arXiv subjects

Pablo Barenbaum

Publications and source records attributed to Pablo Barenbaum.

16 recordsLinked to original sources

arXiv2026

Verifiers and Generators: Epistemic Semantics for Intuitionistic Logic (Long Version)

This paper explores epistemic realizability, a form of realizability in which the property that a piece of data constitutes evidence for a logical proposition is semi-decidable. In this framework, each proposition A is assigned a verifier} program that checks whether a datum X is a realizer for A, and a dual generator program that behaves as a generic realizer for X. We propose epistemic realizability interpretations for minimal logic, second-order intuitionistic logic, and higher-order intuitionistic logic, proving that each system is sound and complete under the proposed semantics.

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arXiv2026

A Classical Linear $\lambda$-Calculus based on Contraposition

We present a novel linear $\lambda$-calculus for Classical Multiplicative Exponential Linear Logic (\MELL) along the lines of the propositions-as-types paradigm. Starting from the standard term assignment for Intuitionistic Multiplicative Linear Logic (IMLL), we observe that if we incorporate linear negation, its involutive nature implies that both $A\multimap B$ and $B^\perp\multimap A^\perp$ should have the same proofs. The introduction of a linear modus tollens rule, stating that from $B^\perp\multimap A^\perp$ and $A$ we may conclude $B$, allows one to recover classical MLL. Furthermore, a term assignment for this elimination rule, {the study of proof normalization in a $\lambda$-calculus with this elimination rule} prompts us to define the novel notion of contra-substitution $t \{ a \backslash\!\backslash s \}$. Introduced alongside linear substitution, contra-substitution denotes the term that results from "grabbing" the unique occurrence of $a$ in $t$ and "pulling" from it, in order to turn the term $t$ inside out (much like a sock) and then replacing $a$ with $s$. We call the one-sided natural deduction presentation of classical MLL, the $\lambda_{\rm MLL}$-calculus. Guided by the behavior of contra-substitution in the presence of the exponentials, we extend it to a similar presentation for MELL. We prove that this calculus is sound and complete with respect to MELL and that it satisfies the standard properties of a typed programming language: subject reduction, confluence and strong normalization. Moreover, we show that several well-known term assignments for classical logic can be encoded in $\lambda_{\rm MLL}$.

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arXiv2025

Strong normalization through idempotent intersection types: a new syntactical approach

It is well-known that intersection type assignment systems can be used to characterize strong normalization (SN). Typical proofs that typable lambda-terms are SN in these systems rely on semantical techniques. In this work, we study $\Lambda_\cap^e$, a variant of Coppo and Dezani's (Curry-style) intersection type system, and we propose a syntactical proof of strong normalization for it. We first design $\Lambda_\cap^i$, a Church-style version, in which terms closely correspond to typing derivations. Then we prove that typability in $\Lambda_\cap^i$ implies SN through a measure that, given a term, produces a natural number that decreases along with reduction. Finally, the result is extended to $\Lambda_\cap^e$, since the two systems simulate each other.

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arXiv2025

Sharing and Linear Logic with Restricted Access (Extended Version)

The two Girard translations provide two different means of obtaining embeddings of Intuitionistic Logic into Linear Logic, corresponding to different lambda-calculus calling mechanisms. The translations, mapping A -> B respectively to !A -o B and !(A -o B), have been shown to correspond respectively to call-by-name and call-by-value. In this work, we split the of-course modality of linear logic into two modalities, written "!" and "$\bullet$". Intuitively, the modality "!" specifies a subproof that can be duplicated and erased, but may not necessarily be "accessed", i.e. interacted with, while the combined modality "$!\bullet$" specifies a subproof that can moreover be accessed. The resulting system, called MSCLL, enjoys cut-elimination and is conservative over MELL. We study how restricting access to subproofs provides ways to control sharing in evaluation strategies. For this, we introduce a term-assignment for an intuitionistic fragment of MSCLL, called the $\lambda!\bullet$-calculus, which we show to enjoy subject reduction, confluence, and strong normalization of the simply typed fragment. We propose three sound and complete translations that respectively simulate call-by-name, call-by-value, and a variant of call-by-name that shares the evaluation of its arguments (similarly as in call-by-need). The translations are extended to simulate the Bang-calculus, as well as weak reduction strategies.

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arXiv2024

Useful Evaluation: Syntax and Semantics (Technical Report)

This work provides the first inductive definition of useful CBV evaluation. For that, we first restrict the substitution operation in the Value Substitution Calculus to be linear, yielding the LCBV strategy. We then further restrict substitution in LCBV, so that substitution contributes to the progress of the computation. This optimisation is the UCBV strategy, and its notion of substitution is sensitive to the surrounding evaluation context, so it is non-trivial to capture it inductively. Moreover, we show that UCBV is a sound and complete implementation of LCBV, optimised to implement useful evaluation. As a further contribution, we show that an existing notion of usefulness in the literature, namely the GLAMoUr abstract machine, implements the UCBV strategy with polynomial overhead in time. This establishes that UCBV is time-invariant, i.e., that the number of reduction steps to normal form in UCBV can be used as a measure of time complexity. Defining UCBV leads us to the first semantic model of useful CBV evaluation through system U, a non-idempotent intersection type system. Our main result is a characterisation of termination for useful CBV evaluation via system U: a term is typable in system U if and only if it terminates in UCBV. Additionally, system U provides a quantitative interpretation for UCBV, offering exact step-count information for program evaluation. Even though the specification of the operational semantics of UCBV is highly complex, system U is notably simple. As far as we know, system U is one of the scarce quantitative type systems capturing exactly the substitution step-count for a call-by-value strategy.

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arXiv2024

Hybrid Intersection Types for PCF (Extended Version)

Intersection type systems have been independently applied to different evaluation strategies, such as call-by-name (CBN) and call-by-value (CBV). These type systems have been then generalized to different subsuming paradigms being able, in particular, to encode CBN and CBV in a unique unifying framework. However, there are no intersection type systems that explicitly enable CBN and CBV to cohabit together without making use of an encoding into a common target framework. This work proposes an intersection type system for PCF with a specific notion of evaluation, called PCFH. Evaluation in PCFH actually has a hybrid nature, in the sense that CBN and CBV operational behaviors cohabit together. Indeed, PCFH combines a CBV-like operational behavior for function application with a CBN-like behavior for recursion. This hybrid nature is reflected in the type system, which turns out to be sound and complete with respect to PCFH: not only typability implies normalization, but also the converse holds. Moreover, the type system is quantitative, in the sense that the size of typing derivations provides upper bounds for the length of the reduction sequences to normal form. This type system is then refined to a tight one, offering exact information regarding the length of normalization sequences. This is the first time that a sound and complete quantitative type system has been designed for a hybrid computational model.

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arXiv2023

A Diamond Machine For Strong Evaluation

Abstract machines for strong evaluation of the $\lambda$-calculus enter into arguments and have a set of transitions for backtracking out of an evaluated argument. We study a new abstract machine which avoids backtracking by splitting the run of the machine in smaller jobs, one for argument, and that jumps directly to the next job once one is finished. Usually, machines are also deterministic and implement deterministic strategies. Here we weaken this aspect and consider a light form of non-determinism, namely the diamond property, for both the machine and the strategy. For the machine, this introduces a modular management of jobs, parametric in a scheduling policy. We then show how to obtain various strategies, among which leftmost-outermost evaluation.

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arXiv2023

Two Decreasing Measures for Simply Typed Lambda-Terms (Extended Version)

This paper defines two decreasing measures for terms of the simply typed lambda-calculus, called the W-measure and the Tm-measure. A decreasing measure is a function that maps each typable lambda-term to an element of a well-founded ordering, in such a way that contracting any beta-redex decreases the value of the function, entailing strong normalization. Both measures are defined constructively, relying on an auxiliary calculus, a non-erasing variant of the lambda-calculus. In this system, dubbed the m-calculus, each beta-step creates a "wrapper" containing a copy of the argument that cannot be erased and cannot interact with the context in any other way. Both measures rely crucially on the observation, known to Turing and Prawitz, that contracting a redex cannot create redexes of higher degree, where the degree of a redex is defined as the height of the type of its lambda-abstraction. The W-measure maps each lambda-term to a natural number, and it is obtained by evaluating the term in the m-calculus and counting the number of remaining wrappers. The Tm-measure maps each lambda-term to a structure of nested multisets, where the nesting depth is proportional to the maximum redex degree.

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arXiv2022

Reductions in Higher-Order Rewriting and Their Equivalence

Proof terms are syntactic expressions that represent computations in term rewriting. They were introduced by Meseguer and exploited by van Oostrom and de Vrijer to study equivalence of reductions in (left-linear) first-order term rewriting systems. We study the problem of extending the notion of proof term to higher-order rewriting, which generalizes the first-order setting by allowing terms with binders and higher-order substitution. In previous works that devise proof terms for higher-order rewriting, such as Bruggink's, it has been noted that the challenge lies in reconciling composition of proof terms and higher-order substitution (\b{eta}-equivalence). This led Bruggink to reject "nested" composition, other than at the outermost level. In this paper, we propose a notion of higher-order proof term we dub rewrites that supports nested composition. We then define two notions of equivalence on rewrites, namely permutation equivalence and projection equivalence, and show that they coincide. We also propose a standardization procedure, that computes a canonical representative of the permutation equivalence class of a rewrite.

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arXiv2022

Proofs and Refutations for Intuitionistic and Second-Order Logic (Extended Version)

The lambda-PRK-calculus is a typed lambda-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend lambda-PRK to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order lambda-PRK, and we study canonicity results.

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arXiv2021

A Constructive Logic with Classical Proofs and Refutations (Extended Version)

We study a conservative extension of classical propositional logic distinguishing between four modes of statement: a proposition may be affirmed or denied, and it may be strong or classical. Proofs of strong propositions must be constructive in some sense, whereas proofs of classical propositions proceed by contradiction. The system, in natural deduction style, is shown to be sound and complete with respect to a Kripke semantics. We develop the system from the perspective of the propositions-as-types correspondence by deriving a term assignment system with confluent reduction. The proof of strong normalization relies on a translation to System F with Mendler-style recursion.

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arXiv2021

Semantics of a Relational λ-Calculus (Extended Version)

We extend the λ-calculus with constructs suitable for relational and functional-logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions. In order to be able to unify λ-expressions and still obtain a confluent theory, we depart from related approaches, such as λProlog, in that we do not attempt to solve higher-order unification. Instead, abstractions are decorated with a location, which intuitively may be understood as its memory address, and we impose a simple coherence invariant: abstractions in the same location must be equal. This allows us to formulate a confluent small-step operational semantics which only performs first-order unification and does not require strong evaluation (below lambdas). We study a simply typed version of the system. Moreover, a denotational semantics for the calculus is proposed and reduction is shown to be sound with respect to the denotational semantics.

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arXiv2019

Factoring Derivation Spaces via Intersection Types (Extended Version)

In typical non-idempotent intersection type systems, proof normalization is not confluent. In this paper we introduce a confluent non-idempotent intersection type system for the lambda-calculus. Typing derivations are presented using proof term syntax. The system enjoys good properties: subject reduction, strong normalization, and a very regular theory of residuals. A correspondence with the lambda-calculus is established by simulation theorems. The machinery of non-idempotent intersection types allows us to track the usage of resources required to obtain an answer. In particular, it induces a notion of garbage: a computation is garbage if it does not contribute to obtaining an answer. Using these notions, we show that the derivation space of a lambda-term may be factorized using a variant of the Grothendieck construction for semilattices. This means, in particular, that any derivation in the lambda-calculus can be uniquely written as a garbage-free prefix followed by garbage.

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arXiv2016

A Strong Distillery

Abstract machines for the strong evaluation of lambda-terms (that is, under abstractions) are a mostly neglected topic, despite their use in the implementation of proof assistants and higher-order logic programming languages. This paper introduces a machine for the simplest form of strong evaluation, leftmost-outermost (call-by-name) evaluation to normal form, proving it correct, complete, and bounding its overhead. Such a machine, deemed Strong Milner Abstract Machine, is a variant of the KAM computing normal forms and using just one global environment. Its properties are studied via a special form of decoding, called a distillation, into the Linear Substitution Calculus, neatly reformulating the machine as a standard micro-step strategy for explicit substitutions, namely linear leftmost-outermost reduction, i.e., the extension to normal form of linear head reduction. Additionally, the overhead of the machine is shown to be linear both in the number of steps and in the size of the initial term, validating its design. The study highlights two distinguished features of strong machines, namely backtracking phases and their interactions with abstractions and environments.

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arXiv2014

Distilling Abstract Machines (Long Version)

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between big-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.

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arXiv2010

Superdevelopments for Weak Reduction

We study superdevelopments in the weak lambda calculus of Cagman and Hindley, a confluent variant of the standard weak lambda calculus in which reduction below lambdas is forbidden. In contrast to developments, a superdevelopment from a term M allows not only residuals of redexes in M to be reduced but also some newly created ones. In the lambda calculus there are three ways new redexes may be created; in the weak lambda calculus a new form of redex creation is possible. We present labeled and simultaneous reduction formulations of superdevelopments for the weak lambda calculus and prove them equivalent.

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