Masako Takahashi. Parallel reductions in -calculus. Technical report, Department of Information Science, Tokyo Institute of Technology, 1993. Internal report. 19  Home/Search   Document Not in Database   Summary   Related Articles   Check

....any terms t 1 and t 2 such that t 1 t 2 , there exists t 2 such that t 1 . 1 , t 2 . 2 and t 1 =U t 2 . Proof. This is a consequence of Proposition 1, Theorem 1 and Proposition 2. The Church Rosser property can be proved using the usual methods (for an elegant one, see [Tak93]) Theorem 4 (Church Rosser property) For any terms t 1 and t 2 of EPECC such that t 1 and t 2 are equivalent, there exists t such that t 1 . and t 2 . We conjecture subject reduction can be proved using the same techniques as [Luo90] for ECC: Conjecture 1 (Subject Reduction ....

M. Takahashi. Parallel reductions in -calculus. Technical report, Department of Information Science, Tokyo Institute of Technology, 1993. Internal report.

....on the type of the variable being substituted, and on the interaction between rules and subtyping. 5 3.1. 2 Proving the Church Rosser Property The Church Rosser property for lambda calculi with only reduction is generally proved on untyped terms using the Tait Martin Lf method as described in [Tak93]. Unfortunately, as we show below, the Church Rosser property for . does not hold for untyped terms in . A priori, this does not preclude us from using such a proof method for proving the ChurchRosser property on untyped terms. Indeed, in his study of reduction for the Calculus of ....

M. Takahashi. Parallel reductions in -calculus. Technical report, Department of Information Science, Tokyo Institute of Technology, 1993. Internal report. 17

....in the empty environment for the module language is inhabited in the base language, that is that every proposition provable within the module system is provable in the base proof language. For both reduction notions, confluence properties are proved with the standard Tait and Martin Lof s method [Tak93] Subject reduction for and ae is proved the usual way (substitution property and study of possible types of a functor) In this proof, we have in particular to prove the following proposition: Proposition 1 If E M modtype and E (functor(x i : M 0 )m) x i ) M then E (functor(x i : M ....

M. Takahashi. Parallel reductions in -calculus. Technical report, Department of Information Science, Tokyo Institute of Technology, 1993. Internal report.

....relation, which states that if M can be partially computed into two different terms N 1 and N 2 , then there exists a third term N such that both N 1 and N 2 can be computed into N . In order to verify this property, we use Takahashi s variant of Martin Lof and Tait s parallel reduction method [85]. The parallel reduction method lays on the introduction of an auxiliary strategy for computing a term, in which all the redexes appearing in the term are contracted at the same time, in a single reduction step. Let us introduce first such notion of simultaneous computation. Definition 3.5.1 ....

M. Takahashi. Parallel reductions in -calculus. Technical report, Department of Information Science, Tokyo Institute of Technology, 1993. Internal report. 185

....of theorem 6, we have: Theorem 7 (Conservativity of the module extension) In the empty environment, a type T of a PTS is inhabited if and only if it is inhabited in its module extension. For both reduction notions, confluence properties are proved with the standard Tait and Martin Lof s method [Tak93] Subject reduction for and ae is proved as usual (substitution property and study of possible types of a functor) In this proof, we have in particular to prove the following proposition: Proposition 1 If E M modtype and E ( functor(x : M 0 )m) x i ) M then E ( functor(x : M 0 )m) ....

M. Takahashi. Parallel reductions in -calculus. Technical report, Department of Information Science, Tokyo Institute of Technology, 1993. Internal report. 18

....long computation times. In this paper, we will consider a multilevel (ML) solution algorithm for Markov chains, which was introduced in [5] The method is based on the principle of iterative aggregation and disaggregation, a well established numerical solution technique for Markov chains [7, 16, 14]. This principle utilizes a coarse level level correction that is multiplicative, rather than additive, i.e. newly obtained coarse level values are used as a factor by which fine level approximations are rescaled. Furthermore, the aggregation itself, or the coarsening strategy is ....

....strategy. We use a greedy algorithm to determine the aggregates whose complexity is linear in the number of edges of the Markov graph. 4 Interpretation as a multigrid method The multilevel method is based on the iterative aggregation disaggregation strategy, which dates at least from 1975 [16] and whose equations are derived in a natural way by probability arguments. In this section, we will show that the scheme can be written as a classical algebraic multigrid algorithm and point out the particular choices of multigrid components that the ML scheme represents. We begin by ....

Y. Takahashi: A Lumping Method for Numerical Calculations of Stationary Distributions of Markov Chains, Research Report No. B-18, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1975.

....aggregation makes the assumption that the number of aggregate states, N , is much less than the number of states, n, i.e. N n. In our algorithm we generally assume N = n 2 . In addition, much of the related work assumes that the Markov chains being solved are generalized birth death processes [17, 15], or that the Markov chains are nearly completely decomposable systems [5, 6] In the latter case the solution is usually an approximation often accompanied by bounds on the error. We refer the reader to [13] for descriptions of these special Markov chain structures and for a more comprehensive ....

....of these special Markov chain structures and for a more comprehensive list of references. Our work differs in that it does not require any special structure in the Markov chain, and the result is exact, not an approximation. The work that most strongly resemble ours is the algorithm of Takahashi [17] and its variants [13] We subsequently use the terminology derived in the previous section. The Takahashi algorithm starts with an initial iterate for the fine level chain. The fine level chain of n states is then aggregated into a coarse Markov chain of N states, where N n using equation ....

Y. Takahashi: A Lumping Method for Numerical Calculations of Stationary Distributions of Markov Chains, Research Report No. B-18, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1975.

....long computation times. In this paper we will consider the multi level (ML) solution algorithm for Markov chains, which was introduced in [4] The method is based on the principle of iterative aggregation and disaggregation, a well established numerical solution technique for Markov chains [6, 14, 13]. It is shown that the method is equivalent to an algebraic multigrid scheme which uses the Galerkin method for the coarse level operator and is of Full Approximation scheme (FAS) type. The novelty of the method stems from the definition of the prolongation operator, which is solution dependent ....

....with the largest coupling coefficient to a common aggregate whilst respecting a user defined upper limit for the size of each aggregate. 4 Interpretation as a multigrid method The multi level method is based on the iterative aggregation disaggregation strategy, which dates at least from 1975 [14] and whose equations are derived in a natural way by probability arguments. In this section we will show that the scheme can be written as a classical multigrid algorithm and point out the particular choices of multigrid components that the ML scheme represents. We begin by considering the ....

Y. Takahashi: A Lumping Method for Numerical Calculations of Stationary Distributions of Markov Chains, Research Report No. B-18, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1975.

....whether the excellent performance can be realized for all Markov chains. In this work we investigate the utility of the ML algorithm for solving Nearly Completely Decomposable (NCD) Markov chains. For this class of Markov chains special purpose iterative aggregation disaggregation (IAD) algorithms [3, 11] have been shown to perform well. Thus, demonstration of the utility of an algorithm for solving NCD chains necessitates comparison with one of these IAD schemes. We include the Koury McAllister Stewart (KMS) 3] algorithm as a representative example. We also include the Gauss Seidel (GS) method ....

.... i (11) ffl Apply fine level correction to obtain new iterate p (i 1) p (i 1) p = C( p; p ) j p p (12) In this two level form the method is similar to well known iterative aggregation disaggregation (IAD) methods such as those of Koury, McAllister and Stewart [3] and of Takahashi [11]. The multi level algorithm is obtained by recursive application of the two level algorithm to obtain a solution to the aggregated equation (9) and is described in algorithmic form in figure 3. We use the subscript l to denote level of representation (l = lmax finest level, i.e. the original ....

Y. Takahashi: A Lumping Method for Numerical Calculations of Stationary Distributions of Markov Chains, Research Report No. B-18, Department of Information Sciences, Tokyo Institute

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Masako Takahashi. Parallel reductions in -calculus. Technical report, Department of Information Science, Tokyo Institute of Technology, 1993. Internal report. 19

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