# Beta reduction is invariant, indeed

@article{Accattoli2014BetaRI,
title={Beta reduction is invariant, indeed},
author={Beniamino Accattoli and Ugo Dal Lago},
journal={Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)},
year={2014}
}
• Published 14 July 2014
• Mathematics
• Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is λ-calculus a reasonable machine? Is there a way to measure the computational complexity of a λ-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of λ-calculus: the length of a leftmost-outermost derivation to…

## Figures from this paper

Beta reduction is invariant, indeed
• Mathematics
CSL-LICS
• 2014
The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties, and the first complete positive answer to this long-standing problem of λ-calculus.
(Leftmost-Outermost) Beta Reduction is Invariant, Indeed
• Mathematics
Log. Methods Comput. Sci.
• 2016
The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties, and the first complete positive answer to this long-standing problem.
A reasonable time measure for the weak call-by-value lambda calculus
We provide a formalization of a generalized version of the time measure for the weak call-by-value lambda calculus as introduced by Dal Lago and Martini in 2008. We will use the language L as
The Useful MAM, a Reasonable Implementation of the Strong λ-Calculus
This paper presents a new abstract machine for the strong $\lambda$$-calculus based on useful sharing, the Useful Milner Abstract Machine, and proves that it reasonably implements leftmost-outermost evaluation. Church Meets Cook and Levin • Damiano Mazza • Mathematics, Computer Science 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) • 2016 The notion of affine approximation is addressed, which offers the possibility of using order-theoretic arguments, in contrast to the machine-level arguments employed in standard proofs, and an interesting relationship between approximations and intersection types is presented. Church Meets Cook and Levin • Mathematics, Computer Science • 2016 The notion of affine approximation is addressed, which offers the possibility of using order-theoretic arguments, in contrast to the machine-level arguments employed in standard proofs, and an interesting relationship between approximations and intersection types is presented. A Strong Distillery • Computer Science APLAS • 2015 This paper introduces a machine for the simplest form of strong evaluation, leftmost-outermost (call-by-name) evaluation to normal form, proving it correct, complete, and bounding its overhead. The Complexity of Abstract Machines This paper provides an unusual introduction to abstract machines, based on the complexity of their overhead with respect to the length of the implemented strategies, and is conceived to be a tutorial, focusing on the case study of implementing the weak head (call-by-name) strategy. Optimality and the Linear Substitution Calculus • Computer Science FSCD • 2017 This work lifts the theory of optimal reduction to a decomposition of the lambda calculus known as the Linear Substitution Calculus, and proposes a notion of redex family obtained by adapting Lévy labels to support these two distinctive features of LSC. A Compositional Cost Model for the λ-calculus • J. Laird • Computer Science 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) • 2021 A (time) cost model for the λ-calculus based on a natural presentation of its game semantics, where the cost of computing a finite approximant to the denotation of a term (its evaluation tree) is the size of its smallest derivation in the semantics. ## References SHOWING 1-10 OF 55 REFERENCES Beta reduction is invariant, indeed • Mathematics CSL-LICS • 2014 The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties, and the first complete positive answer to this long-standing problem of λ-calculus. Beta Reduction is Invariant, Indeed (Long Version) • Mathematics ArXiv • 2014 This paper presents the first complete positive answer to this long-standing problem, completely machine-independent and based over a standard notion in the theory of$\lambda\$-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model.
The weak lambda calculus as a reasonable machine
• Computer Science
Theor. Comput. Sci.
• 2008
On the Invariance of the Unitary Cost Model for Head Reduction (Long Version)
• Computer Science, Mathematics
RTA
• 2012
Invariance is proved by way of a linear calculus of explicit substitutions, which allows to nicely decompose any head reduction step in the lambda calculus into more elementary substitution steps, thus making the combinatorics of head-reduction easier to reason about.
On Constructor Rewrite Systems and the Lambda-Calculus
• Computer Science
ICALP
• 2009
We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate
Parallel beta reduction is not elementary recursive
• Computer Science
POPL '98
• 1998
The computational paradigms of superposition of values and of higher-order sharing are identified, appealing to compelling analogies with quantum mechanics and SIMD-parallelism.
Distilling abstract machines
• Computer Science
ICFP 2014
• 2014
The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic, and shows that the LSC is a complexity-preserving abstraction of abstract machines.
A nonstandard standardization theorem
• Mathematics, Computer Science
POPL 2014
• 2014
This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in lambda-calculus and linear logic proof-nets, and relies on Gonthier, Lévy, and Melliès' axiomatic theory for standardization.
The fine-structure of lambda calculus
This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application, and provides a type operator which gives a representative type for a typeable tenll.
Maximal Sharing in the Lambda Calculus with letrec
• Computer Science
ICFP 2014
• 2014
This work shows how a maximal degree of sharing can be obtained for programs expressed as terms in the lambda calculus with letrec, and introduces a notion of 'maximal compactness' for λletrec-terms among all terms with the same infinite unfolding.