Proof System Research Articles

Strong (D)QBF Dependency Schemes via Implication-free Resolution Paths

We suggest a general framework to study dependency schemes for dependency quantified Boolean formulas (DQBF). As our main contribution, we exhibit a new infinite collection of implication-free DQBF dependency schemes that generalise the reflexive resolution path dependency scheme. We establish soundness of all these schemes, implying that they can be used in any DQBF proof system. We further explore the power of QBF and DQBF resolution systems parameterised by implication-free dependency schemes and show that the hierarchical structure naturally present among the dependency schemes translates isomorphically to a hierarchical structure of parameterised proof systems with respect to p-simulation. As a special case, we demonstrate that our new schemes are exponentially stronger than the reflexive resolution path dependency scheme when used in Q-resolution, thus resulting in the strongest QBF dependency schemes known to date.

Cut-free labelled calculi and decidability for intuitionistic sentential logic with identity

Abstract In this paper we consider the intuitionistic non-Fregean sentential calculus with Suszko’s identity ($\mathsf{ISCI}$). After recalling the basic concepts of the logic and its associated Hilbert proof system, we introduce a new sound and complete class of models for $\mathsf{ISCI}$, called Topological Beth (${\mathsf{TB}}$) models, that can be viewed as algebraic counterparts (and extensions) of sheaf-theoretic topological models of intuitionistic logic. From this semantical study we define a family of sound and cut-free complete labelled calculi that capture both the Kripke and the ${\mathsf{TB}}$ semantics. Using a key property of the forcing relation in ${\mathsf{TB}}$ models, called regularity, we show termination and decidability results. Finally we discuss the automation of the proof search in one of the labelled calculi and its implementation.

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. ’99] also on proof size. [Alekhnovich and Razborov’03] established that if the clause-variable incidence graph of a conjunctive normal form (CNF) formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop their techniques to show that if one can “cluster” clauses and variables in a way that “respects the structure” of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. We also show how a weaker structural condition is sufficient to obtain lower bounds on width for the resolution proof system, and give a unified treatment that highlights similarities and differences between resolution and polynomial calculus (PC) lower bounds. As a corollary of our main technical theorem, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov’02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution (PCR). Thus, while onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis’93], both FPHP and onto-PHP formulas are hard even when restricted to bounded-degree expanders.

Proof Complexity of Propositional Model Counting

Recently, the proof system MICE for the model counting problem #SAT was introduced by Fichte, Hecher and Roland (SAT’22). As demonstrated by Fichte et al., the system MICE can be used for proof logging for state-of-the-art #SAT solvers. We perform a proof-complexity study of MICE. For this we first simplify the rules of MICE and obtain a calculus MIC E ′ that is polynomially equivalent to MICE. We then establish an exponential lower bound for the number of proof steps in MIC E ′ (and hence also in MICE) for a specific family of CNFs. We also explain a tight connection between MIC E ′ proofs and decision DNNFs.

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

The width complexity measure plays a central role in resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most used method for proving size lower bounds, and it is known that for the mentioned proof systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the cutting planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2009 under the name of CP cutwidth. In this article, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs \(G\) and \(H\) , we show a direct connection between the Weisfeiler–Leman differentiation number \(\mathsf{WL}(G,H)\) of the graphs and the width of a CP refutation for the corresponding isomorphism formula \(\mathrm{Iso}(G,H)\) . In particular, we show that if \(\mathsf{WL}(G,H)\leq k\) , then there is a CP refutation of \(\mathrm{Iso}(G,H)\) with width \(k\) , and if \(\mathsf{WL}(G,H) \gt k\) , then there are no CP refutations of \(\mathrm{Iso}(G,H)\) with width \(k-2\) . Similar results are known for other proof systems, like Resolution, Sherali–Adams, or Polynomial Calculus. We also obtain polynomial-length CP refutations from our width bound for isomorphism formulas for graphs with constant Weisfeiler–Leman dimension. Furthermore, we notice that a length lower bound for refuting graph isomorphism formulas in the subsystem of tree-like cutting planes with polynomially bounded coefficients follows from known results.

Automating Unrealizability Logic: Hoare-Style Proof Synthesis for Infinite Sets of Programs

Automated verification of all members of a (potentially infinite) set of programs has the potential to be useful in program synthesis, as well as in verification of dynamically loaded code, concurrent code, and language properties. Existing techniques for verification of sets of programs are limited in scope and unable to create or use interpretable or reusable information about sets of programs. The consequence is that one cannot learn anything from one verification problem that can be used in another. Unrealizability logic (UL), proposed by Kim et al. as the first Hoare-style proof system to prove properties over sets of programs (defined by a regular tree grammar), presents a theoretical framework that can express and use reusable insight. In particular, UL features nonterminal summaries ---inductive facts that characterize recursive nonterminals (analogous to procedure summaries in Hoare logic). In this work, we design the first UL proof synthesis algorithm, implemented as Wuldo. Specifically, we decouple the problem of deciding how to apply UL rules from the problem of synthesizing/checking nonterminal summaries by computing proof structure in a fully syntax-directed fashion. We show that Wuldo, when provided nonterminal summaries, can express and prove verification problems beyond the reach of existing tools, including establishing how infinitely many programs behave on infinitely many inputs. In some cases, Wuldo can even synthesize the necessary nonterminal summaries. Moreover, Wuldo can reuse previously proven nonterminal summaries across verification queries, making verification 1.96 times as fast as when summaries are instead proven from scratch.

Capybara and Tsubaki: Verifiable Random Functions from Group Actions and Isogenies

In this work, we introduce two post-quantum Verifiable Random Function (VRF) constructions based on abelian group actions and isogeny group actions with a twist. The former relies on the standard group action Decisional Diffie-Hellman (GA-DDH) assumption. VRFs serve as cryptographic tools allowing users to generate pseudorandom outputs along with publicly verifiable proofs. Moreover, the residual pseudorandomness of VRFs ensures the pseudorandomness of unrevealed inputs, even when multiple outputs and proofs are disclosed. Our work aims at addressing the growing demand for post-quantum VRFs, as existing constructions based on elliptic curve cryptography (ECC) or classical DDH-type assumptions are vulnerable to quantum threats. In our contributions, our two VRF constructions, rooted in number-theoretic pseudorandom functions, are both simple and secure over the random oracle model. We introduce a new proof system for the factorization of group actions and set elements, serving as the proofs for our VRFs. The first proposal is based on the standard GA-DDH problem, and for its security proof, we introduce the (group action) master Decisional Diffie-Hellman problem over group actions, proving its equivalence to the standard GA-DDH problem. In the second construction, we leverage quadratic twists to enhance efficiency, reducing the key size and the proof sizes, expanding input size. The scheme is based on the square GA-DDH problem. Moreover, we employ advanced techniques from the isogeny literature to optimize the proof size to 39KB and 34KB using CSIDH-512 without compromising VRF notions. The schemes feature fast evaluations but exhibit slower proof generation. To the best of our knowledge, these constructions represent the first two provably secure VRFs based on isogenies.

Games for hybrid logic from semantic games to analytic calculi

Abstract Game semantics and winning strategies offer a potential conceptual bridge between semantics and proof systems of logics. We illustrate this link for hybrid logic—an extension of modal logic that allows for explicit reference to worlds within the language. We lift a Hintikka-style semantic game to a disjunctive game. The disjunctive game adequately models entailment and validity over classes of frames characterizable in the hybrid language. The search for winning strategies in this game can be reformulated as a sequent-style proof system.

Accelerating large-scale multi-scalar multiplication in Zk-SNARK through exploiting its multilevel parallelism

In the context of zk-SNARK, MSM emerges as a major computational bottleneck, particularly due to its high computational and memory overhead. In this work, we exploit multiple levels of parallelism to effectively accelerate largescale MSM in zk-SNARKs. Firstly, a distributed parameter generation method is proposed in this paper to replace that of centralized method. Based on this methodology, the paper realizes a system with extraordinary scalability, capable of computing large-scale MSMs. Subsequently, the approach presented in this paper elevates the computation of Bellperson, the most prominent zero-knowledge proof system in real-world applications, from a single-node computation to a clustering mode, significantly enhancing its computational performance – a crucial advancement for practical applications. Finally, we implement a multi-level, fully parallelised MSM computing system by leveraging hierarchical sub-task partitioning and cross-node communication optimization, thereby thoroughly exploiting parallelism at diverse granularities. Experimental results show that in the cluster scenario, the proposed approach achieves acceleration ratios of approximately 3.60 and 6.50 times compared to the cutting-edge heterogeneous version Bellperson in dual-node and quad-node settings, respectively. On a single node, the proposed optimization approach achieves an acceleration ratio of 1.38 times compared to the current State-of-the-Art MSM calculation module of cuZK, outperforms the industry-popular Bellman by 186 times and the Bellperson by 1.96 times.

Notions of Proof and Refutation in ‘Gentzensemantik’: Franz von Kutschera as an Early Proponent of (Bilateralist) Proof-Theoretic Semantics

This is a comment on a translation of Franz von Kutschera's paper ‘Ein verallgemeinerter Widerlegungsbegriff für Gentzenkalküle’, which was published in German in 1969. The paper is an important predecessor of what is nowadays called ‘proof-theoretic semantics’, which describes the view that the meaning of logical connectives is determined by the rules governing their use in a proof system. Von Kutschera adopts this view in this paper, and more specifically, a bilateralist view on this subject in that his aim is to give a general framework that provides generalized rule schemata for arbitrary connectives both for proving and refuting and to use this as a reference to prove completeness of certain systems of operators purely proof-theoretically. The main logical system in focus here has been shown to be equivalent to N4, Nelson's constructive logic with strong negation. In order to understand the translated paper better, a contextualizing comment is offered referring and relating it to a preceding paper by von Kutschera.

QBF Merge Resolution is powerful but unnatural

The Merge Resolution proof system (M-Res) for QBFs, proposed by Beyersdorff et al. in 2019, explicitly builds partial strategies inside refutations. The original motivation for this approach was to overcome the limitations encountered in long-distance Q-Resolution proof system (LD-Q-Res), where the syntactic side-conditions, while prohibiting all unsound resolutions, also end up prohibiting some sound resolutions. However, while the advantage of M-Res over many other resolution-based QBF proof systems was already demonstrated, a comparison with LD-Q-Res itself had remained open. In this paper, we settle this question. We show that M-Res has an exponential advantage over not only LD-Q-Res, but even over LQU$^+$-Res and IRM, the most powerful among currently known resolution-based QBF proof systems. Combining this with results from Beyersdorff et al. 2020, we conclude that M-Res is incomparable with LQU-Res and LQU$^+$-Res. Our proof method reveals two additional and curious features about M-Res: (i) M-Res is not closed under restrictions, and is hence not a natural proof system, and (ii) weakening axiom clauses with existential variables provably yields an exponential advantage over M-Res without weakening. We further show that in the context of regular derivations, weakening axiom clauses with universal variables provably yields an exponential advantage over M-Res without weakening. These results suggest that M-Res is better used with weakening, though whether M-Res with weakening is closed under restrictions remains open. We note that even with weakening, M-Res continues to be simulated by eFrege $+$ $\forall$red (the simulation of ordinary M-Res was shown recently by Chew and Slivovsky).

Resolution Over Linear Equations: Combinatorial Games for Tree-like Size and Space

We consider the proof system Res \((\oplus)\) introduced by Itsykson and Sokolov (Ann. Pure Appl. Log.’20), which is an extension of the resolution proof system and operates with disjunctions of linear equations over \({\mathbb {F}}_2\) . We study characterizations of tree-like size and space of Res \((\oplus)\) refutations using combinatorial games. Namely, we introduce a class of extensible formulas and prove tree-like size lower bounds on it using Prover–Delayer games, as well as space lower bounds. This class is of particular interest since it contains many classical combinatorial principles, including the pigeonhole, ordering, and dense linear ordering principles. Furthermore, we present the width-space relation for Res \((\oplus)\) generalizing the results by Atserias and Dalmau (J. Comput. Syst. Sci.’08) and their variant of Spoiler–Duplicator games.

E-waste Management Using Blockchain Technology

Introduction; Bridging the digital divide requires the provision of affordable, fair and quality ICT. With nearly two-thirds of the world’s population still offline, there is a need to provide affordable web access for everyone. For developing countries, increasing the popularity of information and communication technology has become the most important factor in reducing poverty. The danger of electrical and electronic waste disposal contains hazardous substances, but most of the electrical and electronic equipment is still disposed of in an unhealthy environment in the field development area, affecting the level of contamination in Water, Air and Soil ultimately affecting people’s health. Eliminating E-waste responsibility and protecting the environment is a challenge for countries. Smart cities can solve environmental problems through proper waste management for improving human health, protecting water resources, and reducing pollution. Objective; In this paper, we explore how blockchain technology can help smart cities to manage E-waste by providing consistency, immutability, transparency, and accountability control in a distributed, reliable, and secure manner. We discussed the advantages of blockchain technology in various aspects of E-waste management, such as instant tracking and monitoring, E-waste disposal and E-waste management regulation compliance, proper disposal management, E-waste management, and material handling, etc. All examples of disposal services, but in our study we have found that there is no fool proof system to check the disposal of E-waste whether it has been disposed off Fully or Partially. We mainly focused on the tracking of E-waste management system for 100% safe and eco-friendly disposal from the originating point of E-waste to end disposal point of total disposal. Methods; For this, we have used machine-learning model to find the existing percentage of disposal of E-waste at the end-point which reveals that it is never 100%. And partial disposal of E-waste means we have still partial E-waste around us in different forms, which will be a threat for the society to be indulged in hazardous after effects of randomly dumping E-waste.Results; After this we have modelled a Disposal Tracking System(DTS) using blockchain technology to create an E-waste data storage as Decentralized Shareable Ledger (DSL) which records the quantity and state of E-waste data from its originating point to a different level of disposal unit and finally reflect the balance of E-waste data as NIL at the end of last disposal point.Conclusion; This system will helpful for safe and ecofriendly E-waste management and it provides complete transparency and traceability of E-waste during the life cycle of complete disposal. After implementation of this system any district or Block or Village authority can ensure to its citizens for E-waste hazards free environment and safety of natural resources

Komparasi Sistem Pembuktian Hukum Acara Pidana Antara Indonesia Dengan Republik Rakyat Cina

Indonesia regulates its criminal proof system in the Indonesian Criminal Procedure Code, while China governs its proof system under the Criminal Procedure Law of the People's Republic of China. Although both Indonesia and China follow a Civil Law system, there are similarities and differences in their criminal proof systems. This study employs a normative legal research approach with a descriptive-analytical nature, utilizing secondary data analyzed qualitatively and drawing deductive conclusions. The research findings indicate that the similarity between the criminal proof systems of Indonesia and China lies in the presence of proof hearings during the trial process in both countries. The difference is found in the types of evidence admissible in court: Indonesia recognizes five types of evidence, whereas China acknowledges seven types.

Grokking the Sequent Calculus (Functional Pearl)

The sequent calculus is a proof system which was designed as a more symmetric alternative to natural deduction. The 𝜆𝜇𝜇-calculus is a term assignment system for the sequent calculus and a great foundation for compiler intermediate languages due to its first-class representation of evaluation contexts. Unfortunately, only experts of the sequent calculus can appreciate its beauty. To remedy this, we present the first introduction to the 𝜆𝜇𝜇-calculus which is not directed at type theorists or logicians but at compiler hackers and programming-language enthusiasts. We do this by writing a compiler from a small but interesting surface language to the 𝜆𝜇𝜇-calculus as a compiler intermediate language.

PENGATURAN PEMBUKTIAN TERBALIK TERBALIK DALAM UNDANG-UNDANG KORUPSI

The criminal act of corruption not only harms state finances, but also constitutes a violation of the social and economic rights of society at large, so that the criminal act of corruption needs to be classified as a crime whose eradication must be carried out in an extraordinary manner, to further guarantee legal certainty, avoid diversity of legal interpretations and providing protection for the social and economic rights of the community, as well as fair treatment in eradicating criminal acts of corruption. The statutory regulations that regulate criminal acts of corruption are Law no. 28 of 1999 concerning the Administration of a State that is Clean and Free from KKN, Law no. 31 of 1999 in conjunction with Law no. 20 of 2001 concerning Eradication of Corruption Crimes, Law no. 30 of 2002 concerning the Corruption Eradication Commission, and finally with the ratification of the United Nations Convention Against Corruption, 2003 (United Nations Convention Against Corruption, 2003) through Law no. 7 of 2006. The theories used to support the preparation of this book are the Theory of the Rule of Law, the Hierarchy Theory of Legal Norms, the Theory of Authority, the Concept of Corruption and Corruption Crimes and the Concept of Reverse Evidence. The approaches used in preparing this book are: statutory approach, conceptual research approach, case approach. Legal Materials use primary legal materials, in the form of secondary law and other legal materials. Primary legal materials include court decisions that have permanent legal force and laws related to corruption. Secondary legal materials are books, unpublished scientific works, journals and newspapers. Other legal materials are internet sites regulating the reverse evidence system in the provisions of Article 12B of Law Number 20 of 2001, Articles 37, 37A and 38B of Law Number 31 of 1999 in conjunction with Law No. 20 of 2001 and the Criminal Procedure Code. is limited or balanced, and uses a negative proof system according to law (negatief wettelijk overtuiging). The Public Prosecutor is still obliged to prove his charges. In the future, it is necessary to carry out an evaluation of the regulation of Corruption Crimes, it is necessary to strictly regulate the substance of the legal system of reverse evidence so that its application can be implemented effectively and is not just a rule contained in the norms.

Dependency Schemes in CDCL-Based QBF Solving: A Proof-Theoretic Study

In Quantified Boolean Formulas QBFs, dependency schemes help to detect spurious or superfluous dependencies that are implied by the variable ordering in the quantifier prefix but are not essential for constructing countermodels. This detection can provably shorten refutations in specific proof systems, and is expected to speed up runs of QBF solvers. The proof system QCDCL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt{QCDCL}$$\\end{document} recently defined by Beyersdorff and Boehm (LMCS 2023) abstracts the reasoning employed by QBF solvers based on conflict-driven clause-learning (CDCL) techniques. We show how to incorporate the use of dependency schemes into this proof system, either in a preprocessing phase, or in the propagations and clause learning, or both. We then show that when the reflexive resolution path dependency scheme Drrs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt{D}^{\ exttt{rrs}}$$\\end{document} is used, a mixed picture emerges: the proof systems that add Drrs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt{D}^{\ exttt{rrs}}$$\\end{document} to QCDCL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt{QCDCL}$$\\end{document} in these three ways are not only incomparable with each other, but are also incomparable with the basic QCDCL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt{QCDCL}$$\\end{document} proof system that does not use Drrs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt{D}^{\ exttt{rrs}}$$\\end{document} at all, as well as with several other resolution-based QBF proof systems. A notable fact is that all our separations are achieved through QBFs with bounded quantifier alternation.

Encoding TLA+ proof obligations safely for SMT

The TLA+ Proof System (TLAPS) allows users to verify proofs with the support of automated theorem provers, including SMT solvers. To increase trust in TLAPS, we revisited the encoding of TLA+ for SMT, whose implementation had become too complex. Our approach is based on a first-order axiomatization with E-matching patterns. The new encoding is available with TLAPS and achieves performances similar to the previous version, despite its simpler design.

Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the "source data" I, and the other is for the "interface data" O. Such a specification is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a transformation (or "query" below) that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability "up to unique isomorphism". The latter connection uses a variation of a result of Gaifman concerning "relatively categorical" theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

ESTARK: extending STARKs with arguments

STARK is a widely used transparent proof system that uses low-degree tests for proving the correctness of a computer program. STARK consumes an intermediate representation known as AIR that is more appropriate for programs with a relatively short and structured description. However, an AIR is not able to succinctly express non-equality constraints, leading to the incorporation of unwanted polynomials. We present the eSTARK protocol, a new probabilistic proof that generalizes the STARK family through the introduction of a more generic intermediate representation called eAIR. We describe eSTARK in the polynomial IOP model, which combines the optimized version of the STARK protocol with the incorporation of three arguments into the protocol. We also explain various techniques that enhance the vanilla STARK complexity, including optimizations applied to polynomial computations, and analyze the tradeoffs between controlling the constraint degree either at the representation of the AIR or inside the eSTARK itself.