Proof System Research Articles

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.

The Complex(ity) Landscape of Checking Infinite Descent

Cyclic proof systems, in which induction is managed implicitly, are a promising approach to automatic verification. The soundness of cyclic proof graphs is ensured by checking them against a trace-based Infinite Descent property. Although the problem of checking Infinite Descent is known to be PSPACE-complete, this leaves much room for variation in practice. Indeed, a number of different approaches are employed across the various cyclic proof systems described in the literature. In this paper, we study criteria for Infinite Descent in an abstract, logic-independent setting. We look at criteria based on Büchi automata encodings and relational abstractions, and determine their parameterized time complexities in terms of natural dimensions of cyclic proofs: the numbers of vertices of the proof-tree graphs, and the vertex width—an upper bound on the number of components (e.g., formulas) of a sequent that can be simultaneously tracked for descent. We identify novel algorithms that improve upon the parameterised complexity of the existing algorithms. We implement the studied criteria and compare their performance on various benchmarks.

Monotonicity and the Precision of Program Analysis

It is widely known that the precision of a program analyzer is closely related to intensional program properties, namely, properties concerning how the program is written. This explains, for instance, the interest in code obfuscation techniques, namely, tools explicitly designed to degrade the results of program analysis by operating syntactic program transformations. Less is known about a possible relation between what the program extensionally computes, namely, its input-output relation, and the precision of a program analyzer. In this paper we explore this potential connection in an effort to isolate program fragments that can be precisely analyzed by abstract interpretation, namely, programs for which there exists a complete abstract interpretation. In the field of static inference of numeric invariants, this happens for programs, or parts of programs, that manifest a monotone (either non-decreasing or non-increasing) behavior. We first formalize the notion of program monotonicity with respect to a given input and a set of numerical variables of interest. A sound proof system is then introduced with judgments specifying whether a program is monotone relatively to a set of variables and a set of inputs. The interest in monotonicity is justified because we prove that the family of monotone programs admits a complete abstract interpretation over a specific class of non-trivial numerical abstractions and inputs. This class includes all non-relational abstract domains that refine interval analysis (i.e., at least as precise as the intervals abstraction) and that satisfy a topological convexity hypothesis.

Brain Tumor Detection in MRI Images Using CNN With U-NET

Human brain tumors are exceptionally hazardous and devil of the advanced period, prompting definite death. Likewise, when a brain tumor progresses, the patient's life turns out to be more muddled. Early tumor diagnosis is subsequently vital for save the patient's life and expanding their personal satisfaction. In this way, further developed brain tumor recognizable proof is required in the clinical space. Magnetic resonance imaging (MRI)programmed ID of human brain tumors is fundamental for a few indicative and remedial applications. The ongoing techniques, for example, wavelet transform, random forest, fuzzy C-means, and artificial neural networks (ANN), may identify brain tumors, however they need additional opportunity to execute (in minutes) and have less accuracy. In this work, we give a better technique to distinguishing brain cancers in people that utilizes principal component analysis (PCA) and super pixels related to the format based K-means (TK) calculation to rapidly track down growths more. To increment exactness, we will expand this and use CNN with U-Net. To start with, we use PCA and super pixels to remove key qualities that guide in the exact recognition of cerebrum cancers. Then, a channel that guides in expanding exactness is utilized to improve the image. To recognize the mind growth, the TK-means grouping strategy is utilized to lead picture division. The discoveries of the examination exhibit that, in contrast with other current strategies, the proposed detection system for brain tumor recognizable proof in attractive reverberation imaging accomplishes higher exactness and more limited execution times (measured in seconds).

Reversal of Burden Proof as a Reform of Criminal Law Eradicate Money Laundering

One of the efforts of the Indonesian government in reducing money laundering is to pay special attention in terms of proof, namely with the reverse proof system. Over time, the reversed proof system in the Corruption Law is still regulated even though the Corruption Law has undergone many changes. Although it has been included in positive law, in practice this proof is not applied thoroughly and there are still obstacles in its application. This research seeks to answer in what cases the reverse burden of proof is applied and what are the obstacles in its application. This type of research is normative juridical. The results of the study resulted in the merging of corruption cases with money laundering can be considered to provide its own advantages in handling corruption cases. More actors are ensnared including corporations. Maximum punishment, streamlining the return of state assets can impoverish corruptors. Legal culture that has not adapted to the new system, reverse evidence is not clearly regulated in the law.

Algebraic Attacks on RAIN and AIM Using Equivalent Representations

Designing novel symmetric-key primitives for advanced protocols like secure multiparty computation (MPC), fully homomorphic encryption (FHE) and zero-knowledge proof systems (ZK), has been an important research topic in recent years. Many such existing primitives adopt quite different design strategies from conventional block ciphers. Notable features include that many of these ciphers are defined over a large finite field, and that a power map is commonly used to construct the nonlinear component due to its efficiency in these applications as well as its strong resistance against the differential and linear cryptanalysis. In this paper, we target the MPC-friendly ciphers AIM and RAIN used for the post-quantum signature schemes AIMer (CCS 2023 and NIST PQC Round 1 Additional Signatures) and Rainier (CCS 2022), respectively. Specifically, we can find equivalent representations of 2-round RAIN and full-round AIM, respectively, which make them vulnerable to either the polynomial method, or the crossbred algorithm, or the fast exhaustive search attack. Consequently, we can break 2-round RAIN with the 128/192/256-bit key in only 2111/2170/2225 bit operations. For full-round AIM with the 128/192/256-bit key, we could break them in 2136.2/2200.7/2265 bit operations, which are equivalent to about 2115/2178/2241 calls of the underlying primitives. In particular, our analysis indicates that AIM does not reach the required security levels by the NIST competition.

The failure of cut-elimination in cyclic proof for first-order logic with inductive definitions

Abstract A cyclic proof system is a proof system whose proof figure is a tree with cycles. The cut-elimination in a proof system is fundamental. It is conjectured that the cut-elimination in the cyclic proof system for first-order logic with inductive definitions does not hold. This paper shows that the conjecture is correct by giving a sequent not provable without the cut rule but provable in the cyclic proof system.

Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

Sustainable and Round-Optimized Group Authenticated Key Exchange in Vehicle Communication

Vehicle authentication is an essential component validating the vehicle’s identity and ensuring the integrity of transformed data for intelligent transport vehicles (ITS) in the vehicular ad hoc network (VANET). Easy to deploy and operate privacy-enhancing vehicle authentication mechanisms are the mainstay for the widespread ITS in the VANET. Very recently, VANET security architectures are constituting by IEEE 1609.2 group, NoW project, the SeVeCom project. However, these approaches heavily depend on the consuming public key infrastructure (PKI) and certification authorities (CA). In this work, walking along the research line, we attempt to design authentication protocols with two diverse factors for Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) networks, respectively, without depending on the stumbling block PKI/CA. In addition, a smooth projective hash function (SPHF) (a.k.a., a special case of the designated-verifier zero-knowledge proof system) guarantees any recipient can confirm the authenticity and integrity of the received messages without knowing the authentication factors. Thus, to optimize the communication round, SPHF is used to design a (group) two-factor authenticated key exchange (AKE) with low-interactive communication rounds. The proof-of-concept implementation indicates that the computation and communication overheads introduced by our solution are acceptable in real-world deployments. The security of the proposed approach is validated using Bellare-Pointcheval-Rogaway (BPR) model along with the experimental evaluation and the theoretical analysis.

Examining Schnorrs protocol in the context of zero-knowledge proofs

The rise of technology has brought with it a heightened awareness of the necessity to shield personal data and maintain exclusive access to specific knowledge. A notable solution that emerged from this consciousness is Zero-Knowledge Proofs (ZKPs) and, more specifically, Schnorrs Protocol. Historically, Zero-Knowledge Proofs have a compelling lineage, tracing their roots back to the fervent discussions among cryptographers aiming to achieve a balance between information sharing and privacy. ZKPs are cryptographic methods that allow one party to prove to another that a statement is true, without revealing any specific information about the statement itself. In the midst of these developments, Schnorrs Protocol emerged as a renowned interactive proof system. It possesses an intuitive structure that has made it pivotal in the enhancement of digital security. The typical flow of Schnorrs Protocol begins with the prover sending a commitment to the verifier. The verifier then sends a random challenge back to the prover, who, in turn, produces a response. Whats captivating is that the verifier can ascertain the validity of the proof without gaining insight into the underlying secret. Interactive Schnorrs Protocol involves real-time back-and-forth communication between the prover and verifier. On the other hand, the non-interactive version eliminates this need by using a cryptographic hash function, thereby streamlining the process.

Defining Logical Systems via Algebraic Constraints on Proofs

Abstract We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof system for a target logic by enriching a proof system for another, typically simpler, logic with an algebra of constraints that act as correctness conditions on the latter to capture the former; e.g. one may use Boolean algebra to give constraints in a sequent calculus for classical propositional logic to produce a sequent calculus for intuitionistic propositional logic. The idea behind such forms of decomposition is to obtain a tool for uniform and modular treatment of proof theory and to provide a bridge between semantics logics and their proof theory. The paper discusses the theoretical background of the project and provides several illustrations of its work in the field of intuitionistic and modal logics: including, a uniform treatment of modular and cut-free proof systems for a large class of propositional logics; a general criterion for a novel approach to soundness and completeness of a logic with respect to a model-theoretic semantics; and a case study deriving a model-theoretic semantics from a proof-theoretic specification of a logic.

ON THE EXISTENCE OF STRONG PROOF COMPLEXITY GENERATORS

AbstractCook and Reckhow [5] pointed out that $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:•There exists a p-time function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets.We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for g. We define a new hardness property of generators, the $\bigvee $ -hardness, and show that one specific gadget generator is the $\bigvee $ -hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite $\mathcal {N}\mathcal {P}$ sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite $\mathcal {N}\mathcal {P}$ sets.

Extensible Proof Systems for Infinite-State Systems

This article revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.

On vanishing sums of roots of unity in polynomial calculus and sum-of-squares

We introduce a novel take on sum-of-squares that is ableto reason with complex numbers and still make use of polynomial inequalities.This proof system might be of independent interest since itallows to represent multivalued domains both with Boolean and Fourierencoding. We show degree and size lower bounds in this system for anatural generalization of knapsack: the vanishing sums of roots of unity.These lower bounds naturally apply to polynomial calculus as-well.

Binary Generalized PCP for Two Periodic Morphisms is Decidable in Polynomial Time

We show that the binary generalized Post’s Correspondence Problem is decidable in polynomial time for the case where both morphisms are periodic. Our result is based on combinatorial properties and we have formalized the proofs and verified correctness of our algorithm using the Isabelle/HOL formal proof system.