Proof Certificates Research Articles

Verifying a Sequent Calculus Prover for First-Order Logic with Functions in Isabelle/HOL

We describe the design, implementation and verification of an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we formally verify its soundness and completeness in Isabelle/HOL using an existing abstract framework for coinductive proof trees. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do the same when building a countermodel from a failed proof. Finally, we formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover can generate human-readable SeCaV proofs which are also machine-verifiable proof certificates. The abstract framework we rely on requires us to fix a stream of proof rules in advance, independently of the formula we are trying to prove. We discuss the efficiency implications of this and the difficulties in mitigating them.

A SAT Solver and Computer Algebra Attack on the Minimum Kochen-Specker Problem (Student Abstract)

The problem of finding the minimum three-dimensional Kochen–Specker (KS) vector system, an important problem in quantum foundations, has remained open for over 55 years. We present a new method to address this problem based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS). Our approach improved the lower bound on the size of a KS system from 22 to 24. More importantly, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a proof size of 41.6 TiB for order 23. The efficiency is due to the powerful combination of SAT solvers and CAS-based orderly generation.

Generating and Exploiting Automated Reasoning Proof Certificates

Moving toward a full suite of proof-producing automated reasoning tools with SMT solvers that can produce full, independently checkable proofs for real-world problems.

Mostly Automated Proof Repair for Verified Libraries

The cost of maintaining formally specified and verified software is widely considered prohibitively high due to the need to constantly keep code and the proofs of its correctness in sync—the problem known as proof repair . One of the main challenges in automated proof repair for evolving code is to infer invariants for a new version of a once verified program that are strong enough to establish its full functional correctness. In this work, we present the first proof repair methodology for higher-order imperative functions, whose initial versions were verified in the Coq proof assistant and whose specifications remained unchanged. Our proof repair procedure is based on the combination of dynamic program alignment, enumerative invariant synthesis, and a novel technique for efficiently pruning the space of invariant candidates, dubbed proof-driven testing , enabled by the constructive nature of Coq’s proof certificates. We have implemented our approach in a mostly-automated proof repair tool called Sisyphus. Given an OCaml function verified in Coq and its unverified new version, Sisyphus produces a Coq proof for the new version, discharging most of the new proof goals automatically and suggesting high-confidence obligations for the programmer to prove for the cases when automation fails. We have evaluated Sisyphus on 10 OCaml programs taken from popular libraries, that manipulate arrays and mutable data structures, considering their verified original and unverified evolved versions. Sisyphus has managed to repair proofs for all those functions, suggesting correct invariants and generating a small number of easy-to-prove residual obligations.

Generating Proof Certificates for a Language-Agnostic Deductive Program Verifier

Previous work on rewriting and reachability logic establishes a vision for a language-agnostic program verifier, which takes three inputs: a program, its formal specification, and the formal semantics of the programming language in which the program is written. The verifier then uses a language-agnostic verification algorithm to prove the program correct with respect to the specification and the formal language semantics. Such a complex verifier can easily have bugs. This paper proposes a method to certify the correctness of each successful verification run by generating a proof certificate. The proof certificate can be checked by a small proof checker. The preliminary experiments apply the method to generate proof certificates for program verification in an imperative language, a functional language, and an assembly language, showing that the proposed method is language-agnostic.

Hubungan Bimbingan Perkawinan Pranikah dengan Tingginya Angka Penceraian:


 The decision number of the Director General of Islamic Guidance Long term 2018 on the Execution of Early Marriage Mentoring plans to both Diminish the Quantity of Separations and Make a Cheerful and Tranquil Family. However, the annual divorce rate in Medan continues to rise. In accordance with the Director General of Islamic Guidance's decision No. 379 of 2018 in the KUA of Medan, as well as the results of carrying out early advice in light of the decision made by the chief general of Islamic Direction No. 379 in 2018 to decrease the divorce rate in Medan. Qualitative (empirical) field research was used in this study. The study came to the conclusion that premarital counseling isn't as effective as it should be; Instead, many married couples take a marriage certificate course. It was thought that couples who wanted to get married needed to show a proof certificate to get premarital counseling. Additionally, only the authority to provide premarital counseling and mediation services to couples filing for divorce with the Religious Court ought to be granted by the government to KUA.
 Keywords: Divorce, KUA, premarital counseling, and Medan

Verified Propagation Redundancy and Compositional UNSAT Checking in CakeML

Modern SAT solvers can emit independently-checkable proof certificates to validate their results. The state-of-the-art proof system that allows for compact proof certificates is propagation redundancy (textsf{PR}). However, the only existing method to validate proofs in this system with a formally verified tool requires a transformation to a weaker proof system, which can result in a significant blowup in the size of the proof and increased proof validation time. This article describes the first approach to formally verify textsf{PR} proofs on a succinct representation. We present (i) a new Linear PR (LPR) proof format, (ii) an extension of the DPR-trim tool to efficiently convert textsf{PR} proofs into LPR format, and (iii) cake_lpr, a verified LPR proof checker developed in CakeML. We also enhance these tools with (iv) a new compositional proof format designed to enable separate (parallel) proof checking. The LPR format is backwards compatible with the existing LRAT format, but extends LRAT with support for the addition of textsf{PR} clauses. Moreover, cake_lpr is verified using CakeML ’s binary code extraction toolchain, which yields correctness guarantees for its machine code (binary) implementation. This further distinguishes our clausal proof checker from existing checkers because unverified extraction and compilation tools are removed from its trusted computing base. We experimentally show that: LPR provides efficiency gains over existing proof formats; cake_lpr ’s strong correctness guarantees are obtained without significant sacrifice in its performance; and the compositional proof format enables scalable parallel proof checking for large proofs.

Tinjauan Yuridis Sertifikat Hak Milik Atas Tanah Berdasarkan Undang-Undang Nomor 5 Tahun 1960 Tentang Peraturan Dasar Pokok-Pokok Agraria

According to Article 19 paragraph (2) letter c of Law No. 5 of 1960 concerning Basic Regulations on Agrarian, with the granting of a certificate of proof of the rights over the fields of the land that is with the issuance of the certificate of land rights for land rights, including property rights to land, the state has provided the guarantee and certainty in the land sector. For holders of the rights to land, have a certificate shall have legal certainty and the protection of the law. Compared with written evidence of the other, a certificate is proof of a strong right, meaning that the holder of land rights that the name listed in the certificate should be regarded as true until proven otherwise in a court with the other evidence. On the other hand a letter of proof of the right or the certificate of land that can serve to create orderly land law. Nevertheless, the applicability than the certificate-the certificate is often questioned its effectiveness in providing certainty and legal protection. if the certificate is really protecting the rights of the subjects of the law or protect the object under such a certificate, or even just can be used as proof certificate course because it often happens that there is a certificate that is presented in the trial can be formally recognized but can't be used to protect the subject and object of the law

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.

WITHDRAWN: Physical unclonable function based proof-carrying approximate circuits

Abstract Ensuring security without compromising the performance of a system and adaptability is fundamentally a very challenging task for approximate Circuits research groups and practitioners to trade off the computational precision against hardware, latency or energy consumption improvements. Internet protocol core suppliers who want to make such circuits have been able to persuade customers and of ability to origin the details of the accurate estimate. This paper Proposed a proof-carrying based on Physical Unclonable Function (PUF) as a response. The seller produces an estimated IP core combined with a certificate that shows the precision of the reliability of all the specific ratings. The proof certificate is packaged and shipped off to the buyer with the estimated IP heart. To activate such feature the PUF response from the chip will be used. The IP core approximation accuracy can be formally verified by the purchaser for formal verification at a portion of the usual computational cost.

Blockchain based Tamper Proof Certificates

Document is the main proof for showing identity of a person and nowadays there has been many cases related to document tampering. Most of the documents like marksheet, birth certificate, identity card, etc. can be easily tampered and produced, sometimes the tampered document looks much more real and it becomes difficult for the organization or the verifier to verify the document. This research paper will try to overcome the issues with the help of blockchain technology. Blockchain is a decentralized peer-to-peer mechanism that provides protection, trustworthiness, legitimacy, immutability, and transparency. In this proposed system hashing technique is used for encryption. By using this technology people can request for the necessary documents and verifier can easily verify the documents using hash value.

Proof-Carrying Approximate Circuits

Approximate circuits (AxCs) tradeoff computational accuracy against improvements in hardware area, delay, or energy consumption. IP core vendors who wish to create such circuits need to convince consumers of the resulting approximation quality. As a solution, we propose proof-carrying AxCs. The vendor creates an approximate IP core together with a certificate that proves the approximation quality. The proof certificate is bundled with the approximate IP core and sent off to the consumer. The consumer can formally verify the approximation quality of the IP core at a fraction of the typical computational cost for formal verification. In this brief, we first make the case for proof-carrying AxCs and then demonstrate the feasibility of the approach by a set of synthesis experiments using an exemplary approximation framework.

A Proof via Finite Elements for Schiffer’s Conjecture on a Regular Pentagon

A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartlomiej Siudeja it was shown that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. The case for the regular pentagon is more challenging, and has resisted a completely analytic attack. In this paper, we present a validated numerical method to prove this case, which involves iteratively bounding eigenvalues for a sequence of subdomains of the triangle. We use a learning algorithm to find and optimize this sequence of subdomains, making it straightforward to check our computations with standard software. Our proof has a short proof certificate, is checkable without specialized software and is adaptable to other situations.

A Semantic Framework for Proof Evidence

Theorem provers produce evidence of proof in many different formats, such as proof scripts, natural deductions, resolution refutations, Herbrand expansions, and equational rewritings. In implemented provers, numerous variants of such formats are actually used: consider, for example, such variants of or restrictions to resolution refutations as binary resolution, hyper-resolution, ordered-resolution, paramodulation, etc. We propose the foundational proof certificates (FPC) framework for defining the semantics of a broad range of proof evidence. This framework allows both producers of proof certificates and the checkers of those certificates to have a clear formal definition of the semantics of a wide variety of proof evidence. Employing the FPC framework will allow one to separate a proof from its provenance and to allow anyone to construct their own proof checker for a given style of proof evidence. The foundation on which FPC relies is that of proof theory, particularly recent work into focused proof systems: such proof systems provide protocols by which a checker extracts information from the certificate (mediated by the so called clerks and experts) as well as performs various deterministic and non-deterministic computations. While we shall limit ourselves to first-order logic in this paper, we shall not limit ourselves in many other ways. The FPC framework is described for both classical and intuitionistic logics and for proof structures as diverse as resolution refutations, natural deduction, Frege proofs, and equality proofs.

Proof Certificates for Equality Reasoning

The kinds of inference rules and decision procedures that one writes for proofs involving equality and rewriting are rather different from proofs that one might write in first-order logic using, say, sequent calculus or natural deduction. For example, equational logic proofs are often chains of replacements or applications of oriented rewriting and normal forms. In contrast, proofs involving logical connectives are trees of introduction and elimination rules. We shall illustrate here how it is possible to check various equality-based proof systems with a programmable proof checker (the kernel checker) for first-order logic. Our proof checker's design is based on the implementation of focused proof search and on making calls to (user-supplied) clerks and experts predicates that are tied to the two phases found in focused proofs. It is the specification of these clerks and experts that provide a formal definition of the structure of proof evidence. As we shall show, such formal definitions work just as well in the equational setting as in the logic setting where this scheme for proof checking was originally developed. Additionally, executing such a formal definition on top of a kernel provides an actual proof checker that can also do a degree of proof reconstruction. We shall illustrate the flexibility of this approach by showing how to formally define (and check) rewriting proofs of a variety of designs.

Proof Outlines as Proof Certificates: A System Description

We apply the foundational proof certificate (FPC) framework to the problem of designing high-level outlines of proofs. The FPC framework provides a means to formally define and check a wide range of proof evidence. A focused proof system is central to this framework and such a proof system provides an interesting approach to proof reconstruction during the process of proof checking (relying on an underlying logic programming implementation). Here, we illustrate how the FPC framework can be used to design proof outlines and then to exploit proof checkers as a means for expanding outlines into fully detailed proofs. In order to validate this approach to proof outlines, we have built the ACheck system that allows us to take a sequence of theorems and apply the proof outline "do the obvious induction and close the proof using previously proved lemmas".

A framework for proof certificates in finite state exploration

Model checkers use automated state exploration in order to prove various properties such as reachability, non-reachability, and bisimulation over state transition systems. While model checkers have proved valuable for locating errors in computer models and specifications, they can also be used to prove properties that might be consumed by other computational logic systems, such as theorem provers. In such a situation, a prover must be able to trust that the model checker is correct. Instead of attempting to prove the correctness of a model checker, we ask that it outputs its "proof evidence" as a formally defined document--a proof certificate--and that this document is checked by a trusted proof checker. We describe a framework for defining and checking proof certificates for a range of model checking problems. The core of this framework is a (focused) proof system that is augmented with premises that involve "clerk and expert" predicates. This framework is designed so that soundness can be guaranteed independently of any concerns for the correctness of the clerk and expert specifications. To illustrate the flexibility of this framework, we define and formally check proof certificates for reachability and non-reachability in graphs, as well as bisimulation and non-bisimulation for labeled transition systems. Finally, we describe briefly a reference checker that we have implemented for this framework.

Compositional certified resource bounds

This paper presents a new approach for automatically deriving worst-case resource bounds for C programs. The described technique combines ideas from amortized analysis and abstract interpretation in a unified framework to address four challenges for state-of-the-art techniques: compositionality, user interaction, generation of proof certificates, and scalability. Compositionality is achieved by incorporating the potential method of amortized analysis. It enables the derivation of global whole-program bounds with local derivation rules by naturally tracking size changes of variables in sequenced loops and function calls. The resource consumption of functions is described abstractly and a function call can be analyzed without access to the function body. User interaction is supported with a new mechanism that clearly separates qualitative and quantitative verification. A user can guide the analysis to derive complex non-linear bounds by using auxiliary variables and assertions. The assertions are separately proved using established qualitative techniques such as abstract interpretation or Hoare logic. Proof certificates are automatically generated from the local derivation rules. A soundness proof of the derivation system with respect to a formal cost semantics guarantees the validity of the certificates. Scalability is attained by an efficient reduction of bound inference to a linear optimization problem that can be solved by off-the-shelf LP solvers. The analysis framework is implemented in the publicly-available tool C4B. An experimental evaluation demonstrates the advantages of the new technique with a comparison of C4B with existing tools on challenging micro benchmarks and the analysis of more than 2900 lines of C code from the cBench benchmark suite.

The Structure of Differential Invariants and Differential Cut Elimination

The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closed-form solutions only exist for very simple differential equations, proof certificates have been proposed for more scalable verification. Search procedures for these proof certificates are still rather ad-hoc, though, because the problem structure is only understood poorly. We investigate differential invariants, which define an induction principle for differential equations and which can be checked for invariance along a differential equation just by using their differential structure, without having to solve them. We study the structural properties of differential invariants. To analyze trade-offs for proof search complexity, we identify more than a dozen relations between several classes of differential invariants and compare their deductive power. As our main results, we analyze the deductive power of differential cuts and the deductive power of differential invariants with auxiliary differential variables. We refute the differential cut elimination hypothesis and show that, unlike standard cuts, differential cuts are fundamental proof principles that strictly increase the deductive power. We also prove that the deductive power increases further when adding auxiliary differential variables to the dynamics.

Secure sealed-bid online auctions using discreet cryptographic proofs

This work describes the design and implementation of an auction system using secure multiparty computation techniques. Our aim is to produce a system that is practical under actual field constraints on computation, memory, and communication. The underlying protocol is privacy-preserving, that is, the winning bid is determined without information about the losing bids leaking to either the auctioneer or other bidders. Practical implementation of the protocol is feasible using circuit-based cryptographic proofs along with additively homomorphic bit commitment. Moreover, we propose the development of a Proof Certificate standard. These certificates convey sufficient information to recreate the cryptographic proofs and verify them offline.