We present a generalization of the ideal model for recursive polymorphic types. Types are defined as sets of terms instead of sets of elements of a semantic domain. Our proof of the existence of types (computed by fixpoint of a typing operator) does not rely on metric properties, but on the fact that the identity is the limit of a sequence of projection terms. This establishes a connection with the work of Pitts on relational properties of domains. This also suggests that ideals are better understood as closed sets of terms defined by orthogonality with respect to a set of contexts.

Abstract. We propose ⊤⊤-lifting as a technique for extending operational predicates to Moggi’s monadic computation types, independent of the choice of monad. We demonstrate the method with an application to Girard-Tait reducibility, using this to prove strong normalisation for the computational metalanguage λml. The particular challenge with reducibility is to apply this semantic notion at computation types when the exact meaning of “computation ” (stateful, side-effecting, nondeterministic, etc.) is left unspecified. Our solution is to define reducibility for continuations and use that to support the jump from value types to computation types. The method appears robust: we apply it to show strong normalisation for the computational metalanguage extended with sums, and with exceptions. Based on these results, as well as previous work with local state, we suggest that this “leap-frog ” approach offers a general method for raising concepts defined at value types up to observable properties of computations. 1

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Security properties are profitably expressed using notions of contextual equivalence, and logical relations are a powerful proof technique to establish contextual equivalence in typed lambda calculi, see e.g. Sumii and Pierce's logical relation for a cryptographic lambda-calculus. We clarify Sumii and Pierce's approach, showing that the right tool is prelogical relations, or lax logical relations in general: relations should be lax at encryption types, notably. To explore the difficult aspect of fresh name creation, we use Moggi's monadic lambdacalculus with constants for cryptographic primitives, and Stark's name creation monad. We define logical relations which are lax at encryption and function types but strict (non-lax) at various other types, and show that they are sound and complete for contextual equivalence at all types.

Abstract. A semantic formulation of Lindley and Stark’s ⊤⊤-lifting is given. We first illustrate our semantic formulation of the ⊤⊤-lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤⊤-lifting as the lifting of strong monads across bifibrations with lifted symmetric monoidal closed structures. 1

A variety of results which enable model checking of important classes of infinite-state systems are based on exploiting the property of data independence. The literature contains a number of definitions of variants of data independence which are by syntactic restrictions in particular formalisms. More recently, data independence was defined for labelled transition systems using logical relations, enabling results about data independent systems to be proved without reference to a particular syntax. In this paper, we show that the semantic definition is suciently strong for this purpose. More precisely, it was known that any syntactically data independent symbolic LTS denotes a semantically data independent family of LTSs, but here we show that the converse also holds.

The nu-calculus is an extension of the simply-typed lambda calculus with a ground type for names. It is a useful formal system to study dynamically generated names. In this report, we first present briefly the syntax and the operational semantics of the nu-calculus. We also introduce a metalanguage in the style of Moggi's computational lambda-calculus and give the interpretation of the nu-calculus in such a metalanguage.

Pitts and Stark's nu-calculus is a typed lambda-calculus which forms a basis for the study of interaction between higher-order functions and dynamically created names. A similar approach has received renewed attention recently through Sumii and Pierce's cryptographic lambda-calculus, which deals with security protocols. Logical relations are a powerful tool to prove properties of such a calculus, notably observational equivalence. While Pitts and Stark construct a logical relation for the nu-calculus, it rests heavily on operational aspects of the calculus and is hard to be extended. We propose an alternative Kripke logical relation for the nu-calculus, which is derived naturally from the categorical model of the nu-calculus and the general notion of Kripke logical relation. This is also related to the Kripke logical relation for the name creation monad by Goubault-Larrecq et al. (CSL'2002), which the authors claimed had similarities with Pitts and Stark's logical relation. We show that their Kripke logical relation for names is strictly weaker than Pitts and Stark's. We also show that our Kripke logical relation, which extends the de nition of Goubault-Larrecq et al., is equivalent to Pitts and Stark's up to rst-order types; our de nition rests on purely semantic constituents, and dispenses with the detours through operational semantics that Pitts and Stark use.

Abstract. String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higherdimensional rewriting system provided by polygraphs, and describe its links with standard rewriting theory (in particular string and term rewriting systems). After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2-categories and introduce a framework in which a 2-dimensional rewriting system admits a finite number of critical pairs. Recent developments in category theory have established higher-dimensional categories as a fundamental theoretical setting in order to study situations arising in various areas of mathematics, physics and computer science. A nice survey of these can be found in [2],

According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Real programs, however, exhibit computational effects. In this paper, we develop a framework for extending the notion of relational parametricity to languages with effects.