Formal Small-Step Verification of a Call-by-Value Lambda Calculus Machine

Kunze, Fabian; Smolka, Gert; Forster, Yannick

doi:10.1007/978-3-030-02768-1_15

Cited by 4 publications

(6 citation statements)

References 19 publications

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“…The tightness of the provable gap then depends on the timeefficiency of the interpreter in use. As mentioned, the self-interpreter given in Section 6.1 is too inefficient and we want to extract the interpreters described in [18] and [9] to L.…”

Section: Discussionmentioning

confidence: 99%

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A certifying extraction with time bounds from Coq to call-by-value $λ$-calculus

Forster¹,

Kunze²

2019

Preprint

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We provide a plugin extracting Coq functions of simple polymorphic types to the (untyped) call-byvalue λ-calculus L. The plugin is implemented in the MetaCoq framework and entirely written in Coq. We provide Ltac tactics to automatically verify the extracted terms w.r.t a logical relation connecting Coq functions with correct extractions and time bounds, essentially performing a certifying translation and running time validation. We provide three case studies: A universal L-term obtained as extraction from the Coq definition of a step-indexed self-interpreter for L, a many-reduction from solvability of Diophantine equations to the halting problem of L, and a polynomial-time simulation of Turing machines in L.

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Section: Discussionmentioning

confidence: 99%

“…Note that the implementation of eva is very naive and needs steps exponential in n, we thus omit its time complexity. 8 A more reasonable implementation could be obtained by extracting the heap-based abstract machine from [18] to L.…”

Section: Step-indexed L-interpretermentioning

confidence: 99%

A certifying extraction with time bounds from Coq to call-by-value $λ$-calculus

Forster¹,

Kunze²

2019

Preprint

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“…On closed terms, the number of steps to a normal form agrees with the number of steps needed in the version in [16]. We keep the definitions short and use the same notations as in [14], where more details can be found. We define the syntax of the λ-calculus using a de Bruijn representation of terms [9]: s, t, u, v : Ter ::= n | st | λs where n : N.…”

Section: Call-by-value λ-Calculus Lmentioning

confidence: 99%

“…On closed terms, the number of steps to a normal form agrees with the number of steps needed in the version in [16]. We keep the definitions short and use the same notations as in [14],…”

Section: Call-by-value λ-Calculus Lmentioning

confidence: 99%

“…In order to analyse the two mentioned strategies on a more semantic level than just as implementations on Turing machines we introduce two abstract machines implementing these strategies -based on substitutions and based on heaps. The machines are variants of the ones presented in [14]. Both machines will take O( s T ) abstract steps to evaluate a term s, but differ in the size of intermediate states and in the complexity of their respective implementations as Turing machines, which we construct in Section 4.…”

Section: Abstract Machinesmentioning

confidence: 99%

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The weak call-by-value λ-calculus is reasonable for both time and space

Forster

Kunze

Roth

2019

Proc. ACM Program. Lang.

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We study the weak call-by-value λ-calculus as a model for computational complexity theory and establish the natural measures for time and space -the number of beta-reductions and the size of the largest term in a computation -as reasonable measures with respect to the invariance thesis of Slot and van Emde Boas [STOC 84]. More precisely, we show that, using those measures, Turing machines and the weak call-by-value λ-calculus can simulate each other within a polynomial overhead in time and a constant factor overhead in space for all computations that terminate in (encodings) of "true" or "false". We consider this result as a solution to the long-standing open problem, explicitly posed by Accattoli [ENTCS 18], of whether the natural measures for time and space of the λ-calculus are reasonable, at least in case of weak call-by-value evaluation.Our proof relies on a hybrid of two simulation strategies of reductions in the weak call-by-value λ-calculus by Turing machines, both of which are insufficient if taken alone. The first strategy is the most naive one in the sense that a reduction sequence is simulated precisely as given by the reduction rules; in particular, all substitutions are executed immediately. This simulation runs within a constant overhead in space, but the overhead in time might be exponential. The second strategy is heap-based and relies on structure sharing, similar to existing compilers of eager functional languages. This strategy only has a polynomial overhead in time, but the space consumption might require an additional factor of log n, which is essentially due to the size of the pointers required for this strategy. Our main contribution is the construction and verification of a space-aware interleaving of the two strategies, which is shown to yield both a constant overhead in space and a polynomial overhead in time. ACM Subject Classification Theory of computation → Computational complexity and cryptography; Mathematics of computing → Lambda calculusKeywords and phrases invariance thesis, lambda calculus, weak call-by-value reduction, time and space complexity, abstract machines Supplement Material A Coq formalisation of the results in Section 2 and Section 3 is available at https://ps.uni-saarland.de/extras/wcbv-reasonable.

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Towards a Verified Cost Model for Call-by-Push-Value

Chen

2022

Companion Proceedings of the 2022 ACM SIGPLAN International Conference on Systems, Programming, Languages, and Applications: So

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Formal Small-Step Verification of a Call-by-Value Lambda Calculus Machine

Cited by 4 publications

References 19 publications

A certifying extraction with time bounds from Coq to call-by-value $λ$-calculus

A certifying extraction with time bounds from Coq to call-by-value $λ$-calculus

The weak call-by-value λ-calculus is reasonable for both time and space

Towards a Verified Cost Model for Call-by-Push-Value

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