Abstract. Ludics is peculiar in the panorama of game semantics: we first have the definition of interaction-composition and then we have semantical types, as a set of strategies which "behave well" and react in the same way to a set of tests. The semantical types which are interpretations of logical formulas enjoy a fundamental property, called internal completeness, which characterizes ludics and sets it apart also from realizability. Internal completeness entails standard full completeness as a consequence.A growing body of work start to explore the potential of this specific interactive approach. However, ludics has some limitations, which are consequence of the fact that in the original formulation, strategies are abstractions of MALL proofs. On one side, no repetitions are allowed. On the other side, the proofs tend to rely on the very specific properties of the MALL proof-like strategies, making it difficult to transfer the approach to semantical types into different settings.In this paper, we provide an extension of ludics which allows repetitions and show that one can still have interactive types and internal completeness. From this, we obtain full completeness w.r.t. a polarized version of MELL. In our extension, we use less properties than in the original formulation, which we believe is of independent interest. We hope this may open the way to applications of ludics approach to larger domains and different settings.

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Abstract. Gödel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs. Our purpose is to look for a connection between these two disciplines. Following a previous work [3], we consider an extension of the original ludics with contraction and universal nondeterminism, which play dual roles, in order to capture a polarized fragment of linear logic and thus a constructive variant of classical propositional logic.We then prove a completeness theorem for proofs in this extended setting: for any behaviour (formula) A and any design (proof attempt) P , either P is a proof of A or there is a model M of A ⊥ which defeats P . Compared with proofs of full completeness in game semantics, ours exhibits a striking similarity with proofs of Gödel's completeness, in that it explicitly constructs a countermodel essentially using König's lemma, proceeds by induction on formulas, and implies an analogue of Löwenheim-Skolem theorem.

Abstract. Gödel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs. Our purpose is to look for a connection between these two disciplines. Following a previous work [3], we consider an extension of the original ludics with contraction and universal nondeterminism, which play dual roles, in order to capture a polarized fragment of linear logic and thus a constructive variant of classical propositional logic.We then prove a completeness theorem for proofs in this extended setting: for any behaviour (formula) A and any design (proof attempt) P , either P is a proof of A or there is a model M of A ⊥ which defeats P . Compared with proofs of full completeness in game semantics, ours exhibits a striking similarity with proofs of Gödel's completeness, in that it explicitly constructs a countermodel essentially using König's lemma, proceeds by induction on formulas, and implies an analogue of Löwenheim-Skolem theorem.

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