Relational parametricity and control

We study the equational theory of Parigot’s secondorder 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”.

[1]  Ryu Hasegawa,et al.  Categorical data types in parametric polymorphism , 1994, Mathematical Structures in Computer Science.

[2]  Ralph Matthes,et al.  Parigot's Second Order λμ-Calculus and Inductive Types , 2000 .

[3]  Hayo Thielecke,et al.  Categorical Structure of Continuation Passing Style , 1997 .

[4]  J. Y. Girard,et al.  Interpretation fonctionelle et elimination des coupures dans l'aritmetique d'ordre superieur , 1972 .

[5]  Masahito Hasegawa.  Semantics of Linear Continuation-Passing in Call-by-Name , 2004, FLOPS.

[6]  Edmund Robinson,et al.  Premonoidal categories and notions of computation , 1997, Mathematical Structures in Computer Science.

[7]  Martin Hofmann.  Completeness of Continuation Models for λ μ-Calculus , 2002 .

[8]  Yoshihiko Kakutani.  Duality between Call-by-Name Recursion and Call-by-Value Iteration , 2002, CSL.

[9]  C.-H. Luke Ong,et al.  A Curry-Howard foundation for functional computation with control , 1997, POPL '97.

[10]  Izumi Takeuti.  An Axiomatic System of Parametricity , 1998, Fundam. Informaticae.

[11]  Martín Abadi,et al.  A Logic for Parametric Polymorphism , 1993, TLCA.

[12]  Uday S. Reddy,et al.  Parametricity as a notion of uniformity in reflexive graphs , 2002 .

[13]  Michel Parigot,et al.  Proofs of strong normalisation for second order classical natural deduction , 1997, Journal of Symbolic Logic.

[14]  Gilles Barthe,et al.  CPS translating inductive and coinductive types , 2002, PEPM '02.

[15]  Ken-etsu Fujita,et al.  Galois Embedding from Polymorphic Types into Existential Types , 2005, TLCA.

[16]  Paul Taylor.  SOBER SPACES AND CONTINUATIONS , 2002 .

[17]  Andrzej Filinski.  Declarative Continuations: an Investigation of Duality in Programming Language Semantics , 1989, Category Theory and Computer Science.

[18]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[19]  Martín Abadi,et al.  Formal parametric polymorphism , 1993, POPL '93.

[20]  Gordon Plotkin,et al.  Type Theory and Recursion Extended Abstract , 2003, LICS 2003.

[21]  Peter Selinger,et al.  Control categories and duality: on the categorical semantics of the lambda-mu calculus , 2001, Mathematical Structures in Computer Science.

[22]  Some Remarks on Control Categories , 2003 .

[23]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[24]  Thomas Streicher,et al.  Classical logic, continuation semantics and abstract machines , 1998, Journal of Functional Programming.

[25]  John C. Reynolds,et al.  Towards a theory of type structure , 1974, Symposium on Programming.

[26]  Masahito Hasegawa,et al.  Parameterizations and Fixed-Point Operators on Control Categories , 2005, Fundam. Informaticae.

[27]  Makoto Tatsuta,et al.  Strong normalization proof with CPS-translation for second order classical natural deduction , 2003, J. Symb. Log..

[28]  Michel Parigot,et al.  Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction , 1992, LPAR.

[29]  Philip Wadler,et al.  Theorems for free! , 1989, FPCA.

[30]  Bart Jacobs,et al.  Categorical Logic and Type Theory , 2001, Studies in logic and the foundations of mathematics.

[31]  John C. Reynolds,et al.  Types, Abstraction and Parametric Polymorphism , 1983, IFIP Congress.

[32]  Claudio V. Russo,et al.  Operational Properties of Lily, a Polymorphic Linear Lambda Calculus with Recursion , 2001, HOOTS.

[33]  Hayo Thielecke,et al.  Answer Type Polymorphism in Call-by-Name Continuation Passing , 2004, ESOP.