Dialectica Categories and Games with Bidding

This paper presents a construction which transforms categorical models of additive-free propositional linear logic, closely based on de Paiva\u27s dialectica categories and Oliva\u27s functional interpretations of classical linear logic. The construction is defined using dependent type theory, which proves to be a useful tool for reasoning about dialectica categories. Abstractly, we have a closure operator on the class of models: it preserves soundness and completeness and has a monad-like structure. When applied to categories of games we obtain \u27games with bidding\u27, which are hybrids of dialectica and game models, and we prove completeness theorems for two specific such models

Functional Interpretations of Intuitionistic Linear Logic

We present three different functional interpretations of intuitionistic linear logic ILL and show how these correspond to well-known functional interpretations of intuitionistic logic IL via embeddings of IL into ILL. The main difference from previous work of the second author is that in intuitionistic linear logic (as opposed to classical linear logic) the interpretations of !A are simpler and simultaneous quantifiers are no longer needed for the characterisation of the interpretations. We then compare our approach in developing these three proof interpretations with the one of de Paiva around the Dialectica category model of linear logic

The Dialectica Models of Type Theory

This thesis studies some constructions for building new models of Martin-Löf type theory out of old. We refer to the main techniques as gluing and idempotent splitting. For each we give general conditions under which type constructors exist in the resulting model. These techniques are used to construct some examples of Dialectica models of type theory. The name is chosen by analogy with de Paiva's Dialectica categories, which semantically embody Gödel's Dialectica functional interpretation and its variants. This continues a programme initiated by von Glehn with the construction of the polynomial model of type theory. We complete the analogy between this model and Gödel's original Dialectica by using our techniques to construct a two-level version of this model, equipping the original objects with an extra layer of predicates. In order to do this we have to carefully build up the theory of finite sum types in a display map category. We construct two other notable models. The first is a model analogous to the Diller-Nahm variant, which requires a detailed study of biproducts in categories of algebras. To make clear the generalization from the categories studied by de Paiva, we illustrate the construction of the Diller-Nahm category in terms of gluing an indexed system of types together with a system of predicates. Following this we develop the general techniques needed for the type-theoretic case. The second notable model is analogous to the Dialectica category associated to the error monad as studied by Biering. This model has only weak dependent products. In order to get a model with full dependent products we use the idempotent splitting construction, which generalizes the Karoubi envelope of a category. Making sense of the Karoubi envelope in the type-theoretic case requires us to face up to issues of coherence in our models. We choose the route of making sure all of the constructions we use preserve strict coherence, rather than applying a general coherence theorem to produce a strict model afterwards. Our chosen method preserves more detailed information in the final model.EPSRC studentshi

Cartesian Closed Dialectica Categories

AbstractWhen Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove (relative) consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations (e.g. Kleene’s realizability interpretation and Kreisel’s modified realizability) Gödel’s Dialectica interpretation gives rise to category theoretic constructions that serve both as new models for logic and semantics and as tools for analysing and understanding various aspects of the Dialectica interpretation itself.Gödel’s Dialectica interpretation gives rise to the Dialectica categories (described by V. de Paiva in [V.C.V. de Paiva, The Dialectica categories, in: Categories in Computer Science and Logic (Boulder, CO, 1987), in: Contemp. Math., vol. 92, Amer. Math. Soc., Providence, RI, 1989, pp. 47–62] and J.M.E. Hyland in [J.M.E. Hyland, Proof theory in the abstract, Ann. Pure Appl. Logic 114 (1–3) (2002) 43–78, Commemorative Symposium Dedicated to Anne S. Troelstra (Noordwijkerhout, 1999)]). These categories are symmetric monoidal closed and have finite products and weak coproducts, but they are not Cartesian closed in general. We give an analysis of how to obtain weakly Cartesian closed and Cartesian closed Dialectica categories, and we also reflect on what the analysis might tell us about the Dialectica interpretation