Dialectica interpretation
In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic logic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T.  It was developed by Kurt Gödel to provide a consistency proof of arithmetic. The name of the interpretation comes from the journal Dialectica, where Gödel's paper was published in a 1958 special issue dedicated to Paul Bernays on his 70th birthday.

Motivation 
Via the Gödel–Gentzen negative translation, the consistency of classical Peano arithmetic had already been reduced to the consistency of intuitionistic Heyting arithmetic. Gödel's motivation for developing the dialectica interpretation was to obtain a relative consistency proof for Heyting arithmetic (and hence for Peano arithmetic).

Intuitionistic logic  

The interpretation has two components: a formula translation and a proof translation. The formula translation describes how each formula  of Heyting arithmetic is mapped to a quantifier-free formula  of the system T, where  and  are tuples of fresh variables (not appearing free in ). Intuitively,  is interpreted as . The proof translation shows how a proof of  has enough information to witness the interpretation of , i.e. the proof of  can be converted into a closed term  and a proof of  in the system T.

Formula translation 
The quantifier-free formula  is defined inductively on the logical structure of  as follows, where  is an atomic formula:

Proof translation (soundness) 
The formula interpretation is such that whenever  is provable in Heyting arithmetic then there exists a sequence of closed terms  such that  is provable in the system T. The sequence of terms  and the proof of  are constructed from the given proof of  in Heyting arithmetic. The construction of  is quite straightforward, except for the contraction axiom  which requires the assumption that quantifier-free formulas are decidable.

Characterisation principles 
It has also been shown that Heyting arithmetic extended with the following principles
 Axiom of choice 
 Markov's principle
 Independence of premise for universal formulas
is necessary and sufficient for characterising the formulas of HA which are interpretable by the Dialectica interpretation. 
The choice axiom here is formulated for any 2-ary predicate in the premise and an existence claim with a variable of function type in its conclusion.

Extensions 
The basic dialectica interpretation of intuitionistic logic has been extended to various stronger systems. Intuitively, the dialectica interpretation can be applied to a stronger system, as long as the dialectica interpretation of the extra principle can be witnessed by terms in the system T (or an extension of system T).

Induction 
Given Gödel's incompleteness theorem (which implies that the consistency of PA cannot be proven by finitistic means) it is reasonable to expect that system T must contain non-finitistic constructions. Indeed, this is the case. The non-finitistic constructions show up in the interpretation of mathematical induction. To give a Dialectica interpretation of induction, Gödel makes use of what is nowadays called Gödel's primitive recursive functionals, which are higher-order functions with primitive recursive descriptions.

Classical logic 
Formulas and proofs in classical arithmetic can also be given a Dialectica interpretation via an initial embedding into Heyting arithmetic followed by the Dialectica interpretation of Heyting arithmetic. Shoenfield, in his book, combines the negative translation and the Dialectica interpretation into a single interpretation of classical arithmetic.

Comprehension 
In 1962 Spector
 extended Gödel's Dialectica interpretation of arithmetic to full mathematical analysis, by showing how the schema of countable choice can be given a Dialectica interpretation by extending system T with bar recursion.

Linear logic 
The Dialectica interpretation has been used to build a model of Girard's refinement of intuitionistic logic known as linear logic, via the so-called Dialectica spaces. Since linear logic is a refinement of intuitionistic logic, the dialectica interpretation of linear logic can also be viewed as a refinement of the dialectica interpretation of intuitionistic logic.

Although the linear interpretation in Shirahata's work  validates the weakening rule  (it is actually an interpretation of affine logic), de Paiva's  dialectica spaces interpretation does not validate weakening for arbitrary formulas.

Variants 
Several variants of the Dialectica interpretation have been proposed since. Most notably the Diller-Nahm variant (to avoid the contraction problem) and Kohlenbach's monotone and Ferreira-Oliva bounded interpretations (to interpret weak Kőnig's lemma). 
Comprehensive treatments of the interpretation can be found at,  
 and.

References 

Proof theory
Intuitionism
1958 introductions