## Induction and recursion on the partial real line with applications to Real PCF (1997)

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Venue: | Theoretical Computer Science |

Citations: | 6 - 1 self |

### BibTeX

@ARTICLE{Escardo97inductionand,

author = {Martín Hötzel Escardo and Thomas Streicher},

title = {Induction and recursion on the partial real line with applications to Real PCF},

journal = {Theoretical Computer Science},

year = {1997},

volume = {210},

pages = {157}

}

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### Abstract

The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify that Real PCF programs meet their specification. They resemble the so-called Peano axioms for natural numbers. The theory is based on a domain-equation-like presentation of the partial unit interval. The principles are applied to show that Real PCF is universal in the sense that all computable elements of its universe of discourse are definable. These elements include higher-order functions such as integration operators. Keywords: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion [28,21]. The idea is that singleton intervals represent total rea...

### Citations

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876 |
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- 1966
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Citation Context ...: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion =-=[28,21]-=-. The idea is that singleton intervals represent total real numbers, and that the remaining intervals represent (properly) partial real numbers. This is justified by the fact that the singleton map x ... |

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Citation Context ...d sections at the same time. Preliminary ideas on recursion and induction on the real line appeared in [8], which considers uniform spaces. Our axioms are formulated in the framework of domain theory =-=[1]-=-. Domain theory allows one to derive induction principles and recursion schemes from canonical solutions of domain equations [20,34,24]. Domain equations model recursive definitions of data types, suc... |

397 |
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Citation Context ...f a simulation and a more general coinduction principle for establishing inequalities. 6 Applications to the programming language Real PCF Real PCF [9] is an extension of the programming language PCF =-=[29,22]-=- with data types for the partial unit interval and the partial real line. For simplicity and without essential loss of generality, we only discuss the partial unit interval. A sketch of the treatment ... |

321 |
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- 1980
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267 |
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229 |
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- 1990
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169 |
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- 1982
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Citation Context ...orm spaces. Our axioms are formulated in the framework of domain theory [1]. Domain theory allows one to derive induction principles and recursion schemes from canonical solutions of domain equations =-=[20,34,24]-=-. Domain equations model recursive definitions of data types, such as lists and trees. Since the partial real line is not an algebraic domain, it is not the canonical solution of any domain equation i... |

121 |
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- 1972
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Citation Context ...the partial unit interval, (6) Applications to the programming language Real PCF. 1 Domain theory Our main reference to domain theory is [1]. For its connections with topology see [33,16] (the papers =-=[27]-=- and [31] contain interesting technical and intuitive computational interpretations of topological concepts). Here we establish terminology and recall basic facts. Readers who are familiar with domain... |

92 |
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- 1990
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Citation Context ...unctors. We establish new results about the notion of an inductive retraction introduced in [10], which generalizes canonical solutions of domain equations by means of ideas similar to those of Freyd =-=[14,15]-=-. In particular, we introduce the notion of a biquotient of a bifree algebra, and we show that the inductive retractions are the biquotients of the bifree algebras. An interesting observation is that ... |

62 |
Remarks on algebraically compact categories
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- 1991
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Citation Context ...unctors. We establish new results about the notion of an inductive retraction introduced in [10], which generalizes canonical solutions of domain equations by means of ideas similar to those of Freyd =-=[14,15]-=-. In particular, we introduce the notion of a biquotient of a bifree algebra, and we show that the inductive retractions are the biquotients of the bifree algebras. An interesting observation is that ... |

49 |
T ω as a universal domain
- Plotkin
- 1978
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Citation Context ...ypes and the existential quantifier is universal by means of the technique introduced in [36]. Here are the main steps of the technique: (i) Take a universal domain U of PCF, for example [N → B] (see =-=[23]-=-). (ii) Show that for every domain D in the extended language there is a definable retraction D rD ⇆ sD U with rD ◦ sD = idD. (iii) Given d ∈ D computable, sD(d) ∈ U is computable because sD is comput... |

46 | PCF extended with real numbers
- Escardó
- 1996
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Citation Context ... real line into the partial real line endowed with its Scott topology. The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF =-=[9,10]-=-, including computation of integrals [4]. To appear in Theoretical Computer Science 28 November 1997sWe introduce induction principles and recursion schemes for the partial unit interval (the domain o... |

43 | A domain-theoretic approach to computability on the real line, Theoretical Computer Science 210
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- 1999
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Citation Context ...was conjectured in [11] that a real valued function of real variables is computable iff it has a computable extension to a partial real valued function of partial real variables. It has been shown in =-=[5]-=- that this is indeed the case. Organization This paper is the full version of the extended abstracts [10,12]. It is organized in the following sections: (1) Domain theory, (2) The partial real line, (... |

37 | A coinduction principle for recursive data types based on bisimulation
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- 1996
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Citation Context ...tial unit interval via “co-Peano axioms” based on coinduction and coiteration. Although we don’t have such a characterization yet, a coinduction principle related to the ideas of Smyth [32] and Fiore =-=[13]-=- immediately follows by considering the bifree T-algebra. A bisimulation on the partial unit interval is a binary relation ∼⊆ I × I such that x ∼ y implies that left(x) = left(y) and pred a(x) ∼ pred ... |

37 |
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- 1981
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Citation Context ...orm spaces. Our axioms are formulated in the framework of domain theory [1]. Domain theory allows one to derive induction principles and recursion schemes from canonical solutions of domain equations =-=[20,34,24]-=-. Domain equations model recursive definitions of data types, such as lists and trees. Since the partial real line is not an algebraic domain, it is not the canonical solution of any domain equation i... |

35 |
Set Theory and Logic
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Citation Context ...pecified, up to isomorphism, by the so-called Peano axioms, which are essentially the properties that we informally considered above for the sake of motivation. This idea is made formal in e.g. Stoll =-=[35]-=-, where unary systems are used as a tool (a unary system is a set X together with an element x ∈ X and a function s : X → X). In the following definition, the domain D generalizes the partial unit int... |

32 |
Effectively given domains
- Smyth
- 1977
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Citation Context ... Here we consider a further extension of the language with recursive types. The universality result depends on a notion of computability. In domain theory this is achieved via effective presentations =-=[6,30]-=-. It is straightforward to show that there exists an effective presentation of the partial real line that makes the primitive operations of the language computable. For example, any standard enumerati... |

29 |
Lattice theory, data types and semantics
- Scott
- 1972
(Show Context)
Citation Context ...: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion =-=[28,21]-=-. The idea is that singleton intervals represent total real numbers, and that the remaining intervals represent (properly) partial real numbers. This is justified by the fact that the singleton map x ... |

27 | Properly injective spaces and function spaces
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- 1998
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Citation Context ...o maximal elements are dense. In any case, R belongs to a larger class of domains, which are characterized precisely as the injective spaces over finitary subspace embeddings, which include j : R → R =-=[7]-=-. But in this paper we prefer to give a proof of the above special case of the extension property from first principles.) Among all extensions there is a canonical extension If : R → R given by If(x) ... |

23 |
domains and predicate transformers: a topological view
- Power
- 1983
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Citation Context ...al unit interval, (6) Applications to the programming language Real PCF. 1 Domain theory Our main reference to domain theory is [1]. For its connections with topology see [33,16] (the papers [27] and =-=[31]-=- contain interesting technical and intuitive computational interpretations of topological concepts). Here we establish terminology and recall basic facts. Readers who are familiar with domain theory c... |

21 |
A type theoretical alternative to CUCH
- Scott
- 1993
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Citation Context ...f a simulation and a more general coinduction principle for establishing inequalities. 6 Applications to the programming language Real PCF Real PCF [9] is an extension of the programming language PCF =-=[29,22]-=- with data types for the partial unit interval and the partial real line. For simplicity and without essential loss of generality, we only discuss the partial unit interval. A sketch of the treatment ... |

20 |
PCF Extended with Real Numbers: a domain-theoretic approach to higher-order exact real number computation
- Escardó
(Show Context)
Citation Context ...eral result does not tell the full story about definability of computable first order functions (over the partial unit interval). By means of a more direct method of proof similar to that of [22], in =-=[11]-=- it is shown that the existential quantifier is not needed to obtain the definability result at first-order types. Conclusions We have defined and studied inductive retractions. We have shown that the... |

17 | Basic category theory - Poigné - 1992 |

11 | Induction and recursion on the real line
- Escardó
- 1995
(Show Context)
Citation Context ...l number constructors. This is related to binary expansions and, perhaps surprisingly, to Dedekind sections at the same time. Preliminary ideas on recursion and induction on the real line appeared in =-=[8]-=-, which considers uniform spaces. Our axioms are formulated in the framework of domain theory [1]. Domain theory allows one to derive induction principles and recursion schemes from canonical solution... |

9 |
Integration in Real PCF (extended abstract
- Edalat, Escardó
- 1996
(Show Context)
Citation Context ...ed with its Scott topology. The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF [9,10], including computation of integrals =-=[4]-=-. To appear in Theoretical Computer Science 28 November 1997sWe introduce induction principles and recursion schemes for the partial unit interval (the domain of closed subintervals of the unit interv... |

9 |
PCF extended with ∃ is universal
- Real
- 1996
(Show Context)
Citation Context ... real line into the partial real line endowed with its Scott topology. The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF =-=[9,10]-=-, including computation of integrals [4]. To appear in Theoretical Computer Science 28 November 1997sWe introduce induction principles and recursion schemes for the partial unit interval (the domain o... |

9 |
Post-graduate Lectures in advanced domain theory
- Domains
- 1980
(Show Context)
Citation Context ...orm spaces. Our axioms are formulated in the framework of domain theory [1]. Domain theory allows one to derive induction principles and recursion schemes from canonical solutions of domain equations =-=[20,34,24]-=-. Domain equations model recursive definitions of data types, such as lists and trees. Since the partial real line is not an algebraic domain, it is not the canonical solution of any domain equation i... |

8 |
A universality theorem for PCF with recursive types, parallel-or and
- Streicher
- 1994
(Show Context)
Citation Context ...tion induced by them automatically gives rise to coinduction and corecursion. The techniques discussed here in a more general setting were also applied in conjunction with the technique introduced in =-=[36]-=- to show that Real PCF extended with a certain computable existential quantifier is universal [10], in the sense that all computable elements of its universe of discourse are definable. These elements... |

8 | PCF extended with 9 is universal - Real - 1996 |

6 |
When are two effectively given domains identical
- Kanda, Park
- 1979
(Show Context)
Citation Context ... wonders whether a cleverer choice of an effective presentation would change the induced set of computable elements and functions, and this is indeed the case in general, as Kanda and Park have shown =-=[19]-=-. We apply the presentation of the partial real line as an inductive retract to show that there is a unique effective presentation of the partial real line that makes the primitive (and hence all) ope... |

5 |
I-categories and duality
- Smyth
- 1992
(Show Context)
Citation Context ...o be defined by structural recursion and corecursion respectively. Structural recursion generalizes primitive recursion on the natural numbers, whereas structural corecursion generalizes minimization =-=[32]-=-. 1.5 Effectively given domains The reader is referred to [30]. The references [6,24] consider only algebraic domains, which exclude the partial real line. On the other hand, [5] considers a weaker ve... |

4 |
Computability concepts for programming languages
- Egli, Constable
- 1976
(Show Context)
Citation Context ... Here we consider a further extension of the language with recursive types. The universality result depends on a notion of computability. In domain theory this is achieved via effective presentations =-=[6,30]-=-. It is straightforward to show that there exists an effective presentation of the partial real line that makes the primitive operations of the language computable. For example, any standard enumerati... |

4 |
Induction and recursion on the partial real line via biquotients of bifree algebras
- Escardó, Streicher
- 1997
(Show Context)
Citation Context ...ble extension to a partial real valued function of partial real variables. It has been shown in [5] that this is indeed the case. Organization This paper is the full version of the extended abstracts =-=[10,12]-=-. It is organized in the following sections: (1) Domain theory, (2) The partial real line, (3) Peanolike axioms for the partial unit interval, (4) Generalized domain equations, (5) A generalized domai... |

2 | Induction and recursion on the partial real line via biquotients of bifree algebras (extended abstract - Escard'o, Streicher - 1997 |