Total order

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d 3 and n > 2(d -1) be given. Let p = [p 1 , p 2 , . . . , p d-1 ] with p 1 1, p 2 1, . . . , p d-1 1. Let C(p) be the caterpillar obtained from the stars S p 1 , S p 2 , . . . , S p d-1 and the path P d-1 by identifying the root of S p i with the i-vertex of P d-1 . Let We prove that the algebraic connectivity of C(p) ∈ C is bounded above by is the unique caterpillar in C attaining the upper bound and that if d is odd then the upper bound cannot be achieved. Finally, for 1 k d-1 2 , we give a total ordering by algebraic connectivity on C k = {C (1, . . . , 1, p k , 1, . . . , 1, p d-k , 1, . . . , 1) ∈ C : p k p d-k } .

One way to implement a fault-tolerant a service is by replicating it at sites that fail independently. One of the replication techniques is active replication where each request is executed by all the replicas. Thus, the effects of failures can be completely masked resulting in an increase of the service availability. In order to preserve consistency among replicas, replicas must exhibit a deterministic behavior, what has been traditionally achieved by restricting replicas to be singlethreaded. However, this approach cannot be applied in some setups, like transactional systems, where it is not admissible to process transactions sequentially. In this paper, we present a deterministic scheduling algorithm for multithreaded replicas in a transactional framework. To ensure replica determinism requests to replicated servers are submitted by means of reliable and totally ordered multicast. Internally, a deterministic scheduler ensures that all threads are scheduled in the same way at all replicas what guarantees replica consistency £ This work has been partially funded by the Spanish Research Council (CICYT), contract number TIC98-1032-C03-01 and the Madrid Regional Research Council (CAM), contract number CAM-07T/0012/1998. This paper is organized as follows. First, the system model is described in Section 2. Then, the different sources of nondeterminism are identified in Section 3. Section 4 presents a deterministic scheduling algorithm for multithreaded replicas (MTRDS). The correctness of the MTRDS algorithm is proven in Section 5. Section 6 presents some implementation issues to support the algorithm. Finally, we compare our approach to other works and present our conclusions. The system consists of a set of nodes interconnected by means of a network. We assume the nodes to be fail-silent and that there are no network partitions. We do not consider neither malicious failures (i.e., byzantine failures), nor the non-determinism introduced by software interrupts.

Contemporary workflow management systems are driven by explicit process models, i.e., a completely specified workflow design is required in order to enact a given workflow process. Creating a workflow design is a complicated time-consuming process and typically, there are discrepancies between the actual workflow processes and the processes as perceived by the management. Therefore, we propose a technique for rediscovering workflow models. T his technique uses workflow logs to discover the workflow process as it is actually being executed. The workflow log contains information about events taking place. We assume that these events are totally ordered and each event refers to one task being executed for a single case. This information can easily be extracted from transactional information systems (e.g., Enterprise Resource Planning systems such as SAP and Baan). The rediscovering technique proposed in this paper can deal with noise and can also be used to validate workflow processes by uncovering and measuring the discrepancies between prescriptive models and actual process executions.

We construct the non-standard complex (and real) numbers using the ultrapower method in the spirit of Cauchy's construction of the real numbers. We show that the nonstandard complex numbers are a non-archimedean, algebraically closed field, and that the non-standard real numbers are a totally ordered, real-closed, non-archimedean field. We explore the various types of non-standard numbers, and develop the non-standard completeness results (Saturation Principle, Supremum Completeness of Bounded Internal Sets, etc) for * R. We give non-standard characterizations for such usual topological objects as open, closed, bounded, and compact sets in terms of monads. We also consider such traditional topics of real analysis as limits, continuity, uniform continuity, convergence, uniform convergence, etc. in a non-standard setting. In both topology and real analysis we reduce (and in some cases eliminate) the number of quantifiers in the non-standard setting.

We consider the problem of partial order production: arrange the elements of an unknown totally ordered T into a target partially ordered set S, by comparing a minimum number of pairs in T . Special cases of this problem include sorting by comparisons, selection, multiple selection, and heap construction. We give an algorithm performing IT LB + o(IT LB) + O(n) comparisons in the worst case. Here, n denotes the size of the ground sets, and IT LB denotes a natural information-theoretic lower bound on the number of comparisons needed to produce the target poset. The overall complexity of our algorithm is polynomial. This answers a question of Yao (SICOMP, 1989). Our strategy is to extend the poset S to a weak order W whose corresponding information-theoretic lower bound is provably not much larger than that for S. Taking W instead of S as a target poset, we then solve the problem by applying a multiple selection algorithm that performs not much more than IT LB comparisons. We base our analysis on the entropy of the target poset S, a quantity that can be efficiently computed and provides a good estimate of IT LB since the latter is, up to a linear error term, n times the entropy of S.

We examine the problem of using LOCC to distinguish a known pure state from an unknown (possibly mixed) state, bounding the error probability from above and below. We study the asymptotic rate of detecting multiple copies of the pure state and show that, if the overlap of the two states is great enough, then they can be distinguished asymptotically as well with LOCC as with global measurements; otherwise, the maximal Schmidt coefficient of the pure state is sufficient to determine the asymptotic error rate.

We explore the question of entanglement as a universal resource for implementing quantum measurements by local operations and classical communication (LOCC). We show that for most multipartite systems (consisting of three or more subsystems), there is no entangled state from the same space that can enable all measurements by LOCC. This is in direct contrast to the bipartite case, where a maximally entangled state is an universal resource. Our results are obtained by considering the problem of local state discrimination using entanglement as a resource; and a key ingredient in our proofs is showing an equivalence between the problem of local state discrimination and that of unambiguous local state transformation. We also present partial characterization of universal resource states in multipartite systems.

We study so-called betweennesses induced by trees and forests. Algorithmic problems related to betweennesses are typically hard. They have been studied as relaxations of ordinal embeddings and occur for instance in psychometrics and molecular biology. Our contributions are hardness results, efficient algorithms, and structural insights such as complete axiomatic characterizations.

GlobData is a project that aims to design and implement a middleware tool offering the abstraction of a global object database repository. This tool, called Copla, supports transactional access to geographically distributed persistent objects independent of their location. Additionally, it supports replication of data according to different consistency criteria. For this purpose, Copla implements a number of consistency protocols offering different tradeoffs between performance and fault-tolerance. This paper presents the work on strong consistency protocols for the Glob-Data system. Two protocols are presented: a voting protocol and a nonvoting protocol. Both these protocols rely on the use of atomic broadcast as a building block to serialize conflicting transactions. The paper also introduces the total order protocol being developed to support large-scale replication.

Assuming data domains are partially ordered, we apply Paredaens' and Bancilhon's Theorem to examine the expressiveness of the extended relational algebra (the PORA), which allows the ordering predicate to be used in formulae of the selection operator (σ). The PORA expresses exactly the set of all possible relations which are invariant under order-preserving automorphism of databases. Our main result shows that there is a one-toone correspondence between three hierarchies of: (1) computable queries, (2) query languages and (3) partially ordered domains.

For many years, the intuitions underlying partial-order planning were largely taken for granted. Only in the past few years has there been renewed interest in the fundamental principles underlying this paradigm. In this paper, we present a rigorous comparative analysis of partial-order and total-order planning by focusing on two specific planners that can be directly compared. We show that there are some subtle assumptions that underly the wide-spread intuitions regarding the supposed efficiency of partial-order planning. For instance, the superiority ofpartial-order planning can depend critically upon the search strategy and the structure of the search space. Understanding the underlying assumptions is crucial for constructing efficient planners.

In this paper we compare the utility of different commitment strategies in planning. Under a "least commitment strategy", plans are represented as partial orders and operators are ordered only when interactions are detected. We investigate claims of the inherent advantages of planning with partial orders, as compared to planning with total orders. By focusing our analysis on the issue of operator ordering commitment, we are able to carry out a rigorous comparative analysis of two planners. We show that partial-order planning can be more efficient than total-order planning, but we also show that this is not necessarily so.

We study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain with totally ordered (discrete or continuous) state space. The number of steps in the chain can be random and unbounded. The method can be used in particular to get a low-variance unbiased estimator of the expected total cost up to some random stopping time, when statedependent costs are paid at each step. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial.

A range-sum query is very popular and becomes important in finding trends and in discovering relationships between attributes in diverse database applications. It sums over the selected cells of an OLAP data cube where target cells are decided by the specified ...

Mathematical morphology offers popular image processing tools, successfully used for binary and grayscale images. Recently, its extension to color images has become of interest and several approaches were proposed. Due to various issues arising from the vectorial nature of the data, none of them imposed as a generally valid solution. We propose a probabilistic pseudo-morphological approach, by estimating two pseudo-extrema based on Chebyshev inequality. The framework embeds a parameter which allows controlling the linear versus non-linear behavior of the probabilistic pseudo-morphological operators. We compare our approach for grayscale images with the classical morphology and we emphasize the impact of this parameter on the results. Then, we extend the approach to color images, using principal component analysis. As validation criteria, we use the estimation of the color fractal dimension, color textured image segmentation and color texture classification. Furthermore, we compare our proposed method against two widely used approaches, one morphological and one pseudo-morphological.

This paper focuses on interactive Knowledge Discovery pro-cesses in the context of understanding an activity from behavioural data. Data mining provides patterns experts have to interpret and synthesize as new knowledge. Discovering patterns is an analysis task while build-ing new symbolic knowledge is a synthesis task. A previous trace based approach (Abstract) offered a first answer to support analysis. This paper goes one step forward in supporting the synthesis task. We mod-ify an algorithm of automata discovery in order to involve the user in the mining process, exploiting his expert knowledge about the observed activity. We chose the α-algorithm (Van Der Aalst et al.) developed for Petri nets discovery in a workflow management context. The modified algorithm is described and illustrated, showing how to use intermediate data to converge interactively to a satisfying automata. Finally, we dis-cuss the use of this approach to contribute to a new knowledge mining process.

We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of 2-design POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basic) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilizing their graph-theoretical properties. Our paper is the first step towards an abstract definition of a generalised qplex.

We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of 2-design POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basic) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilizing their graph-theoretical properties. Our paper is the first step towards an abstract definition of a generalised qplex.

In 1963, W. Charles Holland proved that every lattice-ordered group can be embedded in the lattice-ordered group of all order-preserving permutations of a totally ordered set. In this article we examine the context and proof of this result and survey some of the many consequences of the ideas involved in this important theorem.

Poset game, which includes some famous games. e.g., Nim and Chomp as sub-games, is an important two-player impartial combinatorial game. The rule of the game is as follows: For a given poset (partial ordered set), each player intern chooses an element and the selected element and it's descendants (elements succeeding it) are all removed from the poset. A player who choose the last element is the winner. On the complexity of poset game, although it is clearly in PSPACE, it have not known whether it is in P or NP-hard. Recently a weighted poset game, which is a generalization of poset game, have been presented and it was found that some sub-games of it can be solved in polynomial-time. The complexity of this game is also open. This paper shows that weighted poset game is PSPACE-complete even if the weights are restricted in {1, -1}, the dag, which represents the poset, is bipartite, and the length of each path in the dag is at most two.

The paper develops a general theory of orderability of quandles with a focus on link quandles of tame links and gives some general constructions of orderable quandles. We prove that knot quandles of many fibered prime knots are right-orderable, whereas link quandles of most nontrivial torus links are not right-orderable. As a consequence, we deduce that the knot quandle of the trefoil is neither left nor right orderable. Further, it is proved that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The paper also explores interconnections between orderability of quandles and that of their enveloping groups. The results establish that orderability of link quandles behave quite differently than that of corresponding link groups.

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.

In many situations, cooperating agents have different status with respect to an activity. Often a coalition can undertake the activity only if sufficiently many agents, or agents of sufficient seniority participate in it. The concept of a simple game, introduced by von Neumann & Morgenstern (1944), is flexible enough to model a large class of such situations. In this dissertation, we consider two important classes of simple games called the classes of weighted simple games,

We have investigated the crystal structure and phase transitions of new series of ordered double perovskite oxides Ca 2Àx Sr x LaSbO 6 and Ca 2Àx Sr x SmSbO 6 (x = 0, 0.5, 1). Rietveld refinements of the X-ray powder diffraction data at room temperature shows that these compounds crystallize in monoclinic symmetry with P2 1 /n space group, with cation distribution [Ca 1Àx Sr x R] A CaSbO 6 general formula. The La compounds have a total ordering between La and Ca in A-and B-sites, respectively; however, in Sm compounds, the degree of Sm-Ca ordering has been found decreasing with an increase in the Sr content. A linear increasing of the lattice parameters and the volume, at room temperature, has been found with increasing of Sr concentration. The temperature evolution of the structure has revealed that the compounds Ca 1.5 Sr 0.5 LaS-bO 6 , CaSrLaSbO 6 and CaSrSmSbO 6 present two phase transitions, the first one being discontinuous and the second one continuous as a following phase-transition sequence: P2 1 =n ! R 3 ! Fm 3m. In Ca 2 LaSbO 6 , Ca 2 SmSbO 6 and Ca 1.5 Sr 0.5 SmSbO 6 , no indication of any phase transitions from room temperature to 1770 K was observed.

Abstract. Effective information systems require the existence of explicit process models; a completely specified process design needs to be developed in order to enact a given business process. This development is time consuming and the resulting models are often subjective and incomplete. We propose a method that discovers the process model from process logs where process events are recorded as they have been executed over time. We induce rules that predict causal, exclusive, and parallel relations between process events. ...

Given a loset I, every surjective map p: A ---> I endows the set A with a structure of preordered set by "replacing" the elements of I with their inverse images via p considered as "bubbles" (sets endowed with an equivalence relation), lifting the structure of loset on A, and "agglutinating" this structure with the bubbles. Every bubbling A of a structure of loset I is a structure of preordered set A (not necessarily complete) whose preorder has negatively transitive asymmetric part and every such structure on a given set A can be obtained by bubbling up of certain structure of a loset I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of bubbles. As a consequence of this characterization, under certain natural topological conditions on the preordered set A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.

We consider the problem of partial order production: arrange the elements of an unknown totally ordered set T into a target partially ordered set S, by comparing a minimum number of pairs in T. Special cases include sorting by comparisons, selection, multiple selection, and heap construction. We give an algorithm performing IT LB + o(IT LB) + O(n) comparisons in the worst case. Here, n denotes the size of the ground sets, and IT LB denotes a natural information-theoretic lower bound on the number of comparisons needed to produce the target partial order. Our approach is to replace the target partial order by a weak order (that is, a partial order with a layered structure) extending it, without increasing the information theoretic lower bound too much. We then solve the problem by applying an efficient multiple selection algorithm. The overall complexity of our algorithm is polynomial. This answers a question of Yao (SIAM J. Comput. 18, 1989). We base our analysis on the entropy of the target partial order, a quantity that can be efficiently computed and provides a good estimate of the information-theoretic lower bound.

An n-ary associative function is called reducible if it can be written as a composition of a binary associative function. We summarize known results when the function is defined on a chain and is nondecreasing. Our main result shows that associative idempotent and nondecreasing functions are uniquely reducible.

Let G be a finite subgroup of unitary matrices acting on the space of Nqubits. We associate with G a uniform quantum channel QU from the space on N-qubits to itself. We give a quantum algorithm to approximate this channel by considering a set of generators on G. Under suitable assumptions this approximation is BPQ. We then apply this approximation to study the orbit equivalence of two density matrices under the action of G. We show that for some special cases of G and two pure states the orbit equivalence in BPQ, if a specific quantum observation can be implemented efficiently. We discuss the application of our problem to the graph isomorphism problem.

Quantum information and computing has an interesting notion of proofs in complexity theory. In this project, we limit ourselves to QMA, a quantum analogue of the NP class with bounded error polynomial time verifier rather than a deterministic one. The notion of proof and an efficient verification is extended to the quantum setting along with a growing list of complete problems and different variants of the QMA (which might or might not be equal to the class QMA). As a preliminary study, we have studied an introduction to the QMA covering the quantum proofs, definitions of efficient quantum verification, the formalization of the class QMA illustrated using the group non-membership problem in quantum setting. Few QMA complete problems like (a, b)-QCS (Quantum Circuit Satisfiability) and the first QMA-complete problem, the local Hamiltonian problem are discussed. We would like to look further into the proof of the QMA completeness for local hamiltonian problem by [VW15] which is intuit...

A totally ordered monoid-or tomonoid , for short-is a commutative semigroup with identity S equipped with a total order ≤ S that is translation invariant, i.e., that satisfies: ∀x, y, z ∈ S x ≤ S y ⇒ x + z ≤ S y + z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice Z n that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell.

Synchronous cooperative systems (SCS) bring together users that are geographically distributed and connected through a network to carry out a task. Examples of SCS include Tele- Immersion and Tele-Conferences. In SCS, the coordination is the core of the system, and it has been defined as the act of managing interdependencies between activities performed to achieve a goal. Some of the main problems that SCS present deal with the management of constraints between simultaneous activities and the execution ordering of these activities. In order to resolve these problems, orderings based on Lamport-s happened-before relation have been used, namely, causal, Δ-causal, and causal-total orderings. They mainly differ in the degree of asynchronous execution allowed. One of the most important orderings is the causal order, which establishes that the events must be seen in the cause-effect order as they occur in the system. In this paper we show that for certain SCS (e.g. videoconferences, tele-...

This paper examines variation in language development based on production data from three Greek-speaking children. Variation suggests that children employ more than one grammar during the acquisition process. This naturally raises the question of how 'unwanted' grammars gradually give way to the one that relates to the adult/target grammar. To account for variation, we implement partial ordering (Anttila 1997a, 1997b) to Tzakosta's (2004) Multiple Parallel Grammars model of language development. More specifically, we propose that, in the intermediate stage of acquisition, constraint permutation of the initial Markedness » Faithfulness ranking leads to grammar explosion. We view the resulting grammars as partial orders that contain sets of totally ranked grammars (subgrammars). The pivotal claim is that only those subgrammars that are typologically closer to the target one will eventually survive. This is stated as the Grammar Inclusion Hypothesis. The theoretical gain of the proposed model is that it provides a principled basis to define developmental paths and also to distinguish between smart and non-smart paths. The latter are partial orders that do not contain the target grammar as a total order and hence are doomed to extinction. The former, on the other hand, are partial orders that contain at least one total order that relates to the target grammar and, crucially, connect the running state of acquisition with the end state of language development. Our hypothesis finds empirical support by both inter-child and intra-child language acquisition data.

The all nearest smaller values problem is de ned as follows. Let A = (a 1 ; a 2 ; : : :; a n) be n elements drawn from a totally ordered domain. For each a i , 1 i n, nd the two nearest elements in A that are smaller than a i (if such exist): the left nearest smaller element a j (with j < i) and the right nearest smaller element a k (with k > i). We give an O(loglog n) time optimal parallel algorithm for the problem on a CRCW PRAM. We apply this algorithm to achieve optimal O(loglog n) time parallel algorithms for four problems: (i) Triangulating a monotone polygon, (ii) Preprocessing for answering range minimum queries in constant time, (iii) Reconstructing a binary tree from its inorder and either preorder or postorder numberings, (vi) Matching a legal sequence of parentheses. We also show that any optimal CRCW PRAM algorithm for the triangulation problem requires (loglog n) time.

Cure kinetic model is an integral part of composite process simulation, which is used to predict the degree of curing and the amount of the generated heat. The parameters involved in kinetic models are usually determined empirically from isothermal or dynamic differential scanning calorimetry (DSC) data. In this work, DSC and rheological techniques were used to investigate some of the kinetic parameters of cure reactions of carbon/F161 epoxy prepreg and to evaluate the cure cycle used to manufacture polymeric composites for aeronautical applications. As a result, it was observed that the F161 prepreg presents cure kinetic with total order 1.2-1.9.

The present paper proposes and evaluates some extensions to the CORBA Event Service, based on the use of the Lightweight Reliable Multicast Protocol (LRMP). These extensions address some limitations of the CORBA Event Service, namely multicasting, reliability and total ordering. The extensions were developed and tested in a prototype environment created for this purpose, and are transparent and fully compatible with the standard OMG Event Service, enabling the user to decide for the use of standard IIOP or IP multicasting.

Let [n] = {1, 2,. .. , n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn = {α ∈ Pn : (f or all x, y ∈ Dom α) |xα − yα| ≤ |x − y|} be the subsemigroup of partial contraction mappings on [n]. We have shown that the semigroup CPn and some of its subsemigroups are nonregular left abundant semigroups for all n but not right abundant for n ≥ 4.

In the bureaucratic moments that Michael Herzfeld describes, functionaries of the nation-state deny the contingent points of possible connection with their hapless subjects yet draw on a common symbolic language to do so-thus is the institution reproduced. Such ethnographic moments implicitly rely on other moments, those informed by their expressed lack of total order: illicit activities, games, and competitive verbal performance; cultural intimacy is generated in such occasions of private fallibility. This article explores what happens when a different kind of institution, a virtual world maker, attempts to colonize such disorder to govern legitimately.

This paper is concerned with the form of typed name binding used by the FreshML family of languages. Its characteristic feature is that a name binding is represented by an abstract (name,value)-pair that may only be deconstructed via the generation of fresh bound names. The paper proves a new result about what operations on names can co-exist with this construct. In FreshML the only observation one can make of names is to test whether or not they are equal. This restricted amount of observation was thought necessary to ensure that there is no observable difference between alpha-equivalent name binders. Yet from an algorithmic point of view it would be desirable to allow other operations and relations on names, such as a total ordering. This paper shows that, contrary to expectations, one may add not just ordering, but almost any relation or numerical function on names without disturbing the fundamental correctness result about this form of typed name binding (that object-level alpha...

Abstmt. The following general theorem is proven: Given a partially ordered set and a group Gf prmu tations among itu elements which preserves the order relation, there is a set of elements no twc? c&red acalled an independpnt set, or an antichain) of maximal size which consists of mmplete orbits under the group. The result is spplicd to several examplco including the lattice of partitions of a set. The maximal size (3f an anticham is also shown to Se the solution of a certain linear program dc fined by the parzial order. Some generalLatio;7s of the main resurult are also described.

We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3−SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT , but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.

A known geometrical problem is to find low density zones (voids) in planar point sets and to represent them as polygons. In this paper we recall the concept of terminal-edge region to identify subvoid candidates over a triangulation, present a linear algorithm to find subvoids taking as input a Delaunay triangulation, and show that this new strategy can be naturally parallelized using GPU computing. We also show preliminary experimental results.

We extend a planning algorithm to cover simple forms of arithmetics. The operator preconditions can refer to the values of numeric variables and the operator postconditions can modify the values of numeric variables. The basis planning algorithm is based on techniques from propositional satisfiability testing and does not restrict to forward or backward chaining. When several operations affect a numeric variable by increasing and decreasing its value in parallel, the effects have to be combined in a meaningful way. This problem is especially acute in planning algorithms that maintain an incomplete state description of every time point of a plan execution. The approach we take requires that for operators that are executed in parallel, all linearizations of the operations to total orders behave equivalently. We provide an efficient and general solution to the problem.

We extend a planning algorithm to cover simple forms of arithmetics. The operator preconditions can refer to the values of numeric variables and the operator postconditions can modify the values of numeric variables. The basis planning algorithm is based on techniques from propositional satisfiability testing and does not restrict to forward or backward chaining. When several operations affect a numeric variable by increasing and decreasing its value in parallel, the effects have to be combined in a meaningful way. This problem is especially acute in planning algorithms that maintain an incomplete state description of every time point of a plan execution. The approach we take requires that for operators that are executed in parallel, all linearizations of the operations to total orders behave equivalently. We provide an efficient and general solution to the problem.