pinecone/core-2020-05-10-deduplication huggingface dataset:
Complete Geometric Query Languages

AbstractWe extend Chandra and Harel's seminal work on computable queries for relational databases to a setting in which also spatial data may be present, using a constraint-based data model. Concretely, we introduce both coordinate-based and point-based query languages that are complete in the sense that they can express precisely all computable queries that are generic with respect to certain classes of transformations of space, corresponding to certain geometric interpretations of spatial data. The languages we introduce are obtained by augmenting basic languages with a “while” construct. We also show that the respective basic point-based languages are complete, relative to the subclass of the corresponding generic queries consisting of those that are expressible in the relational calculus with real polynomial constraints
Constraint-Generating Dependencies

AbstractTraditionally, dependency theory has been developed for uninterpreted data. Specifically, the only assumption that is made about the data domains is that data values can be compared for equality. However, data is often interpreted and there can be advantages in considering data as such, for instance, obtaining more compact representations as is done in constraint databases. This paper considers dependency theory in the context of interpreted data. Specifically, it studies constraint-generating dependencies. These are a generalization of equality-generating dependencies where equality requirements are replaced by constraints on an interpreted domain. The main technical results in the paper are a general decision procedure for the implication and consistency problems for constraint-generating dependencies and complexity results for specific classes of such dependencies over given domains. The decision procedure proceeds by reducing the dependency problem to a decision problem for the constraint theory of interest and is applicable as soon as the underlying constraint theory is decidable. The complexity results are, in some cases, directly lifted from the constraint theory; in other cases, optimal complexity bounds are obtained by taking into account the specific form of the constraint decision problem obtained by reducing the dependency implication problem
Reasoning about Strings in Databases

AbstractIn order to enable the database programmer to reason about relations over strings of arbitrary length, we introduce alignment calculus, a modal extension of the relational calculus. In addition to relations, a state in the model consists of a two-dimensional array where the strings are aligned on top of each other. The basic modality in the language (a transpose, or “slide”) rearranges this alignment, and more complex formulae can be formed using a syntax reminiscent of regular expressions, in addition to the usual connectives and quantifiers. It turns out that the computational counterpart of the string-based portion of the logic is the class of multitape two-way finite state automata, which are devices particularly well suited for the implementation of string matching. A computational counterpart of the full logic is obtained from relational algebra by performing selection with these devices. Safety of formulae in alignment calculus implies that new strings generated from old ones have to be of bounded length. While an undecidable property in general, this boundedness is decidable for an important subclass of formulae. As far as expressive power is concerned, alignment calculus includes previous proposals for querying string databases and gives full Turing computability. The language can be restricted to define exactly the regular sets and the sets in each level of the polynomial-time hierarchy above P
Petri Net Languages and Infinite Subsets of Nm

AbstractFamilies of Petri net languages are usually defined by varying the type of transition labeling and the class of subsets of Nm to be used as sets of final markings (m is the number of places). So far three main classes of subsets have been studied: the trivial class containing as single element Nm, the class of finite subsets of Nm, and the class of ideals (or covering subsets) of Nm. In this paper we extend the known hierarchy of Petri net languages by considering the classes of semi-cylindrical, star-free, recognizable, rational (or semilinear) subsets of Nm. We compare the related Petri net languages. For arbitrarily labeled and for λ-free labeled Petri net languages, the above hierarchy collapses: one does not increase the generality by considering semilinear accepting sets instead of the usual finite ones. However, for free-labeled and for deterministic Petri net languages, we show that one gets new distinct subclasses of languages, for which several decidability problems become solvable. We establish as intermediate results some properties of star-free subsets of general monoids
Analysis of Two Gradient-Based Algorithms for On-Line Regression

AbstractIn this paper we present a new analysis of two algorithms, Gradient Descent and Exponentiated Gradient, for solving regression problems in the on-line framework. Both these algorithms compute a prediction that depends linearly on the current instance, and then update the coefficients of this linear combination according to the gradient of the loss function. However, the two algorithms have distinctive ways of using the gradient information for updating the coefficients. For each algorithm, we show general regression bounds for any convex loss function. Furthermore, we show special bounds for the absolute and the square loss functions, thus extending previous results by Kivinen and Warmuth. In the nonlinear regression case, we show general bounds for pairs of transfer and loss functions satisfying a certain condition. We apply this result to the Hellinger loss and the entropic loss in case of logistic regression (similar results, but only for the entropic loss, were also obtained by Helmbold et al. using a different analysis.) Finally, we describe the connection between our approach and a general family of gradient-based algorithms proposed by Warmuth et al. in recent works
Models of Nondeterministic Regular Expressions

AbstractNondeterminism is a direct outcome of interactions and is, therefore a central ingredient for modelling concurrent systems. Trees are very useful for modelling nondeterministic behaviour. We aim at a tree-based interpretation of regular expressions and study the effect of removing the idempotence law X+X=X and the distribution law X•(Y+Z)=X•Y+X•Z from Kleene algebras. We show that the free model of the new set of axioms is a class of trees labelled over A. We also equip regular expressions with a two-level behavioural semantics. The basic level is described in terms of a class of labelled transition systems that are detailed enough to describe the number of equal actions a system can perform from a given state. The abstract level is based on a so-called resource bisimulation preorder that permits ignoring uninteresting details of transition systems. The three proposed interpretations of regular expressions (algebraic, denotational, and behavioural) are proven to coincide. When dealing with infinite behaviours, we rely on a simple version of the ω-induction and obtain a complete proof system also for the full language of nondeterministic regular expressions
Robust Learning Aided by Context

AbstractEmpirical studies of multitask learning provide some evidence that the performance of a learning system on its intended targets improves by presenting to the learning system related tasks, also called contexts, as additional input. Angluin, Gasarch, and Smith, as well as Kinber, Smith, Velauthapillai, and Wiehagen, have provided mathematical justification for this phenomenon in the inductive inference framework. However, their proofs rely heavily on self-referential coding tricks; that is, they directly code the solution of the learning problem into the context. Fulk has shown that for the Ex- and Bc-anomaly hierarchies, such results, which rely on self-referential coding tricks, do not hold robustly. In this work we analyze robust versions of learning aided by context and show that—in contrast to Fulk's result above—context also aids learning robustly. Also studied is the difficulty of the functional dependence between the intended target tasks and useful associated contexts
Structural Results about Exact Learning with Unspecified Attribute Values

AbstractThis paper deals with the UAV learning model of Goldman, Kwek, and Scott, where “UAV” is the acronym for “unspecified attribute values.” As they, we consider exact earning within the UAV framework. A smooth transition between exact learning in the UAV setting and standard exact learning is obtained by putting a fixed bound r on the number of unspecified attribute values per instance. For r=0, we obtain the standard model. For r=n (the total number of attributes), we obtain the (unrestricted) UAV model. Between these extremes, we find the hierarchies (UAV-MQr)0⩽r⩽n, (UAV-EQr)0⩽r⩽n, and (UAV-ARB-EQr)0⩽r⩽n. Our main results are as follows. We present various lower bounds on the number of ARB-EQs and UAV-MQs in terms of the Vapnik–Chervonenkis dimension of the concept class. We show, furthermore, that a natural extension of Angluin's Sunflower Lemma is still applicable in the exact UAV learning model. Our UAV Sunflower Lemma allows the establishment of exponentially large lower bounds on the necessary number of UAV-MQs for several popular concept classes. On the other hand, we can show that slight simplifications of these classes are efficiently learnable using only a few UAV-MQs. Finally, we investigate the inherent structure of the aforementioned three hierarchies and the relations between them. It turns out that query type UAV-EQr−1 is strictly stronger than UAV-EQr (for each constant r). The analogous result for UAV-ARB-EQ is valid. Furthermore, UAV-MQr+ω(logn) is strictly stronger than UAV-MQr. We also determine the relation between query types chosen from different hierarchies
Replication and Consistency in a Distributed Environment

AbstractThe issue of data replication is considered in the context of a restricted system model motivated by certain distributed data-warehousing applications. A new replica management protocol is defined for this model in which global serializability is ensured, while message overhead and deadlock frequency are less than in previously published work. The advantages of the protocol arise from its use of a lazy approach to update secondary copies of replicated data and the use of a new concept, virtual sites, to reduce the potential for conflict among global transactions
Graph Isomorphism and Identification Matrices: Sequential Algorithms

AbstractA number of properties on identification matrices are presented here. For example, we prove that adjacency matrices are identification matrices for all bipartite graphs. We also study the application of the theory of identification matrices to solving the graph isomorphism problem, a famous open problem. We show that, given two graphs represented by two identification matrices with respect to a certain relation, isomorphism can be decided efficiently if at least one matrix satisfies the consecutive 1's property or a relaxed property thereof. Graphs which have identification matrices satisfying the consecutive 1's property include, among others, proper interval graphs and doubly convex bipartite graphs. This work leads to the first efficient isomorphism testing algorithms for certain classes of graphs and more efficient algorithms for some other classes of graphs. The algorithms for some classes of graphs including convex bipartite graphs run in linear time and are optimal
Petri Nets and Regular Processes

AbstractWe consider the following problems: (a) Given a labelled Petri net and a finite automaton, are they equivalent?; (b) Given a labelled Petri net, is it equivalent to some (unspecified) finite automaton? These questions are studied within the framework of trace and bisimulation equivalences, in both their strong and weak versions. (In the weak version a special τ action—likened to an ε-move in automata theory—is considered to be nonobservable.) We demonstrate that (a) is decidable for strong and weak trace equivalence and for strong bisimulation equivalence, but undecidable for weak bisimulation equivalence. On the other hand, we show that (b) is decidable for strong bisimulation equivalence, and undecidable for strong and weak trace equivalence, as well as for weak bisimulation equivalence
Sample Complexity of Model-Based Search

AbstractWe consider the problem of searching a domain for points that have a desired property, in the special case where the objective function that determines the properties of points is unknown and must be learned during search. We give a parallel to PAC learning theory that is appropriate for reasoning about the sample complexity of this problem. The learner queries the true objective function at selected points, and uses this information to choose models of the objective function from a given hypothesis class that is known to contain a correct model. These models are used to focus the search on more promising areas of the domain. The goal is to find a point with the desired property in a small number of queries. We define an analog to VC dimension, needle dimension, to be the size of the largest sample in which any single point could have the desired property without the other points' values revealing this information. We give an upper bound on sample complexity that is linear in needle dimension for a natural type of search protocol and a linear lower bound for a class of constrained problems. We also describe the relationship between needle dimension and VC dimension, explore connections between model-based search and active concept learning (including several novel positive results in active learning), and consider a scale-sensitive version of needle dimension. Several simple examples illustrate the dependence of needle dimension on features of search problems
Isolation, Matching, and Counting Uniform and Nonuniform Upper Bounds

AbstractWe show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting the number of accepting paths of a nondeterministic logspace machine can be done in NL/poly, if the number of paths is small. This clarifies the complexity of the class FewL. Using derandomization techniques, we then improve this to show that this counting problem is in NL. Determining if our other theorems hold in the uniform setting remains an important open question, although we provide evidence that they do. More precisely, if there are problems in DSPACE(n) requiring exponential-size circuits, then all of our results hold in the uniform setting
On Arithmetic Branching Programs

AbstractThe model of arithmetic branching programs is an algebraic model of computation generalizing the model of modular branching programs. We show that, up to a polynomial factor in size, arithmetic branching programs are equivalent to complements of dependency programs, a model introduced by Pudlák and Sgall. Using this equivalence we prove that dependency programs are closed under conjunction over every field, answering an open problem of theirs. Furthermore, we show that span programs, an algebraic model of computation introduced by Karchmer and Wigderson, are at least as strong as arithmetic programs; every arithmetic program can be simulated by a span program of size not more than twice the size of the arithmetic program. Using the above results we give a new proof that NL/poly⊆⊕L/poly, first proved by Wigderson [25]. Our simulation of NL/poly is more efficient, and it holds for logspace counting classes over every field
Hard Sets Are Hard to Find

AbstractWe investigate the frequency of complete sets for various complexity classes within EXP under several polynomial-time reductions in the sense of resource-bounded measure. We show that these sets are scarce: The sets that are ⩽pnα−tt-complete for NP, the levels of the polynomial-time hierarchy, and PSPACE have p2-measure zero for any constant α<1; The ⩽pnc−T-complete sets for EXP have p2-measure zero for any constant c; Assuming MA≠EXP, the ⩽ptt-complete sets for EXP have p-measure zero. A key ingredient is the Small Span Theorem, which states that for any set A in EXP at least one of its lower span (i.e., the sets that reduce to A) or its upper span (i.e., the sets that A reduces to) has p2-measure zero. Previous to our work, the best published theorem along these lines held for ⩽pbtt-reductions. We establish it for ⩽pno(1)-tt-reductions
Complexity Limitations on Quantum Computation

AbstractWe use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP:BQP is low for PP, i.e., PPBQP=PP; There exists a relativized, world, where P=BQP and the polynomial-time hierarchy is infinite; There exists a relativized world, where BQP does not have complete sets; There exists a relativized world, where P=BQP, but P≠UP∩coUP and one-way functions exist. This gives a relativized answer to an open question of Simon
Computational Indistinguishability: A Sample Hierarchy

AbstractWe consider the existence of pairs of probability ensembles that may be efficiently distinguished given k samples, but cannot be efficiently distinguished given k′<k samples. It is well known that in any such pair of ensembles it cannot be that both are efficiently computable (and that such phenomena cannot exist for nonuniform classes of distinguishers, say, polynomial-size circuits). It was also known that there exist pairs of ensembles that may be efficiently distinguished based on two samples, but cannot be efficiently distinguished, based on a single sample. In contrast, it was not known whether the distinguishing power increases when one moves from two samples to polynomially-many samples. We show the existence of pairs of ensembles which may be efficiently distinguished given k+1 samples but cannot be efficiently distinguished given k samples, where k can be any function bounded above by a polynomial in the security parameter. In the course of establishing the above result, we prove several technical lemmas regarding polynomials and graphs. We believe that these may be of independent interest
On Membership Comparable Sets

AbstractA set A is k(n) membership comparable if there is a polynomial-time computable function that, given k(n) instances of A of length at most n, excludes one of the 2k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(logn) membership comparable, then the NP-hard promise problem UniqueSAT can be solved in polynomial time. Our result settles an open question, suggested by Buhrman, Fortnow, and Torenvliet, and extends the work of Ogihara, Beigel, Kummer, Stephan, and Agrawal and Arvind. These authors showed that if SAT is clogn membership comparable for c<1, then NP=P, and that if SAT is O(logn) membership comparable, then UniqueSAT∈DTIME[2log2n]. Our proof also shows that if SAT is o(n) membership comparable, then UniqueSAT can be solved in deterministic time 2o(n). Our main technical tool is an algorithm of Madhu Sudan (building on the work of Ar, Lipton, Rubinfeld, and Sudan) to reconstruct polynomials from noisy data through the use of bivariate polynomial factorization
Arthur–Merlin Games in Boolean Decision Trees

AbstractIt is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N. Nisan, SIAM J. Comput.20, No. 6 (1991), 999–1007). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree, we address structural properties of Arthur–Merlin games in this model and prove some lower bounds. We consider two cases of interest, the first when the length of communication between the players is limited and the second, if it is not. While in the first case we can carry over the relations between the corresponding Turing complexity classes, in the second case we observe in contrast with Turing complexity that a one-round Merlin–Arthur protocol is as powerful as a general interactive proof system and, in particular, can simulate a one-round Arthur–Merlin protocol. Moreover, we show that sometimes a Merlin–Arthur protocol can be more efficient than an Arthur–Merlin protocol and than a Merlin–Arthur protocol with limited communication. This is the case for a boolean function whose set of zeroes is a code with high minimum distance and a natural uniformity condition. Such functions provide an example when the Merlin–Arthur complexity is 1 with one-sided error ε∈(23, 1), but at the same time the nondeterministic decision tree complexity is Ω(n). The latter should be contrasted with another fact we prove. Namely, if a function has Merlin–Arthur complexity 1 with one-sided error probability ε∈(0, 23], then its nondeterministic complexity is bounded by a constant. Other results of the paper include connections with the block sensitivity and related combinatorial properties of a boolean function
Space-Bounded Quantum Complexity

AbstractThis paper investigates the computational power of space-bounded quantum Turing machines. The following facts are proved for space-constructible space bounds s satisfying s(n)=Ω(logn): 1.Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Turing machine (PTM) run- ning in space O(s). No assumptions on the probability of error or running time for the QTM are required, although it is assumed that all transition amplitudes of the QTM are rational. 2.Any PTM that runs in space s and halts absolutely (i.e., has finite worst-case running time) can be simulated by a QTM running in space O(s). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the case of unbounded error, the QTM may be taken to halt absolutely. We therefore have that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power and, furthermore, that any QTM running in space s can be simulated deterministically in NC2(2s)⊆DSPACE(s2)∩DTIME(2O(s)). We also consider quantum analogues of nondeterministic and one-sided error probabilistic space-bounded classes and prove some simple facts regarding these classes
Testing Problems with Sublearning Sample Complexity

AbstractWe study the problem of determining, for a class of functions H, whether an unknown target function f is contained in H or is “far” from any function in H. Thus, in contrast to problems of learning, where we must construct a good approximation to f in H on the basis of sample data, in problems of testing we are only required to determine the existence of a good approximation. Our main results demonstrate that, over the domain [0, 1]d for constant d, the number of examples required for testing grows only as O(s1/2+δ) (where δ is any small constant), for both decision trees of size s and a special class of neural networks with s hidden units. This is in contrast to the Ω(s) examples required for learning these same classes. Our tests are based on combinatorial constructions demonstrating that these classes can be approximated by small classes of coarse partitions of space and rely on repeated application of the well-known birthday paradox