# 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} }

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…

## 44 Citations

Beta reduction is invariant, indeed

- MathematicsCSL-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

- MathematicsLog. 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

- Mathematics
- 2015

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

- Computer ScienceWoLLIC
- 2016

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

- Mathematics, Computer Science2016 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 ScienceAPLAS
- 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

- Computer ScienceWPTE@FSCD
- 2016

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 ScienceFSCD
- 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

- Computer Science2021 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.

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Beta reduction is invariant, indeed

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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)

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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.

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