pinecone/core-2020-05-10-deduplication huggingface dataset:
Conjunctive-Query Containment and Constraint Satisfaction

AbstractConjunctive-query containment is recognized as a fundamental problem in database query evaluation and optimization. At the same time, constraint satisfaction is recognized as a fundamental problem in artificial intelligence. What do conjunctive-query containment and constraint satisfaction have in common? Our main conceptual contribution in this paper is to point out that, despite their very different formulation, conjunctive-query containment and constraint satisfaction are essentially the same problem. The reason is that they can be recast as the following fundamental algebraic problem: given two finite relational structures A and B, is there a homomorphism h:A→B? As formulated above, the homomorphism problem is uniform in the sense that both relational structures A and B are part of the input. By fixing the structure B, one obtains the following nonuniform problem: given a finite relational structure A, is there a homomorphism h:A→B? In general, nonuniform tractability results do not uniformize. Thus, it is natural to ask: which tractable cases of nonuniform tractability results for constraint satisfaction and conjunctive-query containment do uniformize? Our main technical contribution in this paper is to show that several cases of tractable nonuniform constraint-satisfaction problems do indeed uniformize. We exhibit three nonuniform tractability results that uniformize and, thus, give rise to polynomial-time solvable cases of constraint satisfaction and conjunctive-query containment. We begin by examining the tractable cases of Boolean constraint-satisfaction problems and show that they do uniformize. This can be applied to conjunctive-query containment via Booleanization; in particular, it yields one of the known tractable cases of conjunctive-query containment. After this, we show that tractability results for constraint-satisfaction problems that can be expressed using Datalog programs with bounded number of distinct variables also uniformize. Finally, we provide a new proof for the fact that tractability results for queries with bounded treewidth uniformize as well, via a connection with first-order logic with a bounded number of distinct variables
The Lob–Pass Problem

AbstractWe consider a new variant of the online learning model in which the goal of an agent is to choose his or her actions so as to maximize the number of successes, while learning about his or her reacting environment through those very actions. In particular, we consider a model of tennis play, in which the only actions that the player can take are a pass and a lob, and the opponent is modeled by two linear (probabilistic) functions fL(r)=a1r+b1 and fP(r)=a2r+b2, specifying the probability that a lob (and a pass, respectively) will win a point when the proportion of lobs played in the past trials is r. We measure the performance of a player in this model by his or her expected regret, namely how many fewer points the player expects to win as compared to the ideal player (one that knows the two probabilistic functions) as a function of t, the total number of trials, which is unknown to the player a priori. Assuming that the probabilistic functions satisfy the “matching shoulders condition,” i.e., fL(0)=fP(1), we obtain a variety of upper bounds for assumptions and restrictions of varying degrees, ranging from O(logt), O(t1/2), O(t3/5), O(t2/3) to O(t5/7) as well as a matching lower bound of order Ω(logt) for the first case. When the total number of trials t is given to the player in advance, the upper bounds can be improved significantly. An extended abstract describing part of this work has appeared in N. Abe and J. Takeuchi, 1993, in “Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory,” pp. 422–428
On the Difference of Horn Theories

AbstractIn this paper, we consider computing the difference between two Horn theories. This problem may arise, for example, if we take care of a theory change in a knowledge base. In general, the difference of Horn theories is not Horn. Therefore, we consider Horn approximations of the difference in terms of Horn cores (i.e., weakest Horn theories included in the difference) and the Horn envelope (i.e., the strongest Horn theory containing the difference), which have been proposed and analyzed extensively in the literature. We study the problem under the familiar representation of Horn theories by Horn CNFs, as well as under the recently proposed model-based representation in terms of the characteristic models. For all problems and representations, polynomial time algorithms or proofs of intractability for the propositional case are provided; thus, our work gives a complete picture of the tractability–intractability frontier in the propositional Horn theories
On the No-Z-Cycle Property in Distributed Executions

AbstractGiven a checkpoint and communication pattern of a distributed execution, the No Z-Cycle property (NZC) states that a dependency between a checkpoint and itself does not exist. In other words, a noncausal sequence of messages that starts after a checkpoint and terminates before that checkpoint does not exist. From an operational point of view, this property corresponds to the fact that each checkpoint belongs to at least one consistent global checkpoint. So it could be used, for example, for restarting a distributed application after the occurrence of a failure. In this paper we derive a characterization of the NZC property (previously an open problem). It identifies a subset of Z-cycles, namely core Z-cycles (CZCs), that has to be empty in order that the checkpoint and communication pattern of the execution satisfies the NZC property. Then, we present a communication-induced checkpointing protocol that prevents CZCs on-the-fly. This protocol actually removes the common causal part to any CZC. Finally we propose a taxonomy of communication-induced checkpointing protocols that ensure the NZC property
Agnostic Learning of Geometric Patterns

AbstractP. W. Goldberg, S. A. Goldman, and S. D. Scott (Mach. Learning25, No. 1 (1996), 51–70) discussed how the problem of recognizing a landmark from a one-dimensional visual image might be mapped to that of learning a one-dimensional geometric pattern and gave a PAC algorithm to learn that class. In this paper, we present an efficient online agnostic learning algorithm for learning the class of constant-dimensional geometric patterns. Our algorithm can tolerate both classification and attribute noise. By working in higher dimensional spaces we can represent more features from the visual image in the geometric pattern. Our mapping of the data to a geometric pattern and, hence, our learning algorithm are applicable to any data representable as a constant-dimensional array of values, e.g., sonar data, temporal difference information, amplitudes of a waveform, or other pattern recognition data. To our knowledge, these classes of patterns are more complex than any class of geometric patterns previously studied. Also, our results are easily adapted to learn the union of fixed-dimensional boxes from multiple-instance examples. Finally, our algorithms are tolerant of concept shift, where the target concept that labels the examples can change over time
A Lower Bound for Primality

AbstractRecent work by Bernasconi, Damm, and Shparlinski showed that the set of square-free numbers is not in AC0 and raised as an open question whether similar (or stronger) lower bounds could be proved for the set of prime numbers. We show that the Boolean majority function is AC0-Turing reducible to the set of prime numbers (represented in binary). From known lower bounds on Maj (due to Razborov and Smolensky) we conclude that primality cannot be tested in AC0[p] for any prime p. Similar results are obtained for the set of square-free numbers and for the problem of computing the greatest common divisor of two numbers
Linear Gaps between Degrees for the Polynomial Calculus Modulo Distinct Primes

AbstractThis paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the modp counting principles, p⩾2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds for the polynomial calculus for k-CNF formulas. As a consequence, it follows that the Gröbner basis algorithm used as a heuristic for k-SAT, requires exponential time in the worst case. Moreover, our lower bounds distinguish linearly between proofs over fields of characteristic q and r, q≠r, and more generally distinguish linearly the rings Zq and Zr where q and r do not have the identical prime factors
On the Complexity of k-SAT

AbstractThe k-SAT problem is to determine if a given k-CNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve k-SAT for k⩾3. Here exponential time means 2δn for some δ>0. In this paper, assuming that, for k⩾3, k-SAT requires exponential time complexity, we show that the complexity of k-SAT increases as k increases. More precisely, for k⩾3, define sk=inf{δ:there exists 2δn algorithm for solving k-SAT}. Define ETH (Exponential-Time Hypothesis) for k-SAT as follows: for k⩾3, sk>0. In this paper, we show that sk is increasing infinitely often assuming ETH for k-SAT. Let s∞ be the limit of sk. We will in fact show that sk⩽(1−d/k)s∞ for some constant d>0. We prove this result by bringing together the ideas of critical clauses and the Sparsification Lemma to reduce the satisfiability of a k-CNF to the satisfiability of a disjunction of 2εnk′-CNFs in fewer variables for some k′⩾k and arbitrarily small ε>0. We also show that such a disjunction can be computed in time 2εn for arbitrarily small ε>0
Computational Arithmetic Geometry I. Sentences Nearly in the Polynomial Hierarchy

AbstractWe consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I.Given a polynomial f∈Z[v, x, y], decide the sentence ∃v∀x∃yf(v, x, y)=?0, with all three quantifiers ranging over N (or Z). II.Given polynomials f1, …, fm∈Z[x1, …, xn] with m⩾n, decide if there is a rational solution to f1=…=fm=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PNPNP, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems, which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of cryptosystems
Graph Ramsey Theory and the Polynomial Hierarchy

AbstractIn the Ramsey theory of graphs F→(G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H. Arrowing, the problem of deciding whether F→(G, H), lies in Πp2=coNPNP and it was shown to be coNP-hard by Burr [Bur90]. We prove that Arrowing is Πp2-complete, simultaneously settling a conjecture of Burr and providing a rare natural example of a problem complete for a higher level of the polynomial hierarchy. We also show that Strong Arrowing, the induced subgraph version of Arrowing, is Πp2-complete, and that the complexity of not arrowing stars is the same as that of finding a perfect matching
The Communication Complexity of Pointer Chasing

AbstractWe study the k-round two-party communication complexity of the pointer chasing problem for fixed k. C. Damm, S. Jukna and J. Sgall (1998, Comput. Complexity7, 109–127) showed an upper bound of O(nlog(k−1)n) for this problem. We prove a matching lower bound; this improves the lower bound of Σ(n) shown by N. Nisan and A. Widgerson (1993, SIAM J. Comput.22, 211–219), and yields a corresponding improvement in the hierarchy results derived by them and by H. Klauck (1998, in “Proceeding of the Thirteenth Annual IEEE Conference on Computational Complexity,” pp. 141–152) for bounded-depth monotone circuits. We consider the bit version of this problem, and show upper and lower bounds. This implies that there is an abrupt jump in complexity, from linear to superlinear, when the number of rounds is reduced to k/2 or less. We also consider the s-paths version (originally studied by H. Klauck) and show an upper bound. The lower bounds are based on arguments using entropy. One of the main contributions of this work is a transfer lemma for distributions with high entropy; this should be of independent interest
Quantum Simulations of Classical Random Walks and Undirected Graph Connectivity

AbstractWhile it is straightforward to simulate a very general class of random processes space-efficiently by non-unitary quantum computations (e.g., quantum computations that allow intermediate measurements to occur), it is not currently known to what extent restricting quantum computations to be unitary affects the space required for such simulations. This paper presents a method by which a limited class of random processes—random walks on undirected graphs—can be simulated by unitary quantum computations in a space-efficient (and time-efficient) manner. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in a quantum analogue of randomized logspace that disallows intermediate measurements
Asymptotically Optimal Bounds for OBDDs and the Solution of Some Basic OBDD Problems

AbstractOrdered binary decision diagrams (OBDDs) are nowadays the most common dynamic data structure or representation type for Boolean functions. Among the many areas of application are verification, model checking, and computer aided design. For many functions it is easy to estimate the OBDD size but asymptotically optimal bounds are only known in simple situations. In this paper, methods for proving asymptotically optimal bounds are presented and applied to the solution of some basic problems concerning OBDDs. The largest size increase by a synthesis step of π-OBDDs followed by an optimal reordering is determined as well as the largest ratio of the size of deterministic finite automata, quasi-reduced OBDDs, and zero-suppressed BDDs compared to the size of OBDDs. Moreover, the worst case OBDD size of functions with a given number of 1-inputs is investigated
On ACTL Formulas Having Linear Counterexamples

AbstractIn case an ACTL formula φ fails over a transition graph M, it is most useful to provide a counterexample, i.e., a computation tree of M, witnessing the failure. If there exists a single path in M which by itself witnesses the failure of φ, then φ has a linear counterexample. We show that, given M and φ, where M⊭φ, it is NP-hard to determine whether there exists a linear counterexample. Moreover, it is PSPACE-hard to decide whether an ACTL formula φ always admits a linear counterexample if it fails. This means that there exists no simple characterization of the ACTL formulas that guarantee linear counterexamples. Consequently, we study templates of ACTL formulas, i.e., skeletons of modal formulas whose atoms are disregarded. We identify the (unique) maximal set LIN of templates whose instances (obtained by replacing atoms with arbitrary pure state formulas) always guarantee linear counterexamples. We show that for each ACTL formula φ which is an instance of a template γ★∈LIN, and for each Kripke structure M such that M⊭φ, a single path of M witnessing the failure by itself can be computed in polynomial time
Synthesizing Learners Tolerating Computable Noisy Data

AbstractAn index for an r.e. class of languages (by definition) generates a sequence of grammars defining the class. An index for an indexed family of recursive languages (by definition) generates a sequence of decision procedures defining the family. F. Stephan's model of noisy data is employed, in which, roughly, correct data crops up infinitely often and incorrect data only finitely often. In a computable universe, all data sequences, even noisy ones, are computable. New to the present paper is the restriction that noisy data sequences be, nonetheless, computable. This restriction is interesting since we may live in a computable universe. Studied, then, is the synthesis from indices for r.e. classes and for indexed families of recursive languages of various kinds of noise-tolerant language-learners for the corresponding classes or families indexed, where the noisy input data sequences are restricted to being computable. Many positive results, as well as some negative results, are presented regarding the existence of such synthesizers. The main positive result is: grammars for each indexed family can be learned behaviorally correctly from computable, noisy, positive data. The proof of another positive synthesis result yields, as a pleasant corollary, a strict subset-principle or telltale style characterization, for the computable noise-tolerant behaviorally correct learnability of grammars from positive and negative data, of the corresponding families indexed
The Stability of Saturated Linear Dynamical Systems Is Undecidable

AbstractWe prove that several global properties (global convergence, global asymptotic stability, mortality, and nilpotence) of particular classes of discrete time dynamical systems are undecidable. Such results had been known only for point-to-point properties. We prove these properties undecidable for saturated linear dynamical systems, and for continuous piecewise affine dynamical systems in dimension 3. We also describe some consequences of our results on the possible dynamics of such systems
Parallel Output-Sensitive Algorithms for Combinatorial and Linear Algebra Problems

AbstractThis paper gives output-sensitive parallel algorithms whose performance depends on the output size and are significantly more efficient tan previous algorithms for problems with sufficiently small output size. Inputs are n×n matrices over a fixed ground field. Let P(n) and M(n) be the PRAM processor bounds for O(logn) time multiplication of two degree n polynomials, and n×n matrices, respectively. Let T(n) be the time bounds, using M(n) processors, for testing if an n×n matrix is nonsingular, and if so, computing its inverse. We compute the rankR of a matrix in randomized parallel time O(logn+T(R)logR) using nP(n)+M(R) processors (P(n)+RP(R) processors for constant displacement rank matrices, e.g., Toeplitz matrices). We find a maximum linearly independent subset (MLIS) of an n-set of n-dimensional vectors in time O(T(n)logn) using M(n) randomized processors and we also give output-sensitive algorithms for this problem. Applications include output-sensitive algorithms for finding: (i) a size R maximum matching in an n-vertex graph using time O(T(R)logn) and nP(n)/T(R)+RM(R) processors, and (ii) a maximum matching in an n-vertex bipartite graph, with vertex subsets of sizes n1⩽n2, using time O(T(n1)logn) and nP(n)/T(n1)+n1M(n1) processors
The Descriptive Complexity Approach to LOGCFL

AbstractBuilding upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers in first-order logic with linear order. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of arithmetical predicates for plus and times (equivalently, in the absence of the BIT predicate), we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's hardest context-free language is LOGCFL-complete under quantifier-free reductions without arithmetic. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with predicates for plus and times than without. Considering a particular groupoidal quantifier, we prove that first-order logic with the “majority of pairs” quantifier is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers
Verification by Augmented Abstraction: The Automata–Theoretic View

AbstractThis paper deals with the proof method of verification by finitary abstraction (vfa), which presents an alternative approach to the verification of (potentially infinite-state) reactive systems. We assume that the negation of the property to be verified is given by the user in the form of an infinite-state nondeterministic Büchi discrete system (bds). The method consists of a two-step process by which, in a first step, the system and its (negated) specification are combined into a single infinite-state fair discrete system (fds, which is similar to a bds but with Streett acceptance conditions), which is abstracted into a finite-state automaton. The second step uses model checking to establish that the abstracted automaton is infeasible, i.e., has no computations. The vfa method can be considered as a viable alternative to verification by temporal deduction, which, up to now, has been the main method generally applicable for verification of infinite-state systems. The paper presents a general recipe for an fds abstraction, which is shown to be sound, where soundness means that infeasibility of the abstracted fds implies infeasibility of the unabstracted one, implying in turn the validity of the property over the concrete (infinite-state) system. To make the method applicable for the verification of liveness properties, pure abstraction is sometimes no longer adequate. We show that by augmenting the system with an appropriate (and standardly constructible) progress monitor, we obtain an augmented system, whose computations are essentially the same as those of the original system and which may now be abstracted while preserving the desired liveness properties. We refer to the extended method as verification by augmented abstraction (vaa). We then proceed to show that the vaa method is sound and complete for proving all properties whose negations are expressible by a bds. Given that every linear temporal logic (ltl) property can be translated to a bds, this establishes that the vaa method is sound and complete for proving the validity of all ltl properties, including both safety and liveness