361 research outputs found
Relational Parametricity and Control
We study the equational theory of Parigot's second-order
λμ-calculus in connection with a call-by-name continuation-passing
style (CPS) translation into a fragment of the second-order λ-calculus.
It is observed that the relational parametricity on the target calculus induces
a natural notion of equivalence on the λμ-terms. On the other hand,
the unconstrained relational parametricity on the λμ-calculus turns
out to be inconsistent with this CPS semantics. Following these facts, we
propose to formulate the relational parametricity on the λμ-calculus
in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc
An estimation for the lengths of reduction sequences of the -calculus
Since it was realized that the Curry-Howard isomorphism can be extended to
the case of classical logic as well, several calculi have appeared as
candidates for the encodings of proofs in classical logic. One of the most
extensively studied among them is the -calculus of Parigot. In this
paper, based on the result of Xi presented for the -calculus Xi, we
give an upper bound for the lengths of the reduction sequences in the
-calculus extended with the - and -rules.
Surprisingly, our results show that the new terms and the new rules do not add
to the computational complexity of the calculus despite the fact that
-abstraction is able to consume an unbounded number of arguments by virtue
of the -rule
Proving termination of evaluation for System F with control operators
We present new proofs of termination of evaluation in reduction semantics
(i.e., a small-step operational semantics with explicit representation of
evaluation contexts) for System F with control operators. We introduce a
modified version of Girard's proof method based on reducibility candidates,
where the reducibility predicates are defined on values and on evaluation
contexts as prescribed by the reduction semantics format. We address both
abortive control operators (callcc) and delimited-control operators (shift and
reset) for which we introduce novel polymorphic type systems, and we consider
both the call-by-value and call-by-name evaluation strategies.Comment: In Proceedings COS 2013, arXiv:1309.092
Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi
The intuitionistic fragment of the call-by-name version of Curien and
Herbelin's \lambda\_mu\_{\~mu}-calculus is isolated and proved strongly
normalising by means of an embedding into the simply-typed lambda-calculus. Our
embedding is a continuation-and-garbage-passing style translation, the
inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's
\lambda\_mu-calculus. The embedding strictly simulates reductions while usual
continuation-passing-style transformations erase permutative reduction steps.
For our intuitionistic sequent calculus, we even only need "units of garbage"
to be passed. We apply the same method to other calculi, namely successive
extensions of the simply-typed λ-calculus leading to our intuitionistic
system, and already for the simplest extension we consider (λ-calculus
with generalised application), this yields the first proof of strong
normalisation through a reduction-preserving embedding. The results obtained
extend to second and higher-order calculi
Strong normalization results by translation
We prove the strong normalization of full classical natural deduction (i.e.
with conjunction, disjunction and permutative conversions) by using a
translation into the simply typed lambda-mu-calculus. We also extend Mendler's
result on recursive equations to this system.Comment: Submitted to APA
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