Crystal: Integrating Structured Queries into a Tactic Language

We present the language CRStL (Control Rule Strategy Language, pronounce “crystal”) to formulate mathematical reasoning techniques as proof strategies in the context of the proof assistant Ωmega. The language is arranged in two levels, a query language to access mathematical knowledge maintained in development graphs, and a strategy language to annotate the results of these queries with further control information. The two-leveled structure of the language allows the specification of proof techniques in a declarative way. We present the syntax and semantics of CRStL and illustrate its use by examples.     

On Succinct Greedy Drawings of Plane Triangulations and 3-Connected Plane Graphs

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in the Euclidean plane R ² (by Papadimitriou and Ratajczak in Theor. Comp. Sci. 344(1):3–14, 2005). Leighton and Moitra (Discrete Comput. Geom. 44(3):686–705, 2010) recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations require Ω(nlogn) bits to represent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications. In this paper, we show that the classical Schnyder drawing in R ² of plane triangulations is greedy with respect to a simple natural metric function H(u,v) over R ² that is equivalent to Euclidean metric D _E (u,v) (in the sense that $D_{E}(u,v) \leq H(u,v) \leq2\sqrt{2}D_{E}(u,v)$ ). The drawing uses two integer coordinates between 0 and 2n?5, which can be represented by logn bits. We also show that the classical Schnyder drawing in R ² of 3-connected plane graphs is weakly greedy with respect to the same metric function H(?,?). The drawing uses two integer coordinates between 0 and f (where f is the number of internal faces of G).     

PP-Lowness and a Simple Definition of AWPP

Abstract. We show that the counting classes AWPP and APP [FFKL], [L] are more robust than previously thought. Our results identify a sufficient condition for a language to be low for PP, and we show that this condition is at least as weak as other previously studied criteria. We extend a result of Köbler et al. by proving that all sparse co-C ₌ P languages are in APP, and are thus PP-low. Our results also imply that AWPP ?eq APP, and thus APP contains many other established subclasses of PP-low, thereby reducing several different lowness results to membership in APP. We also show that AWPP and APP are Σ ⁰ ₂ -definable classes. Some of our results are reminiscent of amplifying certainty in probabilistic computation.     

A new factorization of the mass matrix for optimal serial and parallel calculation of multibody dynamics

This paper describes a new factorization of the inverse of the joint-space inertia matrix M. In this factorization, M ^?1 is directly obtained as the product of a set of sparse matrices wherein, for a serial chain, only the inversion of a block-tridiagonal matrix is needed. In other words, this factorization reduces the inversion of a dense matrix to that of a block-tridiagonal one. As a result, this factorization leads to both an optimal serial and an optimal parallel algorithm, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors. The novel feature of this algorithm is that it first calculates the interbody forces. Once these forces are known, the accelerations are easily calculated. We discuss the extension of the algorithm to the task of calculating the forward dynamics of a kinematic tree consisting of a single main chain plus any number of short side branches. We also show that this new factorization of M ^?1 leads to a new factorization of the operational-space inverse inertia, Λ ^?1, in the form of a product involving sparse matrices. We show that this factorization can be exploited for optimal serial and parallel computation of Λ ^?1, that is, a serial algorithm with a complexity of O(N) and a parallel algorithm with a time complexity of O(logN) on a computer with O(N) processors.     

Combinatorial PCPs with Efficient Verifiers

The PCP theorem asserts the existence of proofs that can be verified by a verifier that reads only a very small part of the proof. The theorem was originally proved by Arora and Safra (J. ACM 45(1)) and Arora et al. (J. ACM 45(3)) using sophisticated algebraic tools. More than a decade later, Dinur (J. ACM 54(3)) gave a simpler and arguably more intuitive proof using alternative combinatorial techniques. One disadvantage of Dinur’s proof compared to the previous algebraic proof is that it yields less efficient verifiers. In this work, we provide a combinatorial construction of PCPs with verifiers that are as efficient as the ones obtained by the algebraic methods. The result is the first combinatorial proof of the PCP theorem for NEXP (originally proved by Babai et al., FOCS 1990), and a combinatorial construction of super-fast PCPs of Proximity for NP (first constructed by Ben-Sasson et al., CCC 2005).     

Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke counter-model for σ. Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.     

An SDP Primal-Dual Algorithm for Approximating the Lovász-Theta Function

The Lovász ?-function (Lovász in IEEE Trans. Inf. Theory, 25:1–7, 1979) of a graph G=(V,E) can be defined as the maximum of the sum of the entries of a positive semidefinite matrix X, whose trace Tr(X) equals 1, and X _ij =0 whenever {i,j}∈E. This function appears as a subroutine for many algorithms for graph problems such as maximum independent set and maximum clique. We apply Arora and Kale’s primal-dual method for SDP to design an algorithm to approximate the ?-function within an additive error of δ>0, which runs in time $O(\frac{\vartheta ^{2} n^{2}}{\delta^{2}} \log n \cdot M_{e})$ , where ?=?(G) and M _e =O(n ³) is the time for a matrix exponentiation operation. It follows that for perfect graphs G, our primal-dual method computes ?(G) exactly in time O(? ² n ⁵logn). Moreover, our techniques generalize to the weighted Lovász ?-function, and both the maximum independent set weight and the maximum clique weight for vertex weighted perfect graphs can be approximated within a factor of (1+?) in time O(? ^?2 n ⁵logn).     

Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Let ${\mathcal{B}}$ be a centrally symmetric convex polygon of ?² and ‖p?q‖ be the distance between two points p,q∈?² in the normed plane whose unit ball is ${\mathcal{B}}$ . For a set T of n points (terminals) in ?², a ${\mathcal{B}}$ -network on T is a network N(T)=(V,E) with the property that its edges are parallel to the directions of ${\mathcal{B}}$ and for every pair of terminals t _i and t _j , the network N(T) contains a shortest ${\mathcal{B}}$ -path between them, i.e., a path of length ‖t _i ?t _j ‖. A minimum ${\mathcal{B}}$ -network on T is a ${\mathcal{B}}$ -network of minimum possible length. The problem of finding minimum ${\mathcal{B}}$ -networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX’99) in the case when the unit ball ${\mathcal{B}}$ is a square (and hence the distance ‖p?q‖ is the l ₁ or the l _∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun (Symposium on Computational Geometry, pp. 393–402, 2009) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4, 3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum ${\mathcal{B}}$ -network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (Chepoi et al. in Theor. Comput. Sci. 390:56–69, 2008, and APPROX-RANDOM, pp. 40–51, 2005) and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.     

Shorthand Universal Cycles for Permutations

The set of permutations of ??n??={1,??,n} in one-line notation is ??(n). The shorthand encoding of a ₁?a _n ????(n) is a ₁?a _n?1. A shorthand universal cycle for permutations (SP-cycle) is a circular string of length n! whose substrings of length n?1 are the shorthand encodings of ??(n). When an SP-cycle is decoded, the order of ??(n) is a Gray code in which successive permutations differ by the prefix-rotation ?? _i =(1 2 ? i) for i??{n?1,n}. Thus, SP-cycles can be represented by n! bits. We investigate SP-cycles with maximum and minimum ??weight?? (number of ?? _n?1s in the Gray code). An SP-cycle n a n b?n z is ??periodic?? if its ??sub-permutations?? a,b,??,z equal ??(n?1). We prove that periodic min-weight SP-cycles correspond to spanning trees of the (n?1)-permutohedron. We provide two constructions: B(n) and C(n). In B(n) the spanning trees use ??half-hunts?? from bell-ringing, and in C(n) the sub-permutations use cool-lex order by Williams (SODA, 987?C996, 2009). Algorithmic results are: (1)?memoryless decoding of B(n) and C(n), (2)?O((n?1)!)-time generation of B(n) and C(n) using sub-permutations, (3)?loopless generation of B(n)??s binary representation n bits at a time, and (4)?O(n+??(n))-time ranking of B(n)??s permutations where ??(n) is the cost of computing a permutation??s inversion vector. Results (1)?C(4) improve on those for the previous SP-cycle construction D(n) by Ruskey and Williams (ACM Trans. Algorithms 6(3):Art.?45, 2010), which we characterize here using ??recycling??.     

OXPath: A language for scalable data extraction, automation, and crawling on the deep web

The evolution of the web has outpaced itself: A growing wealth of information and increasingly sophisticated interfaces necessitate automated processing, yet existing automation and data extraction technologies have been overwhelmed by this very growth. To address this trend, we identify four key requirements for web data extraction, automation, and (focused) web crawling: (1) interact with sophisticated web application interfaces, (2) precisely capture the relevant data to be extracted, (3) scale with the number of visited pages, and (4) readily embed into existing web technologies. We introduce OXPath as an extension of XPath for interacting with web applications and extracting data thus revealed—matching all the above requirements. OXPath’s page-at-a-time evaluation guarantees memory use independent of the number of visited pages, yet remains polynomial in time. We experimentally validate the theoretical complexity and demonstrate that OXPath’s resource consumption is dominated by page rendering in the underlying browser. With an extensive study of sublanguages and properties of OXPath, we pinpoint the effect of specific features on evaluation performance. Our experiments show that OXPath outperforms existing commercial and academic data extraction tools by a wide margin.     

Effective condition number and its applications

Consider the over-determined system Fx = b where ${{\bf F}\in\mathcal{R}^{m \times n}, m \geq n}$ and rank (F) = r ≤ n, the effective condition number is defined by ${{\rm Cond_{-}eff }= \frac {\|{\bf b}\|}{\sigma_r\|{\bf x}\|}}$ , where the singular values of F are given as σ _max = σ ₁ ≥ σ ₂ ≥ . . . ≥ σ _r > 0 and σ _r+1 = . . . = σ _n = 0. For the general perturbed system (A+ΔA)(x+Δx) = b+Δb involving both ΔA and Δb, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz’s problem by the collocation Trefftz methods (CTM). Motz’s problem is the benchmark of singularity problems. We choose the general particular solutions ${v_L = \sum\nolimits_{k=0}^L d_k (\frac {r}{R_p})^{k+\frac 12}}$ ${{\rm cos}(k +\frac 12)\theta}$ with a radius parameter R _p . The CTM is used to seek the coefficients d _i by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter R _p = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of R _p is misleading. Under the optimal choice R _p = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.     

Invariant-Free Clausal Temporal Resolution

Resolution is a well-known proof method for classical logics that is well suited for mechanization. The most fruitful approach in the literature on temporal logic, which was started with the seminal paper of M. Fisher, deals with Propositional Linear-time Temporal Logic (PLTL) and requires to generate invariants for performing resolution on eventualities. The methods and techniques developed in that approach have also been successfully adapted in order to obtain a clausal resolution method for Computation Tree Logic (CTL), but invariant handling seems to be a handicap for further extension to more general branching temporal logics. In this paper, we present a new approach to applying resolution to PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Hence, we say that the approach presented in this paper is invariant-free. Our method is based on the dual methods of tableaux and sequents for PLTL that we presented in a previous paper. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called trs-resolution, that extends classical propositional resolution. Finally, we prove that trs-resolution is sound and complete. In fact, it finishes for any input formula deciding its satisfiability, hence it gives rise to a new decision procedure for PLTL.     

Query reverse engineering

In this paper, we introduce a new problem termed query reverse engineering (QRE). Given a database \(D\) and a result table \(T\) —the output of some known or unknown query \(Q\) on \(D\) —the goal of QRE is to reverse-engineer a query \(Q'\) such that the output of query \(Q'\) on database \(D\) (denoted by \(Q'(D)\) ) is equal to \(T\) (i.e., \(Q(D)\) ). The QRE problem has useful applications in database usability, data analysis, and data security. In this work, we propose a data-driven approach, TALOS for Tree-based classifier with At Least One Semantics, that is based on a novel dynamic data classification formulation and extend the approach to efficiently support the three key dimensions of the QRE problem: whether the input query is known/unknown, supporting different query fragments, and supporting multiple database versions.     

Local noncuppability in R/M

Given any [c], [a], [d] xxxxxxxxR/M such that [d] ≤ [a] ≤ [c], [a] is locally noncuppable between [c] and [d] if [d] < [a] ≤ [c] and [a] ∨ [b] < [c] for any [b] xxxxxxxxR/M such that [d] ≤ [b] < [c]. It will be shown that given any nonzero [c] xxxxxxxxR/M, there are [a], [d] xxxxxxxxR/M such that [d] < [a] ≤ [c] and [a] is locally noncuppable between [c] and [d].     

ALTEP — A cellular processor for high-speed pattern matching

The design of a high-speed cellular pattern matcher, called theAssociative Linear Text Processor (ALTEP), is presented.ALTEP was originally designed for systems which use signature files as an access method. However, it is also suitable for systems which store the database in fixed length blocks.ALTEP is a linear array of logic cells which respond to commands sent from a central controller over a bus. A text block is loaded into the cells and pattern characters are broadcast to the cells for comparison.ALTEP has the capability of recognizing full regular expressions and is the only cellular logic array which has this capability. It requiresO(p) steps for patterns which do not contain closures andO(len(max(T(P)))) steps for closures, wherep is the the length of the pattern andlen(max(T(P))) is the length of the longest substring in the text which matched the closure.     

A note on balanced immunity

For a finite alphabet ∑ we define a binary relation on \(2^{\Sigma *} \times 2^{2^{\Sigma ^* } } \) , called balanced immunity. A setB ? ∑* is said to be balancedC-immune (with respect to a classC ? 2^Σ* of sets) iff, for all infiniteL εC, $$\mathop {\lim }\limits_{n \to \infty } \left| {L^{ \leqslant n} \cap B} \right|/\left| {L^{ \leqslant n} } \right| = \tfrac{1}{2}$$ Balanced immunity implies bi-immunity and in natural cases randomness. We give a general method to find a balanced immune set'B for any countable classC and prove that, fors(n) =o(t(n)) andt(n) >n, there is aB εSPACE(t(n)), which is balanced immune forSPACE(s(n)), both in the deterministic and nondeterministic case.     

Parallel computation of disease transforms

Distance transforms are an important computational tool for the processing of binary images. For ann ×n image, distance transforms can be computed in time \(\mathcal{O}\) (n) on a mesh-connected computer and in polylogarithmic time on hypercube related structures. We investigate the possibilities of computing distance transforms in polylogarithmic time on the pyramid computer and the mesh of trees. For the pyramid, we obtain a polynomial lower bound using a result by Miller and Stout, so we turn our attention to the mesh of trees. We give a very simple \(\mathcal{O}\) (logn) algorithm for the distance transform with respect to theL ₁-metric, an \(\mathcal{O}\) (log² n) algorithm for the transform with respect to theL _∞-metric, and find that the Euclidean metric is much more difficult. Based on evidence from number theory, we conjecture the impossibility of computing the Euclidean distance transform in polylogarithmic time on a mesh of trees. Instead, we approximate the distance transform up to a given error. This works for anyL _k -metric and takes time \(\mathcal{O}\) (log³ n).     

Iterations for solving linear variational inequalities on domains with nonlinear boundaries

This paper deals with numerical methods for solving linear variational inequalities on an arbitrary closed convex subsetC of ? ⁿ . Although there were numerous iterations studied for the caseC=? ₊ ⁿ , few were proposed for the case whenC is a closed convex subset. The essential difficulty in this case is the nonlinearities ofC's boundaries. In this paper iteration processes are designed for solving linear variational inequalities on an arbitrary closed convex subsetC. In our algorithms the computation of a linear variational inequality is decomposed into a sequence of problems of projecting a vector to the closed convex subsetC, which are computable as long as the equations describing the boundaries are given. In particular, using our iterations one can easily compute a solution whenC is one of the common closed convex subsets such as cube, ball, ellipsoid, etc. The non-accurate iteration, the estimate of the solutions on unbounded domains and the theory of approximating the boundaries are also established. Moreover, a necessary and sufficient condition is given for a vector to be an approximate solution. Finally, some numerical examples are presented, which show that the designed algorithms are effective and efficient. The exposition of this paper is self-contained.     

A novel constant degree and constant congestion DHT scheme for peer-to-peer networks

1 Introduction and related work In recent years, peer-to-peer computing has attracted significant attention from both industry field and academic field[1-3]. The core component of many proposed peer-to- peer systems is the distributed hash table (DHT) schemes[4,5] that use a hash table-like interface to publish and look up data objects. Many proposed DHT schemes[6-15] are based on some traditional interconnection to- pology: Chord[6], Tapestry[7,8], Pastry[9] are based on hypercube topolog…     

Bisection forAx=λBx with matrices of variable band width

Some extensions of the bisection method and of the inverse vector iteration for the general eigenvalue problemAx=λBx with symmetric matrices are given. A version with restricted pivoting is applied to sparse matricesA andB in which case the decomposition ofA-μB can be performed within an extended envelope with respect to the envelopeA andB. The effect of these refinements is illustrated by an example.