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arxiv: 2604.15290 · v3 · submitted 2026-04-16 · 💻 cs.PL

Recognition: unknown

Pure Borrow: Linear Haskell Meets Rust-Style Borrowing

Hiromi Ishii, Yusuke Matsushita

Pith reviewed 2026-05-10 08:54 UTC · model grok-4.3

classification 💻 cs.PL
keywords linear typesborrowingLinear Haskellmutable referencesRustpurityconfluenceparallel mutation
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The pith

Pure Borrow shows that Linear Haskell can support Rust-style non-local borrowing while preserving purity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Linear Haskell adds linear types to control resource usage in a purely functional setting, yet it had no clear way to allow flexible, splittable borrowing like Rust without breaking purity or requiring monads. Pure Borrow supplies this capability through a library that introduces affine mutable references usable inside pure code, supporting parallel state mutation, free splitting and dropping of borrows, and no need for ownership hand-back. A sympathetic reader would care because the result keeps Haskell's lazy evaluation, first-class polymorphism, and leak freedom while adding imperative-style mutation that avoids the restrictions of IO or ST monads. The authors back the claim with a working implementation, a parallel-computing case study, and a formal metatheory built on a new history-based model of borrowing.

Core claim

Pure Borrow is a library and formal framework that embeds Rust-style borrowing into Linear Haskell. It permits affine mutable references to be created, borrowed, split, and dropped non-locally inside pure computations, enabling parallel mutation without monadic wrappers. The framework guarantees purity, lazy evaluation, first-class polymorphism, and leak freedom. Safety, leak freedom, and confluence are addressed via a history-based model that tracks borrow lifetimes to enforce the required invariants for splittable, non-communicating ownership transfer.

What carries the argument

The history-based model of borrowing, which records sequences of borrow events to enforce rules for creation, splitting, and dropping of affine references without direct lender-borrower communication.

If this is right

  • Parallel mutable state updates become expressible directly inside pure Haskell functions without IO or ST.
  • Borrowed references can be freely split and discarded without returning ownership to the original lender.
  • Leak freedom and confluence hold even when multiple borrowers operate concurrently on the same reference.
  • First-class polymorphic functions can still be used with borrowed mutable data, unlike typical Rust restrictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same history-tracking approach could be adapted to add similar borrowing to other linearly typed functional languages.
  • It opens a route to imperative data structures inside pure code that remain compatible with Haskell's laziness and equational reasoning.
  • Future extensions might support more advanced patterns such as conditional reborrowing while keeping the purity guarantees.

Load-bearing premise

The history-based model correctly encodes the semantics required to prove that non-local splittable borrowing remains safe, leak-free, and confluent in Linear Haskell.

What would settle it

A concrete Pure Borrow program that type-checks under the history model yet produces a data race, use-after-free, or non-confluent result when executed would falsify the safety and confluence claims.

Figures

Figures reproduced from arXiv: 2604.15290 by Hiromi Ishii, Yusuke Matsushita.

Figure 1
Figure 1. Figure 1: ST monad APIs and an example usage [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Core Linear Haskell APIs and an example usage before Pure Borrow. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Rust-style borrowing illustrated. Each Ferris corresponds to a borrower. Ferris illustrations by Karen [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Core APIs of Pure Borrow and an example usage. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pure Borrow APIs for parallelism and an example usage, modifying [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reborrowing API and an example usage, modifying [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Core APIs for lifetimes. newtype BO𝛼 ::Lifetime a; pure :: a ⊸ BO𝛼 a (≫=) :: BO𝛼 a ⊸ (a ⊸ BO𝛼 b) ⊸ BO𝛼 b; parBO :: BO𝛼 a ⊸ BO𝛼 b ⊸ BO𝛼 (a, b) execBO :: Now𝛼 ⊸ BO𝛼 a ⊸ (Now𝛼 , a) runBO :: Linearly =◦ (∀𝛼. BO𝛼 (End𝛼 → a)) ⊸ a runBO bo = newLifetime λnow → do◦ (now, f) ← execBO now bo Ur end ← endLifetime now; f end sexecBO :: Now𝛽 ⊸ BO𝛽∧𝛼 a ⊸ BO𝛼 (Now𝛽 , a) srunBO :: Linearly =◦ (∀𝛽. BO𝛽∧𝛼 (End𝛽 → a)) ⊸ BO𝛼 … view at source ↗
Figure 8
Figure 8. Figure 8: Core APIs for the BO monad. Static as the maximum element. We introduce two basic tokens for lifetimes, the live lifetime token Now𝛼 and the dead lifetime token End𝛼 . 8 The former exclusively asserts that 𝛼 is ongoing, and the latter persistently asserts that 𝛼 has ended. With newLifetime, we can get NowAl ι for a fresh id ι. Then with endLifetime, we can consume that to get EndAl ι . The idea of these to… view at source ↗
Figure 9
Figure 9. Figure 9: Core APIs for borrowing. getAt :: 𝛽 ≤ 𝛼 ⇒ Int → Borrow𝛼 k (Vector a) ⊸ BO𝛽 (Borrow𝛼 k a) swapAt :: 𝛽 ≤ 𝛼 ⇒ Int → Int → Mut𝛼 (Vector a) ⊸ BO𝛽 (Mut𝛼 (Vector a)) updateAt :: 𝛽 ≤ 𝛼 ⇒ Int → (a ⊸ BO𝛽 (b, a)) ⊸ Mut𝛼 (Vector a) ⊸ BO𝛽 (b, Mut𝛼 (Vector a)) modifyAt :: 𝛽 ≤ 𝛼 ⇒ Int → (a ⊸ a) ⊸ Mut𝛼 (Vector a) ⊸ BO𝛽 (Mut𝛼 (Vector a)) modifyAt i f mvec = do ((), mvec) ← updateAt i (λ a → pure ((), f a)) mvec pure mvec … view at source ↗
Figure 10
Figure 10. Figure 10: Core APIs for accessing vectors through borrowers. [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Core APIs for copying. splitPair :: Borrow𝛼 k (a, b) ⊸ (Borrow𝛼 k a, Borrow𝛼 k b) splitEither :: Borrow𝛼 k (Either a b) ⊸ Either (Borrow𝛼 k a) (Borrow𝛼 k b) splitAt :: Int → Borrow𝛼 k (Vector a) ⊸ (Borrow𝛼 k (Vector a), Borrow𝛼 k (Vector a)) [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Core APIs for subtyping. joinMut :: Borrow𝛽 k (Mut𝛼 a) ⊸ Borrow𝛽∧𝛼 k a reborrow :: Mut𝛼 a ⊸ (Mut𝛽∧𝛼 a, Lend𝛽 (Mut𝛼 a)) reborrowing :: Mut𝛼 a ⊸ (∀𝛽. Mut𝛽∧𝛼 a ⊸ BO𝛽∧𝛼 ′ r) ⊸ BO𝛼 ′ (r, Mut𝛼 a) sharing :: Mut𝛼 a ⊸ (∀𝛽. Share𝛽∧𝛼 a → BO𝛽∧𝛼 ′ r) ⊸ BO𝛼 ′ (r, Mut𝛼 a) copyAtMut :: (Copyable a, 𝛽 ≤ 𝛼) ⇒ Int → Mut𝛼 (Vector a) → BO𝛽 (Ur a, Mut𝛼 (Vector a)) reborrow mut = withLinearly mut λmut → let !(mut′ , lend) = bo… view at source ↗
Figure 14
Figure 14. Figure 14: Primitive and derived operators for reborrowing. [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: A naïve parallel implementation of quicksort and an illustration. [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The implementation of divide function. Work-stealing parallel quicksort. While the previous implementation is concise, it is rather naïve because parBO is a structural parallel composition that waits for both arguments to finish, which can lead to extra waiting time and degraded performance. To address this issue, we develop a general, modular, safe, and efficient API for work-stealing parallelization of … view at source ↗
Figure 17
Figure 17. Figure 17: Work-stealing parallelism. Pure Borrow can provide good basic building blocks for more sophisticated APIs, and we do not claim that it is the best API for work-stealing parallelism. DivideConquer wraps a function for each step of divide-and-conquer, which takes a mutable borrower, executes in the BO monad, and returns Result, which is either Done or Continues with divided parallelized pieces of work. Give… view at source ↗
Figure 18
Figure 18. Figure 18: The performance of parallel quicksort implementations. [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Program syntax. Lifetime variable α, β Lifetime id ι Lifetime 𝛼, 𝛽 F al ι | static | 𝛼 ∧ 𝛽 | α Multiplicity variable π, μ Multiplicity 𝜋, 𝜇 F 1 | ω | Î π Type variable 𝑋, 𝑌 Variable 𝑥 F α | ι | π | 𝑋 Type constructor F Borrower constructor 𝐵 F Mut | Share Type Type ∋ 𝑇 ,𝑈 F ∀𝑥.𝑇 | 𝑋 | 𝑇 →𝜋 𝑈 | F 𝑇¯ | Int | Linearly | Ref 𝑇 | Now𝛼 | End𝛼 | 𝐵 𝛼 𝑇 | Lend𝛼 𝑇 | BO𝛼 𝑇 𝑇 ⊸ 𝑈 ≜ 𝑇 →1 𝑈 𝑇 → 𝑈 ≜ 𝑇 →ω 𝑈 Data type dec… view at source ↗
Figure 20
Figure 20. Figure 20: Syntax for the type system. The biggest difference from §3 is that, for mutable state, our language supports only references Ref and omits vectors Vector for simplicity. Note that we can still emulate the functionality of Vector a by a list of references [Ref a]. 5.2 Type System We also formalize the type system. This is largely straightforward, following the approach of Bernardy et al. [2018] [PITH_FULL… view at source ↗
Figure 21
Figure 21. Figure 21: Core syntax for the two operational semantics. [PITH_FULL_IMAGE:figures/full_fig_p016_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Syntax for histories in denotational semantics. [PITH_FULL_IMAGE:figures/full_fig_p017_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Reduction example in mutative and denotational semantics. [PITH_FULL_IMAGE:figures/full_fig_p017_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Linearity and uniqueness axes in OxCaml and Pure Borrow. [PITH_FULL_IMAGE:figures/full_fig_p020_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Subtyping rules. 𝛤 ≼ 𝛤 𝛤 ≼ 𝛤 ′ 𝛤 ′ ≼ 𝛤 ′′ 𝛤 ≼ 𝛤 ′′ 𝜇 ≤ 𝜋 𝑇 ≼ 𝑈 x ::𝜋 𝑇 , 𝛤 ≼ x ::𝜇 𝑈 , 𝛤 x ::ω 𝑇 , 𝛤 ≼ 𝛤 [PITH_FULL_IMAGE:figures/full_fig_p026_25.png] view at source ↗
Figure 27
Figure 27. Figure 27: Operations on typing contexts. data () where ():: () data Bool where True:: Bool, False:: Bool data Ur 𝑋 where Ur:: 𝑋 → Ur 𝑋 data (𝑋, 𝑌) where (,) :: 𝑋 ⊸ 𝑌 ⊸ (𝑋, 𝑌) [PITH_FULL_IMAGE:figures/full_fig_p027_27.png] view at source ↗
Figure 29
Figure 29. Figure 29: Typing rules. Typing rules. The typing rules are listed in [PITH_FULL_IMAGE:figures/full_fig_p027_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: More syntax for the two operational semantics. [PITH_FULL_IMAGE:figures/full_fig_p028_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Parallel 𝐻 · 𝐻 ′ and sequential 𝐻 /𝐻 ′ composition of histories. x = 𝐾ˆ[y] ∈ 𝐸ˆ 𝐸ˆ; 𝑀; y loop 𝐸ˆ; 𝑀; x loop x = 𝐾¤ [y] ∈ 𝐸¤ 𝐸¤; y loop 𝐸¤; x loop [PITH_FULL_IMAGE:figures/full_fig_p029_31.png] view at source ↗
Figure 33
Figure 33. Figure 33: Basic reduction rules. version of the borrowed object. Notably, when the lender reclaims the ownership of the borrowed object by dred-reclaim, the latest version of the object is restored from the history. For this, we introduce the restoration-by-history predicate restore𝐻 𝑝 𝐸¤ x;y 𝐸¤′ , inductively defined by the rules listed in Fig. 36c. The predicate adds new items to the environment 𝐸¤ to get 𝐸¤′ , r… view at source ↗
Figure 34
Figure 34. Figure 34: Reduction rules for basic operators. li = ∅, y = 𝑣ˆ ∈ 𝐸ˆ x = newRef li y, 𝐸ˆ; 𝑀; x → x = Ref ℓ, 𝐸ˆ; ℓ ↦→ y, 𝑀; x y = Ref ℓ ∈ 𝐸ˆ x = freeRef y, 𝐸ˆ; ℓ ↦→ z, 𝑀; x → x = z, 𝐸ˆ; 𝑀; x (a) For mutative semantics. li = ∅, y = 𝑣¤ ∈ 𝐸¤ x = newRef li y, 𝐸¤; x → x = Ref y, 𝐸¤; x y = Ref z ∈ 𝐸¤ x = freeRef y, 𝐸¤; x → x = z, 𝐸¤; x (b) For denotational semantics [PITH_FULL_IMAGE:figures/full_fig_p031_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Basic reference-related reduction rules. [PITH_FULL_IMAGE:figures/full_fig_p031_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: Borrow-related reduction rules. able to update (we omit the superscript 𝑃 if it is empty). This information is vital to justify parBO, ensuring that the borrow paths updated by the two monad arguments are mutually disjoint. A resource context also contains borrow ids δ, variable-term bindings x = 𝑡ˆ ∝ 𝑡¤, x = 𝑣¤, and points-to tokens ℓ ↦→ x from the environments and memory. Importantly, bindings x = 𝑣ˆ ∝ … view at source ↗
Figure 37
Figure 37. Figure 37: Basic monadic reduction rules. A resource context can contain ghost resources gho to represent various auxiliary information. The meanings of ghost resources are as follows. A linearity witness linear is used for Linearly. A live lifetime token nowι.𝑖;𝑛 𝐻;𝑃 owns the part .𝑖 of the lifetime al ι, asserting that it is alive with the partial history 𝐻, has 𝑛 children tokens, and has the right to perform upda… view at source ↗
Figure 38
Figure 38. Figure 38: Reference-related monadic reduction rules. [PITH_FULL_IMAGE:figures/full_fig_p034_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: Syntax for the association system. a variable x typed 𝑇 . When 𝑞¯ is non-empty, it means that the object is also reborrowed under 𝑞¯. When 𝑞¯ is empty, we omit the last part ‘;𝑞¯’ of the subscript. We have tokens for a mutable borrower mut𝑝¯ , a shared borrower share𝑝¯ , and a lender lendδ 𝑣ˆ;x . We also have tokens for a (local) deposit depo𝑝¯ and a global deposit gdepoδ x;𝑇 ;𝐻 for the borrowed contents.… view at source ↗
Figure 40
Figure 40. Figure 40: Selected association rules. Assocaition rules. Selected rules for the association judgment are shown in [PITH_FULL_IMAGE:figures/full_fig_p035_40.png] view at source ↗
read the original abstract

A promising approach to unifying functional and imperative programming paradigms is to localize mutation using linear or affine types. Haskell, a purely functional language, was recently extended with linear types by Bernardy et al., in the name of Linear Haskell. However, it remained unknown whether such a pure language could safely support non-local borrowing in the style of Rust, where each borrower can be freely split and dropped without direct communication of ownership back to the lender. We answer this question affirmatively with Pure Borrow, a novel framework that realizes Rust-style borrowing in Linear Haskell with purity. Notably, it features parallel state mutation with affine mutable references inside pure computation, unlike the IO and ST monads and existing Linear Haskell APIs. It also enjoys purity, lazy evaluation, first-class polymorphism and leak freedom, unlike Rust. We implement Pure Borrow simply as a library in Linear Haskell and demonstrate its power with a case study in parallel computing. We formalize the core of Pure Borrow and build a metatheory that works toward establishing safety, leak freedom and confluence, with a new, history-based model of borrowing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents Pure Borrow, a library implementation in Linear Haskell that realizes Rust-style non-local, splittable borrowing using affine mutable references inside pure computations. It claims to enable parallel state mutation while preserving purity, lazy evaluation, first-class polymorphism, and leak freedom (unlike Rust or the IO/ST monads). The work includes a parallel-computing case study and a formalization of the core with a metatheory based on a novel history-based model of borrowing, aimed at establishing safety, leak freedom, and confluence.

Significance. If the metatheory holds, the result would be a notable advance in unifying linear/affine types with pure functional programming, offering a library-based path to imperative features without monadic overhead. The history-based model is a fresh semantic device that could inform future linear-type designs; the library implementation and case study provide immediate usability evidence. The paper earns credit for delivering a concrete library artifact and for attempting a metatheory that targets the three key properties simultaneously.

major comments (2)
  1. [§4.2] §4.2, Definition 4.1 (history-based borrowing model): the model encodes borrowing via sequences of access events, yet the invariant for splittable affine references (Lemma 4.5) assumes histories remain single-threaded after splitting; this assumption is load-bearing for the leak-freedom and confluence claims when non-local borrowers mutate in parallel, as required by the case study.
  2. [Theorem 5.3] Theorem 5.3 (confluence): the proof sketch relies on history reconciliation commuting under parallel mutation, but no reduction cases or counter-example analysis are given for the splittable-borrower scenario; without this, the central claim that Pure Borrow is confluent in the presence of non-local borrowing cannot be verified from the provided metatheory.
minor comments (2)
  1. [§3.1] §3.1: the library API is introduced with several combinators, but the linear-type signatures for the key borrowing primitives (e.g., borrow and split) are omitted from the main text and only appear in an appendix; this hinders readability.
  2. [Figure 2] Figure 2: the state-transition diagram for lender/borrower lifetimes is difficult to read at the printed size; labels on the split and drop arrows are too small.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying areas where the metatheory presentation can be strengthened. We agree that the history-based model and confluence proof require additional detail on splittable borrowers under parallel mutation to make the arguments fully verifiable. We will revise the manuscript to address both points.

read point-by-point responses

  1. Referee: [§4.2] §4.2, Definition 4.1 (history-based borrowing model): the model encodes borrowing via sequences of access events, yet the invariant for splittable affine references (Lemma 4.5) assumes histories remain single-threaded after splitting; this assumption is load-bearing for the leak-freedom and confluence claims when non-local borrowers mutate in parallel, as required by the case study.

    Authors: We appreciate this observation. In the history-based model, splitting an affine reference creates independent history segments for each borrower; the single-threaded invariant in Lemma 4.5 applies locally to each segment, while global consistency is restored via reconciliation at the lender upon borrow termination. Parallel mutations are permitted precisely because each segment records only its own accesses. However, the current write-up does not explicitly separate local vs. reconciled histories in the splittable case. We will revise Definition 4.1 and Lemma 4.5 to state the local-segment invariant clearly and add a short paragraph explaining how reconciliation preserves leak freedom and confluence for the parallel case study. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 (confluence): the proof sketch relies on history reconciliation commuting under parallel mutation, but no reduction cases or counter-example analysis are given for the splittable-borrower scenario; without this, the central claim that Pure Borrow is confluent in the presence of non-local borrowing cannot be verified from the provided metatheory.

    Authors: The referee is correct that the proof sketch in Theorem 5.3 is high-level and omits explicit reduction rules and case analysis for splittable borrowers. The underlying argument is that history reconciliation is a commutative operation on disjoint access sequences, which precludes write-write conflicts. To make this verifiable, we will expand the proof with the missing reduction cases (including the parallel split-and-mutate rule) and include a brief counter-example check showing that no confluence violation arises when two splittable borrowers mutate distinct regions. This material will be added to the main text or a dedicated appendix subsection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; novel history-based model introduced independently

full rationale

The paper presents Pure Borrow as a new library and formalization in Linear Haskell, with a newly introduced history-based model of borrowing used to prove safety, leak freedom, and confluence. No equations or derivations reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the metatheory is built from first principles on the new model rather than renaming or importing prior results as forced inputs. The central claims rest on the independent formalization and case study, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger reflects high-level claims; the work relies on the existing Linear Haskell extension and introduces a new history-based borrowing model without independent evidence provided here.

axioms (1)
  • domain assumption Linear types as extended in Linear Haskell by Bernardy et al.
    The framework builds directly on the prior Linear Haskell type system for resource tracking.
invented entities (1)
  • Pure Borrow framework and history-based borrowing model no independent evidence
    purpose: To enable non-local, splittable borrowing with parallel affine mutation inside pure Linear Haskell computations
    Newly postulated in this work to realize the claimed properties; no independent falsifiable evidence is described in the abstract.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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    Pure Borrow: Linear Haskell Meets Rust-Style Borrowing

    Report on the Programming Language Haskell, A Non-strict, Purely Functional Language.ACM SIGPLAN Notices 27, 5 (1992), 1. doi:10.1145/130697.130699 Jane Street. 2025.OxCaml. https://oxcaml.org/ Ralf Jung, Jacques-Henri Jourdan, Robbert Krebbers, and Derek Dreyer. 2018a. RustBelt: Securing the Foundations of the Rust Programming Language.Proc. ACM Program....

    work page doi:10.1145/130697.130699 1992

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    if 𝛽≤𝛼 0 and 𝛽≤𝛼 1, then 𝛽≤𝛼 0 ∧𝛼 1

    Dependent Types and Multi-monadic Effects in F ★. InProceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2016, St. Petersburg, FL, USA, January 20 - 22, 2016, Rastislav Bodík and Rupak Majumdar (Eds.). ACM, 256–270. doi:10.1145/2837614.2837655 The Rust Community. 2017.2094-nll - The Rust RFC Book. https:...

    work page doi:10.1145/2837614.2837655 2016