Extraction from classical proofs using game models
Valentin Blot
University of Bath
United Kingdom
Categories of continuations [4] are the universal models of simply-typed call-by-name λµ-calculus [8], just
like cartesian closed categories for λ-calculus. If C is a bicartesian category and R is an object of C such
that all exponentials RX exist, then the full subcategory RC consisting of the objects of the form RX is a
category of continuations. The interpretation of a simply-typed λµ-calculus in RC requires an object [[ι]] of
C for each base type ι, and ι is then interpreted in RC as R[[ι]] . If we are given a model of a simply-typed
λ-calculus, that is, a cartesian closed category C with an object [ι] for each base type ι of this λ-calculus,
then it is natural to ask whether we can interpret in C a simply-typed λµ-calculus with the same base types
while preserving their interpretation. This requirement boils down to finding a fixed object R and a family
of objects {[[ι]]i |i ∈ I} for each ι in C, such that [ι] ' Πi∈I R[[ι]]i (the family is because C may not have
coproducts). If there is only one base type ι, then an obvious choice is R = [ι] and [[ι]] = {1}. In the general
case, however, there is no such simple solution.
In the context of computational interpretations of logic, there are mainly two possibilities for interpreting
classical logic: indirect and direct interpretations. Indirect interpretations rely on a negative translation from
classical to intuitionistic logic, whereas direct interpretations rely on the observation that control operators
provide a computational interpretation for principles of classical logic [3]. In the case of arithmetic, one of
the purposes of a computational interpretation is to obtain a method for extracting computational content
from proofs. In indirect interpretations, this is achieved by replacing ⊥ with some parameter formula R
in the negative translation, and choosing in the end R to be the existential formula we want to extract
computational content from. In direct interpretations, this is achieved by interpreting the formula ⊥ as a
set of integer rather than as the empty set. In both cases the formula ⊥ of classical arithmetic is interpreted
with the type of natural numbers and the two interpretations are therefore equivalent. This is the logical
counterpart of the ad-hoc choice described in the previous paragraph, which works if the source calculus
has only one base type. In logic, this choice is rather unnatural because the type of atomic formulas is
fundamentally different from the type of natural numbers, even if in the case of first-order arithmetic they
can technically be the same.
It has been shown that relaxing the well-bracketing condition in game semantics [5, 7, 1] gives models of
functional languages with control [6]. This observation gives a general answer to the problem of interpreting
a simply-typed λµ-calculus in the cartesian closed category of unbracketed games while preserving the
interpretation of types. Indeed, it is a very specific property of unbracketed game semantics that every
arena A can be written as Πi∈I RAi where R is the one-move arena, I is the set of roots of A and Ai is the
forest under the root i ∈ I. Therefore, in the category of unbracketed games we can have an interpretation of
classical arithmetic in which the type of natural numbers is interpreted as the usual arena of natural numbers,
while the atomic formulas are interpreted with the one-move arena. We show that this property provides a
new direct realizability interpretation of classical arithmetic which is not equivalent to the indirect method
and uses the control possibilities of the language to provide the extracted value through an exception-like
mechanism. We also show that sequential algorithms [2] enjoy a similar property.
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References
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