arXiv - CS - Symbolic Computation最新文献

{"title":"Probabilistic Shoenfield Machines","authors":"Maksymilian Bujok, Adam Mata","doi":"arxiv-2407.05777","DOIUrl":"https://doi.org/arxiv-2407.05777","url":null,"abstract":"This article provides the theoretical framework of Probabilistic Shoenfield\u0000Machines (PSMs), an extension of the classical Shoenfield Machine that models\u0000randomness in the computation process. PSMs are brought in contexts where\u0000deterministic computation is insufficient, such as randomized algorithms. By\u0000allowing transitions to multiple possible states with certain probabilities,\u0000PSMs can solve problems and make decisions based on probabilistic outcomes,\u0000hence expanding the variety of possible computations. We provide an overview of\u0000PSMs, detailing their formal definitions as well as the computation mechanism\u0000and their equivalence with Non-deterministic Shoenfield Machines (NSM).","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Towards Automated Functional Equation Proving: A Benchmark Dataset and A Domain-Specific In-Context Agent","authors":"Mahdi Buali, Robert Hoehndorf","doi":"arxiv-2407.14521","DOIUrl":"https://doi.org/arxiv-2407.14521","url":null,"abstract":"Automated Theorem Proving (ATP) faces challenges due to its complexity and\u0000computational demands. Recent work has explored using Large Language Models\u0000(LLMs) for ATP action selection, but these methods can be resource-intensive.\u0000This study introduces FEAS, an agent that enhances the COPRA in-context\u0000learning framework within Lean. FEAS refines prompt generation, response\u0000parsing, and incorporates domain-specific heuristics for functional equations.\u0000It introduces FunEq, a curated dataset of functional equation problems with\u0000varying difficulty. FEAS outperforms baselines on FunEq, particularly with the\u0000integration of domain-specific heuristics. The results demonstrate FEAS's\u0000effectiveness in generating and formalizing high-level proof strategies into\u0000Lean proofs, showcasing the potential of tailored approaches for specific ATP\u0000challenges.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"94 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing Clipped Products","authors":"Arthur C. Norman, Stephen M. Watt","doi":"arxiv-2407.04133","DOIUrl":"https://doi.org/arxiv-2407.04133","url":null,"abstract":"Sometimes only some digits of a numerical product or some terms of a\u0000polynomial or series product are required. Frequently these constitute the most\u0000significant or least significant part of the value, for example when computing\u0000initial values or refinement steps in iterative approximation schemes. Other\u0000situations require the middle portion. In this paper we provide algorithms for\u0000the general problem of computing a given span of coefficients within a product,\u0000that is the terms within a range of degrees for univariate polynomials or range\u0000digits of an integer. This generalizes the \"middle product\" concept of Hanrot,\u0000Quercia and Zimmerman. We are primarily interested in problems of modest size\u0000where constant speed up factors can improve overall system performance, and\u0000therefore focus the discussion on classical and Karatsuba multiplication and\u0000how methods may be combined.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"2017 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algorithms for Recursive Block Matrices","authors":"Stephen M. Watt","doi":"arxiv-2407.03976","DOIUrl":"https://doi.org/arxiv-2407.03976","url":null,"abstract":"We study certain linear algebra algorithms for recursive block matrices. This\u0000representation has useful practical and theoretical properties. We summarize\u0000some previous results for block matrix inversion and present some results on\u0000triangular decomposition of block matrices. The case of inverting matrices over\u0000a ring that is neither formally real nor formally complex was inspired by\u0000Gonzalez-Vega et al.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Terminating Differentiable Tree Experts","authors":"Jonathan Thomm, Michael Hersche, Giacomo Camposampiero, Aleksandar Terzić, Bernhard Schölkopf, Abbas Rahimi","doi":"arxiv-2407.02060","DOIUrl":"https://doi.org/arxiv-2407.02060","url":null,"abstract":"We advance the recently proposed neuro-symbolic Differentiable Tree Machine,\u0000which learns tree operations using a combination of transformers and Tensor\u0000Product Representations. We investigate the architecture and propose two key\u0000components. We first remove a series of different transformer layers that are\u0000used in every step by introducing a mixture of experts. This results in a\u0000Differentiable Tree Experts model with a constant number of parameters for any\u0000arbitrary number of steps in the computation, compared to the previous method\u0000in the Differentiable Tree Machine with a linear growth. Given this flexibility\u0000in the number of steps, we additionally propose a new termination algorithm to\u0000provide the model the power to choose how many steps to make automatically. The\u0000resulting Terminating Differentiable Tree Experts model sluggishly learns to\u0000predict the number of steps without an oracle. It can do so while maintaining\u0000the learning capabilities of the model, converging to the optimal amount of\u0000steps.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"We-Math: Does Your Large Multimodal Model Achieve Human-like Mathematical Reasoning?","authors":"Runqi Qiao, Qiuna Tan, Guanting Dong, Minhui Wu, Chong Sun, Xiaoshuai Song, Zhuoma GongQue, Shanglin Lei, Zhe Wei, Miaoxuan Zhang, Runfeng Qiao, Yifan Zhang, Xiao Zong, Yida Xu, Muxi Diao, Zhimin Bao, Chen Li, Honggang Zhang","doi":"arxiv-2407.01284","DOIUrl":"https://doi.org/arxiv-2407.01284","url":null,"abstract":"Visual mathematical reasoning, as a fundamental visual reasoning ability, has\u0000received widespread attention from the Large Multimodal Models (LMMs)\u0000community. Existing benchmarks, such as MathVista and MathVerse, focus more on\u0000the result-oriented performance but neglect the underlying principles in\u0000knowledge acquisition and generalization. Inspired by human-like mathematical\u0000reasoning, we introduce WE-MATH, the first benchmark specifically designed to\u0000explore the problem-solving principles beyond end-to-end performance. We\u0000meticulously collect and categorize 6.5K visual math problems, spanning 67\u0000hierarchical knowledge concepts and five layers of knowledge granularity. We\u0000decompose composite problems into sub-problems according to the required\u0000knowledge concepts and introduce a novel four-dimensional metric, namely\u0000Insufficient Knowledge (IK), Inadequate Generalization (IG), Complete Mastery\u0000(CM), and Rote Memorization (RM), to hierarchically assess inherent issues in\u0000LMMs' reasoning process. With WE-MATH, we conduct a thorough evaluation of\u0000existing LMMs in visual mathematical reasoning and reveal a negative\u0000correlation between solving steps and problem-specific performance. We confirm\u0000the IK issue of LMMs can be effectively improved via knowledge augmentation\u0000strategies. More notably, the primary challenge of GPT-4o has significantly\u0000transitioned from IK to IG, establishing it as the first LMM advancing towards\u0000the knowledge generalization stage. In contrast, other LMMs exhibit a marked\u0000inclination towards Rote Memorization - they correctly solve composite problems\u0000involving multiple knowledge concepts yet fail to answer sub-problems. We\u0000anticipate that WE-MATH will open new pathways for advancements in visual\u0000mathematical reasoning for LMMs. The WE-MATH data and evaluation code are\u0000available at https://github.com/We-Math/We-Math.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SHA-256 Collision Attack with Programmatic SAT","authors":"Nahiyan Alamgir, Saeed Nejati, Curtis Bright","doi":"arxiv-2406.20072","DOIUrl":"https://doi.org/arxiv-2406.20072","url":null,"abstract":"Cryptographic hash functions play a crucial role in ensuring data security,\u0000generating fixed-length hashes from variable-length inputs. The hash function\u0000SHA-256 is trusted for data security due to its resilience after over twenty\u0000years of intense scrutiny. One of its critical properties is collision\u0000resistance, meaning that it is infeasible to find two different inputs with the\u0000same hash. Currently, the best SHA-256 collision attacks use differential\u0000cryptanalysis to find collisions in simplified versions of SHA-256 that are\u0000reduced to have fewer steps, making it feasible to find collisions. In this paper, we use a satisfiability (SAT) solver as a tool to search for\u0000step-reduced SHA-256 collisions, and dynamically guide the solver with the aid\u0000of a computer algebra system (CAS) used to detect inconsistencies and deduce\u0000information that the solver would otherwise not detect on its own. Our hybrid\u0000SAT + CAS solver significantly outperformed a pure SAT approach, enabling us to\u0000find collisions in step-reduced SHA-256 with significantly more steps. Using\u0000SAT + CAS, we find a 38-step collision of SHA-256 with a modified\u0000initialization vector -- something first found by a highly sophisticated search\u0000tool of Mendel, Nad, and Schl\"affer. Conversely, a pure SAT approach could\u0000find collisions for no more than 28 steps. However, our work only uses the SAT\u0000solver CaDiCaL and its programmatic interface IPASIR-UP.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SAT and Lattice Reduction for Integer Factorization","authors":"Yameen Ajani, Curtis Bright","doi":"arxiv-2406.20071","DOIUrl":"https://doi.org/arxiv-2406.20071","url":null,"abstract":"The difficulty of factoring large integers into primes is the basis for\u0000cryptosystems such as RSA. Due to the widespread popularity of RSA, there have\u0000been many proposed attacks on the factorization problem such as side-channel\u0000attacks where some bits of the prime factors are available. When enough bits of\u0000the prime factors are known, two methods that are effective at solving the\u0000factorization problem are satisfiability (SAT) solvers and Coppersmith's\u0000method. The SAT approach reduces the factorization problem to a Boolean\u0000satisfiability problem, while Coppersmith's approach uses lattice basis\u0000reduction. Both methods have their advantages, but they also have their\u0000limitations: Coppersmith's method does not apply when the known bit positions\u0000are randomized, while SAT-based methods can take advantage of known bits in\u0000arbitrary locations, but have no knowledge of the algebraic structure exploited\u0000by Coppersmith's method. In this paper we describe a new hybrid SAT and\u0000computer algebra approach to efficiently solve random leaked-bit factorization\u0000problems. Specifically, Coppersmith's method is invoked by a SAT solver to\u0000determine whether a partial bit assignment can be extended to a complete\u0000assignment. Our hybrid implementation solves random leaked-bit factorization\u0000problems significantly faster than either a pure SAT or pure computer algebra\u0000approach.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"DNLSAT: A Dynamic Variable Ordering MCSAT Framework for Nonlinear Real Arithmetic","authors":"Zhonghan Wang","doi":"arxiv-2406.18964","DOIUrl":"https://doi.org/arxiv-2406.18964","url":null,"abstract":"Satisfiability modulo nonlinear real arithmetic theory (SMT(NRA)) solving is\u0000essential to multiple applications, including program verification, program\u0000synthesis and software testing. In this context, recently model constructing\u0000satisfiability calculus (MCSAT) has been invented to directly search for models\u0000in the theory space. Although following papers discussed practical directions\u0000and updates on MCSAT, less attention has been paid to the detailed\u0000implementation. In this paper, we present an efficient implementation of\u0000dynamic variable orderings of MCSAT, called dnlsat. We show carefully designed\u0000data structures and promising mechanisms, such as branching heuristic, restart,\u0000and lemma management. Besides, we also give a theoretical study of potential\u0000influences brought by the dynamic variablr ordering. The experimental\u0000evaluation shows that dnlsat accelerates the solving speed and solves more\u0000satisfiable instances than other state-of-the-art SMT solvers. Demonstration Video: https://youtu.be/T2Z0gZQjnPw Code: https://github.com/yogurt-shadow/dnlsat/tree/master/code Benchmark https://zenodo.org/records/10607722/files/QF_NRA.tar.zst?download=1","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reasoning About Action and Change","authors":"Florence Dupin de Saint-CyrIRIT-ADRIA, UT3, Andreas HerzigIRIT-LILaC, CNRS, Jérôme LangLAMSADE, PSL, IRIT-ADRIA, Pierre MarquisCRIL","doi":"arxiv-2406.18930","DOIUrl":"https://doi.org/arxiv-2406.18930","url":null,"abstract":"The purpose of this book is to provide an overview of AI research, ranging\u0000from basic work to interfaces and applications, with as much emphasis on\u0000results as on current issues. It is aimed at an audience of master students and\u0000Ph.D. students, and can be of interest as well for researchers and engineers\u0000who want to know more about AI. The book is split into three volumes.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}