Theoretical Computer Science最新文献

{"title":"The Grothendieck computability model","authors":"Luis Gambarte ,&nbsp;Iosif Petrakis","doi":"10.1016/j.tcs.2025.115550","DOIUrl":"10.1016/j.tcs.2025.115550","url":null,"abstract":"<div><div>Translating notions and results from category theory to the theory of computability models of Longley and Normann, we introduce the Grothendieck computability model. We define the first-projection-simulation and prove its basic properties. With the Grothendieck computability model, the category of computability models is shown to be a type-category, in the sense of Pitts, a result that bridges the categorical interpretation of dependent types with the theory of computability models. We also show that the category of computability models is a category with 2-family arrows and a corresponding structure of Sigma-objects. Finally, we introduce the notion of a fibration and opfibration-simulation, and we prove that the first-projection-simulation is a split opfibration-simulation.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1057 ","pages":"Article 115550"},"PeriodicalIF":1.0,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145160022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal gathering of robots in anonymous butterfly networks via leader election","authors":"Serafino Cicerone ,&nbsp;Alessia Di Fonso ,&nbsp;Gabriele Di Stefano ,&nbsp;Alfredo Navarra","doi":"10.1016/j.tcs.2025.115553","DOIUrl":"10.1016/j.tcs.2025.115553","url":null,"abstract":"<div><div>Robots with very weak capabilities placed on the vertices of a graph are required to move toward a common vertex from where they do not move anymore. The task is known as the <em>Gathering</em> problem and it has been extensively studied in the last decade with respect to both general graphs and specific topologies. Most of the challenges faced are due to possible isometries observable from the placement of the robots with respect to the underlying topology. Rings, Grids, and Complete graphs are just a few examples of very regular topologies where the placement of the robots and suitable movements are crucial for succeeding in Gathering. Here we are interested in understanding what can be done in Butterfly graphs where really many isometries are present and most importantly unavoidable by any movement. We propose a Gathering algorithm for the so-called <em>leader</em> configurations, i.e., those where the initial placement of the robots admits the detection (and election) of one robot as the leader. We introduce a non-trivial technique to elect the leader which is of its own interest. We also prove that the proposed Gathering algorithm is asymptotically optimal in terms of synchronous rounds required.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1057 ","pages":"Article 115553"},"PeriodicalIF":1.0,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Toward Grünbaum’s conjecture bounding vertices of degree 4","authors":"Christian Ortlieb","doi":"10.1016/j.tcs.2025.115551","DOIUrl":"10.1016/j.tcs.2025.115551","url":null,"abstract":"<div><div>Given a spanning tree <span><math><mi>T</mi></math></span> of a planar graph <span><math><mi>G</mi></math></span>, the <em>co-tree</em> of <span><math><mi>T</mi></math></span> is the spanning tree of the dual graph <span><math><msup><mi>G</mi><mo>*</mo></msup></math></span> with edge set <span><math><msup><mrow><mo>(</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mi>E</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>*</mo></msup></math></span>. Grünbaum conjectured in 1970 that every planar 3-connected graph <span><math><mi>G</mi></math></span> contains a spanning tree <span><math><mi>T</mi></math></span> such that both <span><math><mi>T</mi></math></span> and its co-tree have maximum degree at most 3.</div><div>While Grünbaum’s conjecture remains open, Schmidt and the author recently improved the upper bound on the maximum degree from 5 (Biedl 2014) to 4.</div><div>In this paper, we modify this approach taking a further step towards Grünbaum’s conjecture. We again obtain a spanning tree <span><math><mi>T</mi></math></span> such that both <span><math><mi>T</mi></math></span> and its co-tree have maximum degree at most 4 and, additionally, an upper bound on the number of vertices of degree 4 of <span><math><mi>T</mi></math></span> and its co-tree.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1056 ","pages":"Article 115551"},"PeriodicalIF":1.0,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145095609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on busy beaver bounds","authors":"Tomás Schitter ,&nbsp;Sergio Abriola ,&nbsp;Nicolás González","doi":"10.1016/j.tcs.2025.115541","DOIUrl":"10.1016/j.tcs.2025.115541","url":null,"abstract":"<div><div>We investigate the relationship between several variations of the Busy Beaver game proposed by Radó (1962), such as the <span><math><mrow><mi>s</mi><mi>p</mi><mi>a</mi><mi>c</mi><mi>e</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span> or the <span><math><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span> functions, establishing new bounds on these functions in terms of each other, as well as some properties about their growth. We also introduce and investigate a new function, <span><math><mrow><mi>b</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>, to this family of noncomputable functions. We give some specific values for <span><math><mrow><mi>b</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>, as well as several results concerning its growth and its relationship to the previously studied Busy Beaver functions. We also investigate growth properties and relationships of these functions when considering Turing Machines with non-binary alphabets with a single blank symbol.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1057 ","pages":"Article 115541"},"PeriodicalIF":1.0,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145159984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uniform-budget solo chess with only rooks or only knights is hard","authors":"Davide Bilò ,&nbsp;Luca Di Donato ,&nbsp;Luciano Gualà ,&nbsp;Stefano Leucci","doi":"10.1016/j.tcs.2025.115544","DOIUrl":"10.1016/j.tcs.2025.115544","url":null,"abstract":"<div><div>We study the Solo-Chess problem which has been introduced in [Aravind et al., FUN 2022]. This is a single-player variant of chess in which the player must clear all but one piece from the board via a sequence captures while ensuring that each piece performs at most as many captures as its <em>budget</em> allows. The time complexity of finding a winning sequence of captures has already been pinpointed for several combinations of piece types and initial budgets. We contribute to a better understanding of the computational landscape of Solo-Chess by closing two problems left open in [Aravind et al., FUN 2022]. Namely, we show that Solo-Chess is hard even when all pieces are restricted to only rooks with budget exactly 2, or only knights with budget exactly 11.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1056 ","pages":"Article 115544"},"PeriodicalIF":1.0,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145095608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bottom-up rebalancing binary search trees by flipping a coin","authors":"Gerth Stølting Brodal","doi":"10.1016/j.tcs.2025.115543","DOIUrl":"10.1016/j.tcs.2025.115543","url":null,"abstract":"<div><div>Rebalancing schemes for dynamic binary search trees are numerous in the literature, where the goal is to maintain trees of low height, either in the worst-case or expected sense. In this paper we study randomized rebalancing schemes for sequences of <span><math><mi>n</mi></math></span> insertions into an initially empty binary search tree, under the assumption that a tree only stores the elements and the tree structure without any additional balance information. Seidel (2009) presented a top-down randomized insertion algorithm, where insertions take expected <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>lg</mi><mn>2</mn></msup><mi>n</mi><mo>)</mo></mrow></math></span> time, and the resulting trees have the same distribution as inserting a uniform random permutation of <span><math><mi>n</mi></math></span> elements into a binary search tree without rebalancing. Seidel states as an open problem if a similar result can be achieved with bottom-up insertions. In this paper we fail to answer this question.</div><div>We consider two simple canonical randomized bottom-up insertion algorithms on binary search trees, assuming that an insertion is given the position where to insert the next element. The subsequent rebalancing is performed bottom-up in expected <span><math><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> time, uses expected <span><math><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> random bits, performs at most two rotations, and the rotations appear with geometrically decreasing probability in the distance from the leaf. For some insertion sequences the expected depth of each node is proved to be <span><math><mrow><mi>O</mi><mo>(</mo><mi>lg</mi><mi>n</mi><mo>)</mo></mrow></math></span>. On the negative side, we prove for both algorithms that there exist simple insertion sequences where the expected depth is <span><math><mrow><mstyle><mi>Ω</mi></mstyle><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span>, i.e., the studied rebalancing schemes are <em>not</em> competitive with (most) other rebalancing schemes in the literature.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115543"},"PeriodicalIF":1.0,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145095233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On shuffling and splitting automata","authors":"Ignacio Mollo Cunningham","doi":"10.1016/j.tcs.2025.115539","DOIUrl":"10.1016/j.tcs.2025.115539","url":null,"abstract":"<div><div>We consider a class of finite state three-tape transducers which models the operation of shuffling and splitting words. We present them as automata over the so-called Shuffling Monoid. These automata can be seen as either shufflers or splitters interchangeably. We prove that functionality is decidable for splitters, and we also show that the equivalence between functional splitters is decidable. Moreover, in the deterministic case, the algorithm for equivalence is polynomial on the number of states of the splitter.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1057 ","pages":"Article 115539"},"PeriodicalIF":1.0,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tight length theorems for multiset extensions of Higman’s lemma","authors":"Vitor Greati ,&nbsp;Revantha Ramanayake","doi":"10.1016/j.tcs.2025.115546","DOIUrl":"10.1016/j.tcs.2025.115546","url":null,"abstract":"<div><div>A well-quasi-ordered (wqo) set generalizes the notion of well-foundedness and is a powerful tool for analyzing the complexity of computational problems through upper bounds on the length of controlled bad sequences, known as length theorems. The finitary multiset extension of a wqo-set induces an ordering on finite multisets over elements of that set, where one multiset precedes another if there exists an injective mapping between their elements that preserves the original ordering. In this work, we refine existing length theorems for the finitary multiset extension of Higman’s ordering over finite alphabets, and we establish a matching lower bound. As a corollary, we obtain tighter length bounds for the majoring extension of Higman’s ordering over finite alphabets. We demonstrate the application of our results in the complexity analysis of noncommutative hypersequent logics.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1057 ","pages":"Article 115546"},"PeriodicalIF":1.0,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A faster algorithm for independent cut","authors":"Vsevolod Chernyshev ,&nbsp;Johannes Rauch ,&nbsp;Dieter Rautenbach ,&nbsp;Liliia Redina","doi":"10.1016/j.tcs.2025.115542","DOIUrl":"10.1016/j.tcs.2025.115542","url":null,"abstract":"<div><div>The previously fastest algorithm for deciding the existence of an independent cut had a runtime of <span><math><mrow><msup><mi>O</mi><mo>*</mo></msup><mrow><mo>(</mo><mn>1</mn><mo>.</mo><msup><mn>4423</mn><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>n</mi></math></span> is the order of the input graph. We improve this to <span><math><mrow><msup><mi>O</mi><mo>*</mo></msup><mrow><mo>(</mo><mn>1</mn><mo>.</mo><msup><mn>4143</mn><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span>. In fact, we prove a runtime of <span><math><mrow><msup><mi>O</mi><mo>*</mo></msup><mrow><mo>(</mo><msup><mn>2</mn><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><msub><mi>α</mi><mstyle><mi>Δ</mi></mstyle></msub><mo>)</mo><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> on graphs of order <span><math><mi>n</mi></math></span> and maximum degree at most <span><math><mstyle><mi>Δ</mi></mstyle></math></span>, where <span><math><mrow><msub><mi>α</mi><mstyle><mi>Δ</mi></mstyle></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo>+</mo><mn>4</mn><mo>⌊</mo><mfrac><mstyle><mi>Δ</mi></mstyle><mn>2</mn></mfrac><mo>⌋</mo></mrow></mfrac></mrow></math></span>. Furthermore, we show that the problem is fixed-parameter tractable on graphs of order <span><math><mi>n</mi></math></span> and minimum degree at least <span><math><mrow><mi>β</mi><mi>n</mi></mrow></math></span> for some <span><math><mrow><mi>β</mi><mo>&gt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, where <span><math><mi>β</mi></math></span> is the parameter.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1054 ","pages":"Article 115542"},"PeriodicalIF":1.0,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145060256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A continuous leakage-amplified IBE scheme with perfect key update","authors":"Zirui Qiao ,&nbsp;Yong Yu ,&nbsp;Yanwei Zhou ,&nbsp;Dong Zheng","doi":"10.1016/j.tcs.2025.115545","DOIUrl":"10.1016/j.tcs.2025.115545","url":null,"abstract":"<div><div>In the practical deployment of cryptographic solutions, diverse applications pose unique challenges in terms of leakage resilience. The one-size-fits-all approach of traditional cryptographic primitives and fixed leakage-resilient ability often fails to satisfy the nuanced security demands of different scenarios. Recognizing this, the concept of a continuous Identity-based Encryption (IBE) scheme has been introduced. This innovative approach allows for the dynamic adjustment of private key lengths to tailor the system’s resistance to various leakage attacks, based on the specific needs of an application. Despite the strides made, the existing implementations of this scheme exhibit limitations, particularly in the aspect of key updates. The current method for key updates is incomplete, only updating some elements of the key. This process relies on a trapdoor mechanism, which results in suboptimal storage efficiency. This paper introduces a new continuous leakage amplified IBE scheme to address these shortcomings. This improved model features a comprehensive key update mechanism that enables users to refresh every element of the private key without utilizing a trapdoor. Our analysis confirms that this enhanced IBE scheme stands out for its security, efficiency, and practicality. Moreover, in pursuit of optimizing performance, we present a novel general construction. This construction illustrates that it is feasible to construct an IBE scheme resilient against chosen-ciphertext attacks, equipped with an impeccable update function, by building upon any semantically secure IBE scheme.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115545"},"PeriodicalIF":1.0,"publicationDate":"2025-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145057284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}