Sequent calculus as a compiler intermediate language

Downen, Paul; Maurer, Luke; Ariola, Zena M.; Jones, Simon Peyton

doi:10.1145/3022670.2951931

Cited by 4 publications

(5 citation statements)

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“…To remedy such code duplication, as well as to increase the expressiveness of the IR to encode complex control flow, LEAN uses join-points [10], [11]. These join-points introduce the ability to create labels within the IR that control flow can jump to.…”

Section: B Join Pointsmentioning

confidence: 99%

Lambda the Ultimate SSA: Optimizing Functional Programs in SSA

Bhat¹,

Grosser²

2022

Preprint

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Static Single Assignment (SSA) is the workhorse of modern optimizing compilers for imperative programming languages. However, functional languages have been slow to adopt SSA and prefer to use intermediate representations based on minimal lambda calculi due to SSA's inability to express higherorder constructs. We exploit a new SSA construct -regionsin order to express functional optimizations via classical SSAbased reasoning. Region optimization currently relies on ad-hoc analyses and transformations on imperative programs. These ad-hoc transformations are sufficient for imperative languages as regions are used in a limited fashion. In contrast, we use regions pervasively to model sub-expressions in our functional IR. This motivates us to systematize region optimizations. We extend classical SSA reasoning to regions for functional-style analyses and transformations. We implement a new SSA+regions based backend for LEAN4, a theorem prover that implements a purely functional, dependently typed programming language. Our backend is feature-complete and handles all constructs of LEAN4's functional intermediate representation λrc within the SSA framework. We evaluate our proposed region optimizations by optimizing λrc within an SSA+regions based framework implemented in MLIR and demonstrating performance parity with the current LEAN4 backend. We believe our work will pave the way for a unified optimization framework capable of representing, analyzing, and optimizing both functional and imperative languages.

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Section: B Join Pointsmentioning

confidence: 99%

Lambda the Ultimate SSA: Optimizing Functional Programs in SSA

Bhat¹,

Grosser²

2022

Preprint

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“…Applicative contexts are thus considered of the restricted shape q • E. contexts. Formally, the syntax (to which we add constants k and co-constants κ) is defined by 18 :…”

Section: Classical Call-by-needmentioning

confidence: 99%

“…Rather than directly proving the normalization of dPA ω , we choose to first give an alternative presentation of the system under the shape of a sequent calculus, which we call dLPA ω . Indeed, sequent calculus presentations of a calculus usually provides good intermediate steps for CPS translations [56,57,18] since they enforce a decomposition of the reduction system into finer-grained rules. To this aim, we first handled separately the difficulties peculiar to the definition of such a calculus: on the one hand, we proved with Herbelin the normalization of a calculus with control operators and lazy evaluation [55]; on the other hand, we defined a classical sequent calculus with dependent types [54].…”

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confidence: 99%

“…(1) reduce dPA ω to an equivalent presentation in a sequent calculus fashion, (2) use the methodology of semantic artifacts to define a CPS or a realizability interpretation. Defining a sequent calculus presentation of a calculus is indeed known to be a good intermediate step for the definition of continuation-passing style translations [56,57,18]. In our case it forces to us to rephrase the contextual rules into abstract-machine like reduction rules which only depend of the top-level command, thus solving the last difficulty evoked above.…”

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confidence: 99%

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A constructive proof of dependent choice in classical arithmetic via memoization

Miquey¹

2019

Preprint

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In a recent paper [29], Herbelin developed dPA ω , a calculus in which constructive proofs for the axioms of countable and dependent choices could be derived via the memoization of choice functions. However, the property of normalization (and therefore the one of soundness) was only conjectured. The difficulty for the proof of normalization is due to the simultaneous presence of dependent types (for the constructive part of the choice), of control operators (for classical logic), of coinductive objects (to encode functions of type N → A into streams (a0, a1, . . .)) and of lazy evaluation with sharing (for memoizing these coinductive objects).Elaborating on previous works, we introduce in this paper a variant of dPA ω presented as a sequent calculus. On the one hand, we take advantage of a variant of Krivine classical realizability that we developed to prove the normalization of classical call-by-need [55]. On the other hand, we benefit from dL tp , a classical sequent calculus with dependent types in which type safety is ensured by using delimited continuations together with a syntactic restriction [54]. By combining the techniques developed in these papers, we manage to define a realizability interpretation à la Krivine of our calculus that allows us to prove normalization and soundness. This paper goes over the whole process, starting from Herbelin's calculus dPA ω until our introduction of its sequent calculus counterpart dLPA ω .

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“…To explore the space of intermediate languages, we use a dual sequent calculus framework which provides a machine-like language suitable for representing functional programs [DMAPJ16]. We begin by reviewing this general framework; specifically: • (Section 2) We introduce the syntax and reduction theory of λµμ-a term assignment for Gentzen's sequent calculus LK [Gen35].…”

Section: Introductionmentioning

confidence: 99%

Compiling With Classical Connectives

Downen,

Ariola

2019

Preprint

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The study of polarity in computation has revealed that an "ideal" programming language combines both call-by-value and call-by-name evaluation; the two calling conventions are each ideal for half the types in a programming language. But this binary choice leaves out call-by-need which is used in practice to implement lazy-by-default languages like Haskell. We show how the notion of polarity can be extended beyond the value/name dichotomy to include call-by-need by adding a mechanism for sharing which is enough to compile a Haskell-like functional language with user-defined types. The key to capturing sharing in this mixed-evaluation setting is to generalize the usual notion of polarity "shifts:" rather than just two shifts (between positive and negative) we have a family of four dual shifts.We expand on this idea of logical duality-"and" is dual to "or;" proof is dual to refutation-for the purpose of compiling a variety of types. Based on a general notion of data and codata, we show how classical connectives can be used to encode a wide range of built-in and user-defined types. In contrast with an intuitionistic logic corresponding to pure functional programming, these classical connectives bring more of the pleasant symmetries of classical logic to the computationally-relevant, constructive setting. In particular, an involutive pair of negations bridges the gulf between the wide-spread notions of parametric polymorphism and abstract data types in programming languages. To complete the study of duality in compilation, we also consider the dual to call-by-need evaluation, which shares the computation within the control flow of a program instead of computation within the information flow.

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Sequent calculus as a compiler intermediate language

Cited by 4 publications

References 46 publications

Lambda the Ultimate SSA: Optimizing Functional Programs in SSA

Lambda the Ultimate SSA: Optimizing Functional Programs in SSA

A constructive proof of dependent choice in classical arithmetic via memoization

Compiling With Classical Connectives

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