Decidability of All Minimal Models (Revised Version - 2012)

This unpublished note is an alternate, shorter (and hopefully more readable) proof of the decidability of all minimal models. The decidability follows from a proof of the existence of a cellular term in each observational equivalence class of a minimal model

An analysis of innocent interaction

We present two abstract machines for innocent interaction. The first, a rather complicated machine, operates directly on innocent strategies. The second, a far simpler machine, requires a “compilation” of the innocent strategies into “cellular” strategies before use. Given two innocent strategies, we get the same final result if we make them interact using the first machine or if we first cellularize them then use the other machine

Cellular strategies and innocent interaction

An analysis of the process of interaction between innocent strategies is presented, leading to a new class of cellular strategies with an associated abstract machine, the CPAM, which radically simplifies the usual machines for innocent interaction. A cellularization process, mapping an innocent strategy to a cellular strategy, is then described which allows us to simulate innocent interaction by first cellularizing both strategies then running them in the CPA

Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism

We give extensional and intensional characterizations of nondeterministic functional programs: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that the extensional and intensional representations of non-deterministic programs are equivalent, by showing how to construct a unique sequential algorithm which computes a given monotone and stable function, and describing the conditions on sequential algorithms which correspond to continuity with respect to each order. We illustrate by defining may and must-testing denotational semantics for a sequential functional language with bounded and unbounded choice operators. We prove that these are computationally adequate, despite the non-continuity of the must-testing semantics of unbounded nondeterminism. In the bounded case, we prove that our continuous models are fully abstract with respect to may and must-testing by identifying a simple universal type, which may also form the basis for models of the untyped lambda-calculus. In the unbounded case we observe that our model contains computable functions which are not denoted by terms, by identifying a further "weak continuity" property of the definable elements, and use this to establish that it is not fully abstract

Decidability of higher-order matching

We show that the higher-order matching problem is decidable using a game-theoretic argument.Comment: appears in LMCS (Logical Methods in Computer Science

Unary PCF is decidable

We show that unary PCF, a very small fragment of Plotkin’s PCF [?], model is effectively presentable. This is in marked contrast to larger fragments, where corresponding results fail [?]. The techniques used are adaptions of those of Padovani [?], who applied them to the minimal model of the simply typed lambda calculus