## Normal form bisimulation for parametric polymorphism (2008)

Venue: | In LICS |

Citations: | 10 - 2 self |

### BibTeX

@INPROCEEDINGS{Lassen08normalform,

author = {Soren B. Lassen and Google Inc and Paul Blain Levy},

title = {Normal form bisimulation for parametric polymorphism},

booktitle = {In LICS},

year = {2008}

}

### OpenURL

### Abstract

This paper presents a new bisimulation theory for parametric polymorphism which enables straightforward coinductive proofs of program equivalences involving existential types. The theory is an instance of typed normal form bisimulation and demonstrates the power of this recent framework for modeling typed lambda calculi as labelled transition systems. We develop our theory for a continuation-passing style calculus, Jump-With-Argument, where normal form bisimulation takes a simple form. We equip the calculus with both existential and recursive types. An “ultimate pattern matching theorem ” enables us to define bisimilarity and we show it to be a congruence. We apply our theory to proving program equivalences, type isomorphisms and genericity. 1

### Citations

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Citation Context ...ories for reasoning about parametric polymorphism is evident and this has been the subject of many research efforts, since Reynolds relational parametricity result for the pure polymorphic λ-calculus =-=[22]-=-. Theories for polymorphic calculi with fixed point recursion or recursive types have encountered a number of difficulties, either complex meta-theories, problems with type recursion, or weak reasonin... |

209 |
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Citation Context ... sum, product, and function types via ultimate patterns is new. Whereas typed normal form bisimulation for monomorphic types [14] appears to be closely related to the game semantics of Hyland and Ong =-=[9]-=-, our polymorphic extension seems not to correspond to existing game models of polymorphism. They are either highly intensional [8, 5] or in the Abramsky-Jagadeesan-Malacaria (AJM) style of game seman... |

119 |
A fully abstract game semantics for general references
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Citation Context ...ut connections with game semantics, by design [15], and it seems likely that our theory could lead to a Hyland-Ong style model of polymorphism, fully abstract in the presence of state—by analogy with =-=[1, 10]-=-. 1.2 Related Work Our work builds on the LTS and typed normal form bisimulation theory for a monomorphic recursively typed CPS calculus in [14]. (See op.cit. for a survey of earlier work on normal fo... |

71 | Categorical Structure of Continuation Passing Style
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(Show Context)
Citation Context ...pe isomorphisms and in Section 7 we prove a genericity property of bisimilarity. 2 Jump-With-Argument Syntax Jump-With-Argument is a continuation-passing style calculus, extending the CPS calculus in =-=[25]-=-. Its types are given by A ::= X | 1 | A × A | ∑ i∈I Ai | ¬A | µX.A | ∃X.A where I is any finite set. Binary sums are, of course, a spedef cial case of finite sums, A1 +A2 = ∑2 i=1Ai, and the empty ty... |

70 | Step-indexed syntactic logical relations for recursive and quantified types
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(Show Context)
Citation Context ...lty and are not covered. This limits the scope of the theory and has motivated work on alternative operationally based approaches, including our work. Ahmed’s step-indexed syntactic logical relations =-=[3]-=- solve the difficulty of recursive types by stratifying the relations by the number of steps available for future evaluation. They form a fully abstract syntactic model for a state-less polymorphic ca... |

56 | Behavioral equivalence in the polymorphic pi-calculus
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Citation Context ...ly more direct without any need for auxiliary “bisimulation up-to” techniques. Many of the ideas we use to model second order types in our LTS appear in the LTSs for polymorphically typed πcalculi in =-=[19, 4]-=- but the combination with recursive, sum, product, and function types via ultimate patterns is new. Whereas typed normal form bisimulation for monomorphic types [14] appears to be closely related to t... |

47 | A bisimulation for type abstraction and recursion
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(Show Context)
Citation Context ...form a fully abstract syntactic model for a state-less polymorphic calculus with recursive types. Gordon’s applicative bisimulation theory [7] and Sumii and Pierce’s relation-sets bisimulation theory =-=[24]-=- for parametric polymorphism are two other operationally based theories with recursive types, relational congruence proofs, and straightforward bisimulation proof rules. Both theories are fully abstra... |

38 | Typed operational reasoning
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(Show Context)
Citation Context ...ce because of the convenience of the associated co-induction proof principle for articulating proofs of bisimilarity, namely by exhibiting a bisimulation. Pitts’ operationally based logical relations =-=[20]-=- is a different syntactic theory for parametric polymorphism with powerful relational proof principles comparable to normal form bisimulation. The theory is fully abstract for a stateless calculus, co... |

32 | Relational Reasoning about Functions and Nondeterminism - Lassen - 1998 |

25 | Sequentiality and the pi-calculus
- Yoshida, Berger
- 2001
(Show Context)
Citation Context ...ly more direct without any need for auxiliary “bisimulation up-to” techniques. Many of the ideas we use to model second order types in our LTS appear in the LTSs for polymorphically typed πcalculi in =-=[19, 4]-=- but the combination with recursive, sum, product, and function types via ultimate patterns is new. Whereas typed normal form bisimulation for monomorphic types [14] appears to be closely related to t... |

23 |
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Citation Context ...by the number of steps available for future evaluation. They form a fully abstract syntactic model for a state-less polymorphic calculus with recursive types. Gordon’s applicative bisimulation theory =-=[7]-=- and Sumii and Pierce’s relation-sets bisimulation theory [24] for parametric polymorphism are two other operationally based theories with recursive types, relational congruence proofs, and straightfo... |

16 | A complete, co-inductive syntactic theory of sequential control and state
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Citation Context ...lculus [14] to type identifiers, open types, and type substitutions. Nonetheless, here we outline the argument in a notationally simpler “relational” formulation, akin to the substitutivity proofs in =-=[13, 23]-=-. We define the relation substitution operation X [S], ∆, Θ, ∆i ‖ ∆o ⊢ x X x ′ ∆, Θi ‖ Θ, Θo ⊢ v, w S v ′ , w ′ ∆, Θi, ∆i ⊣⊢ Θo, ∆o ∆, Θi, ∆i ‖ Θo, ∆o ⊢ w, x[v] X [S] w ′ , x ′ [v ′ ] where S is a pas... |

15 | Games and definability for System F
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(Show Context)
Citation Context ...ppears to be closely related to the game semantics of Hyland and Ong [9], our polymorphic extension seems not to correspond to existing game models of polymorphism. They are either highly intensional =-=[8, 5]-=- or in the Abramsky-Jagadeesan-Malacaria (AJM) style of game semantics [2, 18]. Outline Section 2 introduces JWA and its operational semantics with some examples to illustrate its continuationpassing ... |

15 | Eager normal form bisimulation
- Lassen
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(Show Context)
Citation Context ...tion theory induce normal form bisimilation theories for the direct style source calculi, analogously to the way CPS transforms preserve and reflect normal form bisimilarity in the untyped λ-calculus =-=[12, 13]-=-. For typed calculi, these translations and induced theories factor through a typed normal form bisimulation theory for the Call-By-Push-Value calculus and the stackpassing transform into JWA [16]. Ou... |

11 | A fully abstract trace semantics for general references
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(Show Context)
Citation Context ...ory is not fully abstract for JWA, for the same reason (same counter examples) as in the monomorphic case [14]. We expect that our treatment of polymorphism can be combined with state in the style of =-=[15, 10]-=- and then, if we 1extend the JWA calculus with state, we conjecture that our theory becomes fully abstract. 1.1 Contributions Parametric polymorphism is a very useful and non-trivial addition to the ... |

9 | A game semantics for generic polymorphism
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(Show Context)
Citation Context ... polymorphic extension seems not to correspond to existing game models of polymorphism. They are either highly intensional [8, 5] or in the Abramsky-Jagadeesan-Malacaria (AJM) style of game semantics =-=[2, 18]-=-. Outline Section 2 introduces JWA and its operational semantics with some examples to illustrate its continuationpassing style. Section 3 introduces ultimate patterns and states the ultimate pattern ... |

9 | Evolving games and essential nets for affine polymorphism
- Murawski, Ong
- 2001
(Show Context)
Citation Context ... polymorphic extension seems not to correspond to existing game models of polymorphism. They are either highly intensional [8, 5] or in the Abramsky-Jagadeesan-Malacaria (AJM) style of game semantics =-=[2, 18]-=-. Outline Section 2 introduces JWA and its operational semantics with some examples to illustrate its continuationpassing style. Section 3 introduces ultimate patterns and states the ultimate pattern ... |

8 |
Direct models of the computational lambda-calculus
- Führmann
- 1999
(Show Context)
Citation Context ...at F1 and F2 are “effect-free” in the sense that x:A, j:¬¬¬A ′ ⊢ n j λk.F1〈x, k〉 � F1〈x, λz.jλk.kz〉, x:A ′ , j:¬¬¬A ⊢ n j λk.F2〈x, k〉 � F2〈x, λz.jλk.kz〉 (the CPS equivalent of Führmann’s thunkability =-=[6]-=-). This requirement holds trivially for all our examples. Example 11 If X, Y ⊢ A then ∃X.∃Y.A is isormorphic to ∃Y.∃X.A, because F1 = F2 = λ〈〈X, 〈Y, x〉〉, k〉.k〈Y, 〈X, x〉〉 form an isomorphism. The proof... |

6 |
Head normal form bisimulation for pairs and the λμcalculus (extended abstract
- Lassen
- 2006
(Show Context)
Citation Context ...tion theory induce normal form bisimilation theories for the direct style source calculi, analogously to the way CPS transforms preserve and reflect normal form bisimilarity in the untyped λ-calculus =-=[12, 13]-=-. For typed calculi, these translations and induced theories factor through a typed normal form bisimulation theory for the Call-By-Push-Value calculus and the stackpassing transform into JWA [16]. Ou... |

4 | Game semantics using function inventories. Talk given at Geometry of Computation 2006 - Levy - 2006 |

2 |
Quantification du second ordre en smantique des jeux - Application aux isomorphismes de types
- Lataillade
- 2007
(Show Context)
Citation Context ...ppears to be closely related to the game semantics of Hyland and Ong [9], our polymorphic extension seems not to correspond to existing game models of polymorphism. They are either highly intensional =-=[8, 5]-=- or in the Abramsky-Jagadeesan-Malacaria (AJM) style of game semantics [2, 18]. Outline Section 2 introduces JWA and its operational semantics with some examples to illustrate its continuationpassing ... |

1 |
A Functional/Imperative Synthesis. Semantic Struct. in Computation
- Call-By-Push-Value
- 2004
(Show Context)
Citation Context ...[12, 13]. For typed calculi, these translations and induced theories factor through a typed normal form bisimulation theory for the Call-By-Push-Value calculus and the stackpassing transform into JWA =-=[16]-=-. Our theory is not fully abstract for JWA, for the same reason (same counter examples) as in the monomorphic case [14]. We expect that our treatment of polymorphism can be combined with state in the ... |