The untyped computational λ-calculus and its intersection type discipline

de’Liguoro, Ugo; Treglia, Riccardo

doi:10.1016/j.tcs.2020.09.029

Cited by 5 publications

(5 citation statements)

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“…This work is a development of [14], where we considered a pure untyped computational λ -calculus, namely without operations nor constants. Therefore, the monad T and the respective unit and bind were generic, so that types cannot convey any specific information about the domain T D, nor about effects represented by the monad.…”

Section: Discussion and Related Workmentioning

confidence: 99%

“…2 An untyped imperative λ -calculus: λ imp Imperative extensions of the λ -calculus, both typed and type-free, are usually based on the call-by-value λ -calculus, and enriched with constructs for reading and writing to the store. Aiming at exploring the semantics of side effects in computational calculi, where "impure" functions are modeled by pure ones sending values to computations in the sense of [22], we consider the calculus introduced in [14], to which we add syntax denoting algebraic effect operations à la Plotkin and Power [23,24,25] over a suitable state monad.…”

Section: Introductionmentioning

confidence: 99%

“…Starting with [14], we have considered an untyped variant of Moggi's computational λ -calculus, with two sorts of terms: values denoting points of some domain D, and computations denoting points of T D, where T is some generic monad and D ∼ = D − → T D. The goal was to show how such a calculus can be equipped with an operational semantics and an intersection type system, such that types are invariant under reduction and expansion of computation terms, and convergent computations are characterized by having non-trivial types in the system.…”

Section: Introductionmentioning

confidence: 99%

“…Terms of the shape [V ] and M ⋆ V are written return V and M >>= V in Haskell-like syntax, respectively. With respect to [14] we write [V ] instead of unit V . With respect to the syntax of the "imperative λ -calculus" in [17], we do not have the let construct, nor the application VW among values.…”

Section: Introductionmentioning

confidence: 99%

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From Semantics to Types: the Case of the Imperative lambda-Calculus

de’Liguoro¹,

Treglia²

2021

Electron. Proc. Theor. Comput. Sci.

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We propose an intersection type system for an imperative λ -calculus based on a state monad and equipped with algebraic operations to read and write to the store. The system is derived by solving a suitable domain equation in the category of ω-algebraic lattices; the solution consists of a filter-model generalizing the well known construction for ordinary λ -calculus. Then the type system is obtained out of the term interpretations into the filter-model itself. The so obtained type system satisfies the "type-semantics" property, and it is sound and complete by construction.

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Section: Discussion and Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

From Semantics to Types: the Case of the Imperative lambda-Calculus

de’Liguoro¹,

Treglia²

2021

Electron. Proc. Theor. Comput. Sci.

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“…First, it can be shown that the term M ′ above and ∆!∆ are observational equivalent. Second, the denotational models and type systems studied in [Ehr12,dT20] (which are compatible with λ © ) interpret M ′ in the same ways as ∆!∆, which is a β c -divergent term. It is then reasonable to expect that the two terms have the same operational behavior in λ © .…”

Section: The Essential Role Of σ Steps For Normalizationmentioning

confidence: 99%

On reduction and normalization in the computational core

Faggian¹,

Guerrieri²,

de’Liguoro³

et al. 2021

Preprint

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We study the reduction in a λ-calculus derived from Moggi's computational one, that we call the computational core. The reduction relation consists of rules obtained by orienting three monadic laws. Such laws, in particular associativity and identity, introduce intricacies in the operational analysis. We investigate the central notions of returning a value versus having a normal form, and address the question of normalizing strategies. Our analysis relies on factorization results. * βv V ′ (for some value V ′ ) (5) Such a result (Corollary 1 in [Plo75]) comes from an analysis of the reduction properties of → βv , namely standardization.As we will see, the rules induced by associativity and identity make the behavior of the reduction-and the study of its operational properties-nontrivial in the setting of any monadic λ-calculus. The issues are inherent to the rules coming from the monadic laws ( 2) and ( 3), independently of the syntactic representation of the calculus that internalizes them. The difficulty appears clearly if we want to follow a similar route to [Plo75], as we discuss next.Reduction vs evaluation. Following [Fel88], reduction → © and evaluation → w © of λ © can be defined as the closure of the reduction rules under arbitrary and evaluation contexts, respectively. Consider the following grammars:where the hole can be filled by terms in Com only. Observe that the closure under evaluation context E is precisely weak reduction.Weak reduction of λ © , however, turns out to be non-deterministic, nonconfluent, and its normal forms are not unique. The following is a counterexample to all such properties-see Section 5 for further examples.Such an issue is not specific to the syntax of the computational core. The same phenomena show up with the let-notation. Evaluation contexts are now generated byand examples similar to the one above can be reproduced. We give the details in Example 5.3.Content and contributions. The focus of this paper is an operational analysis of two crucial properties of a term M :(i) M returns a value (i.e. M → * © unit V , for some V value). (ii) M has a normal form (i.e. M → * © N , for some N normal).The cornerstone of our analysis are factorization results (also called semistandardization in the literature): any reduction sequence can be re-organized so to first performing specific steps and then everything else.Via factorization, we establish the following key result, analogous to (5), relating reduction and evaluation:Remark 3.4 (β c and β v ). We are now in place to show how a β v -reduction is simulated by a β c -reduction, possibly with more steps. Observe that the reduction relation → © is only on computations, therefore no computation N will ever reduce to some value V ; however, this is represented by a reduction N → * © !V , where !V is the coercion of the value V into a computation. Moreover, let us assume that M → * © !λx.M ′ . We have:M N ≡ (λz.zN )M → * © 1. → w id is non-deterministic, but it is confluent.2. → w σ is non-deterministic, non-confluent and normal forms ...

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From semantics to types: The case of the imperative λ-calculus

de’Liguoro

Treglia

2023

Theoretical Computer Science

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The untyped computational λ-calculus and its intersection type discipline

Cited by 5 publications

References 36 publications

From Semantics to Types: the Case of the Imperative lambda-Calculus

From Semantics to Types: the Case of the Imperative lambda-Calculus

On reduction and normalization in the computational core

From semantics to types: The case of the imperative λ-calculus

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