Andrea Asperti. On the complexity of beta-reduction. 1996 ACM Symposium on Principles of Programming Languages, pp. 110-118.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of a reduction to be the number of parallel fi reduction steps, plus the lengths of the initial and final terms [FS91] Unfortunately, graph reduction may require interaction of sharing nodes that grows exponentially in the number of parallel fi steps, a result derived independently by Asperti [Asp96], and in a simplified way by the authors [LM96] As a consequence, initial and final terms can be short, parallel reductions can be very few, yet intermediate forms can grow exponentially in these parameters. Mackie has investigated optimizations of the algorithm, however, his result apply to the ....

.... relation inherent in the invariance thesis that characterizes first class machine models; this notion is the computer scientist s refinement of Church s thesis [vEB90] In a recent paper, Asperti has advocated the number of fan interactions as the defining cost metric for calculus reduction [Asp96]. He intuitively asserts that Lamping s abstract algorithm performs no useless work. We agree with the assessment of the efficiency of the abstract algorithm. However, inspired by both the literature on machine simulation, where cost models are not linked to implementation or simulation, as well ....

Andrea Asperti. On the complexity of beta-reduction. 1996 ACM Symposium on Principles of Programming Languages, pp. 110--118.

....related. The essence of these analyses is the identification of a class of terms that require Theta(2 n ) fan interactions to reduce the corresponding term to normal form, while requiring only Theta(n) parallel fi steps. A very similar result was recently and independently derived by Asperti [Asp96] in his analysis of the inherent complexity of fi reduction. ffl We observe that normal order evaluation is an acceptable machine model in the sense of the orthodox Invariance Thesis. Thus the debate over cost models is largely fine structure, or the thesis is too crude. The key observation is ....

....analysis like GAL s context semantics, which can be very expensive. Theorem 2. Optimal evaluators allow Omega Gamma1 n ) useless fan interactions where there are no fi redexes to reduce. Asperti has suggested that we avoid this anomaly by considering only terms that normalize to constants [Asp96]. In this case, all functions are used, and all applications become redexes. But what if we remove this assumption can sharing be implemented with minimal overhead and no implicit foreknowledge that all functions will be applied 4 Optimizations The inefficiency in the reduction of Cn (x:y:xy) ....

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Andrea Asperti. On the complexity of betareduction. 1996 ACM Symposium on Principles of Programming Languages, pp. 110-- 118.

....between sharing nodes. A fundamental and unresolved question about this sharing technology, proposed by Lamping and o ered in modi ed form by others, is to understand the computational complexity of sharing as a function of the real work of reduction. In recent years, various papers [Asp96, LM96, LM97] have begun to address this issue. This question concerning algorithm analysis only begs more global questions that one can pose about the inherent complexity of optimal evaluation and parallel reduction by any implementation technology. In this paper, we take major steps towards resolving such ....

....is only one component of the graph reduction technology that supports nonelementary computation in a trivial number of parallel steps. The other essential phenomenon is higher order sharing, used to construct enormous networks of sharing nodes. A well understood use of sharing appears in [Asp96, LM96], where a linear number of sharing nodes is used to simulate an exponential number of function applications, illustrated in Figure 22. f x k Figure 22: Binary counting and exponential function application. The key idea in this construction, and in more ....

Andrea Asperti. On the complexity of beta-reduction. 1996 ACM Symposium on Principles of Programming Languages, pp. 110-118.

....between sharing nodes. A fundamental and unresolved question about this sharing technology, proposed by Lamping and offered in modified form by others, is to understand the computational complexity of sharing as a function of the real work of fi reduction. In recent years, various papers [Asp96, LM96, LM97] have begun to address this issue. This question concerning algorithm analysis only begs more global questions that one can pose about the inherent complexity of optimal evaluation and parallel fi reduction by any implementation technology. In this paper, we take major steps towards resolving ....

....The real work of the computation becomes communication, performing the j expansion on the lists, and communicating the function to different processors. 6.2 Higher order sharing Higher order sharing constructs enormous networks of sharing nodes. A well understood use of sharing appears in [Asp96, LM96], where a linear number of sharing nodes is used to simulate an exponential number of function applications. The key idea is the pairing of sharing nodes with wires from the circle to star sides, so that we can enter the same graph in two different modes. A network made of 2k sharing nodes ....

Andrea Asperti. On the complexity of betareduction. 1996 ACM Symposium on Principles of Programming Languages, pp. 110--118.

....between sharing nodes. A fundamental and unresolved question about this sharing technology, proposed by Lamping and offered in modified form by others, is to understand the computational complexity of sharing as a function of the real work of fi reduction. In recent years, various papers [Asp96, LM96, LM97] have begun to address this issue. This question concerning algorithm analysis only begs more global questions that one can pose about the inherent complexity of optimal evaluation and parallel fi reduction by any implementation technology. In this paper, we take major steps towards resolving ....

....is only one component of the graph reduction technology that supports nonelementary computation in a trivial number of parallel fi steps. The other essential phenomenon is higher order sharing, used to construct enormous networks of sharing nodes. A well understood use of sharing appears in [Asp96, LM96], where a linear number of sharing nodes is used to simulate an exponential number of function applications, illustrated in Figure 22. f x k Figure 22: Binary counting and exponential function application. The key idea in this construction, and in more ....

Andrea Asperti. On the complexity of beta-reduction. 1996 ACM Symposium on Principles of Programming Languages, pp. 110--118.

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