Coinduction for Exact Real Number Computation

This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio.     

A certified,corecursive implementation of exact real numbers

We implement exact real numbers in the logical framework Coq using streams, i.e., infinite sequences, of digits, and characterize constructive real numbers through a minimal axiomatization. We prove that our construction inhabits the axiomatization, working formally with coinductive types and corecursive proofs. Thus we obtain reliable, corecursive algorithms for computing on real numbers.     

Making a productive use of failure to generate witnesses for coinduction from divergent proof attempts

Coinduction is a proof rule. It is the dual of induction. It allows reasoning about non-well-founded structures such as lazy lists or streams and is of particular use for reasoning about equivalences. A central difficulty in the automation of coinductive proof is the choice of a relation (called a bisimulation). We present an automation of coinductive theorem proving. This automation is based on the idea of proof planning [7]. Proof planning constructs the higher level steps in a proof, using knowledge of the general structure of a family of proofs and exploiting this knowledge to control the proof search. Part of proof planning involves the use of failure information to modify the plan by the use of a proof critic [23] which exploits the information gained from the failed proof attempt. Our approach to the problem was to develop a strategy that makes an initial simple guess at a bisimulation and then uses generalisation techniques, motivated by a critic, to refine this guess, so that a larger class of coinductive problems can be automatically verified. The implementation of this strategy has focused on the use of coinduction to prove the equivalence of programs in a small lazy functional language which is similar to Haskell [22]. We have developed a proof plan for coinduction and a critic associated with this proof plan. These have been implemented in CoCLAM, an extended version of CLAM [9], with encouraging results. The planner has been successfully tested on a number of theorems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Productivity of Edalat-Potts Exact Arithmetic in Constructive Type Theory

In this work we focus on a formalisation of the algorithms of lazy exact arithmetic a la Edalat-Potts in type theory. We choose the constructive type theory extended with coinductive types as our formal verification tool. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output. Furthermore, we show how to employ this semantic information to formalise a restricted case of normalisation algorithm in our type theory.     

Complete sets of cooperations

The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is well known, but commonly elided. In this paper, we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particular, we consider different coalgebra structures that turn the set of infinite streams into the carrier of a final coalgebra. After that we focus on coalgebra structures that are made up using so-called cooperations. We say that a collection of cooperations is complete for a given set X if it gives rise to a coalgebra structure that turns X into the carrier set of a subcoalgebra of a final coalgebra. Any complete set of cooperations yields a coalgebraic proof and definition principle. We exploit this fact and devise a general definition scheme for constants and functions on a set X that is parametrical in the choice of the complete set of cooperations for X.     

I/O scheduling for digital continuous media

A growing set of applications require access to digital video and audio. In order to provide playback of such continuous media (CM), scheduling strategies for CM data servers (CMS) are necessary. In some domains, particularly defense and industrial process control, the timing requirements of these applications are strict and essential to their correct operation. In this paper we develop a scheduling strategy for multiple access to a CMS such that the timing guarantees are maintained at all times. First, we develop a scheduling strategy for the steady state, i.e., when there are no changes in playback rate or operation. We derive an optimal Batched SCAN (BSCAN) algorithm that requires minimum buffer space to schedule concurrent accesses. The scheduling strategy incorporates two key constraints: (1) data fetches from the storage system are assumed to be in integral multiples of the block size, and (2) playback guarantees are ensured for frame-oriented streams when each frame can span multiple blocks. We discuss modifications to the scheduling strategy to handle compressed data like motion-JPEG and MPEG. Second, we develop techniques to handle dynamic changes brought about by VCR-like operations executed by applications. We define a suite of primitive VCR-like operations that can be executed. We show that an unregulated change in the BSCAN schedule, in response to VCR-like operations, will affect playback guarantees. We develop two general techniques to ensure playback guarantees while responding to VCR-like operations: passive and active accumulation. Using user response time as a metric we show that active accumulation algorithms outperform passive accumulation algorithms. An optimal response-time algorithm in a class of active accumulation strategies is derived. The results presented here are validated by extensive simulation studies.     

Fast computation of the Smith form of a sparse integer matrix

We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix . The algorithm treats A as a “black box”—A is only used to compute matrix-vector products and we do not access individual entries in A directly. The algorithm requires about black box evaluations for word-sized primes p and , plus additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. The new algorithm suffers from no “fill-in” or intermediate value explosion, and uses very little additional space. We also present an asymptotically fast algorithm for dense matrices which requires about bit operations, where O(MM(m)) operations are sufficient to multiply two matrices over a field. Both algorithms are probabilistic of the Monte Carlo type — on any input they return the correct answer with a controllable, exponentially small probability of error. Received: March 9, 2000.     

Efficient Modular Squaring Algorithms for Hardware Implementation in GF(p)

ABSTRACT

Some of the most popular public key encryption algorithms use exponentiation as their core operation, which can be mostly broken into several modular squaring operations. In this paper, we present GF(p) modular squaring algorithms and efficiently implement them on hardware. We present different algorithms, two for squaring and one for reduction combined with the squaring, to provide a general modular squaring algorithm. The algorithms are implemented through datapaths that uses redundant Carry-Save Adders, making the computation time independent form the operands precision. The proposed algorithms are compared with each other as well as with the existing modular squaring algorithms. The experimental results are obtained by synthesizing the hardware designs for FPGA Virtex5 chip (xc5vlx50 – ff1153 technology), which showed interesting results and made our ideas very attractive.     

Efficient and Scalable Routing Algorithms for Collective Communication Operations on 2D All-Port Torus Networks

Collective Communication Algorithms for 2D torus networks have been investigated quite extensively in the literature and two broad approaches, namely direct methods and indirect (message combining) methods are recognized in the field. While direct methods minimize the volume of data, the indirect methods reduce the number of message start-ups. Consequently, either a suite of algorithms must be employed for efficiency over a wide range of message lengths and communication operations or algorithms should be able to adapt themselves to the current case, possibly by switching between direct and indirect routing modes as appropriate. In this paper, we propose adaptive routing algorithms for all-port, wormhole routed, synchronous, 2D torus networks optimized for one-to-all broadcast,  gossiping and complete exchange collective communication operations. The proposed algorithms employ completely-connected subnetworks where complete exchange amongst the nodes in the subnetwork can be accomplished in one step only. Combined with suitable 2D plane tiling techniques, the proposed algorithms share the same set of primitive operations and yield superior performance compared to previously proposed methods, either pure or hybridized.     

A survey on algorithms for mining frequent itemsets over data streams

The increasing prominence of data streams arising in a wide range of advanced applications such as fraud detection and trend learning has led to the study of online mining of frequent itemsets (FIs). Unlike mining static databases, mining data streams poses many new challenges. In addition to the one-scan nature, the unbounded memory requirement and the high data arrival rate of data streams, the combinatorial explosion of itemsets exacerbates the mining task. The high complexity of the FI mining problem hinders the application of the stream mining techniques. We recognize that a critical review of existing techniques is needed in order to design and develop efficient mining algorithms and data structures that are able to match the processing rate of the mining with the high arrival rate of data streams. Within a unifying set of notations and terminologies, we describe in this paper the efforts and main techniques for mining data streams and present a comprehensive survey of a number of the state-of-the-art algorithms on mining frequent itemsets over data streams. We classify the stream-mining techniques into two categories based on the window model that they adopt in order to provide insights into how and why the techniques are useful. Then, we further analyze the algorithms according to whether they are exact or approximate and, for approximate approaches, whether they are false-positive or false-negative. We also discuss various interesting issues, including the merits and limitations in existing research and substantive areas for future research.     

Dynamic processor allocation in scalable multiprocessors using boolean algebra *

In this paper, we study a new model for dynamic processor allocation in k-ary n-dimensional mesh or torus multiprocessors. The model uses Boolean functions to represent free processors and allocates processors by applying Boolean operations on the functions. The processor allocation algorithms based on the Boolean model can be implemented easily using binary decision diagrams(BDDs)and related software packages. To enhance the efficiency of the allocation algorithms, a reordering procedure will be introduced to change the ordering of Boolean variables in the BDD representation and thereby change the free subcube composition. Such a change leads to an improved free processor recognition capability. Complexities of the proposed allocation algorithms will be analyzed. Performance of the algorithms will be evaluated using simulation and compared with other approaches.     

Generalized Coiteration Schemata

Coiterative functions can be explained categorically as final coalgebraic morphisms, once coinductive types are viewed as final coalgebras. However, the coiteration schema which arises in this way is too rigid to accommodate directly many interesting classes of circular specifications. In this paper, building on the notion of T-coiteration introduced by the third author and capitalizing on recent work on bialgebras by Turi-Plotkin and Bartels, we introduce and illustrate various generalized coiteration patterns. First we show that, by choosing the appropriate monad T, T-coiteration captures naturally a wide range of coiteration schemata, such as the duals of primitive recursion and course-of-value iteration, and mutual coiteration. Then we show that, in the more structured categorical setting of bialgebras, T-coiteration captures guarded coiterations schemata, i.e. specifications where recursive calls appear guarded by predefined algebraic operations.     

Optimizing the costs of hierarchical quorum consensus

 We study the problem of how to minimize the cost of maintaining consistency among at least N copies of an object in an enviroment where the mix of read and write operations can vary. Quorum consensus requires that read and write operations must assemble appropriate quorums before an operation can succeed. The cost of an operation is proportional to the size of a quorum, and the objective is obviously to minimize the cost while still maintaining consistency. It is known that the quorum size can be reduced by organizing a number of copies into logical structures such as grids and hierarchies. In this paper, we show (a) how methods based on grids and hierarchies can be treated in a common framework, and (b) how these hierarchies can be optimized so as to minimize the cost of consensus. Of course, the optimal solution depends upon the mix of read and write operations that is present. Consequently, given N copies of an object and a ratio of write operations F, our algorithms determine the optimal values for the number of levels in the hierarchy and the read/write quorum sizes at each level. The algorithms, which run in O(N ^1.63) and O(N ²) time, were tested, and the results are reported and discussed. Received September 1, 1992/February 16, 1995     

Regular corecursion in Prolog

Corecursion is the ability of defining a function that produces some infinite data in terms of the function and the data itself, as supported by lazy evaluation. However, in languages such as Haskell strict operations fail to terminate even on infinite regular data, that is, cyclic data.Regular corecursion is naturally supported by coinductive Prolog, an extension where predicates can be interpreted either inductively or coinductively, that has proved to be useful for formal verification, static analysis and symbolic evaluation of programs.In this paper we use the meta-programming facilities offered by Prolog to propose extensions to coinductive Prolog aiming to make regular corecursion more expressive and easier to program with.First, we propose a new interpreter to solve the problem of non-terminating failure as experienced with the standard semantics of coinduction (as supported, for instance, in SWI-Prolog). Another problem with the standard semantics is that predicates expressed in terms of existential quantification over a regular term cannot directly defined by coinduction; to this aim, we introduce finally clauses, to allow more flexibility in coinductive definitions.Then we investigate the possibility of annotating arguments of coinductive predicates, to restrict coinductive definitions to a subset of the arguments; this allows more efficient definitions, and further enhance the expressive power of coinductive Prolog.We investigate the effectiveness of such features by showing different example programs manipulating several kinds of cyclic values, ranging from automata and context free grammars to graphs and repeating decimals; the examples show how computations on cyclic values can be expressed with concise and relatively simple programs.The semantics defined by these vanilla meta-interpreters are an interesting starting point for a more mature design and implementation of coinductive Prolog.     

The continuum as a final coalgebra

We define the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the final coalgebra of either the functor N×X, product with the set of natural numbers, or the functor 1+N×X. This makes an attractive analogy with the definition of N itself as the initial algebra of the functor 1+X, disjoint union with a singleton. We similarly specify Baire space and Cantor space in terms of these final coalgebras. We identify two variants of this approach, a coinductive definition based on final coalgebraic structure in the category of sets, and a direct definition as a final coalgebra in the category of posets. We conclude with some paradoxical discrepancies between continuity and constructiveness in this setting.     

Collective communication operations: experimental results vs. theory

Collective communication operations (CCOs) are one of the most powerful tools for parallel processing on distributed memory architectures. From the theoretical viewpoint there has been a major effort in the design of optimal algorithms for these operations, especially for massive parallel processors (MPPs). However, in spite of the increasing availability of MPPs, there are just a few limited experimental checks of the different theories, so the assessment of their real value is not easy. The aim of the present paper is to address such issues for the most common CCOs, considering practical algorithms that can be included in a generic communication library. The main result is a new algorithm for building a quasi-optimal broadcast tree that is much simpler than, and as efficient as, previously available algorithms. To investigate the advantages and drawbacks of the proposed algorithms, a large set of experimental data has been collected on an IBM SP2 parallel system. The data demonstrate the efficiency of our approach in a number of interesting cases. Finally, all the experimental results have been related to the model used in designing the algorithms. © 1998 John Wiley & Sons, Ltd.     

Coinductive Models of Finite Computing Agents

This paper explores the role of coinductive methods in modeling finite interactive computing agents. The computational extension of computing agents from algorithms to interaction parallels the mathematical extension of set theory and algebra from inductive to coinductive models. Maximal fixed points are shown to play a role in models of observation that parallels minimal fixed points in inductive mathematics. The impact of interactive (coinductive) models on Church's thesis and the connection between incompleteness and greater expressiveness are examined. A final section shows that actual software systems are interactive rather than algorithmic. Coinductive models could become as important as inductive models for software technology as computer applications become increasingly interactive.     

Algorithms on ensemble quantum computers

In ensemble (or bulk) quantum computation, all computations are performed on an ensemble of computers rather than on a single computer. Measurements of qubits in an individual computer cannot be performed; instead, only expectation values (over the complete ensemble of computers) can be measured. As a result of this limitation on the model of computation, many algorithms cannot be processed directly on such computers, and must be modified, as the common strategy of delaying the measurements usually does not resolve this ensemble-measurement problem. Here we present several new strategies for resolving this problem. Based on these strategies we provide new versions of some of the most important quantum algorithms, versions that are suitable for implementing on ensemble quantum computers, e.g., on liquid NMR quantum computers. These algorithms are Shor’s factorization algorithm, Grover’s search algorithm (with several marked items), and an algorithm for quantum fault-tolerant computation. The first two algorithms are simply modified using a randomizing and a sorting strategies. For the last algorithm, we develop a classical-quantum hybrid strategy for removing measurements. We use it to present a novel quantum fault-tolerant scheme. More explicitly, we present schemes for fault-tolerant measurement-free implementation of Toffoli and s_z^1/4,\sigma_{z}^{1/4}, as these operations cannot be implemented “bitwise”, and their standard fault-tolerant implementations require measurement.