## Typed Applicative Structures and Normalization by Evaluation for System F ω

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@MISC{Abel_typedapplicative,

author = {Andreas Abel},

title = {Typed Applicative Structures and Normalization by Evaluation for System F ω},

year = {}

}

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### Abstract

Abstract. We present a normalization-by-evaluation (NbE) algorithm for System F ω with βη-equality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the algorithm. The algorithm is verified through a general model construction using typed applicative structures, called type and object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem.

### Citations

1115 | The Lambda Calculus: Its Syntax and Semantics - Barendregt - 1981 |

237 |
Interprétation Fonctionnelle et Elimination Des Coupures de L’arithmétique D’ordre Supérieur, Thèeses d’état, Université de
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- 1972
(Show Context)
Citation Context ...e syntactical type structure in case of syntactic objects and the structure of type values in case of (object) values. We reorganize a typical normalization proof for System F ω by model construction =-=[Gir72]-=- into our framework of type and object structures. Central to such a normalization proof is the fundamental theorem of typing which states that ⦇t⦈ ∈ [[T ]] for any object t of type T . Herein, ⦇t⦈ in... |

213 |
Intensional Interpretations of Functionals of Finite Type
- Tait
- 1967
(Show Context)
Citation Context ... instance in the structure of syntactical objects. The semantic type [[T ]] is a collection of values, typically a set of normalizing objects, where the function type [[T1 → T2]] is defined à la Tait =-=[Tai67]-=-, i. e., as all values f such that f applied to d inhabits [[T2]] for all d in [[T1]]. The essence of the fundamental theorem is that the (hereditarily) normalizing terms model the typing rules where ... |

207 | Type-directed partial evaluation - Danvy |

167 | Towards a Practical Programming Language Based on Dependent Type Theory
- Norell
- 2007
(Show Context)
Citation Context ...d object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem. 1 Introduction and Related Work In theorem provers, such as Coq [INR07], Agda =-=[Nor07]-=-, Epigram [CAM07], which are based on intensional type theory, checking the validity of proofs or typings relies on deciding equality of types. Types are recognized as equal if they have the same norm... |

69 | A compiled implementation of strong reduction
- Grégoire, Leroy
- 2002
(Show Context)
Citation Context ...l forms (as in Pollack’s constructive engine [Pol94]), closures (as in Coquand’s type checker [Coq96]), tagged functions (Epigram 2 [CAM07]) or virtual machine instructions (compiled reduction in Coq =-=[GL02]-=-). All implementations of values that satisfy the interface given in the following can be used with our NbE algorithm, and in this article we provide a framework to prove all these implementations cor... |

65 | A general formulation of simultaneous inductiverecursive definitions in type theory
- Dybjer
- 2000
(Show Context)
Citation Context ...ive universe and judgmental βη-equality. What is the challenge when stepping up to impredicativity? Predicative type theories allow to define the semantics of types from below via induction-recursion =-=[Dyb00]-=-, and the reification function can be defined by induction on types. This fails in the presence of impredicativity, where one first has to lay out a lattice of semantic type candidates and then define... |

44 | Intuitionistic model constructions and normalization proofs
- Coquand, Dybjer
- 1997
(Show Context)
Citation Context ...by cases on the neutral value H. We continue by constructing a model of the kinding rules which proves soundness of NbE for types. Kinds κ are interpreted as sets Gκ Ξ of pairs (F, T ) glued together =-=[CD97]-=- by reification, i. e., the type value F reifies to syntactic type T up to βη-equality. Function kinds are interpreted via Tait’s function space (see Def. 4), thus, the fundamental theorem of kinding ... |

44 | An algorithm for type-checking dependent types
- Coquand
- 1995
(Show Context)
Citation Context ...ues. This leaves the freedom to implement values in different ways, e. g., β-normal forms, weak head normal forms (as in Pollack’s constructive engine [Pol94]), closures (as in Coquand’s type checker =-=[Coq96]-=-), tagged functions (Epigram 2 [CAM07]) or virtual machine instructions (compiled reduction in Coq [GL02]). All implementations of values that satisfy the interface given in the following can be used ... |

39 | Normalization by evaluation for typed lambda calculus with coproducts - Altenkirch, Dybjer, et al. - 2001 |

37 |
Categorical semantics for higher order polymorphic lambda calculus
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- 1986
(Show Context)
Citation Context ...ject structures, which are sorted applicative structures on type and object level, in order to facilitate generic model constructions for System F ω —which are an alternative to categorical semantics =-=[See87]-=- and Bruce-MeyerMitchell models [BM84]. Using special instances of kind candidate spaces we have shown soundness and completeness of an abstract normalization by evaluation algorithm for types. We hav... |

36 | Definitions by rewriting in the Calculus of Constructions - BLANQUI |

31 | Extensional normalisation and type-directed partial evaluation for typed lambda calculus with sums, in: POPL’04 - Balat, Cosmo, et al. - 2004 |

23 | Closure under alpha-conversion
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(Show Context)
Citation Context ...ω for an abstract domain D of values and type values. This leaves the freedom to implement values in different ways, e. g., β-normal forms, weak head normal forms (as in Pollack’s constructive engine =-=[Pol94]-=-), closures (as in Coquand’s type checker [Coq96]), tagged functions (Epigram 2 [CAM07]) or virtual machine instructions (compiled reduction in Coq [GL02]). All implementations of values that satisfy ... |

22 | Normalization by evaluation for Martin-Löf type theory with one universe
- Abel, Aehlig, et al.
- 2007
(Show Context)
Citation Context ... directed by the shape of type T . NbE has proven a valid tool to structure extensional normalization, especially in the notoriously difficult case of sum types [ADHS01,BCF04,Bar08]. In previous work =-=[ACD07]-=-, we have adapted NbE to a dependent type theory with one predicative universe and judgmental βη-equality. What is the challenge when stepping up to impredicativity? Predicative type theories allow to... |

21 |
The Theory of LEGO
- Pollack
- 1994
(Show Context)
Citation Context ...ard isomorphism, which identifies proofs with programs and propositions with types, is the basis of several type-theoretic theorem provers, such as Coq [INR07], Agda [Nor07], Epigram [CAM07] and LEGO =-=[Pol94a]-=-. In these systems, checking the validity of proofs or typings relies on deciding equality of types. Types are recognized as equal if they have the same normal form, this is why normalization plays a ... |

16 |
Reduction-free normalisation for a polymorphic system
- Altenkirch, Hofmann, et al.
- 1996
(Show Context)
Citation Context ...T89]. Hence, the semantic type structure is not inductive, and reification cannot be defined by induction on types. There are at least two ways out of this dilemma: Altenkirch, Hofmann, and Streicher =-=[AHS96]-=- construct a total normalization function type-wise while building a model for System F. In previous work [Abe08], I have conceived reification as a deterministicrelation between value d and normal f... |

15 |
Epigram reloaded: A standalone typechecker for ETT
- Chapman, Altenkirch, et al.
- 2006
(Show Context)
Citation Context ...es. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem. 1 Introduction and Related Work In theorem provers, such as Coq [INR07], Agda [Nor07], Epigram =-=[CAM07]-=-, which are based on intensional type theory, checking the validity of proofs or typings relies on deciding equality of types. Types are recognized as equal if they have the same normal form. This is ... |

12 | Verifying a semantic βη-conversion test for Martin-Löf type theory
- Abel, Coquand, et al.
- 2008
(Show Context)
Citation Context ...operty of combinatory algebras. The condition (∗) is only necessary since we chose to use eager substitution in the inference rules of F ω , it can be dropped when switching to explicit substitutions =-=[ACD08]-=-. We use “interpretation” and “evaluation” synonymously. Note that while the equations determine the interpretation of constants, variables, and application, there is some freedom in the interpretatio... |

10 |
The semantics of second order polymorphic lambda calculus
- Bruce, Meyer
- 1984
(Show Context)
Citation Context ...cative structures on type and object level, in order to facilitate generic model constructions for System F ω —which are an alternative to categorical semantics [See87] and Bruce-MeyerMitchell models =-=[BM84]-=-. Using special instances of kind candidate spaces we have shown soundness and completeness of an abstract normalization by evaluation algorithm for types. We have gone on to show soundness and comple... |

10 | Building decision procedures in the calculus of inductive constructions
- Blanqui, Jouannaud, et al.
- 2007
(Show Context)
Citation Context ... Thus, it is desirable to add bits of extensionality as long as decidability is preserved. For the CC, different directions have been explored, such as rewriting [Bla05,CWC07] and decision procedures =-=[BJS07]-=-. Our goal is to integrate η-laws. which provide some extensionality for functions, into definitional equality of CC. Normalization by evaluation (NbE) [BS91,Dan99] is a systematic method to perform β... |

10 | A concurrent logical framework I: Judgements and properties - Watkins, Cervesato, et al. - 2003 |

8 | and Helmut Schwichtenberg. An inverse of the evaluation functional for typed lambda-calculus - Berger - 1991 |

5 | Syntactic Metatheory of Higher-Order Subtyping - Abel, Rodriguez - 2008 |

4 |
The Coq Proof Assistant, Version 8.1
- INRIA
(Show Context)
Citation Context ...called type and object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem. 1 Introduction and Related Work In theorem provers, such as Coq =-=[INR07]-=-, Agda [Nor07], Epigram [CAM07], which are based on intensional type theory, checking the validity of proofs or typings relies on deciding equality of types. Types are recognized as equal if they have... |

3 | Towards rewriting in Coq - Chrzaszcz, Walukiewicz-Chrzaszcz - 2007 |

2 | Weak βη-normalization and normalization by evaluation for System F
- Abel
- 2008
(Show Context)
Citation Context ...s. There are at least two ways out of this dilemma: Altenkirch, Hofmann, and Streicher [AHS96] construct a total normalization function type-wise while building a model for System F. In previous work =-=[Abe08]-=-, I have conceived reification as a deterministicrelation between value d and normal form v and their type T , and showed through a model construction that it corresponds to a total function. In this... |

2 | Decidability for non-standard conversions in lambda-calculus - Barral - 2008 |

1 |
volume 1706 of LNCS
- School
- 1999
(Show Context)
Citation Context ...ive universe and judgmental βη-equality. What is the challenge when stepping up to impredicativity? Predicative type theories allow to define the semantics of types from below via induction-recursion =-=[Dyb00]-=-, and the reification function can be defined by induction on types. This fails in the presence of impredicativity, where one first has to lay out a lattice of semantic type candidates and then define... |