Proof System Research Articles

From P Versus NP to Probabilistic and Zero Knowledge Proof Systems

From P Versus NP to Probabilistic and Zero Knowledge Proof Systems

A logarithmic-sized accountable ring signature scheme in the standard model

Ring signature and group signature are two major cryptographic primitives providing users anonymity and authentication simultaneously. While both primitives enable any user to sign messages as a member of a set of users, the feature of ring signature is that a signer can choose a group in an ad-hoc manner, which is called a ring, by itself. Conversely, in group signature, a group membership is managed by some trusted third party, which is called a group manager, and an (appropriate) accountability is provided for the group manager to identify illegal anonymous signers. Accountable ring signature (ARS) is a cryptographic primitive combining the features of ring signature and group signature. ARS allows signers to choose their groups (rings) in an ad-hoc manner, and at the same time, maintain accountability by forcing them to select a designated opener who can identify them.In this paper, we propose the first ARS scheme with O(log⁡n) signature size in the standard model (without depending on the random oracle methodology), where n is the size of a ring. More precisely, we propose a new generic construction of ARS based on (standard) signature, (standard) public key encryption, non-interactive zero-knowledge proof system, somewhere perfectly binding hash function, and public key encryption with non-interactive opening. All building blocks for our ARS scheme can be obtained under the decisional linear (DLIN) assumption over bilinear groups.

Paths, Proofs, and Perfection: Developing a Human-Interpretable Proof System for Constrained Shortest Paths

People want to rely on optimization algorithms for complex decisions but verifying the optimality of the solutions can then become a valid concern, particularly for critical decisions taken by non-experts in optimization. One example is the shortest-path problem on a network, occurring in many contexts from transportation to logistics to telecommunications. While the standard shortest-path problem is both solvable in polynomial time and certifiable by duality, introducing side constraints makes solving and certifying the solutions much harder. We propose a proof system for constrained shortest-path problems, which gives a set of logical rules to derive new facts about feasible solutions. The key trait of the proposed proof system is that it specifically includes high-level graph concepts within its reasoning steps (such as connectivity or path structure), in contrast to, e.g., using linear combinations of model constraints. Thus, using our proof system, we can provide a step-by-step, human-auditable explanation showing that the path given by an external solver cannot be improved. Additionally, to maximize the advantages of this setup, we propose a proof search procedure that specifically aims to find small proofs of this form using a procedure similar to A* search. We evaluate our proof system on constrained shortest path instances generated from real-world road networks and experimentally show that we may indeed derive more interpretable proofs compared to an integer programming approach, in some cases leading to much smaller proofs.

Defeasible Normative Reasoning: A Proof-Theoretic Integration of Logical Argumentation

We present a novel computational approach to resolving conflicts among norms by nonmonotonic normative reasoning (in constrained I/O logics). Our approach extends standard sequent-based proof systems and makes them more adequate to nonmonotonic reasoning by adding to the sequents annotations that keep track of what is known about the defeasible status of the derived sequents. This makes transparent the reasons according to which norms should be applicable or inapplicable, and accordingly the sequents that make use of such norms are accepted or retracted. We also show that this proof theoretic method has tight links to the semantics of formal argumentation frameworks. The outcome of this paper is thus a threefold characterization result that relates, in the context of nonmonotonic normative reasoning, three traditional ingredients of AI-based reasoning methods: maximally consistent sets of premises (in constrained I/O logics), derived sequents (which are accepted in corresponding annotated sequent calculi), and logical arguments (that belong to the grounded extensions of the induced logical argumentation frameworks).

Runtime vs. Extracted Proof Size: An Exponential Gap for CDCL on QBFs

Conflict-driven clause learning (CDCL) is the dominating algorithmic paradigm for SAT solving and hugely successful in practice. In its lifted version QCDCL, it is one of the main approaches for solving quantified Boolean formulas (QBF). In both SAT and QBF, proofs can be efficiently extracted from runs of (Q)CDCL solvers. While for CDCL, it is known that the proof size in the underlying proof system propositional resolution matches the CDCL runtime up to a polynomial factor, we show that in QBF there is an exponential gap between QCDCL runtime and the size of the extracted proofs in QBF resolution systems. We demonstrate that this is not just a gap between QCDCL runtime and the size of any QBF resolution proof, but even the extracted proofs are exponentially smaller for some instances. Hence searching for a small proof via QCDCL (even with non-deterministic decision policies) will provably incur an exponential overhead for some instances.

Prospects of drawing on the results of researches are with application of poligrafe in finishing telling

In the presented article general directions and prospects of drawing on the results of researches are exposed with application of lie-detector in the process of finishing telling in criminal realizations. On the basis of the obtained data about economic feasibilities poligrafe and him software the list of spheres in that drawing on the results of such researches in Ukraine of most effective is exposed. The value of results of poligraf expert researches is reflected in the system of proofs. It is marked that except a tactical value in the process of finishing telling, the conclusion of expert can become founding of acceptance of such important decisions an inquisitional judge in realization, as realization of searches, temporal access to the things and documents, realization of complex of secret inquisitional (of criminal investigation) actions, setting of other judicial examinations and others like that.
 Certainly, that the results of judicial examination got in time with the use of poligrafe not only can come forward as basis for the construction of versions during investigation and determination of corresponding near-term inquisitional (of criminal investigation) actions but also included in the complex of proofs that there is base on a prosecution. It is marked that possibilities of judicial psychological examination with the use of methodology of lie-detector research are the unique method of realization of right for any personality to confirm veracity of the testimonies (including alibi), fix the position of collaboration with investigation, to lead to the openness.
 The list of psychical and physiology aspects, that can be set by an expert during application to the lie-detector for persons that is checked for involvement to offence, is presented. To such aspects, the following is attributed: to accentuation of personality or propensity to the feasance of separate types of offences; signs of pathological sexual perversions (paedophilias, necrophilias); presence of tracks bits are in memory of person about the circumstances of commission of crime; signs of twisting, saying nothing or constructing of information. Vision is exposed in relation to the complex of measures on the increase of prospect of deployment of possibilities of expert technology of the use of lie-detector in the process of finishing telling in criminal realizations.

Hard QBFs for Merge Resolution

We prove the first genuine quantified Boolean formula (QBF) proof size lower bounds for the proof system Merge Resolution (MRes), a refutational proof system for prenex QBFs with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together with information on countermodels, which are syntactically stored in the proofs as merge maps. This makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas that are hard for classical QBF resolution systems. Here, we show the first genuine QBF exponential lower bounds for MRes , thereby uncovering limitations of MRes. Technically, the results are either transferred from bounds from circuit complexity (for restricted versions of MRes) or directly obtained by combinatorial arguments (for full MRes). Our results imply that the MRes approach is largely orthogonal to other QBF resolution models such as the QCDCL resolution systems QRes and QURes and the expansion systems ∀ Exp + Res and IR.

Load-Balanced Parallel Implementation on GPUs for Multi-Scalar Multiplication Algorithm

Multi-scalar multiplication (MSM) is an important building block in most of elliptic-curve-based zero-knowledge proof systems, such as Groth16 and PLONK. Recently, Lu et al. proposed cuZK, a new parallel MSM algorithm on GPUs. In this paper, we revisit this scheme and present a new GPU-based implementation to further improve the performance of MSM algorithm. First, we propose a novel method for mapping scalars into Pippenger’s bucket indices, largely reducing the number of buckets compared to the original Pippenger algorithm. Second, in the case that memory is sufficient, we develop a new efficient algorithm based on homogeneous coordinates in the bucket accumulation phase. Moreover, our accumulation phase is load-balanced, which means the parallel speedup ratio is almost linear growth as the number of device threads increases. Finally, we also propose a parallel layered reduction algorithm for the bucket aggregation phase, whose time complexity remains at the logarithmic level of the number of buckets. The implementation results over the BLS12-381 curve on the V100 graphics card show that our proposed algorithm achieves up to 1.998x, 1.821x and 1.818x speedup compared to cuZK at scales of 221, 222, and 223, respectively.

Verifying Correctness

Yael Tauman Kalai on her career, proof systems, and certifying correctness.

Towards Uniform Certification in QBF

We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution. These results are obtained by taking a technique of Beyersdorff et al. (JACM 2020) that turns strategy extraction into simulation and combining it with new local strategy extraction arguments. This approach leads to simulations that are carried out mainly in propositional logic, with minimal use of the QBF rules. Our proofs therefore provide a new, largely propositional interpretation of the simulated systems. We argue that these results strengthen the case for uniform certification in QBF solving, since many QBF proof systems now fall into place underneath extended QBF Frege.

Weighted, Circular and Semi-Algebraic Proofs

In recent years there has been an increasing interest in studying proof systems stronger than Resolution, with the aim of building more efficient SAT solvers based on them. In defining these proof systems, we try to find a balance between the power of the proof system (the size of the proofs required to refute a formula) and the difficulty of finding the proofs. In this paper we consider the proof systems circular Resolution, Sherali-Adams, Nullstellensatz and Weighted Resolution and we study their relative power from a theoretical perspective. We prove that circular Resolution, Sherali-Adams and Weighted Resolution are polynomially equivalent proof systems. We also prove that Nullstellensatz is polynomially equivalent to a restricted version of Weighted Resolution. The equivalences carry on also for versions of the systems where the coefficients/weights are expressed in unary. The practical interest in these systems comes from the fact that they admit efficient algorithms to find proofs in case these have small width/degree.

Should Decisions in QCDCL Follow Prefix Order?

Quantified conflict-driven clause learning (QCDCL) is one of the main solving approaches for quantified Boolean formulas (QBF). One of the differences between QCDCL and propositional CDCL is that QCDCL typically follows the prefix order of the QBF for making decisions. We investigate an alternative model for QCDCL solving where decisions can be made in arbitrary order. The resulting system QCDCLANY\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{QCDCL}^\ extsf {{A\ iny {\\MakeUppercase {ny}}}}$$\\end{document} is still sound and terminating, but does not necessarily allow to always learn asserting clauses or cubes. To address this potential drawback, we additionally introduce two subsystems that guarantee to always learn asserting clauses (QCDCLUNI-ANY\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{QCDCL}^\ extsf {{U\ iny {\\MakeUppercase {ni}}-A\ iny {\\MakeUppercase {ny}}}}$$\\end{document}) and asserting cubes (QCDCLEXI-ANY\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{QCDCL}^\ extsf {{E\ iny {\\MakeUppercase {xi}}-A\ iny {\\MakeUppercase {ny}}}}$$\\end{document}), respectively. We model all four approaches by formal proof systems and show that QCDCLUNI-ANY\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{QCDCL}^\ extsf {{U\ iny {\\MakeUppercase {ni}}-A\ iny {\\MakeUppercase {ny}}}}$$\\end{document} is exponentially better than QCDCL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathsf{{QCDCL}} $$\\end{document} on false formulas, whereas QCDCLEXI-ANY\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{QCDCL}^\ extsf {{E\ iny {\\MakeUppercase {xi}}-A\ iny {\\MakeUppercase {ny}}}}$$\\end{document} is exponentially better than QCDCL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathsf{{QCDCL}} $$\\end{document} on true QBFs. Technically, this involves constructing specific QBF families and showing lower and upper bounds in the respective proof systems. We complement our theoretical study with some initial experiments that confirm our theoretical findings.

Web Consequence Untangled

AbstractUnder the standard modal explication of consequence, a conclusion is a consequence of some premises just in case necessarily, if the latter are true, so is the former. Notoriously, this explication yields some results that at first glance are counter-intuitive. In particular, a necessary truth is a consequence of arbitrary premises, and premises that cannot all be true together entail arbitrary conclusions. In his paper ‘On Ground and Consequence’ (Synthese, 2021), Benjamin Schnieder introduces a novel notion of web consequence, defined on the basis of the concept of ground, which, he argues, fits our intuitive conception better. Building on his idea, the present paper examines the concept of web consequence in more detail. In particular, I provide three alternative, and simpler semantic characterizations of Schnieder’s propositional logic of web consequence, two within a form of truthmaker semantics, one within a many-valued setting. I then consider some natural variations on that logic and establish their connections to well-known subclassical logics such as FDE, $$\hbox {K}_3$$ K 3 , and LP. Finally, I provide sound and complete tableaux-based proof systems for each of the logics of web consequence so obtained.

IOT Based Smart Solution for Preventing Fall Injuries at Workplace

In India fall injuries are increasing rapidly at workplace not only in the industrial sector even though construction and private sector too. Fall injury is the highest cause of death/ fatal/severe injuries. The severity of working at height incident depending upon the nature of job, height of application, uneven/cluttered surface etc.In this research paper a unique solution has been studied to bring revolutionary change in personal protective equipment which is used while working at height. There are many technologies are available, in chemical industry if any parameter deviate automatically machine will be shut-down and safe operating protocols will be activated but in conventional personal protective equipment it is hardly say there is still area of research where the technological improvement is needed to bring the robust safety system. Air bags are a very good example to correlate the smart PPE, if any passenger vehicle have air bag system collision with other vehicle automatically within milli second the airbag will be activated which saves the life of driver and passenger as well. Similarly Smart PPE has unique provision if PPE compliance is deviated at workplace immediately the violation shall be captured, and warning alert shall be given to user to remind them to compliance. This will help us to improve workplace safety. It is an error proof system that neither required manual surveillance nor manual recording of violations. In this research paper IOT internet of thing based smart safety harness has been studied and the workplace deviation were compared earlier with manual monitoring data to analyze the human error with respect to compliance of personal protective equipment

Gratifikasi Sebagai Bentuk Suap Menurut Undang-Undang Nomor 30 Tahun 2001

Efforts to eradicate criminal acts of corruption have become the mandate of the Indonesian nation and have been outlined in the Decree of the Indonesian People's Consultative Assembly (MPR) Number XI/MPR/1998 concerning the Administration of a State that is Clean and Free of Corruption, Collusion and Nepotism. This mandate was then followed up with Law Number 31 of 1999 concerning the Eradication of Corruption Crimes. The application of a reversal of the burden of proof system in reporting gratuities in the context of eradicating corruption has been implemented by Malaysia and Singapore. The implementation of the system of reversing the burden of proof in reporting gratuities in Law Number 20 of 2001 is expected to prevent criminal acts of corruption more optimally and effectively, at least reducing the corrupt practices that have occurred so far.

Depth lower bounds in Stabbing Planes for combinatorial principles

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Orthologic with Axioms

We study the proof theory and algorithms for orthologic, a logical system based on ortholattices, which have shown practical relevance in simplification and normalization of verification conditions. Ortholattices weaken Boolean algebras while having polynomial-time equivalence checking that is sound with respect to Boolean algebra semantics. We generalize ortholattice reasoning and obtain an algorithm for proving a larger class of classically valid formulas. As the key result, we analyze a proof system for orthologic augmented with axioms. An important feature of the system is that it limits the number of formulas in a sequent to at most two, which makes the extension with axioms non-trivial. We show a generalized form of cut elimination for this system, which implies a sub-formula property. From there we derive a cubic-time algorithm for provability from axioms, or equivalently, for validity in finitely presented ortholattices. We further show that propositional resolution of width 5 proves all formulas provable in orthologic with axioms. We show that orthologic system subsumes resolution of width 2 and arbitrarily wide unit resolution and is complete for reasoning about generalizations of propositional Horn clauses. Moving beyond ground axioms, we introduce effectively propositional orthologic (by analogy with EPR for classical logic), presenting its semantics as well as a sound and complete proof system. Our proof system implies the decidability of effectively propositional orthologic, as well as its fixed-parameter tractability for a bounded maximal number of variables in each axiom. As a special case, we obtain a generalization of Datalog with negation and disjunction.

A Universal, Sound, and Complete Forward Reasoning Technique for Machine-Verified Proofs of Linearizability

We introduce simple, universal , sound , and complete proof methods for producing machine-verifiable proofs of linearizability and strong linearizability. Universality means that our method works for any object type; soundness means that an algorithm can be proved correct by our method only if it is linearizable (resp. strong linearizable); and completeness means that any linearizable (resp. strong linearizable) implementation can be proved so using our method. We demonstrate the simplicity and power of our method by producing proofs of linearizability for the Herlihy-Wing queue and Jayanti’s single-scanner snapshot, as well as a proof of strong linearizability of the Jayanti-Tarjan union-find object. All three of these proofs are machine-verified by TLAPS (the TLA+ Proof System).

A Formalization of Core Why3 in Coq

Intermediate verification languages like Why3 and Boogie have made it much easier to build program verifiers, transforming the process into a logic compilation problem rather than a proof automation one. Why3 in particular implements a rich logic for program specification with polymorphism, algebraic data types, recursive functions and predicates, and inductive predicates; it translates this logic to over a dozen solvers and proof assistants. Accordingly, it serves as a backend for many tools, including Frama-C, EasyCrypt, and GNATProve for Ada SPARK. But how can we be sure that these tools are correct? The alternate foundational approach, taken by tools like VST and CakeML, provides strong guarantees by implementing the entire toolchain in a proof assistant, but these tools are harder to build and cannot directly take advantage of SMT solver automation. As a first step toward enabling automated tools with similar foundational guarantees, we give a formal semantics in Coq for the logic fragment of Why3. We show that our semantics are useful by giving a correct-by-construction natural deduction proof system for this logic, using this proof system to verify parts of Why3's standard library, and proving sound two of Why3's transformations used to convert terms and formulas into the simpler logics supported by the backend solvers.

Calculational Design of [In]Correctness Transformational Program Logics by Abstract Interpretation

We study transformational program logics for correctness and incorrectness that we extend to explicitly handle both termination and nontermination. We show that the logics are abstract interpretations of the right image transformer for a natural relational semantics covering both finite and infinite executions. This understanding of logics as abstractions of a semantics facilitates their comparisons through their respective abstractions of the semantics (rather that the much more difficult comparison through their formal proof systems). More importantly, the formalization provides a calculational method for constructively designing the sound and complete formal proof system by abstraction of the semantics. As an example, we extend Hoare logic to cover all possible behaviors of nondeterministic programs and design a new precondition (in)correctness logic.