Cognitive complexity of geometric proof and geometric calculation: a problem-solving perspective

Abstract Lifting theorems are used to transfer lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure $$A$$ A for some function $$f$$ f , we compose $$f$$ f with a carefully chosen gadget function $$g$$ g and get essentially the same lower bound on a complexity measure $$B$$ B for the lifted function $$f \diamond g$$ f ⋄ g . Lifting theorems have applications in many different areas, such as circuit complexity, communication complexity, proof complexity, etc. One of the main questions in the context of lifting is how to choose a suitable gadget $$g$$ g . Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of $$f$$ f , and it is unclear whether they can be improved to constant-size gadgets. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question: ‘For which gadgets does the lifting result hold?’ in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there are no intermediate cases—for every gadget, there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as $$f\diamond OR \diamond XOR$$ f ⋄ O R ⋄ X O R for some function $$f$$ f .

According to the United States Environmental Protection Agency, transportation accounts for 28% of total U.S. emissions which is 8 billion tons of carbon dioxide, making it the largest contributor to the nation’s greenhouse gas emissions. In an era where sustainability is becoming increasingly crucial, we introduce a novel Carbon-Aware Ant Colony System (CAACS) Algorithm that addresses the Generalized Traveling Salesman Problem while minimizing carbon emissions. We mathematically formulated the sustainable GTSP and developed an innovative approach that leverages the natural efficiency of ant colony pheromone trails to optimize routes, balancing both environmental and economic objectives. We discover new pathways achieving greater improvements in carbon emissions compared to the tradeoff in cost. Through our research, we developed several key concepts and correlations, including: a generalizable carbon emission heuristic that is adaptable to other carbon sources, the correlation that more ants improve solution quality and reduce runtime up to a threshold, and empirical proof of linear time complexity. The CAACS Algorithm identifies routes with carbon emissions less than or equal to the expected amount for 98% of instances in the benchmark datasets and in UPS Package Delivery we found a 0.02 % decrease in cost and 1.07 % decrease in carbon which can scale to millions of tons of carbon dioxide conserved in transportation. To the best of our knowledge, this is the first sustainable algorithm developed for the GTSP since the problem’s introduction in 1969. By integrating sustainability into transportation models, the CAACS Algorithm is a powerful tool for real-world applications, including network design, delivery route planning, and commercial aircraft logistics. Our algorithm’s unique bi-objective optimization represents a significant advancement in sustainable transportation solutions strategically balancing cost and carbon emissions to reduce energy consumption and promote environmental responsibility.

This paper resolves the predefined-time control problem for multi-agent systems under predefined performance metrics and state constraints, addressing critical limitations of traditional methods—notably their inability to enforce strict user-specified deadlines for mission-critical operations, coupled with difficulties in simultaneously guaranteeing transient performance bounds and state constraints while suffering prohibitive stability proof complexity. To overcome these challenges, we propose a predefined performance control methodology that integrates Barrier Lyapunov Functions command-filtered backstepping. The framework rigorously ensures exact convergence within user-defined time independent of initial conditions while enforcing strict state constraints through time-varying BLF boundaries and further delivers quantifiable performance such as overshoot below 5% and convergence within 10 s. By eliminating high-order derivative continuity proofs via command-filter design, stability analysis complexity is reduced by 40% versus conventional backstepping. Stability proofs and dual-case simulations (UAV formation/smart grid) demonstrate over 95% tracking accuracy under disturbances and constraints, validating broad applicability in safety-critical multi-agent systems.

Proof assistants are software tools for formal modeling and verification of software, hardware, design, and mathematical proofs. Due to the growing complexity and scale of formal proofs, compatibility issues frequently arise when using different versions of proof assistants. These issues result in broken proofs, disrupting the maintenance of formalized theories and hindering the broader dissemination of results within the community. Although existing works have proposed techniques to address specific types of compatibility issues, the overall characteristics of these issues remain largely unexplored. To address this gap, we conduct the first extensive empirical study to characterize compatibility issues, using Isabelle as a case study. We develop a regression testing framework to automatically collect compatibility issues from the Archive of Formal Proofs, the largest repository of formal proofs in Isabelle. By analyzing 12,079 collected issues, we identify their types and symptoms and further investigate their root causes. We also extract updated proofs that address these issues to understand the applied resolution strategies. Our study provides an in-depth understanding of compatibility issues in proof assistants, offering insights that support the development of effective techniques to mitigate these issues.

Iterated Lower Bound Formulas: A Diagonalization-Based Approach to Proof Complexity

A systematic analysis of judicial practice reveals that gathering evidence in criminal proceedings related to co-rruption and corruption-related offenses is an exceptionally complex and multidimensional process. This com-plexity stems primarily from the inherently multifaceted nature of corruption in the modern scientific discourse. Additionally, over the past decade, there has been an ongoing effort to redefine and enhance anti-corruption measures within the state. Key institutions such as the Asset Tracing and Management Agency, the Specialized Anti-Corruption Prosecutor’s Office, and the High Anti-Corruption Court of Ukraine play a pivotal role in addres-sing corruption-related offenses. The guiding principles for combating corruption in a legally compromised environment are regarded as top priorities, as they form the foundation for investigative and prosecutorial procedures. These principles define the procedural framework, the specific characteristics of investigative and covert operations, and the primary challenges associated with conducting them. Some core conceptual prin-ciples include: the extreme complexity of proof in criminal proceedings on corruption and corruption-related criminal offenses in Ukraine; compliance with the requirements of national and international human rights standards; the complex nature of investigative (detective) actions and covert investigative (detective) actions; and interaction with the public.

Abstract Schwichtenberg developed the programme called ‘Proofs as Programs’ by measuring the ‘complexity’ of proofs as programs using the Curry-Howard correspondence. In particular, he precisely characterized the complexity of proofs in arithmetic using proof-theoretic tools. In this paper, we extend this result to parameter-free subsystems of Girard’s System F having the strength of $\varPi ^{1}_{1}$-$CA_{0}$, determining the expected complexity of terms in them (instead of proofs). This abstract setting of System F provides a new perspective on Schwichtenberg’s result. His original goal was to determine the complexity of programs, relying on the Curry-Howard correspondence. In contrast, our approach directly determines the complexity of terms in System F, which are essentially programs themselves. Another advantage of this approach should be that our approach for these fragments is more uniform than the original, owing to the abstract setting of System F.

There has been tremendous progress in the past decade in the field of quantified Boolean formulas (QBF), both in practical solving as well as in creating a theory of corresponding proof systems and their proof complexity analysis. Both for solving and for proof complexity, it is important to have interesting formula families on which we can test solvers and gauge the strength of the proof systems. There are currently few such formula families in the literature. We initiate a general programme how to transform computationally hard problems (located in the polynomial hierarchy) into QBFs hard for the main QBF resolution systems Q-Res and QU-Res that relate to core QBF solvers. We illustrate this general approach on three problems from graph theory and logic. This yields QBF families that are provably hard for Q-Res and QU-Res (without any complexity assumptions).

Proof systems can be used for certification of logic problems, and proof complexity can inform us how succinct certificates can be. In the PSPACE complete logic QBF (Quantified Boolean Formulas) refutation proofs often contain information that reproduce the witnesses of the quantified variables. This is known as strategy extraction. There are two known kinds of strategy extraction for proof systems, local strategy extraction and round-based strategy extraction. Formalisation of local strategy extraction was done previously, in this paper we formalise round-based strategy extraction. By formalising the strategy extraction into circuits we can show new p-simulations. P-simulations are processes that allow you to transform proofs from a weaker proof system to a stronger proof system. Thus we solve an open problem in QBF proof complexity that Extended QBF Frege p-simulates LD-Q(\Drrs)-Resolution. LD-Q(\Drrs)-Resolution is the underlying proof system for the solver Qute. This is a positive result for certification. By clarifying the hierarchy of proof systems further suggests the feasibility of using known formats such as Extended QU-Resolution or QRAT to certify QCDCL solvers. The p-simulation is our main result, but we also make other observations from the specifics of the formalisation.

The round complexity of interactive proof systems is a key question of practical and theoretical relevance in complexity theory and cryptography. Moreover, results such as QIP = QIP(3) (STOC'00) show that quantum resources significantly help in such a task. In this work, we initiate the study of round compression of protocols in the bounded quantum storage model (BQSM). In this model, the malicious parties have a bounded quantum memory and they cannot store the all the qubits that are transmitted in the protocol. Our main results in this setting are the following: 1. There is a non-interactive (statistical) witness indistinguishable proof for any language in NP (and even QMA) in BQSM in the plain model. We notice that in this protocol, only the memory of the verifier is bounded. 2. Any classical proof system can be compressed in a two-message quantum proof system in BQSM. Moreover, if the original proof system is zero-knowledge, the quantum protocol is zero-knowledge too. In this result, we assume that the prover has bounded memory. Finally, we give evidence towards the “tightness” of our results. First, we show that NIZK in the plain model against BQS adversaries is unlikely with standard techniques. Second, we prove that without the BQS model there is no 2–message zero-knowledge quantum interactive proof, even under computational assumptions.

The k -dimensional Weisfeiler-Leman ( k -WL) algorithm is a simple combinatorial algorithm that was originally designed as a graph isomorphism heuristic. It naturally finds applications in Babai’s quasipolynomial-time isomorphism algorithm, practical isomorphism solvers, and algebraic graph theory. However, it also has surprising connections to other areas such as logic, proof complexity, combinatorial optimization, and machine learning. The algorithm iteratively computes a coloring of the k -tuples of vertices of a graph. Since Fürer’s linear lower bound [ICALP 2001], it has been an open question whether there is a super-linear lower bound for the iteration number for k -WL on graphs. We answer this question affirmatively, establishing an Ω ( n k /2 )-lower bound for all k .

We investigate the proof complexity of systems based on positive branching programs, i.e. non-deterministic branching programs (NBPs) where, for any 0-transition between two nodes, there is also a 1-transition. Positive NBPs compute monotone Boolean functions, just like negation-free circuits or formulas, but constitute a positive version of (non-uniform) NL, rather than P or NC1, respectively. The proof complexity of NBPs was investigated in previous work by Buss, Das and Knop, using extension variables to represent the dag-structure, over a language of (non-deterministic) decision trees, yielding the system eLNDT. Our system eLNDT+ is obtained by restricting their systems to a positive syntax, similarly to how the 'monotone sequent calculus' MLK is obtained from the usual sequent calculus LK by restricting to negation-free formulas. Our main result is that eLNDT+ polynomially simulates eLNDT over positive sequents. Our proof method is inspired by a similar result for MLK by Atserias, Galesi and Pudl\'ak, that was recently improved to a bona fide polynomial simulation via works of Je\v{r}\'abek and Buss, Kabanets, Kolokolova and Kouck\'y. Along the way we formalise several properties of counting functions within eLNDT+ by polynomial-size proofs and, as a case study, give explicit polynomial-size poofs of the propositional pigeonhole principle.

A transducer is finite-valued if for some bound k, it maps any given input to at most k outputs. For classical, one-way transducers, it is known since the 80s that finite valuedness entails decidability of the equivalence problem. This decidability result is in contrast to the general case, which makes finite-valued transducers very attractive. For classical transducers, it is also known that finite valuedness is decidable and that any k-valued finite transducer can be decomposed as a union of k single-valued finite transducers. In this paper, we extend the above results to copyless streaming string transducers (SSTs), answering questions raised by Alur and Deshmukh in 2011. SSTs strictly extend the expressiveness of one-way transducers via additional variables that store partial outputs. We prove that any k-valued SST can be effectively decomposed as a union of k (single-valued) deterministic SSTs. As a corollary, we obtain equivalence of SSTs and two-way transducers in the finite-valued case (those two models are incomparable in general). Another corollary is an elementary upper bound for checking equivalence of finite-valued SSTs. The latter problem was already known to be decidable, but the proof complexity was unknown (it relied on Ehrenfeucht's conjecture). Finally, our main result is that finite valuedness of SSTs is decidable. The complexity is PSpace, and even PTime when the number of variables is fixed.Comment: 36 pages. This is the TheoretiCS journal version. This article is an extended version of the LICS'24 paper by the same name. Updated to correct metadata

Readers are offered a deep and comprehensive study of the Pythagorean theorem, one of the most famous and important theorems in geometry. The article presents twenty-five different ways to prove this theorem, covering a wide range of mathematical methods and concepts. It actually demonstrates how the Pythagorean theorem can be proven using classical geometric approaches, as well as using modern mathematical tools. Readers will be able to get acquainted with proofs based on metric relations in a triangle, trigonometric properties of angles, elements of vector algebra, properties of a circle, tangent, chord and secant, as well as more complex proofs using differential equations and mechanics. The article will be interesting and useful for a wide range of readers, including high school students, students of mathematical specialties, mathematics teachers and everyone who is interested in the history and development of mathematical knowledge.

Graph and network visualization supports exploration, analysis and communication of relational data arising in many domains: from biological and social networks, to transportation and powergrid systems. With the arrival of AI-based question-answering tools, issues of trustworthiness and explainability of generated answers motivate a significant new role for visualization. In the context of graphs, we see the need for visualizations that can convince a critical audience that an assertion (e. g., from an AI) about the graph under analysis is valid. The requirements for such representations that convey precisely one specific graph property are quite different from standard network visualization criteria which optimize general aesthetics and readability. In this paper, we aim to provide a comprehensive introduction to visual proofs of graph properties and a foundation for further research in the area. We present a framework that defines what it means to visually prove a graph property. In the process, we introduce the notion of a visual certificate, that is, a specialized faithful graph visualization that leverages the viewer's perception, in particular, pre-attentive processing (e. g., via pop-out effects), to verify a given assertion about the represented graph. We also discuss the relationships between visual complexity, cognitive load and complexity theory, and propose a classification based on visual proof complexity. Then, we provide further examples of visual certificates for problems in different visual proof complexity classes. Finally, we conclude the paper with a discussion of the limitations of our model and some open problems.

The malpractice is still far from the reach of criminal procedure. There are no specific regulations on malpractice and there are problems in proving criminal acts of malpractice so that this article uses the theory of evidence to provide legal certainty to fulfill the elements of error and criminal responsibility. This study uses a normative method to answer this problem so that the statute approach and case approach are used. The results of the study show that the problems in proving criminal acts of malpractice and the difficulty of collecting evidence (unus testis nullus testis) are obstacles to assisting the legal process, the complexity of proof in criminal law which is material and problematic in the legal vacuum space limited to the Criminal Code. Contextually, criminal responsibility for malpractice is not specifically regulated in the Criminal Code or medical malpractice, but after the enactment of Law 17 of 2023 concerning Health, it provides hope in protection and helps victims in seeking justice even though cases reported by victims are often in SP3 (Investigation Termination Order).

In this paper, some new quantified propositional proof system is introduced and compared by proof complexities with other quantified and not quantified propositional proof systems. It is proved that the introduced system 1) is polynomially equivalent to its quantifier-free variant and 2) has exponential speed-up by sizes over some variants of the quantified resolution system. As the introduced system has a very simple proof construction strategy, it can be very useful not only in Logic, and therefore in Artificial Intelligence, but also in areas such as Computational Biology and Medical Diagnosis.

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.

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.