Proof Complexity Research Articles

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.

Using Extant Proofs in the Classroom: A Comprehension Activity Structure

ABSTRACT Proof comprehension is an important skill for students to develop in their proof-based courses, yet students are rarely afforded opportunities to develop this skill. In this paper, we describe two implementations of an activity structure that was developed to give students the opportunity to engage with complex proofs and to develop their proof comprehension skills. We share one implementation in an introduction to proof course and one implementation in an abstract algebra course. In particular, we aim to elucidate the details of facilitating these days in class and offer suggestions on how other instructors can adapt this proof reading activity structure in their own classes.

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary research area that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focussed on recent advances in our understanding that the analysis of an appropriately tailored concept of “proof” underlies many of the arguments in algorithms, geometry or combinatorics research that make the core of modern theoretical computer science. These include the analysis of practical Boolean satisfiability (SAT) solving algorithms, the size of linear or semidefinite programming formulations of combinatorial optimization problems, the complexity of solving total NP search problems by local methods, and the complexity of describing winning strategies in two-player round-based games, to name just a few important examples.

QCDCL with cube learning or pure literal elimination – What is best?

Quantified conflict-driven clause learning (QCDCL) is one of the main approaches for solving quantified Boolean formulas (QBF). We formalise and investigate several versions of QCDCL that include cube learning and/or pure-literal elimination, and formally compare the resulting solving variants via proof complexity techniques. Our results show that almost all of the QCDCL variants are exponentially incomparable with respect to proof size (and hence solver running time), pointing towards different orthogonal ways how to practically implement QCDCL.

Mathematical Logic: Proof Theory, Constructive Mathematics

The Workshop ‘Mathematical Logic: Proof Theory, Constructive Mathematics’ focused on proof-theoretic research on the foundations of mathematics, on the extraction of explicit computational content from given proofs in core areas of ordinary mathematics using proof-theoretic methods as well as on topics in proof complexity. The workshop contributed to the following research strands:

Proof Complexity and the Binary Encoding of Combinatorial Principles

Proof Complexity and the Binary Encoding of Combinatorial Principles

Separations in Proof Complexity and TFNP

It is well-known that Resolution proofs can be efficiently simulated by Sherali–Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS). These results have consequences for total \({\text{\upshape \sffamily NP}} \) search problems. First, we characterise the classes \({\text{\upshape \sffamily PPADS}} \) , \({\text{\upshape \sffamily PPAD}} \) , \({\text{\upshape \sffamily SOPL}} \) by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, \({\text{\upshape \sffamily PLS}} \not\subseteq {\text{\upshape \sffamily PPP}} \) , \({\text{\upshape \sffamily SOPL}} \not\subseteq {\text{\upshape \sffamily PPA}} \) , and \({\text{\upshape \sffamily EOPL}} \not\subseteq {\text{\upshape \sffamily UEOPL}} \) . In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical \({\text{\upshape \sffamily TFNP}} \) classes introduced in the 1990s.

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.

Efficient CHAD

We show how the basic Combinatory Homomorphic Automatic Differentiation (CHAD) algorithm can be optimised, using well-known methods, to yield a simple, composable, and generally applicable reverse-mode automatic differentiation (AD) technique that has the correct computational complexity that we would expect of reverse-mode AD. Specifically, we show that the standard optimisations of sparse vectors and state-passing style code (as well as defunctionalisation/closure conversion, for higher-order languages) give us a purely functional algorithm that is most of the way to the correct complexity, with (functional) mutable updates taking care of the final log-factors. We provide an Agda formalisation of our complexity proof. Finally, we discuss how the techniques apply to differentiating parallel functional array programs: the key observations are 1) that all required mutability is (commutative, associative) accumulation, which lets us preserve task-parallelism and 2) that we can write down data-parallel derivatives for most data-parallel array primitives.

Guided Equality Saturation

Rewriting is a principled term transformation technique with uses across theorem proving and compilation. In theorem proving, each rewrite is a proof step; in compilation, rewrites optimize a program term. While developing rewrite sequences manually is possible, this process does not scale to larger rewrite sequences. Automated rewriting techniques, like greedy simplification or equality saturation, work well without requiring human input. Yet, they do not scale to large search spaces, limiting the complexity of tasks where automated rewriting is effective, and meaning that just a small increase in term size or rewrite length may result in failure. This paper proposes a semi-automatic rewriting technique as a means to scale rewriting by allowing human insight at key decision points. Specifically, we propose guided equality saturation that embraces human guidance when fully automated equality saturation does not scale. The rewriting is split into two simpler automatic equality saturation steps: from the original term to a human-provided intermediate guide , and from the guide to the target. Complex rewriting tasks may require multiple guides, resulting in a sequence of equality saturation steps. A guide can be a complete term, or a sketch containing undefined elements that are instantiated by the equality saturation search. Such sketches may be far more concise than complete terms. We demonstrate the generality and effectiveness of guided equality saturation using two case studies. First, we integrate guided equality saturation in the Lean 4 proof assistant. Proofs are written in the style of textbook proof sketches, as a series of calculations omitting details and skipping steps. These proofs conclude in less than a second instead of minutes when compared to unguided equality saturation, and can find complex proofs that previously had to be done manually. Second, in the compiler of the Rise array language, where unguided equality saturation fails to perform optimizations within an hour and using 60 GB of memory, guided equality saturation performs the same optimizations with at most 3 guides, within seconds using less than 1 GB memory.

On the principles of criminalization of interference with the judiciary

Purpose. The purpose of the study is to determine whether the national legislator complies with the basic principles of criminalization of socially dangerous behavior when criminalizing interference with the activities of judicial bodies. Methods. The methodology includes the analysis and generalization of scientific positions on the theory of criminalization, which is justifiably considered to be among the most controversial in criminal law doctrine. The following methods of scientific cognition were used: logical and semantic (for understanding the conceptual apparatus), formal and legal (application of the rules of formal logic in analyzing the construction of the crime under Article 376 of the Criminal Code), and the method of analysis. Results. In the course of the study, the author confirms that the principles of criminalization of interference with the judiciary are not observed: 1) procedural possibility of prosecution. The critically small array of recorded criminal offenses indicates the complexity of proof in this category of cases, which is largely due to the imperfection of the legislative construction of Article 376 of the Criminal Code; 2) certainty and unity of terminology - due to the systemic contradiction in the formulation of prohibited behavior and the use of the evaluative category «unjust decision». The scientific novelty is to confirm the shortcomings of the legislative construction of Article 376 of the Criminal Code, which are the result of violation of certain principles of criminalization of behavior. Practical significance. The results of the study can be used in the course of further regulatory and legal improvement of Article 376 of the Criminal Code, as well as in the development of scientific provisions on criminalization of crimes against justice.

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.

Near-optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement Steps

We prove near-optimal tradeoffs for quantifier depth (also called quantifier rank) versus number of variables in first-order logic by exhibiting pairs of n -element structures that can be distinguished by a k -variable first-order sentence but where every such sentence requires quantifier depth at least n Ω ( k /log k ) . Our tradeoffs also apply to first-order counting logic and, by the known connection to the k -dimensional Weisfeiler–Leman algorithm, imply near-optimal lower bounds on the number of refinement iterations. A key component in our proof is the hardness condensation technique introduced by Razborov in the context of proof complexity. We apply this method to reduce the domain size of relational structures while maintaining the minimal quantifier depth needed to distinguish them in finite variable logics.

A PROOF COMPLEXITY CONJECTURE AND THE INCOMPLETENESS THEOREM

Abstract Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${\mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all proof systems). A propositional version of the construction shows that at least one of the following three statements is true: 1. There is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm). 2. $E \not \subseteq P/poly$ . 3. There exists function h that stretches all inputs by one bit, is computable in sub-exponential time, and its range $Rng(h)$ intersects all infinite ${\text {NP}}$ sets.

Finding (s,d)-hypernetworks in F-hypergraphs is NP-hard

We consider the problem of computing an (s,d)-hypernetwork in an acyclic F-hypergraph. This is a fundamental computational problem arising in directed hypergraphs, and is a foundational step in tackling problems of reachability and redundancy. This problem was previously explored in the context of general directed hypergraphs (containing cycles), where it is NP-hard, and acyclic B-hypergraphs, where a linear time algorithm can be achieved. In a surprising contrast, we find that for acyclic F-hypergraphs the problem is NP-hard, which also implies the problem is hard in BF-hypergraphs. This is a striking complexity boundary given that F-hypergraphs and B-hypergraphs would at first seem to be symmetrical to one another. We provide the proof of complexity and explain why there is a fundamental asymmetry between the two classes of directed hypergraphs.

Optimized Passivity-based Control of a Hybrid Electric Vehicle Source using a Novel Equilibrium Optimizer

Passivity-based control is a well-known strategy for synthesizing stable controllers because it models and manages systems based on their global state space equations. It also permits proving the system's stability by making it passive, which is an undeniable advantage over traditional controllers that have limitations in terms of applications, saturation, and the inability to verify stability. Previous research used nonlinear control based on passivity method to control the energy distribution between the embedded sources, they did not investigate the impact of the dumping matrix parameter, which was typically considered to be a positive random value. In contrast, the results of this investigation show that it significantly affects system behaviour, especially in the transient-state domain. The primary objective of this study is to develop and evaluate an energy management strategy for hybrid electric vehicles powered by Fuel Cells as the primary source and Supercapacitors as the secondary source. The fast response with less overshoot is ensured by optimizing the damping matrix parameter using a well-known optimization technique named Novel Equilibrium Optimizer. The proposed technique is successfully demonstrated in a Matlab® simulation environment, and the control system exhibits robust dynamic behaviour. Finally, despite the complexity of this technique's mathematical proof, it provides a simple, efficient feedback control law for a real electrical vehicle application.

Reform of Criminal Procedure Legislation in Bosnia and Herzegovina - Expectations, Reality and Perspectives

In this paper, the attention is focused on the efficiency of the criminal procedure, with a special focus on the efficiency of the criminal process entities in terms of elucidating and solving a specific criminal matter. In connection with the above, the authors recognized and identified the key procedural problems related to the timely, efficient and legal detection and proof of the criminal act and guilt. Given that a significant period of time has passed since the last general reform of the criminal procedure legislation, sufficient for a critical analysis, a critical review of all phases of the criminal procedure was carried out with the intention of actualizing and problematizing certain legal solutions of a procedural nature (detective activity, investigation concept, standards evidence, evidentiary role of the prosecutor, drawing up/filing of the indictment, complexity of discovery and proof, etc.) on which the efficient and legal conduct and finalization of criminal proceedings directly depend. Also, modern forms of criminality, especially specific forms of organized crime, demand from the legislator the adequacy of the legal norm in terms of achieving a legitimate legal goal related to the effective and energetic fight against crime as a complex social phenomenon and achieving adequate results of criminal justice.

On the difficulty of discovering mathematical proofs

An account of mathematical understanding should account for the differences between theorems whose proofs are “easy” to discover, and those whose proofs are difficult to discover. Though Hilbert seems to have created proof theory with the idea that it would address this kind of “discovermental complexity”, much more attention has been paid to the lengths of proofs, a measure of the difficulty of verifying of a given formal object that it is a proof of a given formula in a given formal system. In this paper we will shift attention back to discovermental complexity, by addressing a “topological” measure of proof complexity recently highlighted by Alessandra Carbone (2009). Though we will contend that Carbone’s measure fails as a measure of discovermental complexity, it forefronts numerous important formal and epistemological issues that we will discuss, including the structure of proofs and the question of whether impure proofs are systematically simpler than pure proofs.

Анализ отдельных положений законодательства Российской Федерации в области защиты авторских прав

The existence of mankind is unthinkable without progressive ideas in the field of medicine,
 education, industrial production, etc. The rhythm of modern life, characterized by a variety of technological solutions, changes in social and political conditions, inevitably entails the complication of social relations.
 At the same time, the current legislation should develop adequately to the trends of these relations. The need for discoveries also requires a different understanding of an inventor’s personality, the results of his creative activity, as well as the development of a mechanism for copyright protection.
 The ongoing researches confirm that today only a small number of copyright holders managed to really protect their copyright subjective rights. The need to find ways to solve the emerging problems determines the relevance of the study.
 The purpose of the study: analysis of the provisions of the legislation of the Russian Federation in the field of copyright protection in a rapidly changing reality, as well as problematic issues that arise in the implementation of these provisions.
 The forms of participation in the civil circulation of objects of copyright in modern reality are becoming more and more significant, which is due to the complexity of judicial proof and the dynamism of legal relations in the field under study.
 The need to improve legislation is due to the importance of consistently updating legal regulatory mechanisms in the context of expanding economic sanctions. In the current conditions of restrictions, government bodies are developing measures aimed at stimulating the processes of import substitution. This circumstance actualizes the demand for domestic developments, inventions, innovative solutions that are important for the development of the country’s economy.
 In the process of research, the following methods were used: comparative legal, formal logic, analogy, interpretation of law, technical legal.
 Based on the results of the study, the authors come to the conclusion that the creation of opportunities for the progressive development of the intellectual sphere (conditions for the implementation of creative ideas and their integration into the economy) begins with the formation of a legal framework, taking into account the balance of interests of the parties.

The impact of heterogeneity and geometry on the proof complexity of random satisfiability

AbstractSatisfiability is considered the canonical NP‐complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large‐scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real‐world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random ‐SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random ‐SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT‐solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time.