Journal of the ACM最新文献

{"title":"On the Zeros of Exponential Polynomials","authors":"Ventsislav Chonev, J. Ouaknine, J. Worrell","doi":"10.1145/3603543","DOIUrl":"https://doi.org/10.1145/3603543","url":null,"abstract":"We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two problems: the Zero Problem, which asks whether an exponential polynomial has a real root, and the Infinite Zeros Problem, which asks whether such a function has infinitely many real roots. Our main result is that for differential equations of order at most 8 the Zero Problem is decidable, subject to Schanuel’s Conjecture, while the Infinite Zeros Problem is decidable unconditionally. We show moreover that a decision procedure for the Infinite Zeros Problem at order 9 would yield an algorithm for computing the Lagrange constant of any given real algebraic number to arbitrary precision, indicating that it will be very difficult to extend our decidability results to higher orders.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"1 1","pages":"1 - 26"},"PeriodicalIF":2.5,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89723916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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":"Co-lexicographically Ordering Automata and Regular Languages - Part I","authors":"Nicola Cotumaccio, Giovanna D’Agostino, Alberto Policriti, Nicola Prezza","doi":"10.1145/3607471","DOIUrl":"https://doi.org/10.1145/3607471","url":null,"abstract":"The states of a finite-state automaton 𝒩 can be identified with collections of words in the prefix closure of the regular language accepted by 𝒩. But words can be ordered, and among the many possible orders a very natural one is the co-lexicographic order. Such naturalness stems from the fact that it suggests a transfer of the order from words to the automaton’s states. This suggestion is, in fact, concrete and in a number of articles automata admitting a total co-lexicographic ( co-lex for brevity) ordering of states have been proposed and studied. Such class of ordered automata — Wheeler automata — turned out to require just a constant number of bits per transition to be represented and enable regular expression matching queries in constant time per matched character. Unfortunately, not all automata can be totally ordered as previously outlined. In the present work, we lay out a new theory showing that all automata can always be partially ordered, and an intrinsic measure of their complexity can be defined and effectively determined, namely, the minimum width p of one of their admissible co-lex partial orders –dubbed here the automaton’s co-lex width . We first show that this new measure captures at once the complexity of several seemingly-unrelated hard problems on automata. Any NFA of co-lex width p : (i) has an equivalent powerset DFA whose size is exponential in p rather than (as a classic analysis shows) in the NFA’s size; (ii) can be encoded using just Θ(log p ) bits per transition; (iii) admits a linear-space data structure solving regular expression matching queries in time proportional to p 2 per matched character. Some consequences of this new parameterization of automata are that PSPACE-hard problems such as NFA equivalence are FPT in p , and quadratic lower bounds for the regular expression matching problem do not hold for sufficiently small p . Having established that the co-lex width of an automaton is a fundamental complexity measure, we proceed by (i) determining its computational complexity and (ii) extending this notion from automata to regular languages by studying their smallest-width accepting NFAs and DFAs. In this work we focus on the deterministic case and prove that a canonical minimum-width DFA accepting a language ℒ–dubbed the Hasse automaton ℋ of ℒ–can be exhibited. ℋ provides, in a precise sense, the best possible way to (partially) order the states of any DFA accepting ℒ, as long as we want to maintain an operational link with the (co-lexicographic) order of ℒ’s prefixes. Finally, we explore the relationship between two conflicting objectives: minimizing the width and minimizing the number of states of a DFA. In this context, we provide an analogue of the Myhill-Nerode Theorem for co-lexicographically ordered regular languages.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135309216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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":"Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs","authors":"Gregory Bodwin, M. Parter","doi":"10.1145/3603542","DOIUrl":"https://doi.org/10.1145/3603542","url":null,"abstract":"The restoration lemma by Afek et al. [3] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original shortest paths. However, the lemma is tiebreaking-sensitive: if one selects a particular canonical shortest path for each node pair, it is no longer guaranteed that one can build replacement paths by concatenating two selected shortest paths. They left as an open problem whether a method of shortest path tiebreaking with this desirable property is generally possible. We settle this question affirmatively with the first general construction of restorable tiebreaking schemes. We then show applications to various problems in fault-tolerant network design. These include a faster algorithm for subset replacement paths, more efficient fault-tolerant (exact) distance labeling schemes, fault-tolerant subset distance preservers and + 4 additive spanners with improved sparsity, and fast distributed algorithms that construct these objects. For example, an almost immediate corollary of our restorable tiebreaking scheme is the first nontrivial distributed construction of sparse fault-tolerant distance preservers resilient to three faults.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"1 1","pages":"1 - 24"},"PeriodicalIF":2.5,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91061623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Strongest Algebraic Program Invariants","authors":"E. Hrushovski, J. Ouaknine, Amaury Pouly, J. Worrell","doi":"10.1145/3614319","DOIUrl":"https://doi.org/10.1145/3614319","url":null,"abstract":"A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial equations over the program variables at each program location. Müller-Olm and Seidl have posed the question of whether one can compute the strongest algebraic invariant of a given polynomial program. In this article, we show that, while strongest algebraic invariants are not computable in general, they can be computed in the special case of affine programs, that is, programs with exclusively linear assignments. For the latter result, our main tool is an algebraic result of independent interest: Given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"8 1","pages":"1 - 22"},"PeriodicalIF":2.5,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74213484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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":"Near-Optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement Steps","authors":"Christoph Berkholz, Jakob Nordström","doi":"https://dl.acm.org/doi/10.1145/3195257","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3195257","url":null,"abstract":"<p>We prove near-optimal trade-offs for quantifier depth (also called quantifier rank) versus number of variables in first-order logic by exhibiting pairs of <i>n</i>-element structures that can be distinguished by a <i>k</i>-variable first-order sentence but where every such sentence requires quantifier depth at least <i>n</i>^{<i>Ω</i>(<i>k</i>/log <i>k</i>)}. Our trade-offs also apply to first-order counting logic, and by the known connection to the <i>k</i>-dimensional Weisfeiler–Leman algorithm imply near-optimal lower bounds on the number of refinement iterations. </p><p>A key component in our proof is the hardness condensation technique introduced by [Razborov ’16] 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.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"11 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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":"Co-lexicographically Ordering Automata and Regular Languages - Part I","authors":"Nicola Cotumaccio, Giovanna D’Agostino, Alberto Policriti, Nicola Prezza","doi":"https://dl.acm.org/doi/10.1145/3607471","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3607471","url":null,"abstract":"&lt;p&gt;The states of a finite-state automaton (mathcal {N} ) can be identified with collections of words in the prefix closure of the regular language accepted by (mathcal {N} ). But words can be ordered, and among the many possible orders a very natural one is the co-lexicographic order. Such naturalness stems from the fact that it suggests a transfer of the order from words to the automaton’s states. This suggestion is, in fact, concrete and in a number of papers automata admitting a &lt;i&gt;total&lt;/i&gt; co-lexicographic (&lt;i&gt;co-lex&lt;/i&gt; for brevity) ordering of states have been proposed and studied. Such class of ordered automata — &lt;i&gt;Wheeler automata&lt;/i&gt; — turned out to require just a constant number of bits per transition to be represented and enable regular expression matching queries in constant time per matched character. &lt;/p&gt;&lt;p&gt;Unfortunately, not all automata can be totally ordered as previously outlined. In the present work, we lay out a new theory showing that all automata can always be &lt;i&gt;partially&lt;/i&gt; ordered, and an intrinsic measure of their complexity can be defined and effectively determined, namely, the minimum width &lt;i&gt;p&lt;/i&gt; of one of their admissible &lt;i&gt;co-lex partial orders&lt;/i&gt;—dubbed here the automaton’s &lt;i&gt;co-lex width&lt;/i&gt;. We first show that this new measure captures &lt;i&gt;at once&lt;/i&gt; the complexity of several seemingly-unrelated hard problems on automata. Any NFA of co-lex width &lt;i&gt;p&lt;/i&gt;: (i) has an equivalent powerset DFA whose size is exponential in &lt;i&gt;p&lt;/i&gt; rather than (as a classic analysis shows) in the NFA’s size; (ii) can be encoded using just &lt;i&gt;Θ&lt;/i&gt;(log &lt;i&gt;p&lt;/i&gt;) bits per transition; (iii) admits a linear-space data structure solving regular expression matching queries in time proportional to &lt;i&gt;p&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt; per matched character. Some consequences of this new parameterization of automata are that PSPACE-hard problems such as NFA equivalence are FPT in &lt;i&gt;p&lt;/i&gt;, and quadratic lower bounds for the regular expression matching problem do not hold for sufficiently small &lt;i&gt;p&lt;/i&gt;. &lt;/p&gt;&lt;p&gt;Having established that the co-lex width of an automaton is a fundamental complexity measure, we proceed by (i) determining its computational complexity and (ii) extending this notion from automata to regular languages by studying their smallest-width accepting NFAs and DFAs. In this work we focus on the deterministic case and prove that a canonical minimum-width DFA accepting a language (mathcal {L} )—dubbed the Hasse automaton (mathcal {H} ) of (mathcal {L} )—can be exhibited. (mathcal {H} ) provides, in a precise sense, the best possible way to (partially) order the states of any DFA accepting (mathcal {L} ), as long as we want to maintain an operational link with the (co-lexicographic) order of (mathcal {L} )’s prefixes. Finally, we explore the relationship between two conflicting objectives: minimizing the width and minimizing the number of states of a DFA. In this context, we provide an analogue of the Myhill-Nerode Theorem for co-lexicogr","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"6 4","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 the Zeros of Exponential Polynomials","authors":"Ventsislav Chonev, Joel Ouaknine, James Worrell","doi":"https://dl.acm.org/doi/10.1145/3603543","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3603543","url":null,"abstract":"<p>We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two problems: the <i>Zero Problem</i>, which asks whether an exponential polynomial has a real root, and the <i>Infinite Zeros Problem</i>, which asks whether such a function has infinitely many real roots. Our main result is that for differential equations of order at most 8 the Zero Problem is decidable, subject to Schanuel’s Conjecture, whilst the Infinite Zeros Problem is decidable unconditionally. We show moreover that a decision procedure for the Infinite Zeros Problem at order 9 would yield an algorithm for computing the Lagrange constant of any given real algebraic number to arbitrary precision, indicating that it will be very difficult to extend our decidability results to higher orders.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"2 6","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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":"Learning Equilibria in Matching Markets with Bandit Feedback","authors":"Meena Jagadeesan, Alexander Wei, Yixin Wang, Michael I. Jordan, Jacob Steinhardt","doi":"https://dl.acm.org/doi/10.1145/3583681","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3583681","url":null,"abstract":"<p>Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) are, unfortunately, of limited value in the learning setting, given that preferences are inherently uncertain and destabilizing while they are being learned. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that “optimism in the face of uncertainty,” the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"24 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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":"Chains, Koch Chains, and Point Sets with Many Triangulations","authors":"Daniel Rutschmann, Manuel Wettstein","doi":"https://dl.acm.org/doi/10.1145/3585535","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3585535","url":null,"abstract":"<p>We introduce the abstract notion of a chain, which is a sequence of <i>n</i> points in the plane, ordered by <i>x</i>-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations.</p><p>We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω (9.08^<i>n</i>) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω (8.65^<i>n</i>) for the maximum number of triangulations of planar point sets.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"27 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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":"Rate-independent Computation in Continuous Chemical Reaction Networks","authors":"Ho-Lin Chen, David Doty, Wyatt Reeves, David Soloveichik","doi":"https://dl.acm.org/doi/10.1145/3590776","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3590776","url":null,"abstract":"<p>Understanding the algorithmic behaviors that are <i>in principle</i> realizable in a chemical system is necessary for a rigorous understanding of the design principles of biological regulatory networks. Further, advances in synthetic biology herald the time when we will be able to rationally engineer complex chemical systems and when idealized formal models will become blueprints for engineering. </p><p>Coupled chemical interactions in a well-mixed solution are commonly formalized as chemical reaction networks (CRNs). However, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. Here, we study the following problem: What functions <i>f : ℝ^k → ℝ</i> can be computed by a CRN, in which the CRN eventually produces the correct amount of the “output” molecule, no matter the rate at which reactions proceed? This captures a previously unexplored but very natural class of computations: For example, the reaction <i>X₁ + X₂ → Y</i> can be thought to compute the function <i>y</i> = min (<i>x₁, x₂</i>). Such a CRN is robust in the sense that it is correct whether its evolution is governed by the standard model of mass-action kinetics, alternatives such as Hill-function or Michaelis-Menten kinetics, or other arbitrary models of chemistry that respect the (fundamentally digital) stoichiometric constraints (what are the reactants and products?). </p><p>We develop a reachability relation based on a broad notion of “what could happen” if reaction rates can vary arbitrarily over time. Using reachability, we define <i>stable computation</i> analogously to probability 1 computation in distributed computing and connect it with a seemingly stronger notion of rate-independent computation based on convergence in the limit <i>t</i> → ∞ under a wide class of generalized rate laws. Besides the direct mapping of a concentration to a nonnegative analog value, we also consider the “dual-rail representation” that can represent negative values as the difference of two concentrations and allows the composition of CRN modules. We prove that a function is rate-independently computable if and only if it is piecewise linear (with rational coefficients) and continuous (dual-rail representation), or non-negative with discontinuities occurring only when some inputs switch from zero to positive (direct representation). The many contexts where continuous piecewise linear functions are powerful targets for implementation, combined with the systematic construction we develop for computing these functions, demonstrate the potential of rate-independent chemical computation.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":"1 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}