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Abstract. We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to may-and-must testing. Our model is based on biordered sets similar to Berry’s bidomains, and stable, monotone functions. However, (as in prior models of unbounded non-determinism) these functions may not be continuous. Working in a biordered setting allows us to exploit the different properties of both extensional and stable orders to construct a Cartesian closed category of sequential, discontinuous functions, with least and greatest fixpoints having strong enough properties to prove computational adequacy. We establish full abstraction of the semantics by showing that it contains a simple, first-order “universal type-object ” within which all types may be embedded using functions defined by (countable) ordinal induction. 1

Stable bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. They represent Berry's bidomains. The representation can proceed in two stages. Bistructures form a categorical model of Girard's linear logic consisting of a linear category together with a comonad. The comonad has a co-Kleisli category which is equivalent to a cartesian-closed full subcategory of Berry's bidomains. A main motivation for bidomains came from the full abstraction problem for Plotkin's functional language PCF. However, although the bidomain model incorporates both the Berry stable order and the Scott pointwise order, its PCF theory (those inequalities on terms which hold in the bidomain model) does not include that of the Scott model. With a simple modification we can obtain a new model of PCF, combining the Berry and Scott orders, which does not have this inadequacy.

Abstract. A lax functor from a bicategory of very general nondeterministic concurrent strategies on concurrent games to the bicategory of profunctors is presented. The lax functor provides a fundamental connection between two approaches to generalizations of domain theory to forms of intensional domain theories, one based on game and strategies, and the other on presheaf categories and profunctors. The lax functor becomes a pseudo functor on the sub-bicategory of rigid strategies which includes ‘simple games ’ (underlying AJM and HO games) and stable spans (specializing to Berry’s stable functions, in the deterministic case). In general, the lax functor illustrates how composition of strategies is obtained from that of profunctors by restricting to ‘reachable ’ elements. The results are based on a new characterization of concurrent strategies, which exhibits concurrent strategies as certain discrete fibrations, or equivalently presheaves, over configurations of the game. Finally, the characterization suggests how to extend the definition of strategy to that of strategy on and between categories with a factorization system, an idea that relates to earlier work on bistructures and bidomains. 1

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We present a proof that the canonical models of the untyped λ-calculus -- with call-by-value and lazy call-by-name evaluation -- in the category of bidomains and continuous and stable functions are fully abstract. This is achieved by showing that bidomains yield a fully abstract model of a version of Plotkin's FPC in which the constructor for sum types is restricted to its unary form -- lifting. It is shown that full abstraction for this model can be reduced to denability for the fragment corresponding to "unary PCF". An algorithm devised by Schmidt-Schau is used to show that the bidomain model of this fragment is fully abstract.

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