Introduction to combinators and &lgr;-calculus
Introduction to combinators and &lgr;-calculus
The mathematics of programming: an inaugural lecture delivered before the Univ. of Oxford on Oct. 17, 1985
Proofs and types
On laziness and optimality in lambda interpreters: tools for specification and analysis
POPL '90 Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
An algorithm for optimal lambda calculus reduction
POPL '90 Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
POPL '90 Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
What is an efficient implementation of the &lgr;-calculus?
Proceedings of the 5th ACM conference on Functional programming languages and computer architecture
A simple proof of a theorem of Statman
Theoretical Computer Science
The geometry of optimal lambda reduction
POPL '92 Proceedings of the 19th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Paths, computations and labels in the &lgr;-calculus
RTA-93 Selected papers of the fifth international conference on Rewriting techniques and applications
From proof-nets to interaction nets
Proceedings of the workshop on Advances in linear logic
Optimality and inefficiency: what isn't a cost model of the lambda calculus?
Proceedings of the first ACM SIGPLAN international conference on Functional programming
On the complexity of beta-reduction
POPL '96 Proceedings of the 23rd ACM SIGPLAN-SIGACT symposium on Principles of programming languages
On global dynamics of optimal graph reduction
ICFP '97 Proceedings of the second ACM SIGPLAN international conference on Functional programming
The optimal implementation of functional programming languages
The optimal implementation of functional programming languages
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
The Calculi of Lambda Conversion. (AM-6) (Annals of Mathematics Studies)
The Calculi of Lambda Conversion. (AM-6) (Annals of Mathematics Studies)
Correct and optimal implementations of recursion in a simple programming language
Journal of Computer and System Sciences
From Hilbert Spaces to Dilbert Spaces: Context Semantics Made Simple
FST TCS '02 Proceedings of the 22nd Conference Kanpur on Foundations of Software Technology and Theoretical Computer Science
From Hilbert space to Dilbert space: context semantics as a language for games and flow analysis
ICFP '03 Proceedings of the eighth ACM SIGPLAN international conference on Functional programming
Sharing Implementations of Graph Rewriting Systems
Electronic Notes in Theoretical Computer Science (ENTCS)
Derivational complexity is an invariant cost model
FOPARA'09 Proceedings of the First international conference on Foundational and practical aspects of resource analysis
Light logics and optimal reduction: Completeness and complexity
Information and Computation
Linear dependent types in a call-by-value scenario
Proceedings of the 14th symposium on Principles and practice of declarative programming
The Cost of Usage in the ?-Calculus
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
Hi-index | 0.00 |
We analyze the inherent complexity of implementing Lévy's notion of optimal evaluation for the α-calculus, where similar redexes are contracted in one step via so-called parallel β-reduction. Optimal evaluation was finally realized by Lamping, who introduced a beautiful graph reduction technology for sharing evaluation contexts dual to the sharing of values. His pioneering insights have been modified and improved in subsequent implementations of optimal reduction. We prove that the cost of parallel β-reduction is not bounded by any Kalmár-elementary recursive function. Not only do we establish that the parallel β-step cannot be a unit-cost operation, we demonstrate that the time complexity of implementing a sequence of n parallel β-steps is not bounded as O (2n), O (22n), O (2n), or in general, O (Kl (n)), where Kl (n) is a fixed stack of l 2's with an n on top. A key insight, essential to the establishment of this non-elementary lower bound, is that any simply typed α-term can be reduced to normal form in a number of parallel β-steps that is only polynomial in the length of the explicitly typed term. The result follows from Statman's theorem that deciding equivalence of typed α-terms is not elementary recursive. The main theorem gives a lower bound on the work that must be done by any technology that implements Lévy's notion of optimal reduction. However, in the significant case of Lamping's solution, we make some important remarks addressing how work done by β-reduction is translated into equivalent work carried out by his bookkeeping nodes. In particular, we identify the computational paradigms of superposition of values and of higher-order sharing, appealing to compelling analogies with quantum mechanics and SIMD-parallelism. Copyright 2001 Academic Press.