We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step semantics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and equivalent nondeterministic denotational semantics (with alternative powerdomains to the Egli-Milner and Smyth constructions), D. Scott’s deterministic denotational semantics, the generalized and Dijkstra’s conservative/liberal predicate transformer semantics, the generalized/total and Hoare’s partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these semantics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarski

We articulate a dialectical argumentation framework for qualitative representation of epistemic uncertainty in scientific domains. The framework is grounded in specific philosophies of science and theories of rational mutual discourse. We study the formal properties of our framework and provide it with a game theoretic semantics. With this semantics, we examine the relationship between the snaphots of the debate in the framework and the long run position of the debate, and prove a result directly analogous to the standard (Neyman-Pearson) approach to statistical hypothesis testing. We believe this formalism for representating uncertainty has value in domains with only limited knowledge, where experimental evidence is ambiguous or conflicting, or where agreement between different stakeholders on the quantification of uncertainty is difficult to achieve. All three of these conditions are found in assessments of carcinogenic risk for new chemicals.

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In game semantics, the higher-order value passing mechanisms of the #-calculus are decomposed as sequences of atomic actions exchanged by a Player and its Opponent. Seen from this angle, game semantics is reminiscent of trace semantics in concurrency theory, where a process is identified to the sequences of requests it generates in the course of time. Asynchronous game semantics is an attempt to bridge the gap between the two subjects, and to see mainstream game semantics as a refined and interactive form of trace semantics. Asynchronous games are positional games played on Mazurkiewicz traces, which reformulate (and generalize) the familiar notion of arena game. The interleaving semantics of #-terms, expressed as innocent strategies, may be analyzed in this framework, in the perspective of true concurrency. The analysis reveals that innocent strategies are positional strategies regulated by forward and backward confluence properties. This captures, we believe, the essence of innocence. We conclude the article by defining a non uniform variant of the #-calculus, in which the game semantics of a #-term is formulated directly as a trace semantics, performing the syntactic exploration or parsing of that #-term.

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.

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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

We give a first-principles description of the context semantics of Gonthier, Abadi, and Levy, a computer-science analogue of Girard's geometry of interaction. We explain how this denotational semantics models λ-calculus, and more generally multiplicative-exponential linear logic (MELL), by explaining the call-by-name (CBN) coding of the λ-calculus, and proving the correctness of readback, where the normal form of a λ-term is recovered from its semantics. This analysis yields the correctness of Lamping's optimal reduction algorithm. We relate the context semantics to linear logic types and to ideas from game semantics, used to prove full abstraction theorems for PCF and other λ-calculus variants.

We recall some of the early occurrences of the notions of interactivity and symmetry in the operational and denotational semantics of programming languages. We suggest some connections with ludics.

We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distributive laws % : ! : ! M G ) G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of back-and-forth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: error-free vs. error-aware, alternated vs. non-alternated, backtracking vs. repetitive, uniform vs. non-uniform.