Birkedal, L. and R. Harper, Relational interpretations of recursive types in an operational setting, Information and Computation 155 (1999), pp. 3--63.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is that the definition of the logical relation (Definition 5.4) proceeds by induction on the structure of types. In the presence of recursive types one has to use some other approach in order to avoid a circular definition; here syntactic versions of the construction of recursively defined domains [4] may be of assistance. A more subtle problem is that some of our definitions (for example the notion of extension of state relations in Definition 5.1) exploit the fact that we restricted attention to memory states with a very simple, flat structure; many of the features listed in Fig. 6 cause ....

L. Birkedal and R. Harper. Relational interpretation of recursive types in an operational setting (Summary). In M. Abadi and T. Ito, editors, Theoretical Aspects of Computer Software, Third International Symposium, TACS'97, Sendai, Japan, September 23 - 26, 1997, Proceedings, volume 1281 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1997.

....soundness proof of the logical relations. Also, introducing recursive types breaks the well definedness of the logical relations. We expect that these limitations can be removed by incorporating the theory of logical relation for # calculus with recursive functions and or recursive types (e.g. [5, 7]) State and Linearity Although real programs often have some kind of state or linearity (in the sense of linear logic) our framework does not take them into account. Thus, it cannot prove the security of a program depending on them. For example, consider an expression p i = new z in #x. let ....

Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting. Information and Computation, 155(1--2):3--63, 1999.

....soundness proof of the logical relations. Also, introducing recursive types breaks the well definedness of the logical relations. We expect that these limitations can be removed by incorporating the theory of logical relation for # calculus with recursive functions and or recursive types (e.g. [5, 7]) State and Linearity Although real programs often have some kind of state or linearity (in the sense of linear logic) our framework does not take them into account. Thus, it cannot prove the security of a program depending on them. For example, consider an expression p i = new z in #x. let ....

L. Birkedal and R. Harper. Relational interpretations of recursive types in an operational setting. Information and Computation, 155(1--2):3--63, 1999.

....architecture would require a very large implementation effort, and for this reason the recursion theoretic approach is not attractive. Compactness of evaluation. The notion of minimal invariance as defined by Pitts [Pit96] and adapted by 15 Birkedal and Harper to an operational setting [BH97] provides a relational interpretation of general recursive types. Like other earlier approaches, these approaches treat terms extensionally and hence appear to be fundamentally different from our approach. We have not investigated generalizing our approach to arbitrary logical relations, but ....

Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting. In Theoretical Aspects of Computer Software, 1997.

....establish such a method, we need to abstract the 34 necessary properties that should hold for the relationship between polymorphic primitives in ml and the corresponding ones in impl and to properly strengthen the logical predicate. Recent results on generalizing logical predicates such as [1, 3, 14] may shed some light on developing such a general technique. Intensional polymorphism. One interesting issue would be to apply the framework presented here to intensional polymorphism of Harper and Morrisett [13] So far we have tacitly assumed that the value denoted by an attribute type of the ....

L. Birkedal and R. Harper. Relational interpretations of recursive types in an operational setting. In Proc. Theoretical Aspects of Computer Software, 1997.

....and Stone 1996. Operational analogues of the domain theoretic concepts of least fixed points, continuity, and the minimal invariant property of recursively defined domains. Mason, Smith, and Talcott 1996, Smith 1998. Applications of operationally based logical relations. Pitts 1997, Birkedal and Harper 1997, Pitts and Stark 1998, Pitts 1998. ....

Birkedal, L. and R. Harper (1997). Relational interpretation of recursive types in an operational setting (Summary). In M. Abadi and T. Ito (Eds.), Theoretical Aspects of Computer Software, 3rd Int. Symp., Volume 1281 of Lecture Notes in Computer Science. Springer-Verlag, Berlin.

....the minimal invariance property of recursively defined domains: see (Pitts 1994; Pitts 1996) Operational versions of the techniques in loc. cit. are possible and can be used to extend the results presented here to cover (a) This suggestion has been taken up for a pure functional language in Birkedal and Harper 1997. Regarding (b) it is well known that the ability to store function values gives rise to more complex behaviour than storing only values of ground types. For example, it becomes possible to encode recursive function definitions. This is reflected in the denotational semantics of such storage by ....

Birkedal, L. and R. Harper (1997). Relational interpretation of recursive types in an operational setting (Summary). In Proc. TACS'97, Lecture Notes in Computer Science. Springer-Verlag, Berlin. To appear.

....bring the full power of type theory to bear on a programming problem. For example, in Crary [30] I use a type theoretic interpretation to expose the relation of power kinds to a nonconstructive set type. Adjusting this interpretation to make the power kind constructive 9 See Birkedal and Harper [11] for a promising approach that may lead to a solution of this problem. 10 However, in some contexts it is not necessary for objects to be first class. For example, Jackson [57] independently used an encoding essentially the same as Pierce and Turner s to implement computational abstract algebra. ....

Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting. In Theoretical Aspects of Computer Software, 1997.

....article. Such results enable one to reuse some familiar denotational methods, such as Scott s induction principle (Scott 1993) For example, the operational analogue of Scott induction is used in proving the Fundamental Property of the logical relation introduced in the article of Pitts Stark. Birkedal and Harper (1997) provide another good example 6 Introduction of the application of such techniques. Methods from the calculus. A glance through recent text books such as those by Gunter (1992) and Mitchell (1996) shows the extent to which the mathematics of the calculus (both the typed and untyped varieties) ....

Birkedal, L. and R. Harper (1997). Relational interpretation of recursive types in an operational setting (Summary). In Proc. TACS'97, Lecture Notes in Computer Science.

....that recursive type will include negative occurrences of the variable of recursion. Type theory may express recursive types with only positive occurrences, but to allow negative occurrences is an open problem. The other main problem arising with a monadic interpretation of state has See Birkedal and Harper (1997) for a promising approach that may lead to a solution of this problem. Conclusion 17 to do with predicativity. If polymorphic functions may be placed into the store, every function type, when monadized to take the store as an argument, will have a level as high as the highest level appearing in ....

Birkedal, L. and Harper, R. (1997) Relational interpretations of recursive types in an operational setting. In Theoretical Aspects of Computer Software.

....the target code. In the case of recursive code and self referential data structures, the finite approximants are the finite unrollings of the recursive code or self referential structure. In the case of recursive types we make use of the syntactic minimal invariant associated with a recursive type [9,5,4] to define the finite approximants of a value. In the rest of this abstract, we sketch the original simply typed closure conversion algorithm and the three translations mentioned above for recursive functions. 2 Simply Typed Closure Conversion We formalize closure conversion for the ....

L. Birkedal and R. Harper. Relational interpretations of recursive types in an operational setting (summary). In Theoretical Aspects of Computer Science, Sendai, Japan, September 1997. (To appear.).

....side effecting expressions take the store as an argument, that recursive type will include negative occurrences of the variable of recursion. Type theory may express recursive types with only positive occurrences, but to allow negative occurrences is an open problem. 7 7 See Birkedal and Harper [3] for a promising approach that may lead to a solution of this problem. The other main problem arising with a monadic interpretation of state has to do with predicativity. If polymorphic functions may be placed into the store, every function type, when monadized to take the store as an argument, ....

Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting. In Theoretical Aspects of Computer Software, 1997.

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Birkedal, L. and R. Harper, Relational interpretations of recursive types in an operational setting, Information and Computation 155 (1999), pp. 3--63.

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Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting (summary). In Theoretical Aspects of Computer Science, pages 458--490, Sendai, Japan, September 1997. 16

....that well typed terms are related to themselves by the relation associated to their type. The constructed relations can be seen to provide a notion of equality of terms and will therefore be called relations for equality and can in fact be used to reason about contextual equivalence [2]. However, we will not pursue that direction in this extended abstract. Instead we shall view these two sections as providing the necessary understanding for constructing a relational interpretation, which we then use to show the correctness of cps transformation in Section 6. In this section we ....

Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting. Unpublished manuscript.

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Birkedal, L., Harper, R.: Relational interpretations of recursive types in an operational setting. In: Theoretical Aspects of Computer Software (TACS). (1997)

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Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting. Information and Computation, 155(1--2):3--63, 1999.

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Lars Birkedal and Robert Harper. Relational interpretations of recursive types in an operational setting (summary). In Theoretical Aspects of Computer Science, volume 1281 of Lecture Notes in Computer Science, pages 458--490. Springer-Verlag, September 1997.

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