## Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory (2003)

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Venue: | In Proc. 17th IEEE Symposium on Logic in Computer Science |

Citations: | 8 - 2 self |

### BibTeX

@INPROCEEDINGS{Simpson03computationaladequacy,

author = {Alex Simpson},

title = {Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory},

booktitle = {In Proc. 17th IEEE Symposium on Logic in Computer Science},

year = {2003}

}

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### Abstract

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo-Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with callby-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.

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Citation Context ...ry of predomains, P - S, whose associated category of partial maps, pP, is algebraically compact. This approach directly follows [40], where it is shown that a model of the simplytyped language PCF [=-=32]-=- can be similarly extracted from any elementary topos S, with natural numbers object, satisfying a stronger Axiom N. The additional goal of the present paper is to show the algebraic compactness of pP... |

372 |
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Citation Context ...ll impredicativity of ECC. In fact, it seems likely that, with appropriate reformulations, the development of this paper could be carried out in the (predicative) context of Martin-Lof's Type Theory [=-=26], mak-=-ing crucial use of universes and so-called \large" elimination rules for inductive types (in particular I). Similarly, it appears that a predicative set theory could be used rather than IZF, for ... |

206 |
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(Show Context)
Citation Context ...s itself to arise as the full subcategory of small objects within a category C with class structure and universal object. The specic toposes we consider will either be Grothendieck toposes, see e.g. [=-=24]-=-, or realizability toposes [13, 15]. There are various results relating such toposes to categories with class structure in [19, Ch. IV], which can be massaged into an appropriate form to obtain embedd... |

184 |
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Citation Context ...ness of every slicesbration of Fam(pP) ! C. The proof of Theorem 1 occupies Sections 4-9. The strategy is to establish a version of the limit-colimit coincidence of classical domain theory (see, e.g. =-=[45]-=-), and apply it to pP. However, a major complication arises. In many models of our setting, the limit-colimit coincidence is false if formulated using diagrams indexed by the natural numbers N, see [2... |

130 |
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Citation Context ...(i.e. a class), |K|, of K-objects, and an internal family, {K(A, B)} A,B : |K| , of K-morphisms indexed by domain and codomain, satisfying the expected axioms for identities and composition, see e.g. =-=[13]-=-. We say that an internal categorysK in C is locally small if the internal family {K(A, B)} A,B : |K| # |K|s|K| is a small map in C. It is small if, in addition, |K| is small. An internal functor, F ,... |

125 | The Type Theoretic Interpretation of Constructive Set Theory: Choice Principles
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Citation Context ...of universes and so-called \large" elimination rules for inductive types (in particular I). Similarly, it appears that a predicative set theory could be used rather than IZF, for example, Aczel's=-= CZF [1]-=-. Such possibilities suggest that it is the conceptual strength of IZF that we are exploiting in this paper, rather than its proof-theoretic strength. Nevertheless, there are two reasons for being con... |

122 |
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(Show Context)
Citation Context ...y to achieve algebraic compactness, many other set theories and type theories appear rich enough to carry out the proofs in this paper. One such theory is the Extended Calculus of Constructions (ECC) =-=[23]-=-, as used, for example, in [35]. However, it appears that one does not need the full impredicativity of ECC. In fact, it seems likely that, with appropriate reformulations, the development of this pap... |

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Citation Context ...# ! t +. It remains to dene the relations. Because they are recursively specied, this takes a considerable amount of work. Although it seems possible to apply Pitts' method of dening relations [31], doing so would require the development of further machinery. Because we already have the technology of suitable categories at our disposal, it seems easier to adapt the techniques of [33, 3]. For ea... |

95 |
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(Show Context)
Citation Context ...exponentials, to interpret function types; and finally, to interpret recursive types, the derived category,spP, of partial maps, induced by L on P, must be algebraically compact in the sense of Freyd =-=[7, 8]-=-, at least with repect to functors defined by type expressions. The above identifies the structure required by a model of FPC, but does not indicate where to find examples of models. Nevertheless, sev... |

91 |
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(Show Context)
Citation Context ... in categories with class structure. Moreover, under mild conditions, AxiomsN is satisfied. Also, by their very definition, realizability models [9, 20, 21, 22, 16, 17] embed in realizability toposes =-=[10, 12]-=-, and hence in categories with class structure [15, Ch. IV]. Again, Axiom N is satisfied. Thus, Theorem 1 gives an account of the construction of solutions to recursive domain equations that applies s... |

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(Show Context)
Citation Context ... x (-)(-) ). Define a morphism FB b # B by b = F i (c si # F l i ). Lemma 6.3 (B, b) is a bifree F -algebra. The proof is by establishing that FB b # B is a specialF -invariant object in the sense of =-=[6, 29]-=-, and that this property is characteristic of bifree F -algebras, again see [6, 29]. This concludes the proof of Proposition 3. We now complete the proof of Theorem 1 by establishing the result below.... |

70 |
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- 1995
(Show Context)
Citation Context ...s of domains to provide a host of \neo-classical" models [3, 6]. A very dierent type of model is given by game-theoretic semantics [25]. Finally, there are a variety of models based on realizabil=-=ity [11, 28, 29, 30, 21, 22, 35]-=-. What has been missing hitherto is a single unifying treatment accounting for the existence of all these types of model. In this paper, we provide the axiomatic basis for such a treatment. In a follo... |

69 |
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(Show Context)
Citation Context ...ss in the natural way. Nevertheless, we remark that the denition also has a simple categorytheoretic formulation using the (double-)strength of the L monad, see, e.g., the denition of bimorphism in [16]. Proposition 4.3 For pointed objects (X 1 ; 1 ), . . . , (X k ; k ) and (Y;s), where k 1, any k-strict map h : X 1 X k - Y is a strict map from (X 1 ; 1 ) (X k ; k ) to (... |

67 |
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(Show Context)
Citation Context ...exponentials, to interpret function types; and finally, to interpret recursive types, the derived category,spP, of partial maps, induced by L on P, must be algebraically compact in the sense of Freyd =-=[7, 8]-=-, at least with repect to functors defined by type expressions. The above identifies the structure required by a model of FPC, but does not indicate where to find examples of models. Nevertheless, sev... |

54 |
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- 1990
(Show Context)
Citation Context ...is section is to isolate a full subcategory of S to act as a category of predomains. This will require imposing further axioms on C. Many axiomatizations have been proposed for this purpose, see e.g. =-=[26, 11, 20, 33, 17, 30, 25, 19]-=-. Here, we follow [30]. As first proposed in [26], the definition of predomain is predicated on a notion of partiality. To implement this, we require a distinguished subobject # # ## . Intuitively # c... |

52 |
and full abstraction for a functional metalanguage with recursive types
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- 1998
(Show Context)
Citation Context ...iomatic domain theory has successfully abstracted the particularities of domains to provide a host of \neo-classical" models [3, 6]. A very dierent type of model is given by game-theoretic semant=-=ics [25]-=-. Finally, there are a variety of models based on realizability [11, 28, 29, 30, 21, 22, 35]. What has been missing hitherto is a single unifying treatment accounting for the existence of all these ty... |

50 |
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- 1967
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48 | Continuity and effectiveness in topoi - Rosolini - 1986 |

40 |
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(Show Context)
Citation Context ...chnically, this is implemented by asking for S to be given as the full subcategory of small objects in a category C with class(ic) structure and universal object, in the sense of [31] (developed from =-=[15]-=-). As our first main result, Theorem 1, we prove that, with such a categorysS, the derived category pP is algebraically compact whenever Axiom N holds. Thus, with enough set-theoretic power to back it... |

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- 1991
(Show Context)
Citation Context ...a Scott showed that such categories can nonetheless live as full subcategories of models of intuitionistic set theory, an observation that led to the subsequent development of synthetic domain theory =-=[36, 14, 28, 46, 22, 40, 35, 27, -=-7]. In this paper, we exploit this idea to obtain algebraically compact categories in a uniform way. Roughly speaking, we start o with a category S of intuitionistic sets that satises one simple axiom... |

35 | An Axiomatisation of Computationally Adequate Domain-Theoretic Models of FPC
- Fiore, Plotkin
- 1994
(Show Context)
Citation Context ...ass of models we work with is given by models of FPC, as in [3, Def. 8.3.1], that have an inductivesxed-point object in the sense of [5, Def. 1.11]. This class includes the domain-theoretic models of =-=[4, 3-=-], and, more generally, all KADT models, as in [5, Def. 1.12]. Let C be any model in this class. As C is a model of FPC, it has, in particular, a dominance , an associated lifting functor L, andsnite ... |

31 |
semantics with partial functions, unpublished lecture notes from
- Plotkin
- 1985
(Show Context)
Citation Context ...er hand, Lemma 9.7 disappears, because P L is small-complete. 10 The language FPC In this section, we give a brief overview of Plotkin's call-by-value recursively typed -calculus, FPC, introduced in [33]. For full details see [3]. We use X;Y; : : : to range over type variables, and ; ; : : : to range over types, which are given by: ::= X j + j j ! j X:: Here the prex X binds X. ... |

26 |
Elementary axioms for categories of classes
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- 1999
(Show Context)
Citation Context ...ry, see e.g. [37]. Technically, this is implemented by asking for S to be given as the full subcategory of small objects in a category C with class(ic) structure and universal object, in the sense of =-=[41, 43]-=- (developed from [19]). As oursrst main result, Theorem 1, we prove that, 3 with such a category S, the derived category pP is algebraically compact whenever Axiom 1 holds. Thus, with enough set-theor... |

22 |
Effective Domains and Intrinsic Structure, in
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- 1990
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Citation Context ...rd. Thus, by Theorem 2, the interpretation of FPC in Mod(A) is computationally adequate. This gives thesrst proof of computational adequacy for an interpretation of FPC in the realizability models of =-=[11, 28, 29, 30, 21, 22]-=-. 15.2 Models of axiomatic domain theory In [3, Def. 8.3.1], an axiomatization of a general categorical notion of model for FPC is given. Moreover, as Theorem 9.2.19 of op. cit., computational adequac... |

20 |
Relating theories of the � -calculus
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Citation Context ...tic framework. Categories that model recursive types have nontrivialsxed-point operators and thus, by a simple argument using classical logic, cannot be full subcategories of the category of sets. In =-=[38]-=-, Dana Scott showed that such categories can nonetheless live as full subcategories of models of intuitionistic set theory, an observation that led to the subsequent development of synthetic domain th... |

20 |
Relating theories of the λ-calculus
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Citation Context ...c framework. Categories that model recursive types have nontrivial fixed-point operators and thus, by a simple argument using classical logic, cannot be full subcategories of the category of sets. In =-=[38]-=-, Dana Scott showed that such categories can nonetheless live as full subcategories of models of intuitionistic set theory, an observation that led to the subsequent development of synthetic domain th... |

17 | Cuboidal sets in axiomatic domain theory
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Citation Context ...ides the classical example of the category of #cpos [24]. More generally, axiomatic domain theory has successfully abstracted the idiosyncracies of domains to provide a host of "neo-classical&quo=-=t; models [2, 4]-=-. A quite different type of model is given by gametheoretic semantics [18]. Finally, while the structure has not previously been exhibited in the form above, it has long been known that there should b... |

17 |
The eective topos
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Citation Context ...ubcategory of small objects within a category C with class structure and universal object. The specic toposes we consider will either be Grothendieck toposes, see e.g. [24], or realizability toposes [=-=13, 15]-=-. There are various results relating such toposes to categories with class structure in [19, Ch. IV], which can be massaged into an appropriate form to obtain embedding results sucient for our needs. ... |

16 | Axiomatic Domain Theory
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Citation Context ...ides the classical example of the category of #cpos [24]. More generally, axiomatic domain theory has successfully abstracted the idiosyncracies of domains to provide a host of "neo-classical&quo=-=t; models [2, 4]-=-. A quite different type of model is given by gametheoretic semantics [18]. Finally, while the structure has not previously been exhibited in the form above, it has long been known that there should b... |

16 | An extension of models of axiomatic domain theory to models of synthetic domain theory
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- 1996
(Show Context)
Citation Context ... [2], an axiomatization of a general order-enriched notion of model for FPC is given, and computational adequacy is proved for any nontrivial model satisfying an additionalsabsoluteness condition. In =-=[4, 3]-=-, a much more general class of enriched models is introduced, although the interpretation of FPC is not explicitly considered. Following the approach of [3], we can accommodate many of the models of [... |

16 | Continuity and E#ectivity in Topoi - Rosolini - 1986 |

12 |
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Citation Context ...main theory. Nevertheless, our applications to axiomatically given classes of models demonstrate that our results should be viewed equally much as a contribution to theseld of axiomatic domain theory =-=[9, 10, 39, 4, 3, 5, 6]-=-. It is the author's view 4 that embedding categories of predomains within models of intuitionistic set theory is the correct approach to obtaining an axiomatic account of domain-theoretic constructio... |

11 | General synthetic domain theory - a logical approach
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(Show Context)
Citation Context ...s of domains to provide a host of \neo-classical" models [3, 6]. A very dierent type of model is given by game-theoretic semantics [25]. Finally, there are a variety of models based on realizabil=-=ity [11, 28, 29, 30, 21, 22, 35]-=-. What has been missing hitherto is a single unifying treatment accounting for the existence of all these types of model. In this paper, we provide the axiomatic basis for such a treatment. In a follo... |

9 |
A uniform account of domain theory in realizability models. To be submitted to special edition
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Citation Context ...s of domains to provide a host of \neo-classical" models [3, 6]. A very dierent type of model is given by game-theoretic semantics [25]. Finally, there are a variety of models based on realizabil=-=ity [11, 28, 29, 30, 21, 22, 35]-=-. What has been missing hitherto is a single unifying treatment accounting for the existence of all these types of model. In this paper, we provide the axiomatic basis for such a treatment. In a follo... |

9 | Computational adequacy in an elementary topos
- Simpson
- 1998
(Show Context)
Citation Context ...a Scott showed that such categories can nonetheless live as full subcategories of models of intuitionistic set theory, an observation that led to the subsequent development of synthetic domain theory =-=[36, 14, 28, 46, 22, 40, 35, 27, -=-7]. In this paper, we exploit this idea to obtain algebraically compact categories in a uniform way. Roughly speaking, we start o with a category S of intuitionistic sets that satises one simple axiom... |

8 |
A presentation of the initial lift algebra
- Jibladze
- 1997
(Show Context)
Citation Context ...tion Lf : LX # LY is defined by (Lf)(e) = {f(x) | x # e}. Further, the endofuctor L carries a monad structure. The unit is singleton {} : X # LX , and the multiplication is union S : LLX # LX . As in =-=[14]-=-, the endofunctor L has a final coalgebra, # : F # LF (necessarily an isomorphism), defined by: F = {c : # N | #n : N. c(n + 1) # c(n)} #(c) = {(n ## c(n + 1)) | c(0)}. Because F is small, there exist... |

8 | Axioms and (counter)examples in synthetic domain theory
- Oosten, Simpson
(Show Context)
Citation Context ...a Scott showed that such categories can nonetheless live as full subcategories of models of intuitionistic set theory, an observation that led to the subsequent development of synthetic domain theory =-=[36, 14, 28, 46, 22, 40, 35, 27, -=-7]. In this paper, we exploit this idea to obtain algebraically compact categories in a uniform way. Roughly speaking, we start o with a category S of intuitionistic sets that satises one simple axiom... |

7 | Elementary axioms for categories of classes (extended abstract
- Simpson
- 1999
(Show Context)
Citation Context ...can be thought of as an object that collects the elements of all classes together within 11 one universal class. In set-theoretic terms, U is simply the class of all sets (and atoms if permitted). In =-=[41, 43]-=- it is shown how the existence of a universal object implies that C contains an internal model of IZF set theory (with Replacement rather than Collection). Henceforth we require that C have a universa... |

6 |
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- 1993
(Show Context)
Citation Context ...ed models axiomatized in [4, 3]. For realizability models, the only existing proof of computational adequacy for a language (implicitly) containing recursive types, applies to just one specific model =-=[1]-=-. We thus obtain the first proof of computational adequacy, for the interpretation of a language with recursive types, in all the realizability models of [9, 20, 21, 22, 16, 17]. Acknowledgements This... |

5 |
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(Show Context)
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3 |
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(Show Context)
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3 | Foundation - Taylor - 1988 |

1 |
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- 2003
(Show Context)
Citation Context ...tly that every Grothendieck topos and every realizability topos arises as the full subcategory of small objects within a category with class structure and universal object. This result will appear in =-=[-=-2]. 15.1 Realizability models A realizability model is specied by a partial combinatory algebra (A; ), which determines a category Mod(A) of modest sets over A, see e.g. [22, x2-3]. In many such categ... |

1 |
Algebraic compactness in an enriched stetting
- Plotkin
- 1995
(Show Context)
Citation Context ... notion of e-p pair to use. A similar issue was addressed by Plotkin, who developed a generalized notion of e-p pair to establish a limit-colimit coincidence in the context of axiomatic domain theory =-=[34-=-]. However, as motivated in Section 3, we have to depart from the standard theorem in another signicant way: our diagrams must be indexed by I not N. Miraculously, the use of I as an indexing object e... |

1 |
Computational adequacy for models of FPC
- Simpson
- 2003
(Show Context)
Citation Context ...ct inductive denition in terms of external structure on pP, as long as the external structure on pP is determined via I and J from the internal structure of pP. This observation will be important in [44]. The external interpretation of FPC we have given is not as pleasant as one might like. For example, the interpretation of closed types satises, [[]] + 0 [[ ]] = [[ + ]] = [[]] + [[ ]] [[]... |

1 |
Relating set theory, toposes and categories of classes
- Streicher
- 2003
(Show Context)
Citation Context ...tly that every Grothendieck topos and every realizability topos arises as the full subcategory of small objects within a category with class structure and universal object. This result will appear in =-=[2]-=-. 15.1 Realizability models A realizability model is specified by a partial combinatory algebra (A, ·), which determines a category Mod(A) of modest sets over A, see e.g. [22, §2–3]. In many such cate... |