IA Scholar Query: A Permutation on Words in a Two Letter Alphabet.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 31 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440INTRODUCTION Restoring the Balance
https://scholar.archive.org/work/qphvvbyjobefbkanfcbdauxyiy
A t some point in the first half of the twentieth century, Sigizmund Krzhizhanovsky (1887-1950)-a writer who, at the height of Stalinist repression, had not abandoned the arguably suicidal project of seeing his work into print-wrote a prospectus for a History of Unwritten Literature. Its opening lines identify the "biological cause" of unwritten literature: the closer an author approaches to death, the greater his accumulated literary to-do list. "As the quantity of themes and materials increases, life decreases in proportion. " Worse yet, the growth of unused inspirations is exponential, since, according to Krzhizhanovsky's "law of unrealization," each idea ramifies into new ones until it has been finalized in writing. Thus the lengthier its period of gestation, "the more themes to which it gives birth," each demanding its own embodiment. The proposed History might seem an effort on the part of a penniless holdover from the pre-Revolutionary period to discover a topic so inherently apolitical that it might possibly make its way through Soviet censorship. But Krzhizhanovsky's project also yields profound insight into the gap between intent and accomplishment, the archive and the canon. Although fundamentally impossible-"to what degree is research into the unrealized realizable?" Krzhizhanovsky asks-the project radically reorients literary culture upon the obligation to pay "some part of that thematic and narrative debt that previous generations did not liquidate" by taking on and carrying out their uncompleted projects. Far from a romantic or modernist quest for originality, writing appears as a kind of ancestor cult whose adherents, the aspiring authors of the present day, endeavor perpetually to perform their fealty to the past. Despite the isolation and obscurity in which its author labored, Krzhizhanovsky's idea is of a piece with the wider literary culture of its time. American fantasist James Branch Cabell's Beyond Life, a 1919 collection of dialogues on aesthetics, takes place in a library containing "the cream of the unwritten books-the masterpieces that were planned and never carried out. " 1 In other ways Krzhizhanovsky's prospectus seems to prefigure the literary future. The very form in which it has come down to us-an idea for a book that fills in for the book itself-anticipates Stanisław Lem's collections of introductions to imaginary works, one of which describes a computer that completes unfinished literary monuments by Franz Kafka and Fyodor Dostoevsky. 2 These examples illustrate the crux of Krzhizhanovsky's brief, long-unpublished prospectus for his History of Unwritten Literature: no matter how long a fragment of an idea persists in darkness, it must find visible form and spark new ideas in an audience beyond the purview of the author. Superficially, the sentiment resembles the famous line from Mikhail Bulgakov'swork_qphvvbyjobefbkanfcbdauxyiySat, 31 Dec 2022 00:00:00 GMTGrammars over the Lambek Calculus with Permutation: Recognizing Power and Connection to Branching Vector Addition Systems with States
https://scholar.archive.org/work/m2zknkte2ncdlgqhonnhhkbgfy
In (Van Benthem, 1991) it is proved that all permutation closures of context-free languages can be generated by grammars over the Lambek calculus with the permutation rule (LP-grammars); however, to our best knowledge, it is not established whether the converse holds or not. In this paper, we show that LP-grammars are equivalent to linearly-restricted branching vector addition systems with states and with additional memory (shortly, lBVASSAM), which are modified branching vector addition systems with states. Then an example of such an lBVASSAM is presented, which generates a non-semilinear set of vectors; this yields that LP-grammars generate more than permutation closures of context-free languages. Moreover, equivalence of LP-grammars and lBVASSAM allows us to present a normal form for LP-grammars and, as a consequence, prove that LP-grammars are equivalent to LP-grammars without product. Finally, we prove that the class of languages generated by LP-grammars is closed under intersection.Tikhon Pshenitsynwork_m2zknkte2ncdlgqhonnhhkbgfySat, 19 Nov 2022 00:00:00 GMTRow monomial matrices and Černy conjecture, short proof
https://scholar.archive.org/work/n4t6x3sam5hbzomh5qdcwe3gjm
The class of row monomial matrices (one unit and rest zeros in every row) with some non-standard operations of summation and usual multiplication is our main object. These matrices generate a space with respect to the mentioned operations. A word w of letters on edges of underlying graph of deterministic finite automaton (DFA) is called synchronizing if w sends all states of the automaton to a unique state J. Černy discovered in 1964 a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n-1)(n-1). The hypothesis, well known today as the Černy conjecture, claims that (n-1)(n-1) is also precise upper bound on the length of such a word for a complete DFA. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. We present the proof of the Černy-Starke conjecture: the deterministic complete n-state synchronizing automaton has synchronizing word of length at most (n-1)(n-1). The proof used connection between dimension of the space and the length of words on paths of edges in underlying graph of automaton.A.N. Trahtmanwork_n4t6x3sam5hbzomh5qdcwe3gjmFri, 18 Nov 2022 00:00:00 GMTIdeals of equations for elements in a free group and context-free languages
https://scholar.archive.org/work/2mkwtagwqrh4zjq3evd7a6w3ki
Let F be a finitely generated free group, and H≤ F a finitely generated subgroup. An equation for an element g∈ F with coefficients in H is an element w(x)∈ H*⟨ x ⟩ such that w(g)=1 in F; the degree of the equation is the number of occurrences of x and x^-1 in the cyclic reduction of w(x). Given an element g∈ F, we consider the ideal ℑ_g⊆ H*⟨ x ⟩ of equations for g with coefficients in H; we study the structure of ℑ_g using context-free languages. We describe a new algorithm that determines whether ℑ_g is trivial or not; the algorithm runs in polynomial time. We also describe a polynomial-time algorithm that, given d∈ℕ, decides whether or not the subset ℑ_g,d⊆ℑ_g of all degree-d equations is empty. We provide a polynomial-time algorithm that computes the minimum degree d_min of a non-trivial equation in ℑ_g. We provide a sharp upper bound on d_min. Finally, we study the growth of the number of (cyclically reduced) equations in ℑ_g and in ℑ_g,d as a function of their length. We prove that this growth is either polynomial or exponential, and we provide a polynomial-time algorithm that computes the type of growth (including the degree of the growth if it's polynomial).Dario Ascariwork_2mkwtagwqrh4zjq3evd7a6w3kiFri, 18 Nov 2022 00:00:00 GMTCo-lexicographically ordering automata and regular languages. Part I
https://scholar.archive.org/work/rvvbtgysgjd3rpxubqb2idhigi
In the present work, we lay out a new theory showing that all automata can always be co-lexicographically 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. 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 analogous of the Myhill-Nerode Theorem for co-lexicographically ordered regular languages.Nicola Cotumaccio, Giovanna D'Agostino, Alberto Policriti, Nicola Prezzawork_rvvbtgysgjd3rpxubqb2idhigiThu, 17 Nov 2022 00:00:00 GMTFast Coalgebraic Bisimilarity Minimization
https://scholar.archive.org/work/3rykej7sfbc45hweawghi4nbny
Coalgebraic bisimilarity minimization generalizes classical automaton minimization to a large class of automata whose transition structure is specified by a functor, subsuming strong, weighted, and probabilistic bisimilarity. This offers the enticing possibility of turning bisimilarity minimization into an off-the-shelf technology, without having to develop a new algorithm for each new type of automaton. Unfortunately, there is no existing algorithm that is fully general, efficient, and able to handle large systems. We present a generic algorithm that minimizes coalgebras over an arbitrary functor in the category of sets as long as the action on morphisms is sufficiently computable. The functor makes at most 𝒪(m log n) calls to the functor-specific action, where n is the number of states and m is the number of transitions in the coalgebra. While more specialized algorithms can be asymptotically faster than our algorithm (usually by a factor of 𝒪(m/n)), our algorithm is especially well suited to efficient implementation, and our tool Boa often uses much less time and memory on existing benchmarks, and can handle larger automata, despite being more generic.Jules Jacobs, Thorsten Wißmannwork_3rykej7sfbc45hweawghi4nbnyThu, 17 Nov 2022 00:00:00 GMTRounding via Low Dimensional Embeddings
https://scholar.archive.org/work/jicnilghmjhsrku2gzxdt63krm
A regular graph G = (V,E) is an (ε,γ) small-set expander if for any set of vertices of fractional size at most ε, at least γ of the edges that are adjacent to it go outside. In this paper, we give a unified approach to several known complexity-theoretic results on small-set expanders. In particular, we show: 1. Max-Cut: we show that if a regular graph G = (V,E) is an (ε,γ) small-set expander that contains a cut of fractional size at least 1-δ, then one can find in G a cut of fractional size at least 1-O(δ/εγ^6) in polynomial time. 2. Improved spectral partitioning, Cheeger's inequality and the parallel repetition theorem over small-set expanders. The general form of each one of these results involves square-root loss that comes from certain rounding procedure, and we show how this can be avoided over small set expanders. Our main idea is to project a high dimensional vector solution into a low-dimensional space while roughly maintaining ℓ_2^2 distances, and then perform a pre-processing step using low-dimensional geometry and the properties of ℓ_2^2 distances over it. This pre-processing leverages the small-set expansion property of the graph to transform a vector valued solution to a different vector valued solution with additional structural properties, which give rise to more efficient integral-solution rounding schemes.Mark Braverman, Dor Minzerwork_jicnilghmjhsrku2gzxdt63krmThu, 17 Nov 2022 00:00:00 GMTDr. Neurosymbolic, or: How I Learned to Stop Worrying and Accept Statistics
https://scholar.archive.org/work/o4q5lwow3nc5nmpl5rhwtm754m
The symbolic AI community is increasingly trying to embrace machine learning in neuro-symbolic architectures, yet is still struggling due to cultural barriers. To break the barrier, this rather opinionated personal memo attempts to explain and rectify the conventions in Statistics, Machine Learning, and Deep Learning from the viewpoint of outsiders. It provides a step-by-step protocol for designing a machine learning system that satisfies a minimum theoretical guarantee necessary for being taken seriously by the symbolic AI community, i.e., it discusses "in what condition we can stop worrying and accept statistical machine learning." Unlike most textbooks which are written for students trying to specialize in Stat/ML/DL and willing to accept jargons, this memo is written for experienced symbolic researchers that hear a lot of buzz but are still uncertain and skeptical. Information on Stat/ML/DL is currently too scattered or too noisy to invest in. This memo prioritizes compactness, citations to old papers (many in early 20th century), and concepts that resonate well with symbolic paradigms in order to offer time savings. It prioritizes general mathematical modeling and does not discuss any specific function approximator, such as neural networks (NNs), SVMs, decision trees, etc. Finally, it is open to corrections. Consider this memo as something similar to a blog post taking the form of a paper on Arxiv.Masataro Asaiwork_o4q5lwow3nc5nmpl5rhwtm754mWed, 16 Nov 2022 00:00:00 GMTM_0,5: Towards the Chabauty-Kim method in higher dimensions
https://scholar.archive.org/work/wr2ya7rghfg5hnvzny5c424rmq
If Z is an open subscheme of Spec ZZ, X is a sufficiently nice Z-model of a smooth curve over QQ, and p is a closed point of Z, the Chabauty-Kim method leads to the construction of locally analytic functions on X(ZZ_p) which vanish on X(Z); we call such functions "Kim functions". At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface M_0,5 should be easier than the previously studied curve M_0,4 since its points are closely related to those of M_0,4, yet they face a further condition to integrality. This is mirrored by a certain "weight advantage" we encounter, because of which, M_0,5 possesses new Kim functions not coming from M_0,4. Here we focus on the case "ZZ[1/6] in half-weight 4", where we provide a first nontrivial example of a Kim function on a surface. Central to our approach to Chabauty-Kim theory (as developed in works by S. Wewers, D. Corwin, and the first author) is the possibility of separating the geometric part of the computation from its arithmetic context. However, we find that in this case the geometric step grows beyond the bounds of standard algorithms running on current computers. Therefore, some ingenuity is needed to solve this seemingly straightforward problem, and our new Kim function is huge.Ishai Dan-Cohen, David Jarossaywork_wr2ya7rghfg5hnvzny5c424rmqWed, 16 Nov 2022 00:00:00 GMTCellular subalgebras of the partition algebra
https://scholar.archive.org/work/a4zmulxchfdfzam54lcwmkbe4m
We describe various diagram algebras and their representation theory using cellular algebras of Graham and Lehrer and the decomposition into half diagrams. In particular, we show the diagram algebras surveyed here are all cellular algebras and parameterize their cell modules. We give a new construction to build new cellular algebras from a general cellular algebra and subalgebras of the rook Brauer algebra that we call the cellular wreath product.Travis Scrimshawwork_a4zmulxchfdfzam54lcwmkbe4mWed, 16 Nov 2022 00:00:00 GMTTesting of Index-Invariant Properties in the Huge Object Model
https://scholar.archive.org/work/ebulwuku6vbpdkkezbn2asgkye
The study of distribution testing has become ubiquitous in the area of property testing, both for its theoretical appeal, as well as for its applications in other fields of Computer Science. The original distribution testing model relies on samples drawn independently from the distribution to be tested. However, when testing distributions over the n-dimensional Hamming cube {0,1}^n for a large n, even reading a few samples is infeasible. To address this, Goldreich and Ron [ITCS 2022] have defined a model called the huge object model, in which the samples may only be queried in a few places. In this work, we initiate a study of a general class of properties in the huge object model, those that are invariant under a permutation of the indices of the vectors in {0,1}^n, while still not being necessarily fully symmetric as per the definition used in traditional distribution testing. We prove that every index-invariant property satisfying a bounded VC-dimension restriction admits a property tester with a number of queries independent of n. To complement this result, we argue that satisfying only index-invariance or only a VC-dimension bound is insufficient to guarantee a tester whose query complexity is independent of n. Moreover, we prove that the dependency of sample and query complexities of our tester on the VC-dimension is tight. As a second part of this work, we address the question of the number of queries required for non-adaptive testing. We show that it can be at most quadratic in the number of queries required for an adaptive tester of index-invariant properties. This is in contrast with the tight exponential gap for general non-index-invariant properties. Finally, we provide an index-invariant property for which the quadratic gap between adaptive and non-adaptive query complexities for testing is almost tight.Sourav Chakraborty, Eldar Fischer, Arijit Ghosh, Gopinath Mishra, Sayantan Senwork_ebulwuku6vbpdkkezbn2asgkyeTue, 15 Nov 2022 00:00:00 GMTA Lego-Brick Approach to Coding for Network Communication
https://scholar.archive.org/work/blizb224cnbt7ihyma35hbgv4i
Coding schemes for several problems in network information theory are constructed starting from point-to-point channel codes that are designed for symmetric channels. Given that the point-to-point codes satisfy certain properties pertaining to the rate, the error probability, and the distribution of decoded sequences, bounds on the performance of the coding schemes are derived and shown to hold irrespective of other properties of the codes. In particular, we consider the problems of lossless and lossy source coding, Slepian--Wolf coding, Wyner--Ziv coding, Berger--Tung coding, multiple description coding, asymmetric channel coding, Gelfand--Pinsker coding, coding for multiple access channels, Marton coding for broadcast channels, and coding for cloud radio access networks (C-RAN's). We show that the coding schemes can achieve the best known inner bounds for these problems, provided that the constituent point-to-point channel codes are rate-optimal. This would allow one to leverage commercial off-the-shelf codes for point-to-point symmetric channels in the practical implementation of codes over networks. Simulation results demonstrate the gain of the proposed coding schemes compared to existing practical solutions to these problems.Nadim Ghaddar and Shouvik Ganguly and Lele Wang and Young-Han Kimwork_blizb224cnbt7ihyma35hbgv4iMon, 14 Nov 2022 00:00:00 GMTTriangulations of prisms and preprojective algebras of type A
https://scholar.archive.org/work/o5kyycckarfdfkfnvs6qxpf4q4
We show that indecomposable two-term presilting complexes over Π_n, the preprojective algebra of A_n, are in bijection with internal n-simplices in the prism Δ_n×Δ_1, the product of an n-simplex with a 1-simplex. We show further that this induces a bijection between triangulations of Δ_n×Δ_1 and two-term silting complexes over Π_n such that bistellar flips of triangulations correspond to mutations of two-term silting complexes. These bijections are shown to compatible with the known bijections involving the symmetric group.Osamu Iyama, Nicholas J. Williamswork_o5kyycckarfdfkfnvs6qxpf4q4Mon, 14 Nov 2022 00:00:00 GMTOn the solutions of universal differential equation by noncommutative Picard-Vessiot theory
https://scholar.archive.org/work/hslucctjmfg2lcnb5lze5hrqku
Basing on the algebraic combinatorics on noncommutative formal series with holomorphic coefficients and Picard-Vessiot theory of noncommutative differential equations, various recursive constructions of sequences converging to solutions of universal differential equation are proposed. These expansions are applied to Knizhnik-Zamolodchikov equations and provide solutions with asymptotic conditions by d\'evissage.V.C. Bui, V. Hoang Ngoc Minh, V. Nguyen Dinh, Q.H. Ngowork_hslucctjmfg2lcnb5lze5hrqkuMon, 14 Nov 2022 00:00:00 GMTA Uniform Sampling Procedure for Abstract Triangulations of Surfaces
https://scholar.archive.org/work/45lplr2hqbfxjmuxoz36mw7umy
We present a procedure to sample uniformly from the set of combinatorial isomorphism types of balanced triangulations of surfaces - also known as graph-encoded surfaces. For a given number n, the sample is a weighted set of graph-encoded surfaces with 2n triangles. The sampling procedure relies on connections between graph-encoded surfaces and permutations, and basic properties of the symmetric group. We implement our method and present a number of experimental findings based on the analysis of 138 million runs of our sampling procedure, producing graph-encoded surfaces with up to 280 triangles. Namely, we determine that, for n fixed, the empirical mean genus g̅(n) of our sample is very close to g̅(n) = n-1/2 - (16.98n -110.61)^1/4. Moreover, we present experimental evidence that the associated genus distribution more and more concentrates on a vanishing portion of all possible genera as n tends to infinity. Finally, we observe from our data that the mean number of non-trivial symmetries of a uniformly chosen graph encoding of a surface decays to zero at a rate super-exponential in n.Rajan Shankar, Jonathan Spreerwork_45lplr2hqbfxjmuxoz36mw7umyMon, 14 Nov 2022 00:00:00 GMTQuantum versus Population Dynamics over Cayley Graphs
https://scholar.archive.org/work/3fwtfxcdkrgqhjecaxu7ur3sqa
Consider a graph whose vertices are populated by identical objects, together with an algorithm for the time-evolution of the number of objects placed at each of the vertices. The discrete dynamics of these objects can be observed and studied using simple and inexpensive laboratory settings. There are many similarities but also many differences between such population dynamics and the quantum dynamics of a particle hopping on the same graph. In this work, we show that a specific decoration of the original graph enables an exact mapping between the models of population and quantum dynamics. As such, population dynamics over graphs is yet another classical platform that can simulate quantum effects. Several examples are used to demonstrate this claim.Emil Prodanwork_3fwtfxcdkrgqhjecaxu7ur3sqaSun, 13 Nov 2022 00:00:00 GMTAmenability of monomial algebras, minimal subshifts and free subalgebras
https://scholar.archive.org/work/22bcflymf5d4bjphlabxgdfv7e
We give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Folner sequences, answering a question due to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization to prove that over projectively simple monomial algebras, every module is exhaustively amenable; we conclude that convolution algebras of minimal subshifts admit the same property. We deduce that any minimal subshift of positive entropy gives rise to a graded algebra which does not satisfy an extension of Vershik's conjecture on amenable groups, proposed by Bartholdi. Finally, we show that non-amenable monomial algebras must contain noncommutative free subalgebras. Examples are given to emphasize the sharpness and necessity of the assumptions in our results.Jason P. Bell, Be'eri Greenfeldwork_22bcflymf5d4bjphlabxgdfv7eSat, 12 Nov 2022 00:00:00 GMTLinking numbers of modular knots
https://scholar.archive.org/work/5fmj52rzh5gifnfjwqjcwafy7y
The modular group PSL(2;Z) acts on the hyperbolic plane HP with quotient the modular surface M, whose unit tangent bundle U is a 3-manifold homeomorphic to the complement of the trefoil knot in the 3-sphere. The hyperbolic conjugacy classes of PSL(2;Z) correspond to the closed oriented geodesics in M. Those lift to the periodic orbits for the geodesic flow in U, which define the modular knots. The linking numbers between modular knots and the trefoil is well understood. Indeed, Etienne Ghys showed in 2006 that they are given by the Rademacher invariant of the corresponding conjugacy classes. The Rademacher function is a homogeneous quasi-morphism of PSL(2;Z) which he had recognised with Jean Barge in 1992 as half the primitive of the bounded euler class. This shed light on the 1987 work of Michael Atiyah concerning the logarithm of the Dedekind eta function which identified it with no less than that six other important functions appearing in diverse areas of mathematics. We are concerned with the linking numbers between modular knots and derive several formulae with arithmetical, combinatorial, topological and group theoretical flavours. In particular we associate to a pair of modular knots a function defined on the character variety of PSL(2;Z), whose limit at the boundary point recovers their linking number. Moreover, we show that the linking number with a modular knot minus that with its inverse yields a homogeneous quasi-morphism on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate.Christopher-Lloyd Simonwork_5fmj52rzh5gifnfjwqjcwafy7yFri, 11 Nov 2022 00:00:00 GMTCoding Schemes Based on Reed-Muller Codes for (d,∞)-RLL Input-Constrained Channels
https://scholar.archive.org/work/v45575b2zzbevmiptarjysoxw4
The paper considers coding schemes derived from Reed-Muller (RM) codes, for transmission over input-constrained memoryless channels. Our focus is on the (d,∞)-runlength limited (RLL) constraint, which mandates that any pair of successive 1s be separated by at least d 0s. In our study, we first consider (d,∞)-RLL subcodes of RM codes, taking the coordinates of the RM codes to be in the standard lexicographic ordering. We show, via a simple construction, that RM codes of rate R have linear (d,∞)-RLL subcodes of rate R·2^-⌈log_2(d+1)⌉. We then show that our construction is essentially rate-optimal, by deriving an upper bound on the rates of linear (d,∞)-RLL subcodes of RM codes of rate R. Next, for the special case when d=1, we prove the existence of potentially non-linear (1,∞)-RLL subcodes that achieve a rate of max(0,R-3/8). This, for R > 3/4, beats the R/2 rate obtainable from linear subcodes. We further derive upper bounds on the rates of (1,∞)-RLL subcodes, not necessarily linear, of a certain canonical sequence of RM codes of rate R. We then shift our attention to settings where the coordinates of the RM code are not ordered according to the lexicographic ordering, and derive rate upper bounds for linear (d,∞)-RLL subcodes in these cases as well. Finally, we present a new two-stage constrained coding scheme, again using RM codes of rate R, which outperforms any linear coding scheme using (d,∞)-RLL subcodes, for values of R close to 1.V. Arvind Rameshwar, Navin Kashyapwork_v45575b2zzbevmiptarjysoxw4Thu, 10 Nov 2022 00:00:00 GMTHierarchies of Minion Tests for PCSPs through Tensors
https://scholar.archive.org/work/7y6by4r2k5d7tnw7pvhjp3dvha
We provide a unified framework to study hierarchies of relaxations for Constraint Satisfaction Problems and their Promise variant. The idea is to split the description of a hierarchy into an algebraic part, depending on a minion capturing the "base level" of the hierarchy, and a geometric part -- which we call tensorisation -- inspired by multilinear algebra. We show that the hierarchies of minion tests obtained in this way are general enough to capture the (combinatorial) bounded width and also the Sherali-Adams LP, Sum-of-Squares SDP, and affine IP hierarchies. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of such hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. Finally, in order to analyse the Sum-of-Squares SDP hierarchy we also characterise the solvability of the standard SDP relaxation through a new minion.Lorenzo Ciardo, Stanislav Živnýwork_7y6by4r2k5d7tnw7pvhjp3dvhaThu, 10 Nov 2022 00:00:00 GMT