Gordin Plotkin and Martn Abadi. A logic for parametric polymorphism. In Proc. LICS'98, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the interaction of recursion and freshness is semantically quite challenging, and was investigated in Part I. In this paper we go one step further and introduce second order quantification in the logic, from which we can define least and greatest fixpoints of formulas, almost along standard lines [14]. Structurally, our logic consists of a collection of left right rules for logical operators, including essentially the standard rules of classical sequent calculus, plus the ones for temporal and spatial operators. In addition, there are special rules about the worlds: they add meaning to the ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In M. Bezem and J. F. Groote, editors, International Conference on Typed Lambda Calculi and Applications, number 664 in Lecture Notes in Computer Science, pages 361--375, Utrecht, The Netherlands, March 1993. Springer-Verlag. TLCA'93.

....terms with that type in the predicative calculus. These sets frequently admit interesting and intuitive descriptions. Some such characterizations of polymorphic types were given by Reynolds [Rey83] Even more results have been described, including what Wadler called Theorems for Free [ACC93, PA93, Wad89] As an example, the type 8X:X)X contains only a single term (up to equivalence) the polymorphic identity function. This is to be expected, as it agrees with our intuitive understanding of uniform computations of the type 8X:X)X. Given an element x of an unknown type X, since the only ....

..... Such a partial recursive function is said to realize f . The re exive graph category PER has the category of PERs over IN and PER morphisms for the vertex category. The edge category of PER is motivated by the relational parametricity identi ed by Bainbridge et.al. BFSS90] and used in [BAC95, PA93] An edge R: A B is a relation between natural numbers satisfying the following. An nRm mBm = n Such a relation R is called a saturated relation. A square from an edge R: A B to R : A B is a pair hf; gi where f : A A g: B B are PER morphisms that have ....

[Article contains additional citation context not shown here]

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications - TLCA '93, LNCS, pages 361-375. Springer-Verlag, 1993.

....the interaction of recursion and freshness is semantically quite challenging, and was investigated in Part I. In this paper we go one step further and introduce second order quantification in the logic, from which we can define least and greatest fixpoints of formulas, almost along standard lines [19]. Structurally, our logic consists of a collection of left right rules for logical operators, including essentially the standard rules of classical sequent calculus, plus the ones for temporal and spatial operators. In addition, there are special rules about the worlds: they add meaning to the ....

.... formula, where the index is 0) We can then easily derive the following inference rules: g; u : AfBg Z) AfCg g; u : B Z) C (MontonL) g u : B Z) C; g u : AfBg Z) AfCg; MontonR) We then define least and greatest fixpoint operators in a style similar to F algebraic encodings [19]. Y:AfY g = 8Y: AfY g Z) Y ) Y Y:AfY g = X: AfXg These definitions turn out to enjoy the expected properties of recursive formulas, in the form of the derivable left and right rules in Figure 16. For example, the derivable rule ( R) corresponds to a coinduction principle. The ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In M. Bezem and J. F. Groote, editors, International Conference on Typed Lambda Calculi and Applications, number 664 in Lecture Notes in Computer Science, pages 361--375, Utrecht, The Netherlands, March 1993. Springer-Verlag. TLCA'93.

....type instances has been John Reynolds Figure 1: Generic list structure notion of relational parametricity [Rey83] which requires that relations between instances be preserved in a suitable sense by generic programs. This has led to numerous further developments, e.g. [MR92, ACC93, PA93]. Relational parametricity is a beautiful and important notion. However, in our view it is not the whole story. In particular: It is a pointwise notion, which gets at genericity indirectly, via a notion of uniformity applied to the family of instantiations of the program, rather than ....

....notion of uniformity applied to the family of instantiations of the program, rather than directly capturing the idea of a program written at the generic level, which necessarily cannot probe the structure of an instance. It is closely linked to strong extensionality principles, as shown e.g. in [ACC93, PA93], whereas the intuition of generic programs not probing the structure of instances is prima facie an intensional notion a constraint on the behaviour of processes. An interestingly di erent analysis of genericity with di erent formal consequences was proposed by Giuseppe Longo, Kathleen Milsted ....

G. Plotkin, M. Abadi. A Logic for Parametric Polymorphism, TLCA'93 Conf. Proc., LNCS, 1993.

....to that of SOL has not been previously available. In a sense, IA is to Idealized Algol what SOL is to polymorphic lambda calculus. Like SOL, it adds types and features that explicitly represent data abstraction. However, while SOL can be faithfully encoded in polymorphic lambda calculus [55], the data abstraction features of IA are more re ned than those expressible in Idealized Algol. The corresponding encoding does not preserve equivalences. Thus, IA is a proper extension. Related work In the earlier work of the author [56, 32] a global state based semantics was de ned ....

.... Z : Z Z serves as the required relation S. The rst step in the above proof shows that the identi cation of behaviorally equivalent implementations is a necessary condition for the identity extension property. The basic reference for parametricity is Reynolds [62] while Plotkin and Abadi [55] de ne a logic for reasoning about parametricity. The notion of existential quanti cation is from [43] but its parametricity semantics discussed above seems new. The idea of simulation relations for implementations dates back to Milner [40] and appears in various sources including [9, 33, 26, ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications - TLCA '93, LNCS, pages 361-375. Springer-Verlag, 1993.

....regardless of the type at which it is applied. In particular, a parametric, polymorphic function can neither branch on nor otherwise analyze its type argument. In many type theories (such as the polymorphic lambda calculus) all polymorphic functions must be parametric. It has long been recognized [10, 12, 8, 3] that in type theories in which all polymorphism is parametric, many properties of polymorphic functions can be determined solely by inspection of their types. For instance, in the polymorphic lambda calculus, any function having the type 8ff: ff ff must be the identity function. In an This ....

Gordon Plotkin and Mart'in Abadi. A logic for parametric polymorphism. In International Conference on Typed Lambda Calculi and Applications, pages 361--375, 1993.

....principle for this encoding provable in U . This weakness has been encountered before. In fact, it is conjectured that it is impossible to encode primitive recursion in System F using equality [21] A stronger equational theory for U , perhaps one incorporating a parametricity principle [19], might solve this problem. However, a simpler way to support primitive recursion would be to include a primitive for primitive recursion directly in the language [12, 18, 3, 4] 4.2 Impredicativity and Non termination Another issue with this encoding is that the target language must have ....

Gordon Plotkin and Martn Abadi. A logic for parametric polymorphism. In International Conference on Typed Lambda Calculi and Applications, pages 361{ 375, 1993.

....the general properties of these categories, e.g. define coproducts, products, etc. 36 . Besides full completeness, parametricity is another quality filter for models of polymorphic functions. In particular, in [Plo93] a logic for linear parametric models has been suggested, in the line of [PA93]. It would be interesting to develop further this approach, and see whether this logic holds on our linear PER models. Longo s genericity ( LMS92] can be viewed as a form of parametricity, in that it amounts to a uniformity property of polymorphic functions w.r.t. their input types. This issue ....

G.Plotkin, M.Abadi. A Logic for Parametric Polymorphism, TLCA'93 Conf. Proc., LNCS, 1993.

.... categories has been found for this hierarchy [Weh99] The next steps will be the development of general structural recursion for the hierarchy in style of this paper, and the development of higher dimensional parametricity to derive program transformations similar to the analysis for system F [Wad93, BFSS90, PA93]. I am currently working on applications exploring the expressive power of dimension three. Preliminary results deal with simply typed lambda calculus, internalisation of algebraic data specifications and examples from Okasakis purely functional data types [Oka98] A challenge is to internalise ....

Gordon Plotkin and Martin Abadi. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications TLCA `93, pages 361--375, LNCS 664, March 1993. Springer Verlag.

....g R 6 f S 6 (5) This indicates that for the data type of functions the coupling should re5 late f to g just if (5) and that is the de nition of logical relation. 2 Logical relations have been studied extensively in the context of polymorphic functional languages [Mit90, PA93] Relational parametricity is a key element in semantics of Algol suitable for data re nement [Ten94, OT95] To deal with a wider variety of program paradigms and constructs, there are general results that give sucient conditions for preservation of simulation; the conditions are expressed in the ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Proceedings of the International Conference on Typed Lambda Calculi and Applications, Utrecht, March 1993.

....Freyd, Scedrov and Scott [BFSS90] Hasegawa [Has91] Pitts [Pit87, Pit89, Pit98] and Wadler [Wad89, Wad91] among others. Formulations of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various combinations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94]. A number of the works cited above have observed some connection between parametricity and algebraic data types [RP90, Has91, BFSS90, ACC93, PA93, PAC94] What is new here is that we don t require a semantic characterization of parametricity, and we don t require a specialized logic. For ....

.... of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various combinations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94] A number of the works cited above have observed some connection between parametricity and algebraic data types [RP90, Has91, BFSS90, ACC93, PA93, PAC94]. What is new here is that we don t require a semantic characterization of parametricity, and we don t require a specialized logic. For instance, Reynolds and Plotkin [RP90] use a categorical model, where the extra constraint posed by parametricity is replaced by the existence of certain ....

[Article contains additional citation context not shown here]

G. Plotkin and M. Abadi, A logic for parametric polymorphism, in M. Bezem and J. F. Groote, editors, Typed Lambda Calculi and Applications, LNCS 664, SpringerVerlag, pp. 361--375, March 1993.

....the general properties of these categories, e.g. define coproducts, products, etc. 36 ffl Besides full completeness, parametricity is another quality filter for models of polymorphic functions. In particular, in [Plo93] a logic for linear parametric models has been suggested, in the line of [PA93]. It would be interesting to develop further this approach, and see whether this logic holds on our linear PER models. Longo s genericity ( LMS92] can be viewed as a form of parametricity, in that it amounts to a uniformity property of polymorphic functions w.r.t. their input types. This issue ....

G.Plotkin, M.Abadi. A Logic for Parametric Polymorphism, TLCA'93 Conf. Proc., LNCS, 1993.

....this concept and its theoretical rigour to a wider spectrum of language principles, and to go beyond the first order boundaries inherent in the universal algebra approach. In this paper we look at Girard Reynolds polymorphic # calculus System F. The accompanying logic of Plotkin and Abadi [30] asserts relational parametricity in Reynolds sense [34, 21] This setting allows an elegant formalisation of abstract data types as existential types [25] and the relational parametricity axiom enables one to derive in the logic that two concrete data types are equal if and only if there ....

....given by grammars T : X T # T #X.T t : x #x:T.t tt #X.t tT where X and x range over type and term variables respectively. Judgements for type and term formation involve type contexts, and term contexts depending on 2 type contexts, e.g. X,x:X#x:X . The logic in [30, 22] for relational parametricity on System F has formulae built using the standard connectives, but now basic predicates are not only equations, but also relation membership statements: # : t =A u) R(t, u) #R#AB.# #R#AB.# where R ranges over relation symbols. We write #[R, X,x] to ....

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G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Proc. of TLCA 93, volume 664 of LNCS, pages 361--375, 1993.

....semantics was first presented by Oles and Reynolds using category theory, because some of the desired uniformity conditions were found to correspond to the categorical notion of naturality. On the other hand, we do not currently have a general categorical understanding of parametricity [3, 43, 12]. Some researchers [45, 32] believe that naturality will prove to be a special case of parametricity, when the latter is finally understood categorically. 7. Non Interference and Passivity David Park had noted [41] that it is virtually impossible to reason about programs in procedural languages ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In M. Bezen and J. F. Groote, editors, Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361--375, Utrecht, The Netherlands, March 1993. Springer-Verlag, Berlin.

....in type theory, where also ideas from algebraic specification have been expressed. This paper investigates specification refinement in a setting consisting of System F and relational parametricity in Reynolds sense [35, 23] as expressed in Plotkin and Abadi s logic for parametric polymorphism [31]. This setting allows an elegant formalisation of abstract data types as existential types [27] Moreover, the relational parametricity axiom enables one to derive in the logic that two concrete data types, i.e. inhabitants of existential type, are equal if and only if there exists a simulation ....

....2 outlines the type theory. In Sect. 3 refinement is introduced in a first order setting, and Sect. 4 generalises to higher order and polymorphism. 2 System F and the Logic for Parametric Polymorphism We briefly recall the parametric calculus System F, and sketch the accompanying logic of [31, 24] for relational parametricity on System F. It is this accompanying logic that bears a relational extension rather than the calculus. See [1] for a more internalised approach. System F has types and terms as follows: T : X j T T j 8X:T t : x j x:T:t j tt j X:t j tT where X and x range over ....

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G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Proc. of TLCA 93, volume 664 of LNCS, pages 361--375, 1993.

....general result about relational parametricity in linear type theory. In suitably parametric models of (intuitionistic) polymorphic calculus types of the form 8ff: T ff ff) ff denote initial T algebras, and this paves the way for a characterization of all second order types in prenex form. Plotkin [1993] in lectures has indicated that the corresponding property in linear polymorphic type theory is that 8ff: T ff Gammaffi ff) ff denotes an initial T algebra, for covariant functors T on a linear category where morphisms correspond to terms of Gammaffi type. It is not immediately obvious ....

....Section 8 we verified an adequacy result to the effect that the translation gets convergence at primitive types right, so the translation can be used to soundly reason about Idealized Algol programs. But the question is irksome, because one might expect that parametricity should imply naturality [Plotkin and Abadi 1993]. Our analysis of equivalence only went as far as second order types. We have not found a counterexample to full abstraction at higher types in Idealized Algol, but we did find an explicit limitation in the model for SCI in Section 7.3. One could consider a more focused study of contextual ....

Plotkin, G. and Abadi, M. 1993. A logic for parametric polymorphism. In M. Bezen and J. F. Groote Eds., Typed Lambda Calculi and Applications, Volume 664 of Lecture Notes in Computer Science (Utrecht, The Netherlands, March 1993), pp. 361--375. SpringerVerlag, Berlin.

....axiomatic descriptions of groups or rings that are common in mathematics. But there are also some di erences with standard mathematical approaches, stemming from their use in computer science. 1 Invariants and bisimilations (or congruences) are also relevant for algebras, see for example [29, 31, 7], and the end of Section 6. 2 1. Coalgebraic speci cations are typically structured, using the object oriented mechanisms of inheritance (with subclasses inheriting from superclasses) and aggregation (with ambient classes having component classes) This is needed because speci cations in ....

....these requirements follow from the structure, they can be generated automatically. That is precisely what is done by the LOOP tool (from [6] As a nal remark we add that predicates and relations which are closed under the operations of an algebra (instead of a coalgebra) also make sense, see [29, 31, 7]. A relation R X X on the carrier set of an algebra a: T (X) X is closed under the operations if: Rel(T ) R) a(x) a(y) R(x; y) In this case one may call R a congruence. But note that such an R need not be an equivalence relation. There is no special name in mathematics for a ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In M. Bezem and J.F. Groote, editors, Typed Lambda Calculi and Applications, number 664 in Lect. Notes Comp. Sci., pages 361-375. Springer, Berlin, 1993.

....Freyd, Scedrov and Scott [BFSS90] Hasegawa [Has91] Pitts [Pit87, Pit89, Pit98] and Wadler [Wad89, Wad91] among others. Formulations of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various cominations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94]. A number of the works cited above have observed some connection between parametricity and algebraic data types [RP90, Has91, BFSS90, ACC93, PA93, PAC94] What is new here is that we don t require a semantic characterization of parametricity, and we don t require a specialized logic. For ....

.... Formulations of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various cominations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94] A number of the works cited above have observed some connection between parametricity and algebraic data types [RP90, Has91, BFSS90, ACC93, PA93, PAC94]. What is new here is that we don t require a semantic characterization of parametricity, and we don t require a specialized logic. For instance, Reynolds and Plotkin [RP90] use a categorical model, where the extra constraint posed by parametricity is replaced by the existence of certain ....

[Article contains additional citation context not shown here]

G. Plotkin and M. Abadi, A logic for parametric polymorphism, in M. Bezem and J. F. Groote, editors, Typed Lambda Calculi and Applications, LNCS 664, Springer-Verlag, pp. 361--375, March 1993.

....implementations, behavioural equivalence and stability. We will express this refinement framework in a type theoretic environment comprised of System F and the assumption of relational parametricity in Reynolds sense [27, 18] as expressed in Plotkin and Abadi s logic for parametric polymorphism [24]. Abstract data types are expressed in the type theory as existential types. The above concepts of specification refinement fall out naturally in this setting. In this, relational parametricity plays an essential role. It gives the equivalence at first order of observational equivalence to ....

....d Algebraic specifications may be complex, built from basic specifications using specification building operators, e.g. 36, 32, 37] But as a starting point for the translation into type theory, we only consider basic specifications. 3 The Type Theory We now sketch the logic in [24, 19] for parametric polymorphism on System F. It is this accompanying logic that bears an extension rather than the type theory. See [1] for a more internalised approach. System F has types and terms as follows. T : X j T T j 8X:T t : x j x:T:t j tt j X:t j tT where X and x range over type and ....

[Article contains additional citation context not shown here]

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Proc. of TLCA 93, volume 664 of LNCS, pages 361--375. Springer, 1993.

...., a uniformity property of polymorphic functions. Roughly, parametricity means that a polymorphic function does not look under a type variable which may be instantiated in a number of different ways. A related statement is that parametric functions work the same way at all types. See [25, 5] for further discussions of this and related notions of parametricity. Thus, parametricity requires much more than mere type correctness of polymorphic functions. For instance, we could conceive of a polymorphic function p of type 8ff : ff ff ff that returns the first projection p[int] xy : ....

....: t i (T ) t i (T 0 ) then the entire definition will respect R. In some cases, such as when ff is not free in the type of M , this will imply that the two declarations are equal. In terms of Mitchell and Plotkin s encoding, such a relation shows the equality of elements of existential type [25]. For instance, we can implement stacks of integers using list[int] as the representation type, or a type (int int) Theta int where the int indicates the top of the stack. The relation used to prove equivalence of the representations relates a list to a pair (f; n) such that f(0) f (n) is ....

[Article contains additional citation context not shown here]

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In M. Bezen and J. F. Groote, editors, Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361--375, Utrecht, The Netherlands, March 1993. Springer-Verlag, Berlin.

....An important idea that comes to light is that classes are abstract data types whose theory corresponds to that of existential types [35] In a sense, IA is to Idealized Algol what SOL is to polymorphic lambda calculus. However, while SOL can be faithfully encoded in polymorphic lambda calculus [45], IA is more constrained than Idealized Algol. The corresponding encoding does not preserve equivalences. Thus, IA is a proper extension. Related work A number of papers [19, 1, 11, 18] discuss object oriented type systems for languages with side effects. It is not clear what contribution ....

....the equivalence class of hZ; pi as hjZ; pji. The relation part 9S: T (R; S) for any relation R: X X 0 is the least relation such that hjZ; pji 9S: T (R; S) hjZ 0 ; p 0 ji ( 9S: Z Z 0 : p T (R; S) p 0 The basic reference for parametricity is Reynolds [53] while Plotkin and Abadi [45] define a logic for reasoning about parametricity. The notion of existential quantification is from [35] but the parametricity semantics is not mentioned there. The idea of simulation relations for abstract type implementations dates back to Milner [33] and appears in various sources including ....

Plotkin, G., and Abadi, M. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications - TLCA '93 (1993), LNCS, Springer-Verlag, pp. 361--375.

....on type parameters oe and , then (the interpretation of) X:M , applied to (the meaning of) oe and , should send related elements of oe and to related elements in the types of the outputs. This is known as relational parametricity, and a syntactic treatment of it is given in [ACC93] and in [PA93]. Another approach to parametricity was proposed by Bainbridge et al. BFSS90] Consider x : X:N . Is it the case that x : X:N depends naturally on X , in the sense of natural transformations of Category Theory Indeed, natural transformations are the core means of expressing uniformity on ....

....and algebraic topology. The rub is that, in general, dinatural transformations do not compose, while terms do; however, the interpretation works well (i.e. it is compositional) on relevant models (see [BFSS90, FGSS88, GSS] in particular on models of relational parametricity as formalized in [PA93]. On essentially similar lines, Freyd suggested a novel notion of structor in order to understand, categorically, the notion of uniformity inherent in second order terms. These attempts suggested brand new constructions and relevant mathematics, but seem still insufficient to fill the essential ....

G. Plotkin and M. Abadi. A Logic for Parametric Polymorphism. In Proceedings of the International Conference on Typed Lambda Calculi and Applications, Utrecht, Netherlands (March 1993). Lecture Notes in Computer Science 664, Springer-Verlag (1993). Edited by M. Bezem and J.F. Groote.

....q ) sortfpg; Listfhg = Listfhg; sortfqg In other words, whenever p and q induce related orderings on A and B (i.e. related by a map h) sortfpg and sortfqg will take related inputs to related outputs. Such free theorems have been investigated for the polymorphic lambda calculus (see e.g. wad89] [pa93], acc93] and are a potential source for investigation in Charity. A third direction of interest is the relationship between our 2 categorical combinators and those of the lambda calculus and combinatory logic ( hls72] hin85] These latter combinators may be viewed as categorical arrow ....

G. Plotkin and M. Abadi, A Logic for Parametric Polymorphism, Lecture Notes in Computer Science 664 (Springer-Verlag, 1993) 361-- 375.

....Invited paper, Fundamenta Informaticae, 22 (1995) 69 92. Longo [MP86,Hase93] to the representation of initial algebras, BB85, ACC93, Hase93] However, the two different views above suggested different technical approaches and results, as well as many open problems. Hase93] ACC93] PA93] and [LMS93] are some of the most recent advances in the two directions. 3. Relational (or invariance) Parametricity. We owe to Strachey the distinction between parametric and ad hoc polymorphism, according to how polymorphic functions depend on their type parameters. A seminal, enlightening ....

....and realizes (Gen) 2 There is no space here to hint at two more, very relevant understandings of parametricity. First, the meaning of (polymorphic) terms as dinatural transformations , proposed in [BFSS90] and further developed by [GSS91] among others. Second, the logic for parametricity in [PA93] a stimulating blend of Logical Frames and Logical Relations. Just note then that the interpretation of second order terms as dinatural transformations may be actually given over PER models. However, the interpretation of is obtained on a slightly different (sub )category. In particular our ....

[Article contains additional citation context not shown here]

G. D. Plotkin, M. Abadi, A logic for parametric polymorphism, TLCA, LNCS, Springer, Berlin, 1993.

....However its polymorphism is of a particularly simple parametric type which could be written (ff ) fi; ff list) fi list where ff and fi stand for any types. Although a complete understanding of the ramifications of this notion of parametricity is not yet available (cf. Freyd et al. 1992b; Plotkin and Abadi, 1993)) Reynolds (1974; 1983) has emphasized the close relationship with representational abstraction. The idea is that a parametric polymorphic function must work in a way that is independent of the types to which it is instantiated. For instance, in the absence of recursion) the only parametric ....

....at this point. This is motivated by the use of a relational condition to constrain values of 8 types in (Reynolds, 1983) The identity relation Delta W plays the same role as the identity relations there. Of course, the foundational difficulties described in (Reynolds, 1983; Reynolds and Plotkin, 1993) do not arise here, because the source collection Sigma, over which indexing is done, is small. 2.3 Recursion The presentation thus far is for a recursion free dialect of Algol. Recursion can be dealt with by using domains in place of sets, as follows. We still use sets, or discretely ordered ....

[Article contains additional citation context not shown here]

Plotkin, G. D. and Abadi, M. (1993). A logic for parametric polymorphism. In Typed Lambda Calculi and Applications, volume 664 of Lect. Notes in Computer Sci., pages 361--375.

....However its polymorphism is of a particularly simple parametric type which could be written (ff ) fi; ff list) fi list where ff and fi stand for any types. Although a complete understanding of the ramifications of this notion of parametricity is not yet available (cf. Freyd et al. 1992b; Plotkin and Abadi, 1993)) Reynolds (1974; 1983) has emphasized the close relationship with representational abstraction. The idea is that a parametric polymorphic function must work in a way that is independent of the types to which it is instantiated. For instance, in the absence of recursion) the only parametric ....

....at this point. This is motivated by the use of a relational condition to constrain values of 8 types in (Reynolds, 1983) The identity relation Delta W plays the same role as the identity relations there. Of course, the foundational difficulties described in (Reynolds, 1983; Reynolds and Plotkin, 1993) do not arise here, because the source collection Sigma, over which indexing is done, is small. 2.3 Recursion The presentation thus far is for a recursion free dialect of Algol. Recursion can be dealt with by using domains in place of sets, as follows. We still use sets, or discretely ordered ....

[Article contains additional citation context not shown here]

Plotkin, G. D. and Abadi, M. (1993). A logic for parametric polymorphism. In Typed Lambda Calculi and Applications, volume 664 of Lect. Notes in Computer Sci., pages 361--375.

....recursive) domains we have established here. The utility of considering actions of type constructors on relations is perhaps best known from the body of work beginning with (Reynolds, 1983) on relational properties of parametric polymorphism: see (Abadi, Cardelli and Curien, 1993) and (Plotkin and Abadi, 1993) and the references therein. It is well known that inductive and co inductive types can be encoded in the Girard Reynolds polymorphic lambda calculus. Hence they inherit relational properties from those of 8 types. Domain theory brings into the picture considerations of partiality and more ....

Plotkin, G. D., and Abadi, M. (1993), A logic for parametric polymorphism, in "Proceedings of the Conference on Typed Lambda Calculus and its Applications, Utrecht", Lecture Notes in Computer Science Vol. 664, pp. 361-375, Springer-Verlag, Berlin.

....R is a complete relation on cpo s. This ensures that the least fixedpoint operator exists in the model. By parametric model we mean what in [13] was referred to as the Identity Extension Lemma: A] I D = I [ A] D This is usually taken as the defining characteristic of Reynolds Parametricity [3, 14, 9], and the definition of 8 is arranged precisely to ensure the identity property (which is included as part of the Isomorphism Lemma) Lemma 1 (Isomorphism Functoriality Lemma) For any type A, the relational action of [ A] is functorial on isomorphisms. That is, i) if Rff is the graph of an ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In M. Bezen and J. F. Groote, editors, Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361--375, Utrecht, The Netherlands, March 1993. Springer-Verlag, Berlin.

.... condition using either Reynolds treatment of abstract types using polymorphic application e.g. ff : x 1 ; xn :M) T ] K 1 : Kn to bind ff and x i to their concrete representations or the treatment later given by Mitchell and Plotkin using existential types (see [Mit86, PA93] Put this way, relational parametricity appears as a systematization and generalization of (often informal) ideas for reasoning about data types, objects, and so on (e.g. Hoa72] and this connection with data abstraction is part of its appeal. But there is also substantial theoretical support ....

....not just relational parametricity, and provide a useful test for any proposed alternative definitions. The importance of these properties can be seen in a number of works, beginning with [Rey83] and continuing in a number of places (e.g. RP93] with probably the most systematic exposition being [PA93] Bainbridge, Freyd, Scedrov and Scott [BFSS90] were the first to define a parametric model satisfying these properties; they achieved this by trimming down the PER model. Of course, for the statements of the properties, and the parametricity condition itself, to make precise sense we need to ....

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G. Plotkin and M. Abadi. A logic for parametric polymorphism. In M. Bezem J. F. Groote, editor, Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361--375. Springer Verlag, 1993.

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Gordon Plotkin and Martn Abadi. A logic for parametric polymorphism. In M. Bezem and J.F. Groote, editors, Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361--375. Springer-Verlag, March 1993.

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G.D. Plotkin and M. Abadi. A logic for parametric polymorphism . Proc. International Conference on Typed Lambda Calculi and Applications . Springer-Verlag.

....recursive type S: the two halves of the isomorphism in and out are not linear. After considering these typed version of the Scott numerals, we may wish to check that they are in fact isomorphic to the standard natural numbers. A direct argument uses many of the datatype constructions studied in [1]: M j N by unfolding j X8R: 1 R) X R) R) since R j (1 R) j X8R: 1 R) Theta (X R) R) by uncurrying j X8R: 1 X) R) R) turning a Theta into a j X: 1 X) as 1 X j R: 1 X) Similarly, we can give Scott versions for other familiar datatypes using covariant ....

Gordon Plotkin and Mart'in Abadi. A logic for parametric polymorphism. To appear in Proceedings of the International Conference on Typed Lambda Calculi and Applications, March 1993.

....like F with just one new rule (a special case of one of the rules of F : Longo, Milstead, Soloviev 1993] The system is weaker than R , and leads to different sorts of results. Finally, Plotkin and Abadi explore an alternative formalization of parametricity closer in spirit to Mairson s [Plotkin, Abadi 1993]. That paper describes a second order logic with an axiom of parametricity; the logic is not an extension of system F, like R , but rather a logic about system F terms. 2. Formal parametricity In this section we describe our formalization of parametricity. We aim at a hypothetical system that ....

G.D. Plotkin and M. Abadi. A logic for parametric polymorphism. Proc. International Conference on Typed Lambda Calculi and Applications. Springer-Verlag.

....validated through the translation. So are the equational rules related to subtyping, with the exception of (Eq Sub Object) Proving the translation of (Eq Sub Object) in the target calculus may require additional principles, such as a bisimulation rule for abstract data types [Aczel, Mendler 1989; Plotkin, Abadi 1993] The translation extends to the full FOb : The coherence problems for arrows and universal quantifiers have been essentially solved [Breazu Tannen, et al. 1991] We do not expect surprises from existential quantifiers. Further work is needed on recursion [Breazu Tannen, Gunter, Scedrov ....

G.D. Plotkin and M. Abadi. A logic for parametric polymorphism. Proc. International Conference on Typed Lambda Calculi and Applications. Lecture Notes in Computer Science 664. Springer-Verlag.

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Gordon Plotkin and Martn Abadi. A logic for parametric polymorphism. In M. Bezem and J.F. Groote, editors, Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361--375. Springer-Verlag, March 1993.

....of the notion of model for R, and then to recast our proof in more abstract terms. The interpretation has been helpful both in understanding R and in thinking about other formal systems for reasoning about polymorphic programs. Several other formal systems come to mind. Following a suggestion of [PA93], we have started to consider a formal system with relations of arities other than 2. Reynolds discussed relations of all arities in his original work, but binary relations have been preferred more recently (e.g. in [BFSS90] in part arbitrarily. It seems interesting to extend the model ....

G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Proceedings of the International Conference on Typed Lambda Calculi and Applications, March 1993, Utrecht, NL, number 664 in Lecture Notes in Computer Science, pages 361--375. Springer-Verlag, 1993.

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Gordin Plotkin and Martn Abadi. A logic for parametric polymorphism. In Proc. LICS'98, 1998.

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Plotkin, G. and Abadi, M., A Logic for Parametric Polymorphism, LICS'98, 42--53, IEEE Press, 1998.

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G. Plotkin, M. Abadi. A Logic for Parametric Polymorphism, TLCA'93 Conf. Proc., LNCS, 1993.

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G. Plotkin and M. Abadi. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications, Lecture Notes in Computer Science 664, pages 361--375. Springer-Verlag, 1993.

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G. D. Plotkin and M. Abadi. A Logic for Parametric Polymorphism. In Proceedings of the Conference on Typed Lambda Calculus and its Applications, Utrecht, 1993.

No context found.

Gordon Plotkin and Martin Abadi. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361-375, 1993.

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G. Plotkin and M. Abadi. A logic for parametric polymorphism. In TLCA '93 [129], pages 361--375. (pp. 5, 70, 122, 123)

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Plotkin, G. and Abadi, M., A Logic for Parametric Polymorphism, LICS'98, 42-53, IEEE Press, 1998.

No context found.

Gordon Plotkin and Martin Abadi. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361-375, 1993.

No context found.

Plotkin, G. and Abadi, M., A Logic for Parametric Polymorphism, LICS'98, 42--53, IEEE Press, 1998.

No context found.

G. D. Plotkin and M. Abadi. A Logic for Parametric Polymorphism. In Proceedings of the Conference on Typed Lambda Calculus and its Applications, Utrecht, 1993, Lecture Notes in Computer Science Vol. 664 (Springer-Verlag, Berlin, 1993) pp 361-375.

No context found.

Plotkin, G. and Abadi, M., A Logic for Parametric Polymorphism, LICS'98, 42--53, IEEE Press, 1998.

No context found.

Gordon Plotkin and Martin Abadi. A logic for parametric polymorphism. In Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 361-375, 1993.

No context found.

G. D. Plotkin and M. Abadi. A Logic for Parametric Polymorphism. In Proceedings of the Conference on Typed Lambda Calculus and its Applications, Utrecht, 1993, Lecture Notes in Computer Science Vol. 664 (Springer-Verlag, Berlin, 1993) pp 361-375.

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