We prove that the class of stable models is incomplete with respect to pure λ-calculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for this case which is much simpler than the original proof by Honsell an Ronchi della Rocca. Moreover, we isolate a very simple finite set, F , of equations and inequations, which has neither a stable nor a continuous model, and which is included in Th(P fs ) and in T

Abstract. The notion of innocent strategy was introduced by Hyland and Ong in order to capture the interactive behaviour of λ-terms and PCF programs. An innocent strategy is defined as an alternating strategy with partial memory, in which the strategy plays according to its view. Extending the definition to nonalternating strategies is problematic, because the traditional definition of views is based on the hypothesis that Opponent and Proponent alternate during the interaction. Here, we take advantage of the diagrammatic reformulation of alternating innocence in asynchronous games, in order to provide a tentative definition of innocence in non-alternating games. The task is interesting, and far from easy. It requires the combination of true concurrency and game semantics in a clean and organic way, clarifying the relationship between asynchronous games and concurrent games in the sense of Abramsky and Melliès. It also requires an interactive reformulation of the usual acyclicity criterion of linear logic, as well as a directed variant, as a scheduling criterion. 1

Bistructures are a generalisation of event structures which allow a representation of spaces of functions at higher types in an orderextensional setting. The partial order of causal dependency is replaced by two orders, one associated with input and the other with output in the behaviour of functions. Bistructures form a categorical model of Girard's classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output. The comonad of the model has an associated co-Kleisli category which is closely related to that of Berry's bidomains (both have equivalent non-trivial full sub-cartesian closed categories).

A degree of parallelism is an equivalence class of Scott-continuous functions which are relatively definable each other with respect to the language PCF (a paradigmatic sequential language). We introduce an infinite ("bi-dimensional") hierarchy of degrees. This hierarchy is inspired by a representation of first order continuous functions by means of a class of hypergraphs. We assume some familiarity with the language PCF and with its continuous model. Keywords: sequentiality, stability, strong stability, logical relations, sequentiality relations. 1 Introduction A natural notion of relative definability in the continuous type hierarchy is given by the following definition: Definition 1 Given two continuous functions f and g, we say that f is less parallel than g (f par g) if there exists a PCF-term M such that [jM j]g = f . A degree of parallelism is a class of the equivalence relation associated to the preorder par . In this paper we deal with degrees of parallelism of first ord...

The Full Abstraction Problem for PCF [14, 12, 4, 8] is one of the longest-standing problems in the semantics of programming languages. There is quite widespread agreement that it is one of the most difficult; there is much less agreement as to what exactly the problem is, or more particularly as to the precise criteria for a solution. The usual formulation is that one wants a "semantic characterization " of the fully abstract model (by which we mean the inequationally fully abstract order-extensional model, which Milner proved to be uniquely specified up to isomorphism by these properties [12]). The problem is to understand what should be meant by a "semantic characterization". Our view is that the essential content of the problem, what makes it important, is that it calls for a semantic characterization of sequential, functional computation at higher types. The phrase "sequential functional computation " deserves careful consideration. On the one hand, sequentiality refers to a computational process extended over time, not a mere function; on the other hand, we want to capture just those sequential computations in which the different parts or "modules " interact with each other in a purely functional fashion.

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Machine translation ## -calculus interpretation ## -calculus Formally, the -calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation.

Abstract. We give a simple order-theoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation of top (error) and bottom (divergence). We show that monotone and bistable functions (which are required to preserve bistably bounded meets and joins) are strongly sequential, and use this fact to prove universality results for the bistable biorder semantics of the simply-typed lambda-calculus (with atomic constants), and an extension with arithmetic and recursion. We also construct a bistable model of SPCF, a higher-order functional programming language with non-local control. We use our universality result for the lambda-calculus to show that the semantics of SPCF is fully abstract. We then establish a direct correspondence between bistable functions and sequential algorithms by showing that sequential data structures give rise to bistable biorders, and that each bistable function between such biorders is computed by a sequential algorithm. 1.

We show that the continuous function model is the unique extensional (but not necessarily pointwise ordered) model of the variant of the applied typed lambda calculus PCF that includes the "parallel or" operation. 1 Introduction Several extensional models of the applied typed lambda calculus PCF are known to exist, including: (i) The continuous function model, which is order-extensional (pointwise ordered) but not equationally fully abstract [Plo]. (A model is equationally fully abstract when terms are identified in the model exactly when they are operationally equivalent.) (ii) The stable function model, which is neither order-extensional nor equationally fully abstract [Ber][BCL]. (iii) The terminal object of the category of equationally fully abstract, extensional models, which is inequationally fully abstract and order-extensional [Mil][Sto2]. (A model is inequationally fully abstract iff one term is less than another in the model exactly when the first is operationally less defin...

The aim of this work is to show how hypergraphs can be used as a sistematic tool in the classication of continous boolean functions according to their degree of parallelism. Intuitively f is \less parallel" than g if it can be dened by a sequential program using g as its only free variable. It turns out that the poset induced by this preorder is (as for the degrees of recursion) a sup-semilattice. Although hypergraphs had already been used in [6] as a tool for studying degrees of parallelism, no general results relating the former to the latter have been proved in that work. We show that the sup-semilattice of degrees has a categorical counterpart: we dene a category of hypergraphs such that every object \represents" a monotone boolean function; nite coproducts in this category correspond to lubs of degrees. Unlike degrees of recursion, where every set has a recursive upper bound, monotone boolean functions may have no sequential upper bound. However the ones which do have a sequential upper bound can be nicely characterised in terms of hypergraphs. These subsequential functions play a major role in the proof of our main result, namely that f is less parallel than g if there exists a morphism between their associated hypergraps. 1