## Recursion on the partial continuous functionals (2006)

Venue: | Logic Colloquium ’05 |

Citations: | 7 - 5 self |

### BibTeX

@INPROCEEDINGS{Schwichtenberg06recursionon,

author = {Helmut Schwichtenberg},

title = {Recursion on the partial continuous functionals},

booktitle = {Logic Colloquium ’05},

year = {2006},

pages = {251--269},

publisher = {Springer}

}

### OpenURL

### Abstract

We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the well-known abbreviation

### Citations

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Citation Context ...his line of research was taken up and developed in a mathematically more satisfactory way by Scott and Ershov [29, 16]. Today this theory is usually presented in the context of abstract domain theory =-=[31, 3]-=-; it is based on classical logic. The present work can be seen as an attempt to develop a constructive theory of formal neighborhoods for continuous functionals, in a direct and intuitive style. The t... |

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Citation Context ... free variables in �x) involving constants D defined by computation rules will be an inductively defined set of tokens ( � U, b), of the type of λ�x M. So we extend the term language of Plotkin’s PCF =-=[25]-=-, by constants defined via “computation rules”. One instance of such rules is the definition of the fixed point operators Yρ of type (ρ ⇒ ρ) ⇒ ρ, by Yρf = f(Yρf). Another instance is the structural re... |

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Citation Context ... axioms – here in the form of which terms are allowed in ∀-elimination – are weak in the sense that these terms involve quantifiers over functionals, so our theory remains predicative. In contrast to =-=[24]-=-, formulas and types are kept separate. This makes it possible to avoid dependent types, which simplifies the theory considerably. More importantly, by separating the logic rules from type theory one ... |

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Citation Context ...e style. The task is to replace abstract domain theory by a more concrete and (in case of finitary free algebras) finitary theory of representations. As a framework we use Scott’s information systems =-=[30, 22, 31]-=-. It turns out 1sthat we only need to deal with “atomic” and “coherent” information systems (abbreviated acis), which simplifies matters considerably. In this setup the basic notion is that of a “toke... |

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Citation Context ...ction on the types, and uses a computability predicate. A similar result in a type-theoretic setting is in Martin-Löf’s notes [23, Second Theorem]. Adequacy theorems have been proved in many contexts =-=[2, 4, 5, 23]-=-. Coquand [14] – building on the work of Martin-Löf [23] and Berger [8] – observed that the adequacy result even holds for untyped languages, hence also for dependently typed ones. The problem of prov... |

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Citation Context ...es, hence also for dependently typed ones. The problem of proving strong normalization for extensions of typed λ-calculi by higher order rewrite rules has been studied extensively in the 3sliterature =-=[32, 17, 33, 12, 1, 8]-=-. Most of these proofs use impredicative methods (e.g., by reducing the problem to strong normalization of second order propositional logic, called system F by Girard [17]). Our definition of the stro... |

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Citation Context ...ghborhood systems, Martin-Löf developed in unpublished (but somewhat distributed) notes [23] a domain theoretic interpretation of his type theory. The intersection type discipline of Coppo and Dezani =-=[5]-=- can be seen as a different style of presenting the idea of a neighborhood system. The desire to have a more general framework for these ideas has lead Martin-Löf, Sambin and others to develop a forma... |

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Citation Context ...s has its roots in work of Kreisel [20] (who used formal neighborhoods) and Kleene [19]. This line of research was taken up and developed in a mathematically more satisfactory way by Scott and Ershov =-=[29, 16]-=-. Today this theory is usually presented in the context of abstract domain theory [31, 3]; it is based on classical logic. The present work can be seen as an attempt to develop a constructive theory o... |

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Citation Context ...standard line of such proofs. Moreover, it extends the result to the present setting. Related work The development of constructive theories of computable functionals of finite type began with Gödel’s =-=[18]-=-. There the emphasis was on particular computable functionals, the structural (or primitive) recursive ones. In contrast to what was done later by Kreisel, Kleene, Scott and Ershov, the domains for th... |

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Citation Context ...es, hence also for dependently typed ones. The problem of proving strong normalization for extensions of typed λ-calculi by higher order rewrite rules has been studied extensively in the 3sliterature =-=[32, 17, 33, 12, 1, 8]-=-. Most of these proofs use impredicative methods (e.g., by reducing the problem to strong normalization of second order propositional logic, called system F by Girard [17]). Our definition of the stro... |

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Citation Context ...resh start, and do full justice to the fundamental notion of computability in finite types, with the partial continuous functionals as underlying domains. The total ones then appear as a dense subset =-=[20, 15, 7, 31, 27, 21]-=-, and seem to be best treated in this way. Computable functionals and logic Types are built from base types by the formation of function types, ρ ⇒ σ. As domains for the base types we choose non-flat ... |

54 |
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Citation Context ...e style. The task is to replace abstract domain theory by a more concrete and (in case of finitary free algebras) finitary theory of representations. As a framework we use Scott’s information systems =-=[30, 22, 31]-=-. It turns out 1sthat we only need to deal with “atomic” and “coherent” information systems (abbreviated acis), which simplifies matters considerably. In this setup the basic notion is that of a “toke... |

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Citation Context ...alities, and does not lend itself well for a theory of computability. A more appropriate semantics for typed languages has its roots in work of Kreisel [20] (who used formal neighborhoods) and Kleene =-=[19]-=-. This line of research was taken up and developed in a mathematically more satisfactory way by Scott and Ershov [29, 16]. Today this theory is usually presented in the context of abstract domain theo... |

38 |
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Citation Context ... Schwichtenberg We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott =-=[28]-=-, under the well-known abbreviation LCF. However, the prime example of such a theory, Per Martin-Löf’s type theory [24] in its present form deals with total (structural recursive) functionals only. An... |

36 |
Total sets and objects in domain theory
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Citation Context ...resh start, and do full justice to the fundamental notion of computability in finite types, with the partial continuous functionals as underlying domains. The total ones then appear as a dense subset =-=[20, 15, 7, 31, 27, 21]-=-, and seem to be best treated in this way. Computable functionals and logic Types are built from base types by the formation of function types, ρ ⇒ σ. As domains for the base types we choose non-flat ... |

31 | Inductively generated formal topologies
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Citation Context ...ttempt of Martin-Löf [23] to give a domain theoretic interpretation of his type theory has not even been published, probably because it was felt that a more general approach – such as formal topology =-=[13]-=- – would be more appropriate. Here we try to make a fresh start, and do full justice to the fundamental notion of computability in finite types, with the partial continuous functionals as underlying d... |

29 |
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Citation Context ...nciple holds. In [20] Kreisel states the important density theorem, which says that any finite functional can be extended to a total one. Full proofs of various versions of the density theorem are in =-=[15, 7, 31, 27, 21]-=-. Here we give a proof for the practically important case where the base domains are not just the flat domain of natural numbers, but non-flat and possibly parametrized free algebras. 3.1 Total and st... |

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Citation Context ...ction on the types, and uses a computability predicate. A similar result in a type-theoretic setting is in Martin-Löf’s notes [23, Second Theorem]. Adequacy theorems have been proved in many contexts =-=[2, 4, 5, 23]-=-. Coquand [14] – building on the work of Martin-Löf [23] and Berger [8] – observed that the adequacy result even holds for untyped languages, hence also for dependently typed ones. The problem of prov... |

13 |
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Citation Context ...es, hence also for dependently typed ones. The problem of proving strong normalization for extensions of typed λ-calculi by higher order rewrite rules has been studied extensively in the 3sliterature =-=[32, 17, 33, 12, 1, 8]-=-. Most of these proofs use impredicative methods (e.g., by reducing the problem to strong normalization of second order propositional logic, called system F by Girard [17]). Our definition of the stro... |

11 |
Schwichtenberg: Refined Program Extraction from Classical Proofs
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Citation Context ...ence to the totality of all predicates, to which the one to be defined belongs. A central application domain for the Minlog proof assistant is program extraction from constructive (and also classical =-=[10]-=-) proofs. This is done by means of a realizability interpretation, which requires – when the formula to be realized is given by an inductively defined predicate – a (possibly non-finitary) free algebr... |

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Citation Context ...resh start, and do full justice to the fundamental notion of computability in finite types, with the partial continuous functionals as underlying domains. The total ones then appear as a dense subset =-=[20, 15, 7, 31, 27, 21]-=-, and seem to be best treated in this way. Computable functionals and logic Types are built from base types by the formation of function types, ρ ⇒ σ. As domains for the base types we choose non-flat ... |

6 |
Schwichtenberg: Program Extraction from Normalization Proofs
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Citation Context ...as, with type parameters) by forming function spaces; this suffices for our intended mathematical applications. For more metamathematical subjects one may also add universe formation processes, as in =-=[9]-=-. Decidable predicates are viewed as boolean valued functions (and hence the rewrite mechanism described below applies to them), and inductive definitions are the common way to introduce undecidable p... |

6 |
Helmut Schwichtenberg. Term rewriting for normalization by evaluation
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Citation Context ...r, this way to deal with computability is too general for concrete applications. In practice, one wants to define computable functionals by recursion equations. We show that and how computation rules =-=[11, 8]-=- can be used to achieve this task. The meaning [λ�x M ] of a term M (with free variables in �x) involving constants D defined by computation rules will be an inductively defined set of tokens ( � U, b... |

6 | Total objects in inductively defined types - Kristiansen, Normann - 1997 |

4 | Continuous semantics for strong normalization
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Citation Context ...r, this way to deal with computability is too general for concrete applications. In practice, one wants to define computable functionals by recursion equations. We show that and how computation rules =-=[11, 8]-=- can be used to achieve this task. The meaning [λ�x M ] of a term M (with free variables in �x) involving constants D defined by computation rules will be an inductively defined set of tokens ( � U, b... |

4 |
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Citation Context ...s has its roots in work of Kreisel [20] (who used formal neighborhoods) and Kleene [19]. This line of research was taken up and developed in a mathematically more satisfactory way by Scott and Ershov =-=[29, 16]-=-. Today this theory is usually presented in the context of abstract domain theory [31, 3]; it is based on classical logic. The present work can be seen as an attempt to develop a constructive theory o... |

4 |
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Citation Context ... problem to strong normalization of second order propositional logic, called system F by Girard [17]). Our definition of the strong computability predicates and also the proof are related to Zucker’s =-=[34]-=- proof of strong normalization of his term system for recursion on the first three number or tree classes. However, Zucker uses a combinatory term system and defines strong computability for closed te... |

3 |
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Citation Context ...iation LCF. However, the prime example of such a theory, Per Martin-Löf’s type theory [24] in its present form deals with total (structural recursive) functionals only. An early attempt of Martin-Löf =-=[23]-=- to give a domain theoretic interpretation of his type theory has not even been published, probably because it was felt that a more general approach – such as formal topology [13] – would be more appr... |

2 |
Konstruktive Interpretation induktiver Definitionen. Master 's thesis, Mathematisches Institut der Universitat Munchen
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Citation Context ...tree classes. However, Zucker uses a combinatory term system and defines strong computability for closed terms only. Following some ideas in an unpublished note of Berger, Benl (in his diploma thesis =-=[6]-=-) transferred this proof to terms in simply typed λ-calculus, possibly involving free variables. Here it is adapted to the present context. Organization of the paper In Section 2 atomic coherent infor... |

1 |
A predicative strong normalization proof for a λ-calculus with interleaving inductive types
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Citation Context ...putation rules are those of the (Gödel) structural recursion operators. We prove their totality, by showing that the rules are strongly normalizing. A predicative proof of this fact has been given in =-=[1]-=-, based on Aczel’s notion of a set-based relation. Our proof is predicative as well, but – being based on an extension of Tait’s method of strong computability predicates – more along the standard lin... |

1 |
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Citation Context ... uses a computability predicate. A similar result in a type-theoretic setting is in Martin-Löf’s notes [23, Second Theorem]. Adequacy theorems have been proved in many contexts [2, 4, 5, 23]. Coquand =-=[14]-=- – building on the work of Martin-Löf [23] and Berger [8] – observed that the adequacy result even holds for untyped languages, hence also for dependently typed ones. The problem of proving strong nor... |

1 |
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Citation Context |