pinecone/core-2020-05-10-deduplication huggingface dataset:
Scalable Secure Storage When Half the System Is Faulty

AbstractIn this paper, we provide a method to safely store a document in perhaps the most challenging settings, a highly decentralized replicated storage system where up to half of the storage servers may incur arbitrary failures, including alterations to data stored in them. Using an error correcting code (ECC), e.g., a Reed–Solomon code, one can take n pieces of a document, replace each piece with another piece of size larger by a factor of nn−2t+1 such that it is possible to recover the original set even when up to t of the larger pieces are altered. For t close to n/2 the space blowup factor of this scheme is close to n, and the overhead of an ECC such as the Reed–Solomon code degenerates to that of a trivial replication code. We show a technique to reduce this large space overhead for high values of t. Our scheme blows up each piece by a factor slightly larger than two using an erasure code which makes it possible to recover the original set using n/2−O(n/d) of the pieces, where d≈80 is a fixed constant. Then we attach to each piece O(d log n/log d) additional bits to make it possible to identify a large enough set of unmodified pieces, with negligible error probability, assuming that at least half the pieces are unmodified and with low complexity. For values of t close to n/2 we achieve a large asymptotic space reduction over the best possible space blowup of any ECC in deterministic setting. Our approach makes use of a d-regular expander graph to compute the bits required for the identification of n/2−O(n/d) good pieces
Degrees of Dowd-Type Generic Oracles

AbstractFor each positive integer r, M. Dowd (1992, Inform. and Comput.96, 65–76) introduced r-generic oracles (we call them r-Dowd oracles; they are different from n-genericity of arithmetical forcing). An oracle D is r-Dowd if every r-query tautology with respect to D is forced by a polynomial-sized portion of D. We propose the study of degrees and complexity of 1-Dowd oracles. Dowd also stated that no r-Dowd oracle is recursively enumerable. However, this is false. We show, among others, the following. There exists a primitive recursive 1-Dowd oracle; For every oracle A, there exists a 1-Dowd oracle D that is Turing-equivalent to A; For every 1-Dowd oracle D, there exists a 1-Dowd oracle E such that E is polynomial time many-one-equivalent to D and E is not 2-Dowd. Problems are formulated, and analogy between the jump-operator and the operation of taking the set of 1-query tautologies is discussed
The Star Problem and the Finite Power Property in Trace Monoids: Reductions beyond C4

AbstractWe deal with the star problem in trace monoids which means to decide whether the iteration of a recognizable trace language is recognizable. We consider trace monoids Kn={a1, b1}*×···×{an, bn}*. Our main result asserts that the star problem is decidable in some trace monoid M iff it is decidable in the biggest Kn submonoid M. Consequently, future research on the star problem can focus on the trace monoids Kn. We develop the main results of the paper for the finite power problem. Then, we establish the link to the star problem by applying the recently shown decidability equivalence between the star problem and the finite power problem (D. Kirsten and G. Richomme, 2001, Theory Comput. Systems34, 193–227)
Conservation and Uniform Normalization in Lambda Calculi with Erasing Reductions

AbstractFor a notion of reduction in a λ-calculus one can ask whether a term satisfies conservation and uniform normalization. Conservation means that single-step reductions of the term preserve infinite reduction paths from the term. Uniform normalization means that either the term will have no reduction paths leading to a normal form or all reduction paths will lead to a normal form. In the classical conservation theorem for ∧I the distinction between the two notions is not clear: uniform normalization implies conservation, and conservation also implies uniform normalization. The reason for this is that ∧I is closed under reduction, due to the fact that reductions never erase terms in ∧I. More generally for nonerasing reductions, the two notions are equivalent on a set closed under reduction. However, when turning to erasing reductions the distinction becomes important as conservation no longer implies uniform normalization. This paper presents a new technique for finding uniformly normalizing subsets of a λ-calculus. This is done by combining a syntactic and a semantic criterion. The technique is demonstrated by several applications. The technique is used to present a new uniformly normalizing subset of the pure λ-calculus; this subset is a superset of ∧I and thus contains erasing K-redexes. The technique is also used to prove strong normalization from weak normalization of the simply typed λ-calculus extended with pairs; this is an extension of techniques developed recently by Sørensen and Xi. Before presenting the technique the paper presents a simple proof of a slightly weaker form of the characterization of perpetual redexes by Bergstra and Klop; this is a step for the later applications of the technique
Test Sets for the Universal and Existential Closure of Regular Tree Languages

AbstractFinite test sets are a useful tool for deciding the membership problem for the universal closure of a given tree language, that is, for deciding whether a term has all its ground instances in the given language. A uniform test set for the universal closure must serve the following purpose: In order to decide membership of a term, it is sufficient to check whether all its test set instances belong to the underlying language. A possible application, and our main motivation, is ground reducibility, an essential concept for many approaches to inductive reasoning. Ground reducibility modulo some rewrite system is membership in the universal closure of the set of reducible ground terms. Here, test sets always exist, and several algorithmic approaches are known. The resulting sets, however, are often unnecessarily large. In this paper we consider regular languages and linear closure operators. We prove that universal as well as existential closure, defined analogously, preserve regularity. By relating test sets to tree automata and to appropriate congruence relations, we show how to characterize, how to compute, and how to minimize ground and non-ground test sets. In particular, optimal solutions now replace previous ad hoc approximations for the ground reducibility problem
Decidability for Left-Linear Growing Term Rewriting Systems

AbstractA term rewriting system is called growing if each variable occurring on both the left-hand side and the right-hand side of a rewrite rule occurs at depth zero or one in the left-hand side. Jacquemard showed that the reachability and the sequentiality of linear (i.e., left-right-linear) growing term rewriting systems are decidable. In this paper we show that Jacquemard's result can be extended to left-linear growing rewriting systems that may have right-nonlinear rewrite rules. This implies that the reachability and the joinability of some class of right-linear term rewriting systems are decidable, which improves the results for right-ground term rewriting systems by Oyamaguchi. Our result extends the class of left-linear term rewriting systems having a decidable call-by-need normalizing strategy. Moreover, we prove that the termination property is decidable for almost orthogonal growing term rewriting systems
A Fully Syntactic AC-RPO

AbstractWe present the first fully syntactic (i.e., non-interpretation-based) AC-compatible recursive path ordering (RPO). It is simple, and hence easy to implement, and its behaviour is intuitive as in the standard RPO. The ordering is AC-total and defined uniformly for both ground and nonground terms, as well as for partial precedences. More important, it is the first one that can deal incrementally with partial precedences, an aspect that is essential, together with its intuitive behaviour, for interactive applications such as Knuth–Bendix completion
Ordered SOS Process Languages for Branching and Eager Bisimulations

AbstractWe present a general and uniform method for defining structural operational semantics (SOS) of process operators by traditional Plotkin-style transition rules equipped with orderings. This new feature allows one to control the order of application of rules when deriving transitions of process terms. Our method is powerful enough to deal with rules with negative premises and copying. We show that rules with orderings, called ordered SOS rules, have the same expressive power as GSOS rules. We identify several classes of process languages with operators defined by rules with and without orderings in the setting with silent actions and divergence. We prove that branching bisimulation and eager bisimulation relations are preserved by all operators in process languages in the relevant classes
Learning Closed Horn Expressions

AbstractThe paper studies the learnability of Horn expressions within the framework of learning from entailment, where the goal is to exactly identify some pre-fixed and unknown expression by making queries to membership and equivalence oracles. It is shown that a class that includes both range restricted Horn expressions (where terms in the conclusion also appear in the condition of a Horn clause) and constrained Horn expressions (where terms in the condition also appear in the conclusion of a Horn clause) is learnable. This extends previous results by showing that a larger class is learnable with better complexity bounds. A further improvement in the number of queries is obtained when considering the class of Horn expressions with inequalities on all syntactically distinct terms
On Complexity of Easy Predictable Sequences

AbstractPredictive complexity is a generalization of Kolmogorov complexity. It corresponds to an “optimal” prediction strategy and gives a natural lower bound to ability of any algorithm to predict elements of a sequence of outcomes. A natural question is studied: how complex can easy-to-predict sequences be? The standard measure of complexity, used in the paper, is Kolmogorov complexity K (which is close to predictive complexity for logarithmic loss function). The difficulty of prediction is measured by the notion of predictive complexity KG for bounded loss function (of nonlogarithmic type). We present an asymptotic relation supx:l(x)=nK(x|n)KG(x)∼1alog n, when n→∞, where a is a constant and l(x) is the length of a sequence x. An analogous asymptotic relation holds for relative complexities K(x|n)/n and KG(x)/n, where n=l(x). To obtain these results we present lower and upper bounds of the cardinality of all sequences of given predictive complexity
Inheritance-Based Subtyping

AbstractClasses play a dual role in mainstream statically typed object-oriented languages, serving as both object generators and object types. In such languages, inheritance implies subtyping. In contrast, the theoretical language community has viewed this linkage as a mistake and has focused on subtyping relationships determined by the structure of object types, without regard to their underlying implementations. In this paper, we explore why inheritance-based subtyping relations are useful, and we describe two different approaches to extending the MOBY programming language with inheritance-based subtyping relations. In addition, we present a typed object calculus that supports both structural and inheritance-based subtyping, and which provides a formal accounting of our extensions to MOBY
A Type System for Lock-Free Processes

AbstractAdvanced type systems for the π-calculus have recently been proposed to guarantee deadlock-freedom in the sense that certain communications will eventually succeed unless the whole process diverges. Although such guarantees are useful for reasoning about the behavior of concurrent programs, there still remains the weakness that the success of a communication is not completely guaranteed due to the possibility of divergence. For example, although a server process that has received a request message cannot discard the request, it is allowed to infinitely delegate the request to other processes, causing a livelock. In this paper, we present a type system which guarantees that certain communications will eventually succeed under fair scheduling, regardless of whether processes diverge. We also present a variant of the type system which guarantees that a communication will succeed within a given number of reduction steps
Mobile Objects as Mobile Processes

AbstractObliq is a lexically scoped, distributed, object-based programming language. In Obliq, the migration of an object is proposed as creating a clone of the object at the target site, whereafter the original object is turned into an alias for the clone. Obliq has only an informal semantics, so there is no proof that this style of migration is safe, i.e., transparent to object clients. In previous work, we introduced Ø, an abstraction of Obliq, where, by lexical scoping, sites have been abstracted away. We used Ø in order to exhibit how the semantics behind Obliq's implementation renders migration unsafe. We also suggested a modified semantics that we conjectured instead to be safe. In this paper, we rewrite our modified semantics of Ø in terms of the π-calculus, and we use it to formally prove the correctness of object surrogation, the abstraction of object migration in Ø
The Complexity of Solving Equations over Finite Groups

AbstractWe study the computational complexity of solving systems of equations over a finite group. An equation over a group G is an expression of the form w1·w2·…·wk=1G, where each wi is either a variable, an inverted variable, or a group constant and 1G is the identity element of G. A solution to such an equation is an assignment of the variables (to values in G) which realizes the equality. A system of equations is a collection of such equations; a solution is then an assignment which simultaneously realizes each equation. We show that the problem of determining if a (single) equation has a solution is NP-complete for all nonsolvable groups G. For nilpotent groups, this same problem is shown to be in P. The analogous problem for systems of such equations is shown to be NP-complete if G is non-Abelian, and in P otherwise. Finally, we observe some connections between these problems and the theory of nonuniform automata
On the Nonapproximability of Boolean Functions by OBDDs and Read-k-Times Branching Programs

AbstractBranching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e.g., verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f. This motivates the investigation of the size of the smallest branching program approximating f. Although several nonapproximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following nonapproximability results are presented.(1) It is proven that two simple and well-known functions from the branching program literature require exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications.(2) The first truly exponential lower bound on the size of approximating syntactic read-k-times branching programs with respect to the uniform distribution and error probability 1/2−2−Ω(n), n the input size, is shown. In order to improve upon the best previous results for error probabilities smaller than 1/3, a strong combinatorial lemma from a paper of Ajtai on linear-length branching programs is exploited
The Small Model Property: How Small Can It Be?

AbstractEfficient decision procedures for equality logic (quantifier-free predicate calculus+the equality sign) are of major importance when proving logical equivalence between systems. We introduce an efficient decision procedure for the theory of equality based on finite instantiations. The main idea is to analyze the structure of the formula and compute accordingly a small domain to each variable such that the formula is satisfiable iff it can be satisfied over these domains. We show how the problem of finding these small domains can be reduced to an interesting graph theoretic problem. This method enabled us to verify formulas containing hundreds of integer and floating point variables that could not be efficiently handled with previously known techniques
Context-Sensitive Rewriting Strategies

AbstractContext-sensitive rewriting is a simple rewriting restriction which is formalized by imposing fixed restrictions on replacements. Such a restriction is given on a purely syntactic basis: it is given on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols. The termination behavior is not only preserved but usually improved and several methods have been developed to formally prove it. In this paper, we investigate the definition, properties, and use of context-sensitive rewriting strategies, i.e., particular, fixed sequences of context-sensitive rewriting steps. We study how to define them in order to obtain efficient computations and to ensure that context-sensitive computations terminate whenever possible. We give conditions enabling the use of these strategies for root-normalization, normalization, and infinitary normalization. We show that this theory is suitable for formalizing the definition and analysis of real computational strategies which are used in programming languages such as OBJ or ELAN
Modelling prehistoric populations : the case of Neolithic Brittany.

The study of prehistoric demography draws inevitably on evidence both imperfect and incomplete, yet is essential for a satisfactory understanding of past communities. It is particularly valuable in addressing controversial questions such as the nature of early farming communities in western Europe, in the period between the adoption of domestic plants and animals and the establishment, centuries or millennia later, of permanent villages and regular field systems. In this article the demography of prehistoric Brittany is considered in light of evidence presented by monumental tombs and stone settings, artifact distributions, palaeoenvironmental determinations, and domestic remains. Whereas the megalithic monuments of Brittany are well known and have been the focus of systematic research since the mid-19th century, relatively little attempt has been made to situate them in their demographic context. The results of this study reveal that while Brittany is, like every region, unique in terms of climate, soils, and social traditions, it shares several significant features of demographic patterning—notably its cyclical character—with other areas of Neolithic Europe. This study also demonstrates how incomplete data sources can be used in combination to provide new interpretations of prehistoric demographic patterning and to frame new research questions. \ud
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Free Products in Prime Characteristic: A Representation of Fuchsian Groups

AbstractIn 1988, S. White proved by means of field theory supplemented by a geometric argument that the real bijectionsx↦x+1 andx↦xd(dan odd prime) generate a free group of rank 2. When these maps are considered in prime characteristicp(so thatx↦x+1 generates a cyclic group of orderp) the geometric argument is no longer available. We show on the one hand that, generally, the geometry is redundant and on the other that, in characteristicp, further algebraic considerations are required to establish a key polynomial lemma. By these means we obtain an analogue of White's theorem for certain (countably) infinite subfieldsLof the algebraic closure of the finite prime field GF(p). For any (odd) primed, not a divisor ofp(p−1), the mapsx↦x+1 andx↦xdgenerate a group ofbijectionsof such a fieldLthat is isomorphic to the free product Z∗(Z/pZ). This implies an explicit natural algebraic faithful representation of the free product as a transitive permutation group on a countable set
Residue Complex for Regular Algebras of Dimension 2

AbstractWe give a construction of a residue complex (a minimal injective resolution) for regular algebras of dimension 2, which are twisted homogeneous coordinate rings of the projective line. Residue complexes for general twisted coordinate rings have been previously constructed by geometric methods. Our method is algebraic based on (non-commutative) localizations of the algebra at orbits of points of the projective line. Along the way we establish a unique factorization in twisted coordinate rings and a partial fraction decomposition result, which we use in our construction
The Representation Ring of the Quantum Double of a Finite Group

AbstractWe present some results about the representation ring of the quantum double of a finite group over fields of arbitrary characteristic. We give a direct sum decomposition of this representation ring into ideals involving Green rings of subgroups. Given characters of such Green rings, we construct characters of the representation ring of the quantum double and show that all characters arise in this fashion. We prove that the characters thereby obtained from Brauer characters separate modules of the quantum double up to composition factors
Kernel Coefficient Ideals in Noetherian Rings

AbstractA new idealI*, the kernel coefficient ideal of a nonprincipal idealI, is introduced in a commutative Noetherian ringR. Various properties of this ideal and its relations with many other standard concepts are studied.I* is also examined in terms of a sequence of subidealsInand the relation type ofIwhenRis a local ring. Several characterizations ofI* are given in terms of the kernels of certain ring homomorphisms, and then it is shown that this new ideal has many nice applications, especially in the study of asymptotic prime divisors
Gradings, Derivations, and Automorphisms of Nearly Associative Algebras

AbstractIn this paper, we examine a class of algebras which includes Lie algebras, Lie color algebras, right alternative algebras, left alternative algebras, antiassociative algebras, and associative algebras. We call this class of algebras (α,β,γ)-algebras and we examine gradings of these algebras by groups with finite support. We generalize various results on associative algebras and finite-dimensional Lie algebras. Two of our main results areTheorem2.2.Let A be a G-graded left(α,β,γ)-algebra and V=⊕g∈GVga G-graded left A-module with finite support, where G is a torsion free abelian group. If A0acts nilpotently on V, then A also acts nilpotently on V.Theorem2.12.Let A be a G-graded(α,β,γ)-algebra with finite support, where G=T×Zmand T is a torsion free abelian group. If the identity component A(0,0)acts nilpotently on A on both sides, then A is solvable.These results are used to examine the invariants of automorphisms and derivations. One such application isCorollary3.3.Let L=⊕g∈GLgbe a Lie color algebra over a field K of characteristic0and let D be a finite-dimensional nilpotent Lie algebra of homogeneous derivations of L which are algebraic as K-linear transformations of L. If LD=0then L is nilpotent.We conclude this paper with counterexamples to various questions which arise naturally in light of our results