Modular Proof Research Articles

Abstract Interpreters: A Monadic Approach to Modular Verification

We argue that monadic interpreters built as layers of interpretations stacked atop the free monad constitute a promising way to implement and verify abstract interpreters in dependently-typed theories such as the one underlying the Coq proof assistant. The approach enables modular proofs of soundness of the resulting interpreters. We provide generic abstract control flow combinators proven correct once and for all against their concrete counterpart. We demonstrate how to relate concrete handlers implementing effects to abstract variants of these handlers, essentially capturing the traditional soundness of transfer functions in the context of monadic interpreters. Finally, we provide generic results to lift soundness statements via the interpretation of stateful and failure effects. We formalize all the aforementioned combinators and theories in Coq, and demonstrate their benefits by implementing and proving correct two illustrative abstract interpreters for a structured imperative language and a toy assembly.

On Tools for Completeness of Kleene Algebra with Hypotheses

In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and we prove completeness for new variants of KAT: KAT extended with a constant for the full relation, KAT extended with a converse operation, and a version of KAT where the collection of tests only forms a distributive lattice.

Modular Denotational Semantics for Effects with Guarded Interaction Trees

We present guarded interaction trees — a structure and a fully formalized framework for representing higher-order computations with higher-order effects in Coq, inspired by domain theory and the recently proposed interaction trees. We also present an accompanying separation logic for reasoning about guarded interaction trees. To demonstrate that guarded interaction trees provide a convenient domain for interpreting higher-order languages with effects, we define an interpretation of a PCF-like language with effects and show that this interpretation is sound and computationally adequate; we prove the latter using a logical relation defined using the separation logic. Guarded interaction trees also allow us to combine different effects and reason about them modularly. To illustrate this point, we give a modular proof of type soundness of cross-language interactions for safe interoperability of different higher-order languages with different effects. All results in the paper are formalized in Coq using the Iris logic over guarded type theory.

Graph-based approximate message passing iterations

Abstract Approximate message passing (AMP) algorithms have become an important element of high-dimensional statistical inference, mostly due to their adaptability and concentration properties, the state evolution (SE) equations. This is demonstrated by the growing number of new iterations proposed for increasingly complex problems, ranging from multi-layer inference to low-rank matrix estimation with elaborate priors. In this paper, we address the following questions: is there a structure underlying all AMP iterations that unifies them in a common framework? Can we use such a structure to give a modular proof of state evolution equations, adaptable to new AMP iterations without reproducing each time the full argument? We propose an answer to both questions, showing that AMP instances can be generically indexed by an oriented graph. This enables to give a unified interpretation of these iterations, independent from the problem they solve, and a way of composing them arbitrarily. We then show that all AMP iterations indexed by such a graph verify rigorous SE equations, extending the reach of previous proofs and proving a number of recent heuristic derivations of those equations. Our proof naturally includes non-separable functions and we show how existing refinements, such as spatial coupling or matrix-valued variables, can be combined with our framework.

SSProve: A Foundational Framework for Modular Cryptographic Proofs in Coq

State-separating proofs (SSP) is a recent methodology for structuring game-based cryptographic proofs in a modular way, by using algebraic laws to exploit the modular structure of composed protocols. While promising, this methodology was previously not fully formalized and came with little tool support. We address this by introducing SSProve, the first general verification framework for machine-checked state-separating proofs. SSProve combines high-level modular proofs about composed protocols, as proposed in SSP, with a probabilistic relational program logic for formalizing the lower-level details, which together enable constructing machine-checked cryptographic proofs in the Coq proof assistant. Moreover, SSProve is itself fully formalized in Coq, including the algebraic laws of SSP, the soundness of the program logic, and the connection between these two verification styles. To illustrate SSProve, we use it to mechanize the simple security proofs of ElGamal and pseudo-random-function–based encryption. We also validate the SSProve approach by conducting two more substantial case studies: First, we mechanize an SSP security proof of the key encapsulation mechanism–data encryption mechanism (KEM-DEM) public key encryption scheme, which led to the discovery of an error in the original paper proof that has since been fixed. Second, we use SSProve to formally prove security of the sigma-protocol zero-knowledge construction, and we moreover construct a commitment scheme from a sigma-protocol to compare with a similar development in CryptHOL. We instantiate the security proof for sigma-protocols to give concrete security bounds for Schnorr’s sigma-protocol.

Construction of feature analysis model for demeanor evidence investigation based on data mining algorithm

Demeanor evidence refers to the authoritative opinions of experts using analytical instruments to collect and interpret the formation of external physiological symptoms of the entire human body. With the development of modern technology, the observation of demeanor has entered a new stage of multimodal quantification. The quantifiable results of demeanor observation with the help of scientific and technological instruments play an important role in judicial work such as search, interrogation, and evidence review. The use of man-made reasoning in the field of legal settling has become increasingly broad. Specialists are giving increasingly more consideration to the securing of modular proof. This paper proposes the development of a component examination model for modular proof examination applications. The technique for this paper is to apply the guileless Bayes strategy, propose a superior information mining calculation, and lay out a model for proof observation and examination. The function of these methods is to systematically explore the basic theoretical issues of demeanor evidence based on the status quo of judicial application of demeanor evidence. Through the prediction of individual demeanor based on data mining algorithm, the evidence analysis model is designed, the neurobiological experiment is carried out, and the demeanor evidence animal stress model is constructed to verify the scientific basis of multidemeanor evidence observation. The experimental results showed that after repeated stimulation of SD rats, the maximum changes in the expression of HSP70 gene and SAA gene were 10.77 and 14.1 respectively, reflecting the high reliability of demeanor evidence biological experiments. The model can improve the accuracy of evidence use, and the correct use of demeanor evidence can realize the true litigation value of intelligent justice and the concept of human rights protection, and promote the construction of intelligent justice.

Subversion Resilient Hashing: Efficient Constructions and Modular Proofs for Crooked Indifferentiability

We consider the problem of constructing secure cryptographic hash functions from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">subverted</i> ideal primitives. Hash functions are used to instantiate Random Oracles in cryptographic protocols. The indifferentiability security notion is a popular tool to certify the structural soundness of a hash design for such instantiations. In CRYPTO 2018, Russell, Tang, Yung, and Zhou introduced the notion of crooked-indifferentiability to extend this paradigm even when the underlying primitive of the hashing mode is subverted. They showed that an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -to- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -bit function implemented using Enveloped XOR construction (EXor) with 3 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> + 1 many independent <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -to- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -bit functions and 3 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -bit random seed can be proven secure asymptotically in the crooked-indifferentiability setting. Unfortunately, known techniques to prove crooked-indifferentiability are extremely complicated, and no practical hashing mode has been analyzed in this setting. • We introduce new techniques to prove crooked-indifferentiability. We establish that upper bounding the subversion probability of a chaining query is sufficient to argue subversion resistance of a standard indifferentiable mode of operation. Our technique links standard indifferentiability and crooked-indifferentiability and circumvents the complications of proving the consistency of the simulator in the crooked setting. • We prove crooked-indifferentiability of the sponge construction when the underlying primitive is modelled as an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -to- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -bit random function. Our proofs only require <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -bit randomly chosen but fixed IV and do not mandate any independent function requirement. The result naturally extends to the Merkle-Damgård domain extension with prefix-free padding. Our results minimize required randomness and solve the main open problem raised by Russell, Tang, Yung, and Zhou.

Tasks in modular proofs of concurrent algorithms

Proving the correctness of distributed or concurrent algorithms is a complex process. Errors in the reasoning are hard to find, calling for computer-checked proof systems like Coq or TLA+. To use these tools, sequential specifications of base objects are required to build modular proofs by composition. Unfortunately, many concurrent objects lack a sequential specification. This article describes a method to transform any task, a specification of a concurrent one-shot distributed problem, into a sequential specification involving two calls, set and get. This enables designers to compose proofs, facilitating modular computer-checked proofs of algorithms built using tasks and sequential objects as building blocks. Moir & Anderson implementation of renaming using splitters, wait-free concurrent objects, is an algorithm designed by composition, but it is not modular. Using our transformation, a modular description of the algorithm is given in TLA+ and mechanically verified using the TLA+ Proof System. As far as we know, this is the first time this algorithm is mechanically verified.

Modular Termination for Second-Order Computation Rules and Application to Algebraic Effect Handlers

We present a new modular proof method of termination for second-order computation, and report its implementation SOL. The proof method is useful for proving termination of higher-order foundational calculi. To establish the method, we use a variation of semantic labelling translation and Blanqui's General Schema: a syntactic criterion of strong normalisation. As an application, we apply this method to show termination of a variant of call-by-push-value calculus with algebraic effects and effect handlers. We also show that our tool SOL is effective to solve higher-order termination problems.

Visibility reasoning for concurrent snapshot algorithms

Visibility relations have been proposed by Henzinger et al. as an abstraction for proving linearizability of concurrent algorithms that obtains modular and reusable proofs. This is in contrast to the customary approach based on exhibiting the algorithm's linearization points. In this paper we apply visibility relations to develop modular proofs for three elegant concurrent snapshot algorithms of Jayanti. The proofs are divided by signatures into components of increasing level of abstraction; the components at higher abstraction levels are shared, i.e., they apply to all three algorithms simultaneously. Importantly, the interface properties mathematically capture Jayanti's original intuitions that have previously been given only informally.

Modular proofs of Gosper's identities

We give unified modular proofs to all of Gosper's identities on the q-constant Πq. We also confirm Gosper's observation that for any distinct positive integers n1,⋯,nm with m≥3, Πqn1, ⋯, Πqnm satisfy a nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gosper's identities. Meanwhile, several results on Πq found by El Bachraoui have been revised. Furthermore, we illustrate a strategy to construct some of Gosper's identities using hauptmoduls for genus zero congruence subgroups.

Verifying replicated data types with typeclass refinements in Liquid Haskell

This paper presents an extension to Liquid Haskell that facilitates stating and semi-automatically proving properties of typeclasses. Liquid Haskell augments Haskell with refinement types —our work allows such types to be attached to typeclass method declarations, and ensures that instance implementations respect these types. The engineering of this extension is a modular interaction between GHC, the Glasgow Haskell Compiler, and Liquid Haskell’s core proof infrastructure. The design sheds light on the interplay between modular proofs and typeclass resolution, which in Haskell is coherent by default (meaning that resolution always selects the same implementation for a particular instantiating type), but in other dependently typed languages is not. We demonstrate the utility of our extension by using Liquid Haskell to modularly verify that 34 instances satisfy the laws of five standard typeclasses. More substantially, we implement a framework for programming distributed applications based on replicated data types (RDTs). We define a typeclass whose Liquid Haskell type captures the mathematical properties RDTs should satisfy; prove in Liquid Haskell that these properties are sufficient to ensure that replicas’ states converge despite out-of-order update delivery; implement (and prove correct) several instances of our RDT typeclass; and use them to build two realistic applications, a multi-user calendar event planner and a collaborative text editor.

Acquiring Plant Features with Optical Sensing Devices in an Organic Strip-Cropping System

The SUREVEG project focuses on improvement of biodiversity and soil fertility in organic agriculture through strip-cropping systems. To counter the additional workforce a robotic tool is proposed. Within the project, a modular proof of concept (POC) version will be produced that will combine detection technologies with actuation on a single-plant level in the form of a robotic arm. This article focuses on the detection of crop characteristics through point clouds obtained with two lidars. Segregation in soil and plants was successfully achieved without the use of additional data from other sensor types, by calculating weighted sums, resulting in a dynamically obtained threshold criterion. This method was able to extract the vegetation from the point cloud in strips with varying vegetation coverage and sizes. The resulting vegetation clouds were compared to drone imagery, to prove they perfectly matched all green areas in said image. By dividing the remaining clouds of overlapping plants by means of the nominal planting distance, the number of plants, their volumes, and thereby the expected yields per row could be determined.

Modularity for decidability of deductive verification with applications to distributed systems

Proof automation can substantially increase productivity in formal verification of complex systems. However, unpredictablility of automated provers in handling quantified formulas presents a major hurdle to usability of these tools. We propose to solve this problem not by improving the provers, but by using a modular proof methodology that allows us to produce decidable verification conditions. Decidability greatly improves predictability of proof automation, resulting in a more practical verification approach. We apply this methodology to develop verified implementations of distributed protocols, demonstrating its effectiveness.

Modular Labelled Sequent Calculi for Abstract Separation Logics

Abstract separation logics are a family of extensions of Hoare logic for reasoning about programs that manipulate resources such as memory locations. These logics are “abstract” because they are independent of any particular concrete resource model. Their assertion languages, called Propositional Abstract Separation Logics (PASLs), extend the logic of (Boolean) Bunched Implications (BBI) in various ways. In particular, these logics contain the connectives * and –*, denoting the composition and extension of resources, respectively. This added expressive power comes at a price, since the resulting logics are all undecidable. Given their wide applicability, even a semi-decision procedure for these logics is desirable. Although several PASLs and their relationships with BBI are discussed in the literature, the proof theory of, and automated reasoning for, these logics were open problems solved by the conference version of this article, which developed a modular proof theory for various PASLs using cut-free labelled sequent calculi. This paper non-trivially improves upon this previous work by giving a general framework of calculi on which any new axiom in the logic satisfying a certain form corresponds to an inference rule in our framework, and the completeness proof is generalised to consider such axioms. Our base calculus handles Calcagno et al.’s original logic of separation algebras by adding sound rules for partial-determinism and cancellativity, while preserving cut-elimination. We then show that many important properties in separation logic, such as indivisible unit, disjointness, splittability, and cross-split, can be expressed in our general axiom form. Thus, our framework offers inference rules and completeness for these properties for free. Finally, we show how our calculi reduce to calculi with global label substitutions, enabling more efficient implementation.

Recalling a witness: foundations and applications of monotonic state

We provide a way to ease the verification of programs whose state evolves monotonically. The main idea is that a property witnessed in a prior state can be soundly recalled in the current state, provided (1) state evolves according to a given preorder, and (2) the property is preserved by this preorder. In many scenarios, such monotonic reasoning yields concise modular proofs, saving the need for explicit program invariants. We distill our approach into the monotonic-state monad , a general yet compact interface for Hoare-style reasoning about monotonic state in a dependently typed language. We prove the soundness of the monotonic-state monad and use it as a unified foundation for reasoning about monotonic state in the F ⋆ verification system. Based on this foundation, we build libraries for various mutable data structures like monotonic references and apply these libraries at scale to the verification of several distributed applications.

A modular reasoning system using uninterpreted predicates for code reuse

This paper proposes a modular proof system based on uninterpreted predicates. The proposed proof system allows modular reasoning about programs with an open-world assumption, which goes beyond behavioral subtyping. The proof system enables modular reasoning about languages with very flexible code reuse mechanisms, such as traits and deltas in the context of object-oriented programming. Whereas related work on incremental proof systems prove soundness in terms of internal consistency, this paper establishes both soundness and relative completeness of the proposed proof system by relating it to a standard proof system for a simple object-oriented language. The applicability of the approach is demonstrated on different code reuse mechanisms: unrestricted class inheritance, delta-oriented programming, and trait-based programming.

A simple soundness proof for dependent object types

Dependent Object Types (DOT) is intended to be a core calculus for modelling Scala. Its distinguishing feature is abstract type members, fields in objects that hold types rather than values. Proving soundness of DOT has been surprisingly challenging, and existing proofs are complicated, and reason about multiple concepts at the same time (e.g. types, values, evaluation). To serve as a core calculus for Scala, DOT should be easy to experiment with and extend, and therefore its soundness proof needs to be easy to modify. This paper presents a simple and modular proof strategy for reasoning in DOT. The strategy separates reasoning about types from other concerns. It is centred around a theorem that connects the full DOT type system to a restricted variant in which the challenges and paradoxes caused by abstract type members are eliminated. Almost all reasoning in the proof is done in the intuitive world of this restricted type system. Once we have the necessary results about types, we observe that the other aspects of DOT are mostly standard and can be incorporated into a soundness proof using familiar techniques known from other calculi.

Profinite techniques for probabilistic automata and the Markov Monoid algorithm

We consider the value 1 problem for probabilistic automata over finite words: it asks whether a given probabilistic automaton accepts words with probability arbitrarily close to 1. This problem is known to be undecidable. However, different algorithms have been proposed to partially solve it; it has been recently shown that the Markov Monoid algorithm, based on algebra, is the most correct algorithm so far. The first contribution of this paper is to give a characterisation of the Markov Monoid algorithm.The second contribution is to develop a profinite theory for probabilistic automata, called the prostochastic theory. This new framework gives a topological account of the value 1 problem, which in this context is cast as an emptiness problem. The above characterisation is reformulated using the prostochastic theory, allowing us to give a simple and modular proof.

Modular embeddings of Teichmüller curves

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.