pinecone/core-2020-05-10-deduplication huggingface dataset:
Proper Learning Algorithm for Functions of k Terms under Smooth Distributions

AbstractIn this paper, we introduce a probabilistic distribution, called a smooth distribution, which is a generalization of variants of the uniform distribution such as q-bounded distribution and product distribution. Then, we give an algorithm that, under the smooth distribution, properly learns the class of functions of k terms given as Fk∘Tkn={g(f1(v), …, fk(v))|g∈Fk, f1, …, fk∈Tn} in polynomial time for constant k, where Fk is the class of all Boolean functions of k variables and Tn is the class of terms over n variables. Although class Fk∘Tkn was shown by Blum and Singh to be learned using DNF as the hypothesis class, it has remained open whether it is properly learnable under a distribution-free setting
Automata, Boolean Matrices, and Ultimate Periodicity

AbstractThis paper presents a Boolean-matrix-based method to automata theory, with an application to the study of regularity-preserving functions. A new characterization of such functions is derived in terms of the property of ultimate periodicity with respect to powers of Boolean matrices. This characterization reveals the intrinsic algebraic nature of regularity-preserving functions. It facilitates a concise proof of known, as well as previously unknown, properties of regularity-preserving functions, leading to the solution of the “subtraction problem,” left open by Kosaraju
A Note on Complexity Measures for Inductive Classes in Constructive Type Theory

AbstractIt is notoriously hard to express computational complexity properties of programs in programming logics based on a semantics which respects extensional function equality. That is a serious impediment to applications of programming logics requiring reasoning about complexity. This paper shows how to use existing mechanisms to define internal computational complexity measures in logics that support inductively defined types, dependent products, and functions. The method exploits a feature of inductive definitions in constructive type theory, namely that implicit proof codes are kept with the objects showing how they are presented in the inductive class. The idea is illustrated by giving a formal inductive definition ofPTimebased on ideas from Leivant's work and on Bellantoni and Cook's approach. Then a complexity measure is defined on elements of this class. This paper discusses the limitations of this idea and the need forfaithfulnessguarantees that link internal complexity classes to the implementation of the logic. The paper concludes with a definition ofresource bounded logicsand a discussion of interesting lines of investigation of these logics which have the potential to make practical uses of results from computational complexity theory in formal reasoning about the efficiency of programs
Decision Trees: Old and New Results

AbstractIn this paper, we prove two general lower bounds for algebraic decision trees which test membership in a set S⊆Rn which is defined by linear inequalities. Let rank(S) be the maximal dimension of a linear sub- space contained in the closure of S (in Euclidean topology). First we show that any decision tree for S which uses products of linear functions (we call such functions mlf-functions) must have depth at least n−rank(S). This solves an open question raised by A. C. Yao and can be used to show that mlf-functions are not really more powerful than simple comparisons between the input variables when computing the largest k out of n elements. Yao proved this result in the special case when products of at most two linear functions are allowed. Our proof also shows that any decision tree for this problem must have exponential size. Using the same methods, we can give an alternative proof of Rabin's theorem, namely that the depth of any decision tree for S using arbitrary analytic functions is at least n−rank(S)
Finite Queries Do Not Have Effective Syntax

AbstractA relational query is called finite, or sometimes safe, iff it yields a finite answer in every database state. The set of finite queries of relational calculus is known to be unsolvable. However, in many cases it is possible to impose syntactical restrictions on the class of queries that guarantee finiteness and do not reduce the expressive power of the calculus. We show that unfortunately this is not always the case, as we construct a recursive domain with decidable theory where any solvable (or enumerable, for that matter) subclass of queries either contains an infinite query, or misses a finite one. We show that, although any domain can always be extended to a domain with an effective syntax for finite queries, such extensions of our domain necessarily have undecidable theories. The reason decidability of domain matters is that this property is, in effect, equivalent to the ability to answer queries effectively. Using the same example, we further show undecidability of the problem of relative finiteness, which is, given a query and a database state, to decide upon finiteness of the query in this state. This settles two long-standing open problems in the theory of relational databases
The Power of Multiobjects

AbstractWe consider shared memory systems that support multiobject operations in which processes may simultaneously access several objects in one atomic operation. We provide upper and lower bounds on the synchronization power (consensus number) of multiobject systems as a function of the type and the number of objects that may be simultaneously accessed in one atomic operation. These bounds imply that known classifications of component objects fail to characterize the synchronization power of their combination. In particular, we show that in the context of multiobjects, fetch & add objects are less powerful than swap objects, which in turn are less powerful than queue objects. This stands in contrast to the fact that swap can be implemented from fetch & add. Herein we introduce a restricted notion of implementation, called direct implementation. We show that, if objects of type Y have a direct implementation from objects of type X, then Y-based multiobjects can also be implemented from X-based multiobjects. Using this observation, we derive results such as: there are no direct implementations of swap or queue objects from any collection of commutative objects (e.g., fetch & add, test & set)
Recursive Computational Depth

AbstractIn the 1980s, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identified the classes of weakly deep and strongly deep sequences and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by defining the class of weakly useful sequences and proving that every weakly useful sequence is strongly deep. The present paper investigates refinements of Bennett's notions of weak and strong depth, called recursively weak depth (introduced by Fenner, Lutz, and Mayordomo) and recursively strong depth (introduced here). It is argued that these refinements naturally capture Bennett's idea that deep objects are those which “contain internal evidence of a nontrivial causal history.” The fundamental properties of recursive computational depth are developed, and it is shown that the recursively weakly (respectively, strongly) deep sequences form a proper subclass of the class of weakly (respectively, strongly) deep sequences. The above-mentioned theorem of Juedes, Lathrop, and Lutz is then strengthened by proving that every weakly useful sequence is recursively strongly deep. It follows from these results that not every strongly deep sequence is weakly useful, thereby answering a question posed by Juedes
Piecemeal Graph Exploration by a Mobile Robot

AbstractWe study how a mobile robot can learn an unknown environment in a piecemeal manner. The robot's goal is to learn a complete map of its environment, while satisfying the constraint that it must return every so often to its starting position (for refueling, say). The environment is modeled as an arbitrary, undirected graph, which is initially unknown to the robot. We assume that the robot can distinguish vertices and edges that it has already explored. We present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph G=(V, E) in which the robot explores every vertex and edge in the graph by traversing at most O(E+V1+o(1)) edges. This nearly linear algorithm improves on the best previous algorithm, in which the robot traverses at most O(E+V2) edges. We also give an application of piecemeal learning to the problem of searching a graph for a “treasure.
Alternation and Bounded Concurrency Are Reverse Equivalent

AbstractNumerous models of concurrency have been considered in the framework of automata. Among the more interesting concurrency models are classical nondeterminism and pure concurrency, the two facets of alternation, and the bounded concurrency model. Bounded concurrency was previously considered to be similar to nondeterminism and pure concurrency in the sense of the succinctness of automata augmented with these features. In this paper we show that, when viewed more broadly, the power (of succinctness) of bounded concurrency is in fact most similar to the power of alternation. Our contribution is that, just like nondeterminism and pure concurrency are “complement equivalent,” bounded concurrency and alternation are “reverse equivalent” over finite automata. The reverse equivalence is expressed by the existence of polynomial transformations, in both directions, between bounded concurrency and alternation for the reverse of the language accepted by the other. It follows, that bounded concurrency is double-exponentially more succinct than DFAs with respect to reverse, while alternation only saves one exponent. This is as opposed to the direct case where alternation saves two exponents and bounded concurrency saves only one. An immediate corollary is that for languages over a one-letter alphabet, bounded concurrency and alternation are equivalent. We complete the picture of succinctness results for these languages by considering the different combinations of the concurrency models using additional lower bounds
Logic with Equality: Partisan Corroboration and Shifted Pairing

AbstractHerbrand's theorem plays a fundamental role in automated theorem proving methods based on tableaux. The crucial step in procedures based on such methods can be described as the corroboration problem or the Herbrand skeleton problem, where, given a positive integer m, called multiplicity, and a quantifier free formula, one seeks a valid disjunction of m instantiations of that formula. In the presence of equality, which is the case in this paper, this problem was recently shown to be undecidable. The main contributions of this paper are two theorems. The first, the Partisan Corroboration Theorem, relates corroboration problems with different multiplicities. The second, the Shifted Pairing Theorem, is a finite tree automata formalization of a technique for proving undecidability results through direct encodings of valid Turing machine computations. These theorems are used in the paper to explain and sharpen several recent undecidability results related to the corroboration problem, the simultaneous rigid E-unification problem and the prenex fragment of intuitionistic logic with equality
The Kleene–Schützenberger Theorem for Formal Power Series in Partially Commuting Variables

AbstractKleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schützenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmański to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schützenberger's and Ochmański's theorems
Complexity and Expressive Power of Deterministic Semantics for DATALOG¬

AbstractDeterministic models are (partial) stable models which are not contradicted by any other stable model; i.e., M is a deterministic model if there is no stable model N such that M∪N is not an interpretation. For instanced the well-founded model, which coincides with the intersection of all partial stable models, is a deterministic model. As the well-founded model is deterministic and unique for each program, well-founded model semantics has been proposed as the canonical deterministic semantics for partial stable models. But the well-founded model is not the unique deterministic model; indeed the family of deterministic (partial stable) models is not, in general, a singleton and admits a minimum (the well-founded model) and a maximum, the max-deterministic model. The latter model is, another candidate for a deterministic semantics. The aim of this paper is to study the complexity and the expressive power of deterministic semantics. In coherence with the deterministic nature of the model, the expressive power of max-deterministic semantics is shown to be able to express problems with unique solutions, whereas the well-founded model only captures a proper subset of the queries computable in polynomial time, the so-called fixpoint queries
Basic Theory of F-Bounded Quantification

AbstractSystem F-bounded is a second-order typed lambda calculus, where the basic features of object-oriented languages can be naturally modelled. F-bounded extends the better known system F⩽, in a way that provides an immediate solution for the treatment of the so-called “binary methods.” Although more powerful than F⩽ and also quite natural, system F-bounded has only been superficially studied from a foundational perspective and many of its essential properties have been conjectured but never proved in the literature. The aim of this paper is to give a solid foundation to F-bounded, by addressing and proving the key properties of the system. In particular, transitivity elimination, completeness of the type checking semi-algorithm, the subject reduction property for βη reduction, conservativity with respect to system F⩽, and antisymmetry of a “full” subsystem are considered, and various possible formulations for system F-bounded are compared. Finally, a semantic interpretation of system F-bounded is presented, based on partial equivalence relations
The Complexity of Universal Text-Learners

AbstractThe present work deals with language learning from text. It considers universal learners for classes of languages in models of additional information and analyzes their complexity in terms of Turing degrees. The following is shown: If the additional information is given by a set containing at least one index for each language from the class to be learned but no index for any language outside the class, then there is a universal learner having the same Turing degree as the inclusion problem for recursively enumerable sets. This result is optimal in the sense that any other successful learner has the same or higher Turing degree. If the additional information is given by the index set of the class of languages to be learned then there is a computable universal learner. Furthermore, if the additional information is presented as an upper bound on the size of some grammar that generates the language, then a high oracle is necessary and sufficient. Finally, it is shown that for the concepts of finite learning and learning from good examples, the index set of the class to be learned gives insufficient information due to the restrictive convergence constraints, these criteria need the jump of the index set instead of the index set itself. So, they have infinite access to the information of the index set in finite time
Robust Behaviorally Correct Learning

AbstractIntuitively, a class of functions is robustly learnable if not only the class itself, but also all of the transformations of the class under natural transformations (such as via general recursive operators) are learnable. Fulk showed the existence of a nontrivial class which is robustly learnable under the criterion Ex. However, several of the hierarchies (such as the anomaly hierarchies for Ex and Bc) do not stand robustly. Fulk left open the question about whether Bc and Ex can be robustly separated. In this paper we resolve this question positively
On the Complexity of Learning for Spiking Neurons with Temporal Coding

AbstractSpiking neurons are models for the computational units in biological neural systems where information is considered to be encoded mainly in the temporal patterns of their activity. In a network of spiking neurons a new set of parameters becomes relevant which has no counterpart in traditional neural network models: the time that a pulse needs to travel through a connection between two neurons (also known as delay of a connection). It is known that these delays are tuned in biological neural systems through a variety of mechanisms. In this article we consider the arguably most simple model for a spiking neuron, which can also easily be implemented in pulsed VLSI. We investigate the Vapnik–Chervonenkis (VC) dimension of networks of spiking neurons, where the delays are viewed as programmable parameters and we prove tight bounds for this VC dimension. Thus, we get quantitative estimates for the diversity of functions that a network with fixed architecture can compute with different settings of its delays. In particular, it turns out that a network of spiking neurons with k adjustable delays is able to compute a much richer class of functions than a threshold circuit with k adjustable weights. The results also yield bounds for the number of training examples that an algorithm needs for tuning the delays of a network of spiking neurons. Results about the computational complexity of such algorithms are also given
Macro Tree Transducers, Attribute Grammars, and MSO Definable Tree Translations

AbstractA characterization is given of the class of tree translations definable in monadic second-order logic (MSO), in terms of macro tree transducers. The first main result is that the MSO definable tree translations are exactly those tree translations realized by macro tree transducers (MTTs) with regular look-ahead that are single use restricted. For this the single use restriction known from attribute grammars is generalized to MTTs. Since MTTs are closed under regular look-ahead, this implies that every MSO definable tree translation can be realized by an MTT. The second main result is that the class of MSO definable tree translations can also be obtained by restricting MTTs with regular look-ahead to be finite copying, i.e., to require that each input subtree is processed only a bounded number of times. The single use restriction is a rather strong, static restriction on the rules of an MTT, whereas the finite copying restriction is a more liberal, dynamic restriction on the derivations of an MTT
Testing Preorders for Probabilistic Processes

AbstractWe present a testing preorder for probabilistic processes based on a quantification of the probability with which processes pass tests. The theory enjoys close connections with the classical testing theory of De Nicola and Hennessy in that whenever a process passes a test with probability 1 (respectively some nonzero probability) in our setting, then the process must (respectively may) pass the test in the classical theory. We also develop an alternative characterization of the probabilistic testing preorders that takes the form of a mapping from probabilistic traces to the interval [0, 1], where a probabilistic trace is an alternating sequence of actions and probability distributions over actions. Finally, we give proof techniques, derived from the alternative characterizations, for establishing preorder relationships between probabilistic processes. The utility of these techniques is demonstrated by means of some simple examples
A Randomized Algorithm for Two Servers on the Line

AbstractIn the k-server problem we wish to minimize, in an online fashion, the movement cost of k servers in response to a sequence of requests. For two servers, it is known that the optimal deterministic algorithm has competitive ratio 2, and it has been a long-standing open problem whether it is possible to improve this ratio using randomization. We give a positive answer to this problem when the underlying metric space is a real line, by providing a randomized online algorithm for this case with competitive ratio at most 15578≈1.987. This is the first algorithm for two servers that achieves a competitive ratio smaller than 2 in a nonuniform metric space with more than three points. We consider a more general problem called the (k, l)-server problem, in which a request is served using l out of k available servers. We show that the randomized 2-server problem can be reduced to the deterministic (2l, l)-server problem. We prove a lower bound of 2 on the competitive ratio of the (4, 2)-server problem. This implies that one unbiased random bit is not sufficient to improve the ratio of 2 for the two-server problem. Then we give a 15578-competitive algorithm for the (6, 3)-server problem on the real line. Our algorithm is simple and memoryless. The solution has been obtained using linear programming techniques that may have applications for other online problems
Tally NP Sets and Easy Census Functions

AbstractWe study the question of whether every P set has an easy (i.e., polynomial-time computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P1⊆FP, where #P1 is the class of functions that count the witnesses for tally NP sets. We prove that every #P1PH function can be computed in FP#P1#P1. Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does. We show that the assumption #P1⊆FP implies P=BPP and PH⊆MODkP for each k⩾2. We also relate a set's property of having an easy census function to other well-studied properties of sets, such as rankability and scalability (the closure of the rankable sets under P-isomorphisms). Finally, we prove that it is no more likely that the census function of any set in P can be approximated (more precisely, can be nα-enumerated in time nβ for fixed α and β) than that it can be precisely computed in polynomial time
Optimal Algorithms for the Coverability, the Subword, the Containment, and the Equivalence Problems for Commutative Semigroups

AbstractIn this paper, we present decision procedures for the coverability, the subword, the containment, and the equivalence problems for commutative semigroups. These procedures require at most space 2c·n, where n is the size of the problem instance, and c is some problem independent constant. Furthermore, we show that the exponential space hardness of the above problems follows from the work of Mayr and Meyer. Thus, the presented algorithms are space optimal. Our results close the gap between the 2c′·n·logn space upper bound, shown by Rackoff for the coverability problem and shown by Huynh for the containment and the equivalence problems, and the exponential space lower bound resulting from the corresponding bound for the uniform word problem established by Mayr and Meyer