Stark, I.: Categorical models for local names. LISP and Symbolic Computation 9(1) (1996) 77--107  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... presented here lies in the calculus , a fragment of ML [MTHM97] introduced by the second author and Stark [PiS93a, Sta95] to explore the properties, with respect to semantic equivalence of programs, of call by value higher order functions and dynamically created names (see also [JeR99] In [Sta96a], Stark studies a model of the calculus based on one of Moggi s dynamic allocation monads [Mog89] in the presheaf category Set , where I is the category of nite ordinals and injective functions between them. Crucial ingredients of the dynamic allocation monad used there are the object of ....

Stark, I. D. B.: Categorical models for local names. Lisp and Symbolic Computation, 9(1):77-107, 1996.

....it should not be possible to examine a type variable in order to select different code based on the value of that type variable. The parametricity of polymorphic functions plays a signi cant role in showing certain program equivalences and similar results [Mil71, Hoa72, Rey83, Wad89, OT95, Pit96, Sta96, Red98] The functions mentioned above are pieces of code as opposed to functions in the usual set theoretic sense. Both meanings for the term function are well established among computer scientists, and we shall freely use both. This ambiguity arises from an understanding of programming where ....

Ian Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77-107, February 1996.

....validated. While remarkable progress has been made in understanding local variables (cf. the collection [15] none of this theory is directly applicable to heap variables because the shape of the heap storage dynamically varies. A number of attacks have been made on the problem: Stark s thesis [25, 24], which deals with dynamic allocation but not pointers, and Ghica s and Levy s theses [4, 5, 7, 8] which address the general semantic structure but not data representation reasoning. The recent paper of Banerjee and Naumann [2] is the rst to address data representation correctness with heap ....

.... [ cls ] W ) 8X :W 9Z :X [ exp ] Z) St(X) 9Y :Z St(Y ) ffaultg] exp ] W ) 8X :W St(X) X) ffaultg [ com] W ) 8X :W St(X) 9Y :X St(Y ) ffaultg The position of the type quanti cations 8 and 9 in the type interpretations has been recognized in earlier work [25, 4, 8]. Intuitively, a command de ned for a world W should be prepared to accept additional locations (represented by X) in its input state, and it might itself allocate new locations during the execution (represented by Y ) The parametricity interpretation of the type quanti ers means that the ....

Stark, I. Categorical models for local names. Lisp and Symbolic Computation 9, 1 (Feb. 1996), 77-107.

....and Stark s operational logical relation up to rstorder types. As preliminaries, Section 2 gives a brief review of the nu calculus and the operational logical relation, more details of which can be found in [11, 12, 14] Section 3 introduces a categorical model Set Set Set for the nu calculus [12, 13] to provide a basis for our Kripke logical relation. We give in Section 4 the de nition of the Kripke logical relation, while before that, we derive another one from Goubault Larrecq, Lasota and Nowak s de nition for name creation monad in [3] and show that this one is strictly weaker than Pitts ....

....nu calculus. Moreover this de nition silently assumes that each term has a canonical form it is indeed the case in nu calculus. This prevents one from extending this relation to non terminating calculi with names. This section introduces the categorical model Set Set Set de ned by Stark in [12, 13] to give the denotational semantics for the nu calculus. This model is a functor category equipped with a strong monad and it provides us not only a denotational model for the behavior of nu calculus expressions, but also a sound basis for a logical relation to be de ned next. 3.1 Categorical ....

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Ian Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77-107, 1996.

....atoms (T[ # ] contains elements of the form #a . d with and d D for modelling this) TD is constructed by quotienting ( # D)# by a suitable equivalence relation, mimicking one of the dynamic allocation monads on a presheaf category used by Stark to model the Pitts Stark # calculus [28]. We omit the details here, and just note that TD carries the following structure. A (mono)morphism # : D TD (this is the unit of the monad) A least element TD. A morphism # : D D ( restriction ) satisfying for all a, b and x TD that #[a] #[b]x) #[b] #[a]x) ....

I. D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, 1996.

....simply typed lambda calculus. Names may be created locally, passed around, and compared with one another, but that is all. The language is reviewed in Section 2; a full description is given by Pitts and Stark in [21, 22] with its operational and denotational semantics studied at some length in [25, 26]. Central to the nu calculus is the use of name abstraction: the expression n:M represents the creation of a fresh name, which is then bound to n within the body of M . So, for example, the expression n:n : n = n generates two new names, bound to n and n , and compares them, finally ....

....[4] and others. Denotational semantics provides another route: if two expressions have equal interpretation in some adequate model, then they are contextually equivalent. For the nu calculus, such operational methods are developed and refined in [21, 22] while categorical models are presented in [26]. Both approaches are treated at length in [25] In principle, methods such as these do give techniques for proving contextual equivalences. In practice however, they are often awkward and can require rather detailed mathematical knowledge. The contribution of this paper is to take two existing ....

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I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, February 1996.

....these references would become dangling references. Or, put another way, the stack discipline of local variables breaks down. A correct type rule for newref is given in the Appendix. Our knowledge of semantics for dynamic storage is rather incomplete. While some semantic models exist [64, 65], it is not yet clear how to integrate them with the reasoning principles presented here. 7. CONCLUSION Reynolds s Idealized Algol is a quintessential foundational system for Algol like languages. By extending it with objects and classes, we hope to provide a similar foundation for ....

Ian Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77-107, February 1996.

....obtain more abstract model which captures data abstractions and the model is extended in [25] to handle classes and objects. Atomic sheaves on the category of finite sets and injections (equivalently, pullback preserving functors) are used to give a model of SCI in [14] and that of nu calculus in [31, 24]. Such atomic sheaves also become the theoretical basis of the approach for handling binders in abstract syntax in [4] The parametricity theory itself arose from an effort to get a model of polymorphic lambda calculus and to formalize logical relations in a categorical setting [29, 10, 30] A ....

Ian Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, February 1996.

.... work presented here lies in the calculus , a fragment of ML [MTHM97] introduced by the second author and Stark [PS93a, Sta95] to explore the properties, with respect to semantic equivalence of programs, of callby value higher order functions and dynamically created names (see also [JR99] In [Sta96a], Stark studies a model of the calculus based on one of Moggi s dynamic allocation monads [Mog89] in the presheaf category Set I , where I is the category of nite ordinals and injective functions between them. Crucial ingredients of the dynamic allocation monad used there are the object of ....

I. D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77-107, 1996.

....Observational equivalence is a more important issue in languages which use a call by value function mechanism that allows for side effects in expressions. It is in this context where semantic aspects of dynamic variables have been studied using a toy language named the nu calculus [PS93b, PS93a, Sta96, Pit92] Nu calculus is a simplified language that offers higher order functions and dynamic name generation. Names are a rudimentary form of dynamic variable that can only be created; assignment or destruction are not supported. Once created, a dynamic variable is associated to a program ....

....literature, to states. Both equivalences fail in the language because the semantic model does not take into account the concept of observational equivalence. This concept will be the object of the next chapter. Most of the equivalences discussed were inspired by axioms proposed in [PS93b, PS93a, Sta96, Pit92] and they are summarized in Table 5.3. CHAPTER 5. SEMANTICS OF THE LANGUAGE 76 #new a: #new b: alloc(a) alloc(b) if a 6= b then diverge j diverge a: exp[ref ] #new b: alloc(b) if b 6= a then diverge j a: exp[ref ] diverge c: comm: #new a: alloc(a) a : p; c; if a = p ; then ....

I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, feb 1996.

....with analogous examples in other formalisms) as a good sign 6. Relation to presheaf models One origin of the work presented here lies in the calculus , a calculus of higher order functions and dynamically created names introduced by the second author and Stark [32, 36] see also [17] In [37], Stark studies a model of the calculus based on one of Moggi s dynamic allocation monads [29] in the presheaf category Set I , where I is the category of finite ordinals and injective functions between them. Crucial ingredients of the dynamic allocation monad used there are the object of ....

I. D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, 1996.

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I. D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, 1996. 10

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I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, February 1996.

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I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 1995. To appear. (p. -5)

....simply typed lambda calculus. Names may be created locally, passed around, and compared with one another, but that is all. The language is reviewed in Section 2; a full description is given by Pitts and Stark in [28, 29] with its operational and denotational semantics studied at some length in [35, 36]. Central to the nu calculus is the use of name abstraction: the expression n:M represents the creation of a fresh name, which is then bound to n within the body of M . So, for example, the expression : n = n generates two new names, bound to n and n , and compares them, finally returning ....

....[9] and others. Denotational semantics provides another route: if two expressions have equal interpretation in some adequate model, then they are contextually equivalent. For the nu calculus, such operational methods are developed and refined in [28, 29] while categorical models are presented in [36]. Both approaches are treated at length in [35] In principle, methods such as these do give techniques for proving contextual equivalences. In practice however, they are often awkward and can require rather detailed mathematical knowledge. The contribution of this paper is to take two existing ....

[Article contains additional citation context not shown here]

I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, February 1996.

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Stark, I.: Categorical models for local names. LISP and Symbolic Computation 9(1) (1996) 77--107

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I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, 1996.

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Stark, I.: Categorical models for local names. LISP and Symbolic Computation 9(1) (1996) 77--107

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I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1), 1996.

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Ian D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77-107, February 1996.

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I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, 1996.

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I. D. B. Stark, Categorical models for local names, Lisp and Symbolic Computation 9 (1) (1996) 77--107.

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I. D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77--107, 1996.

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I. D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9:77--107, 1996.

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Ian D. B. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77-107, February 1996.

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