arXiv subjects

Giulio Guerrieri

Publications and source records attributed to Giulio Guerrieri.

At least 19 recordsLinked to original sources

arXiv2026

Connectivity at the crossroad of intuitionistic and classical polarizations in linear logic

We investigate a property that extends the Danos-Regnier correctness criterion for linear logic proof-structures. The property applies to the correctness graphs of a proof-structure: it states that any such graph is acyclic and the number of its connected components is exactly one more than the number of nodes bottom or weakening. This is known to be necessary but not sufficient in multiplicative exponential linear logic (MELL) to recover a sequent calculus proof from a proof-structure. We present a geometric restriction on proof-structures allowing us to turn this necessary property into a sufficient one, computationally efficient: we can thus introduce the notable fragment VMELL of MELL for which the property is indeed a correctness criterion. The fragment VMELL brings together the classical and intuitionistic polarizations. We translate the bang calculus terms into proof-nets of VMELL, factorize the usual translations in linear logic of the call-by-name and call-by-value lambda-calculi, prove that cut elimination simulates bang reduction, and provide an explicit characterization of the bang calculus terms as proof-nets.

cs.LO↗

arXiv2025

Non-wellfounded parsimonious proofs and non-uniform complexity

In this paper we investigate the complexity-theoretical aspects of cyclic and non-wellfounded proofs in the context of parsimonious logic, a variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. We present non-wellfounded parsimonious proof systems capturing the classes $\mathbf{FP}$ and $\mathbf{FP}/\mathsf{poly}$. Soundness is established via a polynomial modulus of continuity for continuous cut-elimination. Completeness relies on an encoding of polynomial Turing machines with advice within a type assignment system based on parsimonious logic. As a byproduct of our proof methods, we establish a series of characterisation results for various finitary proof systems.

cs.LO↗

arXiv2025

Infinitary Cut-Elimination for Non-Wellfounded Parsimonious Linear Logic

We investigate non-wellfounded proof systems based on parsimonious logic, a weaker variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. Logical consistency is maintained at a global level by adapting a standard progressing criterion. We present an infinitary version of cut-elimination based on finite approximations, and we prove that, in presence of the progressing criterion, it returns well-defined non-wellfounded proofs at its limit. Furthermore, we show that cut-elimination preserves the progressive criterion and various regularity conditions internalizing degrees of proof-theoretical uniformity. Finally, we provide a denotational semantics for our systems based on the relational model.

cs.LO↗

arXiv2025

Closure Conversion, Flat Environments, and the Complexity of Abstract Machines

Closure conversion is a program transformation at work in compilers for functional languages to turn inner functions into global ones, by building closures pairing the transformed functions with the environment of their free variables. Abstract machines rely on similar and yet different concepts of closures and environments. In this paper, we study the relationship between the two approaches. We adopt a very simple λ-calculus with tuples as source language and study abstract machines for both the source language and the target of closure conversion. Moreover, we focus on the simple case of flat closures/environments, that is, with no sharing of environments. We provide three contributions. Firstly, a new simple proof technique for the correctness of closure conversion, inspired by abstract machines. Secondly, we show how the closure invariants of the target language allow us to design a new way of handling environments in abstract machines, not suffering the shortcomings of other styles. Thirdly, we study the machines from the point of view of time complexity, adapting analyses by Accattoli and co-authors. We show that closure conversion decreases various dynamic costs while increasing the size of the initial code. Despite these changes, the overall complexity of the machines before and after closure conversion turns out to be the same.

cs.PL↗

arXiv2024

The Benefits of Diligence

This paper studies the strength of embedding Call-by-Name ({\tt dCBN}) and Call-by-Value ({\tt dCBV}) into a unifying framework called the Bang Calculus ({\tt dBANG}). These embeddings enable establishing (static and dynamic) properties of {\tt dCBN} and {\tt dCBV} through their respective counterparts in {\tt dBANG}. While some specific static properties have been already successfully studied in the literature, the dynamic ones are more challenging and have been left unexplored. We accomplish that by using a standard embedding for the (easy) {\tt dCBN} case, while a novel one must be introduced for the (difficult) {\tt dCBV} case. Moreover, a key point of our approach is the identification of {\tt dBANG} diligent reduction sequences, which eases the preservation of dynamic properties from {\tt dBANG} to {\tt dCBN}/{\tt dCBV}. We illustrate our methodology through two concrete applications: confluence/factorization for both {\tt dCBN} and {\tt dCBV} are respectively derived from confluence/factorization for {\tt dBANG}.

cs.LO↗

arXiv2024

Meaningfulness and Genericity in a Subsuming Framework

This paper studies the notion of meaningfulness for a unifying framework called dBang-calculus, which subsumes both call-by-name (dCbN) and call-by-value (dCbV). We first characterize meaningfulness in dBang by means of typability and inhabitation in an associated non-idempotent intersection type system previously proposed in the literature. We validate the proposed notion of meaningfulness by showing two properties (1) consistency of the theory $\mathcal{H}$ equating meaningless terms and (2) genericity, stating that meaningless subterms have no bearing on the significance of meaningful terms. The theory $\mathcal{H}$ is also shown to have a unique consistent and maximal extension. Last but not least, we show that the notions of meaningfulness and genericity in the literature for dCbN and dCbV are subsumed by the respectively ones proposed here for the dBang-calculus.

cs.LO↗

arXiv2024

Genericity Through Stratification

A fundamental issue in the $λ$-calculus is to find appropriate notions for meaningfulness. It is well-known that in the call-by-name $λ$-calculus (CbN) the meaningful terms can be identified with the solvable ones, and that this notion is not appropriate in the call-by-value $λ$-calculus (CbV). This paper validates the challenging claim that yet another notion, previously introduced in the literature as potential valuability, appropriately represents meaningfulness in CbV. Akin to CbN, this claim is corroborated by proving two essential properties. The first one is genericity, stating that meaningless subterms have no bearing on evaluating normalizing terms. To prove this (which was an open problem), we use a novel approach based on stratified reduction, indifferently applicable to CbN and CbV, and in a quantitative way. The second property concerns consistency of the smallest congruence relation resulting from equating all meaningless terms. While the consistency result is not new, we provide the first direct operational proof of it. We also show that such a congruence has a unique consistent and maximal extension, which coincides with a well-known notion of observational equivalence. Our results thus supply the formal concepts and tools that validate the informal notion of meaningfulness underlying CbN and CbV.

cs.LO↗

arXiv2023

Strong Call-by-Value and Multi Types

This paper provides foundations for strong (that is, possibly under abstraction) call-by-value evaluation for the lambda-calculus. Recently, Accattoli et al. proposed a form of call-by-value strong evaluation for the lambda-calculus, the external strategy, and proved it reasonable for time. Here, we study the external strategy using a semantical tool, namely Ehrhard's call-by-value multi types, a variant of intersection types. We show that the external strategy terminates exactly when a term is typable with so-called shrinking multi types, mimicking similar results for strong call-by-name. Additionally, the external strategy is normalizing in the untyped setting, that is, it reaches the normal form whenever it exists. We also consider the call-by-extended-value approach to strong evaluation shown reasonable for time by Biernacka et al. The two approaches turn out to not be equivalent: terms may be externally divergent but terminating for call-by-extended-value.

cs.LO↗

arXiv2023

Semantic Bounds and Strong Call-by-Value Normalization

This paper explores two topics at once: the use of denotational semantics to bound the evaluation length of functional programs, and the semantics of strong (that is, possibly under abstractions) call-by-value evaluation. About the first, we analyze de Carvalho's seminal use of relational semantics for bounding the evaluation length of lambda-terms, starting from the presentation of the semantics as an intersection types system. We focus on the part of his work which is usually neglected in its many recent adaptations, despite being probably the conceptually deeper one: how to transfer the bounding power from the type system to the relational semantics itself. We dissect this result and re-understand it via the isolation of a simpler size representation property. About the second, we use relational semantics to develop a semantical study of strong call-by-value evaluation, which is both a delicate and neglected topic. We give a semantic characterization of terms normalizable with respect to strong evaluation, providing in particular the first result of adequacy with respect to strong call-by-value. Moreover, we extract bounds about strong evaluation from both the type systems and the relational semantics. Essentially, we use strong call-by-value to revisit de Carvalho's semantic bounds, and de Carvalho's technique to provide semantical foundations for strong call-by-value.

cs.LO↗

arXiv2022

On reduction and normalization in the computational core

We study the reduction in a lambda-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.

cs.LO↗

arXiv2022

The Theory of Call-by-Value Solvability (long version)

The denotational semantics of the untyped lambda-calculus is a well developed field built around the concept of solvable terms, which are elegantly characterized in many different ways. In particular, unsolvable terms provide a consistent notion of meaningless term. The semantics of the untyped call-by-value lambda-calculus (CbV) is instead still in its infancy, because of some inherent difficulties but also because CbV solvable terms are less studied and understood than in call-by-name. On the one hand, we show that a carefully crafted presentation of CbV allows us to recover many of the properties that solvability has in call-by-name, in particular qualitative and quantitative characterizations via multi types. On the other hand, we stress that, in CbV, solvability plays a different role: identifying unsolvable terms as meaningless induces an inconsistent theory.

cs.LO↗

arXiv2022

Strategies for Asymptotic Normalization

We present a technique to study normalizing strategies when termination is asymptotic, that is, it appears as a limit, as opposite to reaching a normal form in a finite number of steps. Asymptotic termination occurs in several settings, such as effectful, and in particular probabilistic computation -- where the limits are distributions over the possible outputs -- or infinitary lambda-calculi -- where the limits are infinitary normal forms such as Boehm trees. As a concrete application, we obtain a result which is of independent interest: a normalization theorem for Call-by-Value (and -- in a uniform way -- for Call-by-Name) probabilistic lambda-calculus.

cs.LO↗

arXiv2022

Gluing resource proof-structures: inhabitation and inverting the Taylor expansion

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing (and deciding in the finite case) those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures. We also prove semi-decidability of the type inhabitation problem for cut-free MELL proof-structures.

cs.LO↗

arXiv2022

Call-by-Value Solvability and Multi Types

This paper provides a characterization of call-by-value solvability using call-by-value multi types. Our work is based on Accattoli and Paolini's characterization of call-by-value solvable terms as those terminating with respect to the solving strategy of the value substitution calculus, a refinement of Plotkin's call-by-value $λ$-calculus. Here we show that the solving strategy terminates on a term $t$ if and only if $t$ is typable in a certain way in the multi type system induced by Ehrhard's call-by-value relational semantics. Moreover, we show how to extract from the type system exact bounds on the length of the solving evaluation and on the size of its normal form, adapting de Carvalho's technique for call-by-name.

cs.LO↗

arXiv2021

A Deep Inference System for Differential Linear Logic

Differential linear logic (DiLL) provides a fine analysis of resource consumption in cut-elimination. We investigate the subsystem of DiLL without promotion in a deep inference formalism, where cuts are at an atomic level. In our system every provable formula admits a derivation in normal form, via a normalization procedure that commutes with the translation from sequent calculus to deep inference.

cs.LO↗

arXiv2021

Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic (long version)

In each variant of the lambda-calculus, factorization and normalization are two key-properties that show how results are computed. Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the lambda-calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV. The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features.

cs.LO↗

arXiv2020

Factorize Factorization

Factorization -- a simple form of standardization -- is concerned with reduction strategies, i.e. how a result is computed. We present a new technique for proving factorization theorems for compound rewriting systems in a modular way, which is inspired by the Hindley-Rosen technique for confluence. Specifically, our technique is well adapted to deal with extensions of the call-by-name and call-by-value lambda-calculi. The technique is first developed abstractly. We isolate a sufficient condition (called linear swap) for lifting factorization from components to the compound system, and which is compatible with beta-reduction. We then closely analyze some common factorization schemas for the lambda-calculus. Concretely, we apply our technique to diverse extensions of the lambda-calculus, among which de' Liguoro and Piperno's non-deterministic lambda-calculus and -- for call-by-value -- Carraro and Guerrieri's shuffling calculus. For both calculi the literature contains factorization theorems. In both cases, we give a new proof which is neat, simpler than the original, and strikingly shorter.

cs.LO↗

arXiv2020

Decomposing Probabilistic Lambda-calculi

A notion of probabilistic lambda-calculus usually comes with a prescribed reduction strategy, typically call-by-name or call-by-value, as the calculus is non-confluent and these strategies yield different results. This is a break with one of the main advantages of lambda-calculus: confluence, which means results are independent from the choice of strategy. We present a probabilistic lambda-calculus where the probabilistic operator is decomposed into two syntactic constructs: a generator, which represents a probabilistic event; and a consumer, which acts on the term depending on a given event. The resulting calculus, the Probabilistic Event Lambda-Calculus, is confluent, and interprets the call-by-name and call-by-value strategies through different interpretations of the probabilistic operator into our generator and consumer constructs. We present two notions of reduction, one via fine-grained local rewrite steps, and one by generation and consumption of probabilistic events. Simple types for the calculus are essentially standard, and they convey strong normalization. We demonstrate how we can encode call-by-name and call-by-value probabilistic evaluation.

cs.LO↗