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| In [[proof theory]], the '''Dialectica interpretation'''<ref>{{cite book
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| | title = Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes
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| | author = Kurt Gödel
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| | publisher = Dialectica
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| | year = 1958
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| | pages = 280–287
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| }}</ref> is a proof interpretation of intuitionistic arithmetic ([[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 special issue dedicated to [[Paul Bernays]] on his 70th birthday.
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| == Motivation ==
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| 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).
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| == Dialectica interpretation of intuitionistic logic ==
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| The interpretation has two components: a formula translation and a proof translation. The formula translation describes how each formula <math>A</math> of Heyting arithmetic is mapped to a quantifier-free formula <math>A_D(x; y)</math> of the system T, where <math>x</math> and <math>y</math> are tuples of fresh variables (not appearing free in <math>A</math>). Intuitively, <math>A</math> is interpreted as <math>\exists x \forall y A_D(x; y)</math>. The proof translation shows how a proof of <math>A</math> has enough information to witness the interpretation of <math>A</math>, i.e. the proof of <math>A</math> can be converted into a closed term <math>t</math> and a proof of <math>A_D(t; y)</math> in the system T.
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| === Formula translation ===
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| The quantifier-free formula <math>A_D(x; y)</math> is defined inductively on the logical structure of <math>A</math> as follows, where <math>P</math> is an atomic formula:
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| : <math>
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| \begin{array}{lcl}
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| (P)_D & \equiv & P \\ | |
| (A \wedge B)_D(x, v; y, w) & \equiv & A_D(x; y) \wedge B_D(v; w) \\
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| (A \vee B)_D(x, v, z; y, w) & \equiv & (z = 0 \rightarrow A_D(x; y)) \wedge (z \neq 0 \to B_D(v; w)) \\
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| (A \rightarrow B)_D(f, g; x, w) & \equiv & A_D(x; f x w) \rightarrow B_D(g x; w) \\
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| (\exists z A)_D(x, z; y) & \equiv & A_D(x; y) \\
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| (\forall z A)_D(f; y, z) & \equiv & A_D(f z; y)
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| \end{array}
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| </math>
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| === Proof translation (soundness) ===
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| The formula interpretation is such that whenever <math>A</math> is provable in Heyting arithmetic then there exists a sequence of closed terms <math>t</math> such that <math>A_D(t; y)</math> is provable in the system T. The sequence of terms <math>t</math> and the proof of <math>A_D(t; y)</math> are constructed from the given proof of <math>A</math> in Heyting arithmetic. The construction of <math>t</math> is quite straightforward, except for the contraction axiom <math>A \rightarrow A \wedge A</math> which requires the assumption that quantifier-free formulas are decidable.
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| === Characterisation principles ===
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| It has also been shown that Heyting arithmetic extended with the following principles
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| * [[Axiom of choice]]
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| * [[Markov's principle]]
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| * [[Independence of premise]] for universal formulas
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| is necessary and sufficient for characterising the formulas of HA which are interpretable by the Dialectica interpretation.
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| == Extensions of basic interpretation ==
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| 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).
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| === Induction ===
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| Given [[Gödel's incompleteness theorem]] (which implies that the consistency of PA cannot be proven by [[Finitism|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 functional]]s, which are [[higher order function]]s with primitive recursive descriptions.
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| === Classical logic ===
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| Formulas and proofs in classical arithmetic can also be given a dialectica interpretation via an initial embedding into Heyting arithmetic followed 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.
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| === Comprehension ===
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| In 1962 Spector
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| <ref>{{cite book | |
| | title = Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics
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| | author = Clifford Spector
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| | publisher = Recursive Function Theory: Proc. Symposia in Pure Mathematics
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| | year = 1962
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| | pages = 1–27
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| }}</ref> 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]].
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| == Dialectica interpretation of linear logic ==
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| 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]].<ref>{{cite book
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| | title = The Dialectica Categories
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| | author = Valeria de Paiva
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| | publisher = University of Cambridge, Computer Laboratory, PhD Thesis, Technical Report 213
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| | year = 1991
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| }}</ref> 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.
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| Although the linear interpretation in <ref>{{cite book
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| | title = The Dialectica interpretation of first-order classical affine logic
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| | author = Masaru Shirahata
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| | publisher = Theory and Applications of Categories, Vol. 17, No. 4
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| | year = 2006
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| | pages = 49–79
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| }}</ref> validates the weakening rule (it is actually an interpretation of [[affine logic]]), the dialectica spaces interpretation does not validate weakening for arbitrary formulas.
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| == Variants of the Dialectica interpretation ==
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| 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 König's lemma]]).
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| Comprehensive treatments of the interpretation can be found at
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| ,<ref>{{cite book
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| | title = Gödel's functional ("Dialectica") interpretation
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| | url = http://math.stanford.edu/~feferman/papers/dialectica.pdf
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| | author = Jeremy Avigad and [[Solomon Feferman]]
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| | publisher = in S. Buss ed., The Handbook of Proof Theory, North-Holland
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| | year = 1999
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| | pages = 337–405
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| }}</ref>
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| <ref>{{cite book
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| | title = Applied Proof Theory: Proof Interpretations and Their Use in Mathematics
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| | author = [[Ulrich Kohlenbach]]
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| | publisher = Springer Verlag, Berlin
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| | year = 2008
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| | pages = 1–536
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| }}</ref> and
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| .<ref>{{cite book
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| | title = Metamathematical Investigation of intuitionistic Arithmetic and Analysis
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| | author = [[Anne S. Troelstra]] (with C.A. Smoryński, J.I. Zucker, W.A.Howard)
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| | publisher = Springer Verlag, Berlin
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| | year = 1973
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| | pages = 1–323
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| }}</ref>
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| == References ==
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| <references />
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| [[Category:Proof theory]]
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| [[Category:Intuitionism]]
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