember names. The core of Lassie is an extensible semantic parser, which allows the user to interactively extend the tactic language through a process of definitional generalization. Defining tactics in Lassie thus does not require any knowledge in implementing custom tactics, while proofs written in Lassie retain the correctness guarantees provided by the HOL4 system. We show through case studies how Lassie can be used in small and larger proofs by novice and more experienced interactive theorem prover users, and how we envision it to ease the learning curve in a HOL4 tutorial.
We present a formalization in Isabelle/HOL of a comprehensive framework for proving the completeness of automatic theorem provers based on resolution, superposition, or other saturation calculi. The framework helps calculus designers and prover developers derive, from the completeness of a calculus, the completeness of prover architectures implementing the calculus. It also helps derive the completeness of calculi obtained by lifting ground (i.e., variable-free) calculi. As a case study, we re-verified Bachmair and Ganzinger's resolution prover RP to show the benefits of modularity.
AProVE is a powerful termination prover for various programming languages, including a termination analysis method for imperative programs specified in the LLVM intermediate representation (IR). The method internally works in three steps: first, it transforms LLVM IR code into a symbolic execution graph; second, the graph is translated into an integer transition system; finally, termination of the transition sys- tem is proved by the back end of AProVE. Since AProVE is unverified software, our aim is to increase its reliability by certifying the generated proofs. To this end, we require formal semantics of all program representations, i.e., for LLVM IR, for symbolic execution graphs and for inte- ger transition systems. As the latter is already available, we define the former ones. We note that our semantics for LLVM IR use arithmetic with unbounded integers. We further verify the first and the second step of AProVE’s termination method, including verified algorithms to check concrete proofs. Since the third step can already be certified, we obtain a complete formally verified method for certifying AProVE’s termination proofs of LLVM IR programs. The whole formalization has been done in Isabelle/HOL and our certifier is available as a Haskell program via code generation.
The first-order theory of rewriting is a decidable theory for finite left-linear right-ground rewrite systems, implemented in FORT. We present a formally verified variant of the decision procedure for the class of linear variable-separated rewrite systems. This variant supports a more expressive theory and is based on the concept of anchored ground tree transducers. The correctness of the decision procedure is verified by a formalization in Isabelle/HOL on top of the Isabelle Formalization of Rewriting (IsaFoR).
The ring of Witt vectors W R over a base ring R is an important tool in algebraic number theory and lies at the foundations of modern p-adic Hodge theory. W R has the interesting property that it constructs a ring of characteristic 0 out of a ring of characteristic p > 1, and it can be used more specifically to construct from a finite field containing ℤ/pℤ the corresponding unramified field extension of the p-adic numbers ℚp (which is unique up to isomorphism).
We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime p, the ring of Witt vectors over ℤ/pℤ is isomorphic to the ring of p-adic integers ℤp. In the process we develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature. Our proofs resemble the informal arguments while being fully rigorous.
Semi-algebraic sets and real analytic functions are fundamental concepts in Real Algebraic Geometry and Real Analysis, respectively. These concepts appear in the study of Differential Equations, where the real analytic solution to a differential equation is known to enter or exit a semi-algebraic set in a predictable way. Motivated to enhance the capability to reason about differential equations in the Prototype Verification System (PVS), a formalization of multivariate polynomials, semi-algebraic sets, and real analytic functions is developed. The way that a real analytic function behaves in a neighborhood around a point where the function meets the boundary of a semi-algebraic set is described and verified. It is further shown that if the function is assumed to be smooth, a slightly weaker assumption than real analytic, the behavior around the boundary of a semi-algebraic set can be very different.
Kolmogorov complexity is an essential tool in the study of algorithmic information theory, and is used in the fields of Artificial Intelligence, cryptography, and coding theory. The formalisation of the theorems of Kolmogorov complexity is also key to proving results in the theory of Intelligent Agents, specifically the results in Universal Artificial Intelligence. In this paper, we present a mechanisation of some of these fundamental results. In particular, building on HOL4's existing model of computability, we provide a formal definition of the complexity of a binary string, and then prove (i) that Kolmogorov complexity is uncomputable; (ii) the Kolmogorov Complexity invariance theorem; (iii) the Kraft and Kolmogorov-Kraft inequalities; and (iv) key Kolmogorov Complexity inequalities.
We investigate the possibility of formalizing quantifiers in proof theory while avoiding, as far as possible, the use of true binding structures, α-equivalence or variable renamings. We propose a solution with two kinds of variables in terms and formulas, as originally done by Gentzen. In this way formulas are first-order structures, and we are able to avoid capture problems in substitutions. However at the level of proofs and proof manipulations, some binding structure seems unavoidable. We give a representation with de Bruijn indices for proof rules which does not impact the formula representation and keeps the whole set of definitions first-order.
We discuss and compare two Coq mechanisations of Sierpinski's result that the generalised continuum hypothesis (GCH) implies the axiom of choice (AC). The first version shows the result, originally stated in first-order ZF set-theory, for a higher-order set theory convenient to work with in Coq. The second version presents a corresponding theorem for Coq's type theory itself, concerning type-theoretic formulations of GCH and AC. Both versions rely on the classical law of excluded middle and extensionality assumptions but we localise the use of axioms where possible.
The generality and pervasiveness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not have a standard, working formalization of category theory. We document our work on solving this dilemma. The formalization revealed a number of potential design choices, and we present, motivate and explain the ones we picked. In particular, we find that alternative definitions or alternative proofs from those found in standard textbooks can be advantageous, as well as "fit" Agda's type theory more smoothly. Some definitions regarded as equivalent in standard textbooks turn out to make different "universe level" assumptions, with some being more polymorphic than others. We also pay close attention to engineering issues so that the library integrates well with Agda's own standard library, as well as being compatible with as many of supported type theories in Agda as possible.