Godel's Dialectica Research Articles

Canonicity and normalization for dependent type theory

We show canonicity and normalization for dependent type theory with a cumulative sequence of universes and a type of Boolean. The argument follows the usual notion of reducibility, going back to Godel's Dialectica interpretation and the work of Tait. A key feature of our approach is the use of a proof relevant notion of reducibility.

Functional Interpretations of Constructive Set Theory in All Finite Types

Godel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit 'interpreting' instances that make the implication valid. For proofs in constructive set theory CZF - , it may not always be possible to find just one such instance, but it must suffice to explicitly name a set consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our A-interpretation theorem for constructive set theory in all finite types CZF ω- shows. By changing to a hybrid interpretation ^q, we show closure of CZF ω- under rules that - in stronger forms - have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZF ω- and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation.

An Analysis of Gödel'sdialecticaInterpretation via Linear Logic

This article presents an analysis of Godel's dialectica interpretation via a refinement of intuitionistic logic known as linear logic. Linear logic comes naturally into the picture once one observes that the structural rule of contraction is the main cause of the lack of symmetry in Godel's interpretation. We use the fact that the dialectica interpretation of intuitionistic logic can be viewed as a composition of Girard's embedding of intuitionistic logic into linear logic followed by de Paiva's dialectica interpretation of linear logic. We then investigate the various properties of the dialectica interpretation, such as the characterisation theorem, and variants of Godel's interpretation within the linear logic context. The role of contraction in extensions to classical logic, arithmetic and analysis is also discussed.

A type-free Gödel interpretation

In 1930, A. Heyting first specified a formal system for part of intuitionistic mathematics. Although his rules were presumably motivated by the “intended interpretation” or meaning of the logical symbols, over the years a number of other possible interpretations have been discovered for which the rules are also valid. In particular, one might mention the realizablity interpretation of Kleene, the (Dialectica) interpretation of Gödel, and various semantic interpretations, such as Kripke models. (Each of these has several variants or close relatives.) Each such interpretation can be regarded as defining precisely a certain “notion of constructivity”, the study of which may illuminate the still rather vague notions which underlie the intended interpretation; or, if one doubts that there is a single interpretation “intended” by all constructivist mathematicians, the study of precisely defined interpretations may help to delineate and distinguish the possibilities.In the last few years, Heyting's systems have been vastly extended, in order to encompass the large and growing body of constructive mathematics. Several kinds of new systems have been put forward and studied. The present author has extended the various realizability interpretations to several of these systems [B1], [B2] and drawn a number of interesting applications. The mathematical content of the present paper is an interpretation in the style of Godel's Dialectica interpretation, but applicable to the new systems put forward by Feferman [Fl]. The original motivation for this work was to obtain certain metamathematical applications: roughly speaking, Markov's rule and its variants.