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Abstract Last issue of MSCS before moving to continuous publishing (first part of a collection devoted to LSFA 2021 and 2022).

Opetopes are algebraic descriptions of shapes corresponding to compositions in higher dimensions. As such, they offer an approach to higher-dimensional algebraic structures, and in particular, to the definition of weak ω-categories, which was the original motivation for their introduction by Baez and Dolan. They are classically defined inductively (as free operads in Leinster's approach, or as zoom complexes in the formalism of Kock et al.), using abstract constructions making them difficult to manipulate with a computer. Here, we present a purely syntactic description of opetopes and opetopic sets as a sequent calculus. Our main result is that well-typed opetopes in our sense are in bijection with opetopes as defined in the more traditional approaches. We expect that the resulting structures can serve as natural foundations for mechanized tools based on opetopes.

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We consider the Curry-Howard-Lambek correspondence for effectful computation and resource management, specifically proposing polarised calculi together with presheaf-enriched adjunction models as the starting point for a comprehensive semantic theory relating logical systems, typed calculi, and categorical models in this context. Our thesis is that the combination of effects and resources should be considered orthogonally. Model theoretically, this leads to an understanding of our categorical models from two complementary perspectives: (i) as a linearisation of CBPV (Call-by-Push-Value) adjunction models, and (ii) as an extension of linear/non-linear adjunction models with an adjoint resolution of computational effects. When the linear structure is cartesian and the resource structure is trivial we recover Levy’s notion of CBPV adjunction model, while when the effect structure is trivial we have Benton’s linear/non-linear adjunction models. Further instances of our model theory include the dialogue categories with a resource modality of Melliès and Tabareau, and the [E]EC ([Enriched] Effect Calculus) models of Egger, Møgelberg and Simpson. Our development substantiates the approach by providing a lifting theorem of linear models into cartesian ones. To each of our categorical models we systematically associate a typed term calculus, each of which corresponds to a variant of the sequent calculi LJ (Intuitionistic Logic) or ILL (Intuitionistic Linear Logic). The adjoint resolution of effects corresponds to polarisation whereby, syntactically, types locally determine a strict or lazy evaluation order and, semantically, the associativity of cuts is relaxed. In particular, our results show that polarisation provides a computational interpretation of CBPV in direct style. Further, we characterise depolarised models: those where the cut is associative, and where the evaluation order is unimportant. We explain possible advantages of this style of calculi for the operational semantics of effects.

The theory presented here is a new, radical step in the research program that started with linear logic and aiming at an interactive, resource and space conscious account of reasoning and programming.

We present the μ -calculus, a syntax for λ-calculus + control operators exhibiting symmetries such as program/context and call-by-name/call-by-value. This calculus is derived from implicational Gentzen's sequent calculus LK, a key classical logical system in proof theory. Under the Curry-Howard correspondence between proofs and programs, we can see LK, or more precisely a formulation called LKμ , as a syntax-directed system of simple types for μ -calculus. For μ -calculus, choosing a call-by-name or call-by-value discipline for reduction amounts to choosing one of the two possible symmetric orientations of a critical pair. Our analysis leads us to revisit the question of what is a natural syntax for call-by-value functional computation. We define a translation of λμ-calculus into μ -calculus and two dual translations back to λ-calculus, and we recover known CPS translations by composing these translations.

Categorical combinators [Curien 1986/1993; Hardin 1989; Yokouchi 1989] and more recently λσ-calculus [Abadi 1991; Hardin and Le´vy 1989], have been introduced to provide an explicit treatment of substitutions in the λ-calculus. We reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper with respect to Curien [1986/1993], Hardin [1989], Abadi [1991], and Hardin and Le´vy [1989] are the following: (1) We present a confluent weak calculus of substitutions, where no variable clashes can be feared; (2) We solve a conjecture raised in Abadi [1991]: λσ-calculus is not confluent (it is confluent on ground terms only). This unfortunate result is “repaired” by presenting a confluent version of λσ-calculus, named the λ Env -caldulus in Hardin and Le´vy [1989], called here the confluent λσ-calculus.

Abstract We exhibit confluent and effectively weakly normalizing (thus decidable) rewriting systems for the full equational theory underlying cartesian closed categories, and for polymorphic extensions of it. The λ-calculus extended with surjective pairing has been well-studied in the last two decades. It is not confluent in the untyped case, and confluent in the typed case. But to the best of our knowledge the present work is the first treatment of the lambda calculus extended with surjective pairing and terminal object via a confluent rewriting system, and is the first solution to the decidability problem of the full equational theory of Cartesian Closed Categories extended with polymorphic types . Our approach yields conservativity results as well. In separate papers we apply our results to the study of provable type isomorphisms, and to the decidability of equality in a typed λ-calculus with subtyping.

ασ-calculus is an extended λ-calculus where substitutions are handled explicitly. It is similar to, and inspired by, Categorical Combinatory logic (CCL). The strong normalization of σ, the subcalculus which computes substitutions, may be inferred from the strong normalization of the similar subsystem SUBST of CCL. We present here an independent proof of the termination of several substitution calculi, including σ and SUBST.

In 1979, Klop (1980), answering a question raised by Mann in 1972, showed that the extension of λ-calculus with subjective pairing is not confluent. We refer to Klop (1980) and Barendregt (1981, revised 1984) for a perspective. The term presented by Klop to provide a counterexample is fairly simple, but the proof of non-confluence, although intuitively quite simple, involves some technical properties. Among others, a suitable standardization result on derivations in the extended system is needed in the proof. Klop's proof was revisited by Bunder (1985), who seemingly used less technical apparatus than Klop, starting with the same term as Klop. Although Bunder's proof does not explicitly use a standardization result, his proof proceeds internally with some rearrangements of derivations, so that it is fair to say that some standardization technique is present in Bunder (1985).

A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds' work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this approach: system ℜ. Girard's system F deals with terms and types; ℜ is an extension of F that deals also with relations between types.

A contribution to the investigation of sequentiality and full abstraction for sequential programming languages, focusing on the language PCF, is presented. Ideas of R. Cartwright and M. Felleisen (1992) on observable sequentiality are fit into the framework of concrete data structures and sequential algorithms. An extension of the category of sequential algorithms is shown to provide an order-extensional model of PCF. The key to this is the presence of errors in the semantic domains. The model of observable algorithms is fully abstract for an extension of PCF. This extension has errors too, as well as a control operation catch as found in languages such as Scheme or CommonLisp.>

A subtyping relation ≤ between types is often accompanied by a typing rule, called subsumption: if a term a has type T and T ≤ U , then a has type U . In presence of subsumption, a well-typed term does not codify its proof of well typing. Since a semantic interpretation is most naturally defined by induction on the structure of typing proofs, a problem of coherence arises: different typing proofs of the same term must have related meanings. We propose a proof-theoretical, rewriting approach to this problem. We focus on F ≤ , a second-order lambda calculus with bounded quantification, which is rich enough to make the problem interesting. We define a normalizing rewriting system on proofs, which transforms different proofs of the same typing judgement into a unique normal proof, with the further property that all the normal proofs assigning different types to a given term in a given environment differ only by a final application of the subsumption rule. This rewriting system is not defined on the proofs themselves but on the terms of an auxiliary type system, in which the terms carry complete information about their typing proof. This technique gives us three different results: — Any semantic interpretation is coherent if and only if our rewriting rules are satisfied as equations. — We obtain a proof of the existence of a minimum type for each term in a given environment. — From an analysis of the shape of normal form proofs, we obtain a deterministic typechecking algorithm, which is sound and complete by construction.

Abstract The λσ-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λσ-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λ-calculus and concrete implementations.

The λσ-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λσ-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λ-calculus and concrete implementations.