F. J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J. Reynolds, editors, Algebraic Methods in Semantics, pages 543--573. Cambridge University Press, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....declaration is to bind the declared variable identifier to any currently unused location for the execution of the block body. This kind of interpretation of local variables is adequate to show the correctness of the usual style of implementation of block structure [9] however, many authors [3, 16, 13, 5, 14, 2, 8] have criticized it as being insufficiently abstract. For example, consider the simple equivalence new . 0; C j C ; when identifier is not free in command C. The equivalence is a consequence of the inaccessibility of the new location to non local entities in Algol like languages. ....

F. J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Methods in Semantics, pages 543--573. Cambridge University Press, Cambridge, 1985.

....of this construct Locality The object created by a local declaration new x in C must be private to C. This causes problems for traditional models based on representing the state in a global, monolithic fashion by a mapping from locations to values. The functor category approaches [22, 33] address this problem by replacing the global state by a functor varying over stages . Irreversibility When a variable is updated, the previous value is lost. Again, models based on representing states as functions find it hard to account for this feature. For good discussions of this point see ....

....of com ) com is isomorphic to the flat natural numbers. 11 Related work There have been two main strands of work addressing the issue of locality of store in programming languages from a semantic point of view. The first, based on the use of functor categories, was pioneered by Reynolds and Oles [22], and has since been considerably refined, notably by O Hearn and Tennent [21] The idea is essentially to take a traditional global state model and parameterize it with respect to store shapes, to account for the allocation and later deallocation of local variables. Stark has also used functor ....

F. J. Oles. Type categories, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Semantics. Cambridge University Press, 1985.

....of this construct Locality The object created by a local declaration new x in C must be private to C. This causes problems for traditional models based on representing the state in a global, monolithic fashion by a mapping from locations to values. The functor category approaches [27, 37] address this problem by replacing the global state by a functor varying over stages . Irreversibility When a variable is updated, the previous value is lost. Again, models based on representing states as functions find it hard to account for this feature. For good discussions of this point see ....

....presentable, for precisely the same reasons. 12 Related Work There have been two main strands of work addressing the issue of locality of store in programming languages from a semantic point of view. The first, based on the use of functor categories, was pioneered by Reynolds and Oles [27], and has since been considerably refined, notably by O Hearn and Tennent [24] The idea is essentially to take a traditional global state model and parameterize it with respect to store shapes, to account for the allocation and later deallocation of local variables. Stark has also used functor ....

F. J. Oles. Type categories, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Semantics. Cambridge University Press, 1985.

....and continuous functions there are similar results for 2 (2 X ) but not for TX = X ) In fact, T does not preserves regular subobjects (i.e. inclusive subsets) More precisely, there is a regular mono m s.t. Tm is not monic. Example 4. 11 Let W be the category of state shapes (see [Ole85]) i.e. an object is a non empty set W , a morphism from W to X is a pair (l; u) where l: X W (l for lookup) and u: W X X (u for update) satisfy the equations u(l(x) x) x, l(u(w; x) w and u(w; u(w 0 ; x) u(w; x) in particular, X = W V (for some V ) and l is the rst ....

F.J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J.C. Reynolds, editors, Algebraic Methods in Semantics, 1985.

....local variables and procedures are not properly captured by Strachey s model of storage allocation; for example, the following equivalence fails: new x in C j C when x is not free in command C. To make the stack discipline a primary feature of a semantic description, Frank Oles and Reynolds [51, 38, 39, 40] developed a possible worlds form of semantics in which store shapes (the way local variables are allocated on the stack) parameterize the meanings of all the state dependent types in the language; expansions (new variable allocations) induce changes of meaning. Thus, a procedure called at a ....

F. J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Methods in Semantics, pages 543--573. Cambridge University Press, Cambridge, England, 1985.

....0 : T (X Y ) let x; y(c 0 in [x] c 0 ) 2.5 Open issues The problem regarding V TLoE is how to construct good models, i.e. models capable of validating (most of) the non logical axioms. We have not been able to adapt models based on functor categories for Algol like languages (see[Ole85, OT92, OT93]) or for dynamic creation of names (see [Mog89, PS93] What follows brie y explains the source of the diculties. A suitable functor category for modeling Algol like languages and languages with dynamic creation of names is Cpo I , where I the the category of nite cardinals and injective maps, ....

F.J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J.C. Reynolds, editors, Algebraic Methods in Semantics, 1985.

.... while [HM94] adapts Gomard s technique to establish correctness for a polymorphic binding time analysis (and introduces further aws in the denotational semantics) The speci c model we propose is based on a functor category. In denotational semantics functor categories have been advocated by [Ole85] to model Algollike languages, and more generally they have been used to model locality and dynamic creation (see [OT92, PS93, FMS96] For this kind of modeling they outperform the more traditional category Cpo of cpos (i.e. posets with lubs of chains and continuous maps) Therefore, they ....

....ne composition in D we are forced to take o terms modulo conversion. 3.2 The static category We de ne b D as the functor category Cpo D op , which is a variant of the more familiar topos of presheaves Set D op . Categories of the form c W (where W is a small category) have been used in [Ole85] for modeling local variables in Algollike languages. c W enjoys the following properties: it has small limits and colimits (computed pointwise) and exponentials; it is Cpo enriched, thus one can interpret x point combinators and solve recursive domain equations by analogy with Cpo; ....

F.J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J.C. Reynolds, editors, Algebraic Methods in Semantics, 1985.

....the semantics of programming languages One suggestion that they do comes from the study of storage allocation in Algol like languages. John Reynolds and Frank Oles have proposed functors over store shapes as a mathematical semantics for languages which admit stack structured storage allocation [Rey81, Ole85]. Some related discussion appears in [Ten85] In addition to taking storage maps as possible worlds, some other possibilities might be sets of declarations (as in our completeness proof) program 29 contexts, or their meanings. Given the di erences between Henkin models and functor ....

F.J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J.C. Reynolds, editors, Algebraic Methods in Semantics, Cambridge University Press, 1985. 31

....see, without AC these axioms can be soundly and sensibly interpreted. 2 Related work The idea of using functor categories to describe binding of variables and freshness seems to be in the air . In a semantical context it has been around for a while, notably in the theory of idealised Algol [16, 15] and in semantic models of the calculus [18, 6] The only place in the literature where functor categories have been used explicitly to justify higher order syntax is [10] It is, however, fair to say that the possibility of using functor categories for HOAS is part of the folklore. Indeed, I ....

F. J. Oles. Type categories, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Semantics, pages 543--574. Cambridge University Press, 1985.

....values stored in them. This essentially frees imperative programming from the limitations suggested by Backus and sets up a truly object based paradigm for thinking about imperative programs. Reynolds s program for the semantics of imperative languages was further developed by Oles and Tennent [22,23,38 40], and continued and expanded in a number of works [13,16,24,18,20,36,17,35] In a separate line of development, a model based more explicitly on a notion of objects has been formulated in [28,29] Reynolds s conception of imperative programming expressed above formed an important pre theoretic ....

F. J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Methods in Semantics, pages 543--573. Cambridge University Press, Cambridge, England, 1985.

....could be executed in a continuation that tests the store in ways that are not expressible in Scheme itself. It is possible to construct denotational semantics in which some equivalences like (1) are true, but they are quite complex, and none extend far enough to encompass real languages [MS88, Ole85, OT95, Ode94] We seek to prove the correctness of such source level transformations and translations in compilers for languages like Scheme and ML. This setting is challenging in a number of respects: ffl We seek methods that are applicable to both untyped and typed languages, including Scheme, ....

....our metalanguage. Their translation uses a monadic style in place of our continuation passing style. Their treatment does not include a store, and it is not clear whether their system can be extended to deal with latently recursively typed languages like Scheme. Most work on store semantics [MS88, Ole85, OT95, Ode94] deals with stores in stack discipline languages, and are complicated by the need to model that restriction. Most of this work depends crucially on the type structure of the language; our work is applicable to untyped or latently typed languages as well. PS93] treats heap allocated ....

F. J. Oles. Type categories, functor categories and block structure. In M. Nivat amd J. C. Reynolds, editor, Algebraic Semantics, pages 543--574. Cambridge University Press, 1985.

....scoped and are executed using a stack discipline. The subtleties of the externally observable behaviour of such locally declared Research partially supported by the EU HCM Research Network on Lambda Calcul Type . state are such that, despite the considerable efforts of a number of researchers [5, 11, 10, 16, 8, 20], no concrete denotational model of Algol has yet been constructed which exactly captures observational equivalence for n th order procedures beyond n = 2 or 3 (depending upon how one counts orders) Nevertheless, useful semantical ideas and techniques have emerged from the effort to construct ....

.... evaluation relation of the form w s; M oe s 0 ; R: 2) Here w is a finite set of global variables we call such sets worlds, because they are an operational trace of the Kripkestyle possible world semantics of block structure using functor categories introduced by Reynolds and Oles [11]. In (2) M and R are elements of IA oe (w) and s; s 0 are states of world w i.e. functions assigning integers to the global variables in w. We write States(w) for the set of all such states. The intended meaning of (2) is that given the initial assignment s of values to the relevant global ....

F. J. Oles. Types algebras, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Methods in Semantics, chapter 15, pages 543--574. Cambridge University Press, 1985.

....of this technique in the context of topos theory and higher order predicate logic. The use of categorical semantics in nonwell pointed categories gives us an increase in ability to construct useful models compared with the more traditional sets and elements approach. The work of Reynolds and Oles [ 1985 ] on the semantics of block structure using functor categories is an example. As the example above shows, functor categories are not well pointed in general. See [ Mitchell and Scott, 1989 ] for a comparison of the categorical and set theoretic approaches in the case of the simply typed lambda ....

F. J. Oles. Types algebras, functor categories and block structure. In Nivat and Reynolds [ 1985 ] , chapter 15, pages 543--574.

.... and considerably extends Sieber s previous demonstration that logical relations give a complete account of sequentiality up to third order in the purely functional language PCF generally considered a hard problem in its own right [Sie92] Following Reynolds work with Oles [Rey81, Ole82, Ole85] models for Idealized Algol using functor categories have been further developed by Tennent, O Hearn and Lent [OT92, OT95, Len93] Functors are important because they capture the fact that the size of the store, as well as its contents, may change over time. Thus an index category of possible ....

F. J. Oles. Type categories, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Semantics, pages 543--574. Cambridge University Press, 1985.

....functions there are similar results for 2 (2 X ) but not for TX = Sigma ( Sigma X ) In fact, T does not preserves regular subobjects (i.e. inclusive subsets) More precisely, there is a regular mono m s.t. Tm is not monic. Example 4. 11 Let W be the category of state shapes (see [Ole85] i.e. ffl an object is a non empty set W , ffl a morphism from W to X is a pair (l; u) where l: X W (l for lookup) and u: W ThetaX X (u for update) satisfy the equations u(l(x) x) x, l(u(w; x) w and u(w; u(w 0 ; x) u(w; x) in particular, X = W ThetaV (for some V ) and l ....

F.J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J.C. Reynolds, editors, Algebraic Methods in Semantics, 1985.

....of this construct Locality The object created by a local declaration new x in C must be private to C. This causes problems for traditional models based on representing the state in a global, monolithic fashion by a mapping from locations to values. The functor category approaches [22, 33] address this problem by replacing the global state by a functor varying over stages . Irreversibility When a variable is updated, the previous value is lost. Again, models based on representing states as functions find it hard to account for this feature. For good discussions of this point see ....

....of com ) com is isomorphic to the flat natural numbers. 11 Related work There have been two main strands of work addressing the issue of locality of store in programming languages from a semantic point of view. The first, based on the use of functor categories, was pioneered by Reynolds and Oles [22], and has since been considerably refined, notably by O Hearn and Tennent [21] The idea is essentially to take a traditional global state model and parameterize it with respect to store shapes, to account for the allocation and later deallocation of local variables. Stark has also used functor ....

F. J. Oles. Type categories, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Semantics. Cambridge University Press, 1985.

....of this construct Locality The object created by a local declaration new x in C must be private to C. This causes problems for traditional models based on representing the state in a global, monolithic fashion by a mapping from locations to values. The functor category approaches [27, 37] address this problem by replacing the global state by a functor varying over stages . Irreversibility When a variable is updated, the previous value is lost. Again, models based on representing states as functions find it hard to account for this feature. For good discussions of this point see ....

....effectively presentable, for precisely the same reasons. 12 Related Work There have been two main strands of work addressing the issue of locality of store in programming languages from a semantic point of view. The first, based on the use of functor categories, was pioneered by Reynolds and Oles [27], and has since been considerably refined, notably by O Hearn and Tennent [24] The idea is essentially to take a traditional global state model and parameterize it with respect to store shapes, to account for the allocation and later deallocation of local variables. Stark has also used functor ....

F. J. Oles. Type categories, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Semantics. Cambridge University Press, 1985.

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F. J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J. Reynolds, editors, Algebraic Methods in Semantics, pages 543--573. Cambridge University Press, 1985.

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F. J. Oles. Type algebras, functor categories and block structure. In M. Nivat and J. Reynolds, editors, Algebraic Methods in Semantics, pages 543--573. Cambridge University Press, 1985. (pp. 57, 93, 125)

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F. J. Oles. Types algebras, functor categories and block structure. In M. Nivat and J. C. Reynolds, editors, Algebraic Methods in Semantics, chapter 15, pages 543--574. Cambridge University Press, 1985.

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