Proof Assistant Research Articles

An Anonymous Verifiable Random Function with Applications in Blockchain

Verifiable random function is a powerful function that provides a noninteractively public verifiable proof for its output. Recently, verifiable random function has found essential applications in designing secure consensus protocols in blockchain. How to construct secure and practical verifiable random functions has also attracted more and more attention. In this paper, we propose a practical anonymous verifiable random function. Security proofs show that the proposed anonymous verifiable random function achieves correctness, anonymity, uniqueness, and pseudorandomness. In addition, we show a concrete application of our proposed anonymous verifiable random function in blockchain to improve the consensus mechanism for Hyperledger fabric. Finally, we implement the proposed anonymous verifiable random function and evaluate its performance. Test results show that the proposed anonymous verifiable random function supports faster computing operations and has a smaller proof size.

Beyond Notations: Hygienic Macro Expansion for Theorem Proving Languages

In interactive theorem provers (ITPs), extensible syntax is not only crucial to lower the cognitive burden of manipulating complex mathematical objects, but plays a critical role in developing reusable abstractions in libraries. Most ITPs support such extensions in the form of restrictive "syntax sugar" substitutions and other ad hoc mechanisms, which are too rudimentary to support many desirable abstractions. As a result, libraries are littered with unnecessary redundancy. Tactic languages in these systems are plagued by a seemingly unrelated issue: accidental name capture, which often produces unexpected and counterintuitive behavior. We take ideas from the Scheme family of programming languages and solve these two problems simultaneously by proposing a novel hygienic macro system custom-built for ITPs. We further describe how our approach can be extended to cover type-directed macro expansion resulting in a single, uniform system offering multiple abstraction levels that range from supporting simplest syntax sugars to elaboration of formerly baked-in syntax. We have implemented our new macro system and integrated it into the new version of the Lean theorem prover, Lean 4. Despite its expressivity, the macro system is simple enough that it can easily be integrated into other systems.

Practical algebraic calculus and Nullstellensatz with the checkers Pacheck and Pastèque and Nuss-Checker

AbstractAutomated reasoning techniques based on computer algebra have seen renewed interest in recent years and are for example heavily used in formal verification of arithmetic circuits. However, the verification process might contain errors. Generating and checking proof certificates is important to increase the trust in automated reasoning tools. For algebraic reasoning, two proof systems, Nullstellensatz and polynomial calculus, are available and are well-known in proof complexity. A Nullstellensatz proof captures whether a polynomial can be represented as a linear combination of a given set of polynomials by providing the co-factors of the linear combination. Proofs in polynomial calculus dynamically capture that a polynomial can be derived from a given set of polynomials using algebraic ideal theory. In this article we present the practical algebraic calculus as an instantiation of the polynomial calculus that can be checked efficiently. We further modify the practical algebraic calculus and gain LPAC (practical algebraic calculus + linear combinations) that includes linear combinations. In this way we are not only able to represent both Nullstellensatz and polynomial calculus proofs, but we are also able to blend both proof formats. Furthermore, we introduce extension rules to simulate essential rewriting techniques required in practice. For efficiency we also make use of indices for existing polynomials and include deletion rules too. We demonstrate the different proof formats on the use case of arithmetic circuit verification and discuss how these proofs can be produced as a by-product in formal verification. We present the proof checkers Pacheck, Pastèque, and Nuss-Checker. Pacheck checks proofs in practical algebraic calculus more efficiently than Pastèque, but the latter is formally verified using the proof assistant Isabelle/HOL. The tool Nuss-Checker is used to check proofs in the Nullstellensatz format.

SeCaV: A Sequent Calculus Verifier in Isabelle/HOL

We describe SeCaV, a sequent calculus verifier for first-order logic in Isabelle/HOL, and the SeCaV Unshortener, an online tool that expands succinct derivations into the full SeCaV syntax. We leverage the power of Isabelle/HOL as a proof checker for our SeCaV derivations. The interactive features of Isabelle/HOL make our system transparent. For instance, the user can simply click on a side condition to see its exact definition. Our formalized soundness and completeness proofs pertain exactly to the calculus as exposed to the user and not just to some model of our tool. Users can also write their derivations in the SeCaV Unshortener, which provides a lighter syntax, and expand them for later verification. We have used both tools in our teaching.

Gradualizing the Calculus of Inductive Constructions

We investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial trade-off between graduality and the key properties of normalization and closure of universes under dependent product that CIC enjoys. Beyond this Fire Triangle of Graduality, we explore the gradualization of CIC with three different compromises, each relaxing one edge of the Fire Triangle. We develop a parametrized presentation of Gradual CIC (GCIC) that encompasses all three variations, and develop their metatheory. We first present a bidirectional elaboration of GCIC to a dependently-typed cast calculus, CastCIC, which elucidates the interrelation between typing, conversion, and the gradual guarantees. We use a syntactic model of CastCIC to inform the design of a safe, confluent reduction, and establish, when applicable, normalization. We study the static and dynamic gradual guarantees as well as the stronger notion of graduality with embedding-projection pairs formulated by New and Ahmed, using appropriate semantic model constructions. This work informs and paves the way towards the development of malleable proof assistants and dependently-typed programming languages.

Proving Reliability of Image Processing Techniques in Digital Forensics Applications

Binary images have found its place in many applications, such as digital forensics involving legal documents, authentication of images, digital books, contracts, and text recognition. Modern digital forensics applications involve binary image processing as part of data hiding techniques for ownership protection, copyright control, and authentication of digital media. Whether in image forensics, health, or other fields, such transformations are often implemented in high-level languages without formal foundations. The lack of formal foundation questions the reliability of the image processing techniques and hence the forensic results loose their legal significance. Furthermore, counter-forensics can impede or mislead the forensic analysis of the digital images. To ensure that any image transformation meet high standards of safety and reliability, more rigorous methods should be applied to image processing applications. To verify the reliability of these transformations, we propose to use formal methods based on theorem proving that can fulfil high standards of safety. To formally investigate binary image processing, in this paper, a reversible formal model of the binary images is defined in the Proof Assistant Coq. Multiple image transformation methods are formalized and their reliability properties are proved. To analyse real-life RGB images, a prototype translator is developed that reads RGB images and translate them to Coq definitions. As the formal definitions and proof scripts can be validated automatically by the computer, this raises the reliability and legal significance of the image forensic applications.

The Impact of Sequencing Human Genome on the New-World Order

When a new discovery brings a seismic shift in our old thinking, what price do we pay? This abstract describes the creation of Life in the Lab. The day we discovered that the RNA, a molecule that codes for Life, can self-replicate, self-organize and self-evolve, it also has the ability not only to store the information like DNA, can catalyze reaction like protein, we know that we embark on a New-World Order to create Life in the Lab. The New World Order would be based on the truth, experimental evidence, facts, reproducible and verifiable results. The Old Word Order of the seven-day creation of Life on Earth was imagined by our elders based on beliefs, fiction, magic, mystery without any evidence without any verifiable proofs. What the New World Order discovered is that the essence of Life is information, and the information is located on four organic molecules called nucleotides and they are Adenine, Thiamine, Guanine and Cytosine. They are found in the nucleus of all living creatures from a tiny blade of grass to mighty Elephant including, Man, Mouse, Monkey and Microbes. The greatest achievement is that we could not only synthesize these molecules in our Labs, but also, we could arrange them in a specific order of three letter called Codon which codes for an Amino acid. The four-nucleotide text could be arranged in sixty-four different combinations to code for all twenty amino acids to make proteins that perform all our body functions. Thousands of proteins interact to make a Cell. Millions of Cells interact to make a Tissue and hundreds of Tissues interact to make an Organ and several Organs interact to make a Human. Once we synthesize RNA molecule in the Lab, we can create novel microbial life forms, a series of biological machines which carry instructions to produce new food, new fuel, and new medicine to treat every disease known to mankind.

Verification of Crashsafe Caching in a Virtual File System Switch

When developing file systems, caching is a common technique to achieve a performant implementation. Integrating write-back caches is not primarily a problem for functional correctness, but is critical for proving crash safety. Since parts of written data are stored in volatile memory, special care has to be taken when integrating write-back caches to guarantee that a power cut during a running operation leads to a consistent state. This article shows how non-order-preserving caches can be added to a virtual file system switch (VFS) and gives a novel crash-safety criterion matching the characteristics of such caches. Broken down to individual files, a power cut can be explained by constructing an alternative run, where all writes since the last synchronization of that file have written a prefix. VFS caches have been integrated modularly into Flashix, a verified file system for flash memory, and both functional correctness and crash-safety of this extension have been verified with the interactive theorem prover KIV.

An integrated tool set for verifying CafeOBJ specifications

CafeOBJ is a language for specifying and verifying a wide variety of software and/or hardware systems. Traditionally, verification has been carried out via proof scores, which consist of reducing goal-related terms in user-defined modules. Although proof scores are semi-formal (the specifier is partially responsible for soundness), their flexibility makes them a useful approach to verification.For the last years, we have developed different formal tools around the CafeInMaude interpreter, a CafeOBJ interpreter implemented in Maude. Besides supporting proof scores, we implemented a theorem prover, a proof script generator from proof scores, and the first stages of a proof script generator and fixer-upper. In this paper, we present (i) an improved and detailed version of our proof script generator and fixer-upper and (ii) a reimplementation of the CafeInMaude interpreter, which supports, among others, parallel execution, an improved tool integration, and an interactive user interface. The benchmarks used to evaluate the tools confirm the usefulness of the approach.

Confluence of left-linear higher-order rewrite theories by checking their nested critical pairs

AbstractUser-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.

Formalization of Double-Word Arithmetic, and Comments on “Tight and Rigorous Error Bounds for Basic Building Blocks of Double-Word Arithmetic”

Recently, a complete set of algorithms for manipulating double-word numbers (some classical, some new) was analyzed [ 16 ]. We have formally proven all the theorems given in that article, using the Coq proof assistant. The formal proof work led us to: (i) locate mistakes in some of the original paper proofs (mistakes that, however, do not hinder the validity of the algorithms), (ii) significantly improve some error bounds, and (iii) generalize some results by showing that they are still valid if we slightly change the rounding mode. The consequence is that the algorithms presented in [ 16 ] can be used with high confidence, and that some of them are even more accurate than what was believed before. This illustrates what formal proof can bring to computer arithmetic: beyond mere (yet extremely useful) verification, correction, and consolidation of already known results, it can help to find new properties. All our formal proofs are freely available.

Evolution of SASyLF 2008-2021

SASyLF was released in 2008 and used as a proof assistant in courses at several universities. It proved itself useful and has continued to be used, and each iteration of use has encouraged further development: fixing bugs and adding enhancements. This paper describes how SASyLF was developed while keeping true to its purpose. Most notable are making substitutions explicit, support of "and" and "or," support for mutual and lexicographic induction, and IDE support.

Natural Language Proof Checking in Introduction to Proof Classes – First Experiences with Diproche

We present and analyze the employment of the Diproche system, a natural language proof checker, within a one-semester mathematics beginners lecture with 228 participants. The system is used to check the students' solution attempts to proving exercises in Boolean set theory and elementary number theory and to give them immediate feedback. The benefits of the employment of the system are assessed via a questionnaire at the end of the semester and via analyzing the solution attempts of a subgroup of the students. Based on our results we develop approaches for future improvements.

Quotients of Bounded Natural Functors

The functorial structure of type constructors is the foundation for many definition and proof principles in higher-order logic (HOL). For example, inductive and coinductive datatypes can be built modularly from bounded natural functors (BNFs), a class of well-behaved type constructors. Composition, fixpoints, and, under certain conditions, subtypes are known to preserve the BNF structure. In this article, we tackle the preservation question for quotients, the last important principle for introducing new types in HOL. We identify sufficient conditions under which a quotient inherits the BNF structure from its underlying type. Surprisingly, lifting the structure in the obvious manner fails for some quotients, a problem that also affects the quotients of polynomial functors used in the Lean proof assistant. We provide a strictly more general lifting scheme that supports such problematic quotients. We extend the Isabelle/HOL proof assistant with a command that automates the registration of a quotient type as a BNF, reducing the proof burden on the user from the full set of BNF axioms to our inheritance conditions. We demonstrate the command's usefulness through several case studies.

Blockchain-based random auditor committee for integrity verification

With the popularity of cloud storage, integrity verification for outsourced data has attracted more and more attention. The majority of existing research works rely on a unique third-party auditor (TPA) or fixed TPAs to audit the integrity of outsourced data, and thus they are vulnerable to DoS and collusion attacks as the TPA is not completely trusted. To conquer this problem, a blockchain-based random auditor committee is proposed to replace the fixed TPAs for the integrity verification (call BRAC scheme for short). The proposed blockchain consensus algorithm utilizes a verifiable random function (VRF) to select the third-party auditor committee (TPAC) which performs contract verification. The leader of TPAs (LTPA) packs the verification proof into a blockchain called verification chain. The main idea is employing the output of a VRF for leader election to make it probabilistic, such that specific nodes cannot be targeted in advance and a certain fraction of the nodes being malicious can be tolerated. We present rigorous security analysis to demonstrate the security of the proposed scheme. Meanwhile, the comprehensive performance evaluation shows that the proposed scheme improves the performance of data integrity verification effectively.

Efficient Publicly Verifiable Proofs of Data Replication and Retrievability Applicable for Cloud Storage

Efficient Publicly Verifiable Proofs of Data Replication and Retrievability Applicable for Cloud Storage

Spatial Isolation Implies Zero Knowledge Even in a Quantum World

Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP , as long as one is willing to make a suitable physical assumption : if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP * model captures this setting. In this work, we study the following question: Does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP * . Our proof consists of constructing a zero knowledge interactive probabilistically checkable proof with a strong algebraic structure, and then lifting it to the MIP * model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory.

A Bi-Directional Extensible Interface Between Lean and Mathematica

We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending translation, our connection allows for the exchange of arbitrary information between the two systems. We show how to make use of the Lean metaprogramming framework to verify certain Mathematica computations, so that the rigor of the proof assistant is not compromised. We also use Mathematica as an untrusted oracle to guide proof search in the proof assistant and interact with a Mathematica notebook from within a Lean session. In the other direction, we import and process Lean declarations from within Mathematica. The proof assistant library serves as a database of mathematical knowledge that the CAS can display and explore.

An extensible equality checking algorithm for dependent type theories

We present a general and user-extensible equality checking algorithm that is applicable to a large class of type theories. The algorithm has a type-directed phase for applying extensionality rules and a normalization phase based on computation rules, where both kinds of rules are defined using the type-theoretic concept of object-invertible rules. We also give sufficient syntactic criteria for recognizing such rules, as well as a simple pattern-matching algorithm for applying them. A third component of the algorithm is a suitable notion of principal arguments, which determines a notion of normal form. By varying these, we obtain known notions, such as weak head-normal and strong normal forms. We prove that our algorithm is sound. We implemented it in the Andromeda 2 proof assistant, which supports user-definable type theories. The user need only provide the equality rules they wish to use, which the algorithm automatically classifies as computation or extensionality rules, and select appropriate principal arguments.

A cost-aware logical framework

We present calf , a c ost- a ware l ogical f ramework for studying quantitative aspects of functional programs. Taking inspiration from recent work that reconstructs traditional aspects of programming languages in terms of a modal account of phase distinctions , we argue that the cost structure of programs motivates a phase distinction between intension and extension . Armed with this technology, we contribute a synthetic account of cost structure as a computational effect in which cost-aware programs enjoy an internal noninterference property: input/output behavior cannot depend on cost. As a full-spectrum dependent type theory, calf presents a unified language for programming and specification of both cost and behavior that can be integrated smoothly with existing mathematical libraries available in type theoretic proof assistants. We evaluate calf as a general framework for cost analysis by implementing two fundamental techniques for algorithm analysis: the method of recurrence relations and physicist’s method for amortized analysis . We deploy these techniques on a variety of case studies: we prove a tight, closed bound for Euclid’s algorithm, verify the amortized complexity of batched queues, and derive tight, closed bounds for the sequential and parallel complexity of merge sort, all fully mechanized in the Agda proof assistant. Lastly we substantiate the soundness of quantitative reasoning in calf by means of a model construction.