J. L. Lawall and H. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? Proc. 1996 ACM SIGPLAN Int'l Conf. Functional Programming, pages 92--101, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....e.g. environments. Because explicit substitution is closer to real implementations, it has the potential to provide a more accurate cost model. This possibility is particularly interesting in light of the difficulty encountered in formulating a useful cost model in terms of graph reduction [LM96, Pey87]. Proof assistants may benefit from explicit substitution, due to the desire to perform substitutions locally and in a formal manner. Local substitutions are needed as follows. Given xx[x: y] one may not be interested in having yy as the result of xx[x: y] but rather only yx[x: y] In other ....

J. L. Lawall and H. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? Proc. 1996 ACM SIGPLAN Int'l Conf. Functional Programming, pages 92--101, 1996.

....e.g. environments. Because explicit substitution is closer to real implementations, it has the potential to provide a more accurate cost model. This possibility is particularly interesting in light of the difficulty encountered in formulating a useful cost model in terms of graph reduction [LM96, Jon87]. Proof assistants may benefit from explicit substitution, owing to the desire to perform substitutions locally and in a formal manner. Local substitutions are needed as follows. Given xx[x: y] one may not be interested in having yy as the result of xx[x: y] but rather only yx[x: y] In other ....

J. L. Lawall and H. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming, pages 92--101, New York, 1996. ACM.

....e.g. environments. Because explicit substitution is closer to real implementations, it has the potential to provide a more accurate cost model. This possibility is particularly interesting in light of the difficulty encountered in formulating a useful cost model in terms of graph reduction [33, 40]. Proof assistants may benefit from explicit substitution, due to the desire to perform substitutions locally and in a formal manner. Local substitutions are needed as follows. Given xx[x: y] one may not be interested in having yy as the result of xx[x: y] but rather only yx[x: y] In other ....

J. L. Lawall and H. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In Proc. 1996 ACM SIGPLAN Int'l Conf. Functional Programming, pp. 92--101, 1996.

....e.g. environments. Because explicit substitution is closer to real implementations, it has the potential to provide a more accurate cost model. This possibility is particularly interesting in light of the difficulty encountered in formulating a useful cost model in terms of graph reduction [LM96, Pey87]. Proof assistants may benefit from explicit substitution, due to the desire to perform substitutions locally and in a formal manner. Local substitutions are needed as follows. Given xx[x: y] one may not be interested in having yy as the result of xx[x: y] but rather only yx[x: y] In other ....

J. L. Lawall and H. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? Proc. 1996 ACM SIGPLAN Int'l Conf. Functional Programming, pages 92--101, 1996.

....e.g. environments. Because explicit substitution is closer to real implementations, it has the potential to provide a more accurate cost model. This possibility is particularly interesting in light of the difficulty encountered in formulating a useful cost model in terms of graph reduction [LM96, Jon87] Proof assistants may benefit from explicit substitution, owing to the desire to perform substitutions locally and in a formal manner. Local substitutions are needed as follows. Given xx[x: y] one may not be interested in having yy as the result of xx[x: y] but rather only yx[x: y] ....

J. L. Lawall and H. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming, pages 92--101, New York, 1996. ACM.

.... number of family reduction is definitely not a good measure of the intrinsic complexity of terms (although the hypothesis P 6= NP suggests that it could be quite difficult to give a direct and formal proof of this claim) Criticisms against this measure had been already raised by many authors [As96, ML96, ML97]. In particular, the total number of interactions in Lamping s abstract system looks as a much more realistic bound to the intrinsic complexity of terms (note that, essentially, the expansion technique consists in transforming fi interactions into fan annihilations) Let us finally remark ....

H. Mairson, J. L. Lawall. Optimality and Inefficiency: What Isn't a Cost Model of the Lambda Calculus? ACM International Conference on Functional Programming, pp. 92--101. 1996.

....to a way of comparing calculus with the computational models used in complexity theory. We think that the interest in the definition of a suitable notion of cost and of complexity classes for sharing computations is even more appealing after the results of Asperti [Asp96] Lawall and Mairson [LM96], and Asperti and Mairson [AM98] showing that L evy families cannot be the cost model of calculus reductions. Acknowledgments I wish to thank Simone Martini who supervised my thesis and always encouraged me persisting, even when things seemed to get everyday more involved; Andrea Asperti who ....

Julia L. Lawall and Harry G. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming, pages 92--101, Philadelphia, Pennsylvania, 1996.

....considerations. We have not studied the efficiency of our approach compared with other approaches. Finding a good measure for the computational complexity of asynchronous and local reductions in proof nets (and calculus) is an important open problem, outside the scope of the present paper (e.g. [Asp96,LM96]) 4 Coherence We state in this section our main results, whose proofs will be presented in Section 5.8. Namely, that the reduction rules fi solve the coherence problem for s structures. This is not trivial, since the rules may be fired in any order (logical and non logical reductions will be ....

Julia L. Lawall and Harry G. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In "Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming, pages 92--101. Philadelphia, 1996. ACM.

....i v Fig. 7. Simplification rules efficiency of our approach compared with the others approaches. Finding a good measure for the computational complexity of asynchronous and local reductions in proof nets (and calculus) is an important open problem, outside the scope of the present paper (e.g. [Asp96, LM96]) 3 Coherence We state in this section our main results, namely that the reduction rules fi solve the coherence problem for nets. This is not trivial, since the rules may be fired in any order (logical and non logical reductions will be in general interleaved) Our proof strategy, ....

Julia L. Lawall and Harry G. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In 1996 ACM International Conference on Functional Programming, 1996.

....to a way to compare calculus with the computational models used in complexity theory. We think that the interest in the definition of a suitable notion of cost and of complexity classes for sharing computations is even more appealing after the results of Asperti [Asp96] and Lawall and Mairson [LM96] showing that L evy families are unlike to be the cost model of calculus reductions. Acknowledgments I thank Simone Martini who supervised my thesis and always encouraged me persisting, even when things seemed to get everyday more involved; Andrea Asperti who introduced me to the intriguing ....

Julia L. Lawall and Harry G. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming, pages 92--101, Philadelphia, Pennsylvania, 24--26 May 1996.

....a charge of 1 for each of these operations. Since dealloc k is a no op , 0 is charged for this operation. As copy makes two copies of a variable of type nat, a cost of 2 is charged. The problem of producing accurate cost models for calculus is difficult, and is an active area of interest. See [ML96, ML97] and its references. The decisions made in the cost model may seem rather arbitrary and ad hoc, and the justification of many of those decisions is presented only informally. Additionally, the model is rather crude. It would be useful to make the implicit closure model that was appealed to more ....

H. Mairson and J. L. Lawall. Optimality and inefficiency: What isn't a cost model of the lambda calculus. In 1996 ACM International Conference on Functional Programming, pages 92--101, 1996. BIBLIOGRAPHY 135

....is so optimal about optimal reduction In collaboration with postdoctoral researchers Lawall and Kucan, and Asperti at U. Bologna, he has investigated the algorithmic problems of graph reduction, shown why the technique can be considered efficient, and proved that the algorithm is correct [LM96, LM97, AM98] Correctness is nontrivial, and a first principles explanation has been lacking. Even the idea of implementation independent cost for languages is not a worked out area: defin18 ing the cost of a language via its compiler makes as much sense as defining the semantics of a language ....

Julia L. Lawall and Harry G. Mairson. Optimality and inefficiency: What isn't a cost model of the lambda calculus? In Proceedings of the

....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 ....

....number of graph reduction steps counts not only interactions between , apply, and sharing nodes, but also interactions involving the croissant and bracket nodes used to manage the indices that control the behavior of the sharing nodes. For more information on how this index management works, see [LM96, AG97]. Finally, in renditions of optimal graph reduction rules, there is some ambiguity defining where reduction ends, and where readback begins. For example, if a graph has no more fi redexes, and thus represents a normalized term, one way to read back the term is to continue graph reduction ....

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Julia L. Lawall and Harry G. Mairson. Optimality and inefficiency: what isn't a cost model of the lambda calculus? 1996 ACM International Conference on Functional Programming, pp. 92--101.

....and thus provide a means to implement L evy s notion of parallel reduction. The varied implementations of this fundamental framework differ in the underlying bookkeeping technology used to control interactions between sharing nodes. For a gentle introduction to and summary of these issues, see [LM96]. A distinguishing characteristic of optimal reduction schemes founded upon graph reduction is their description solely in terms of local reduction rules. This approach simplifies the underlying atomic operations, and facilitates implementation. Reducing the graph representation of a term, ....

....which just look for interacting pairs of nodes, whether or not they are and nodes. An unfortunate consequence of computation via local rules is that optimal graph reduction evaluators can exponentially duplicate graph structure even though there are no more fi redexes in the graph [LM96]. Because paths between and nodes may encounter intermediate sharing nodes linked in perverse ways, the computationally prohibitive context semantics of [GAL92] has been the only known way of doing relevant static analysis of graphs. Optimal graph reduction technology has also obscured the ....

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Julia L. Lawall and Harry G. Mairson. Optimality and inefficiency: what isn't a cost model of the lambda calculus? 1996 ACM International Conference on Functional Programming, pp. 92--101.

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