A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In FPCA, pages 31--41, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1. HM(X) claims to be a general framework for Hindley Milner systems. To support this claim, I show how previous approaches can be seen as instances of the HM(X) system. I consider speci cally systems which deal with records [Oho95, R em89, Gas98] overloading [OWW95, NP95, Jon92] and subtyping [Kae92, AW93, TS96]. Further instances are systems for program analysis such as dimension analysis [Ken96] Details can be found in in Section 3.2 and Chapter 5. 2. The HM(X) systems comes with a novel formulation of the quanti er introduction rule due to Martin Odersky which improves on previous formulations (see ....

....discusses properties of HM(X) type systems and introduces some basic de nitions used throughout my dissertation. Section 3.5 relates HM(X) to previous approaches. I discuss speci cally issues regarding the di erent formulations of the quanti er introduction rule found in previous approaches [Jon92, AW93, Smi91, TS96]. I show how the HM(X) quanti er 13 introduction rule relates to the open or closed world assumption found in overloading systems [OWW95, Smi91] In addition, I consider Ohori s record calculus [Oho95] and the message passing system of Nishimura [Nis98] I also point out the problems in their ....

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Alexander Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In FPCA '93: Conference on Functional Programming Languages and Computer Architecture, Copenhagen, Denmark, pages 31-41, New York, June 1993. ACM Press.

....approaches for defining subtyping: the syntactic approach and the semantic one. The syntactic approach (by far the most widespread) consists in defining the subtyping relation by axiomatizing it in a formal system (a set of inductive or coinductive rules) in the semantic approach (for instance, [1, 3]) instead, one starts with a model of the language and an interpretation of types as subsets of the model, then defines the subtyping relation as the inclusion of denoted sets, and, finally, when the relation is decidable, derives a subtyping algorithm from the semantic definition. The semantic ....

A. Aiken and E. L. Wimmers. Type inclusion constraints and type inference. In Proceedings of the Seventh ACM Conference on Functional Programming and Computer Architecture, pages 31--41, Copenhagen, Denmark, June 93.

....of quoted terms with environments. Because of the extensibility property, the type system has to deal with subtyping, which makes it more difficult to infer principal types. The solution is to use an extension of HindleyMilner type inference known as recursively constrained types, following [AikWim93, Thatte94, EiSmiTri95] Such constraints appear to be a promising way for handling inference and subtyping together. However, an open issue is to design algorithms for simplification of sets of constraints, so that they stay of manageable size even when typing large programs. Some heuristics ....

....appear to be a promising way for handling inference and subtyping together. However, an open issue is to design algorithms for simplification of sets of constraints, so that they stay of manageable size even when typing large programs. Some heuristics borrowed from the Illyria system of [AikWim93] have been incorporated in our prototype, yielding acceptable results for small programs; but no optimal solution is known at the moment. However, encouraging results seem to emerge from the recent work of [Pottier95] An earlier proposal for incorporating dynamic binding in functional languages ....

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Alex Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In Proceedings of the 1993 Conference on Functional Programming Languages and Computer Architecture (Copenhagen, Denmark, June 1993), pp. 31-41.

....variables. For simplicity, we will not consider type schemes and let polymorphism; this dimension is essential for practical languages, but is orthogonal to the issues investigated here, and the technical aspects of type schemes with recursive constraints have already been studied in several places[26, 2, 10]. 4.2 Recursive constraints In presence of subtyping, principal type inference becomes considerably more difficult, because subsumption (the subs rule of the previous section) can in principle be applied at any point: as a result, typing proofs are not unique. Fortunately, the work of Thatte on ....

....In order to be able to infer principal types, he restricted applications of subsumption to some specific places, and used sets of recursive constraints to keep track of the subtyping assumptions involved in a typing proof. Independently of Thatte, a similar approach was taken by Aiken and Wimmers[2] for constraints on sets, and their work was later adapted by Eifrig, Smith and Trifonov[10] for typing object calculi. A set of constraints C is a set of pairs of types or pairs of parameter types, where pairs are written with the symbol , i.e. C j fT 1 U 1 ; T n U n ; P 1 P ; ....

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Alex Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In Proceedings of the 1993.

....to both type checking polymorphic recursion [Myc84, Hen88] and (strong) subsumption [DR90] on feature trees. We extend this correlation by showing how weak subsumption relates to soft typing. In recent years, there has been increasing interest in type analysis for untyped languages (soft typing) [Tha90, AW93, AWL94, CF91, WC94]. By far most of the typing literature is for functional languages. The close relationship between constraint solving and type inference ( Wan87, AW93, PS94] and many others) or more general program analysis (e.g. Hei92, PS94] is well established. Among the abounding literature, the work of ....

....to soft typing. In recent years, there has been increasing interest in type analysis for untyped languages (soft typing) Tha90, AW93, AWL94, CF91, WC94] By far most of the typing literature is for functional languages. The close relationship between constraint solving and type inference ([Wan87, AW93, PS94] and many others) or more general program analysis (e.g. Hei92, PS94] is well established. Among the abounding literature, the work of Aiken, Wimmers et al. seems closest in some respects [AW93, AWL94] In this work, a rich type language containing union, intersection, complement functional ....

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A. Aiken and E. Wimmers. Type Inclusion Constraints and Type Inference. In 6 ACM Conference on Functional Programming and Computer Architecture, pp. 31--41, Copenhagen, Denmark, June 1993.

....a actor s interface. Types are interface like, with multiplicities, therefore without dynamic information, and the type system requires complex operations on a lattice of types. The type inference algorithm is a set constraints algorithm, a well known technique widely used in functional languages [AW93] 3. Najm and Nimour [NN97, NNS99a, NNS99b] propose several versions of a calculus of objects that features dynamically changing interfaces and distinguishes between private and public objects interfaces. For each version of the calculus they develop a typing system handling dynamic method ....

Alexander Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In Functional Programming & Computer Architecture, pages 31-41. ACM Press, 1993.

....1 Introduction Set constraints are inclusions between expressions denoting sets of trees. Such expressions are first order terms with or without set operators. Set constraints have received much attention in constraint based type inference and program analysis for different programming languages [32, 24, 28, 21, 5, 15, 19, 7, 8, 2, 1]. Other applications of set constraints include order sorted unification [37] and constraint logic programming [25] The complexity of the satisfiability problem for various classes of set constraints has been widely studied [17, 20, 10, 3, 6, 18, 4, 11, 12, 3, 35, 29] and was often found to be ....

A. Aiken and E. Wimmers. Type Inclusion Constraints and Type Inference. In 6 Conf. on Functional Programming and Computer Architecture, pp. 31--41, 1993.

.... type inference problems have a natural and simple formulation as the satisfiability problem of an appropriate constraint system (e.g. 29, 32, 42] Constraints were also instrumental in generalizing the ML type system towards record polymorphism [28, 34, 43] overloading [8, 27] and subtyping [1, 12, 32] (see also [26] for further references) Along this line, we use feature trees [3] as the semantic domain of the constraint system underlying our type system. A feature tree is a possibly infinite tree with unordered marked edges (called features) and with marked nodes (called labels) where the ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In Proceedings of ACM Conference on Functional Programming and Computer Architecture, pp. 31--41. ACM Press, New York, June 1993.

....interpreted over possibly empty sets of finite trees. The satisfiability problem for atomic set constraints is also O(n ) This result is implicit in the combined results of [14] and [15] An explicit proof is given in Appendix E of this paper. Set Constraints for Type Analysis. Aiken et al. [3, 4] use constraints over specific sets of trees called types for the type analysis of FL. There is a minimal type 0 which in terms of constraint solving behaves just like the empty set in standard set constraints (although it is not an empty set from the types point of view but contains a ....

A. Aiken and E. Wimmers. Type Inclusion Constraints and Type Inference. In FPCA, pp. 31--41. 1993.

....complexity, computational logics. 1 Introduction Set constraints are logical formulas describing relations between sets of trees [2, 6, 7, 13, 16] Set constraints have received much attention in constraint based type inference and program analysis for different programming languages [3, 12, 15, 17, 20, 27]. Other applications of set constraints include order sorted unification [28] and constraint logic programming [19] Expressiveness and Complexity. Expressiveness and complexity have been widely studied for various classes of set constraint [1, 2, 8, 10, 11, 14, 26] The complexity of their ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In ACM Conf. on Functional Programming and Computer Architecture, pages 31--41, Copenhagen, Denmark, June 1993. ACM Press.

....this problem is even DEXPTIME hard. 1 Introduction Set constraints are logical formulas describing relations between sets of trees [2, 5, 6, 13, 16] Set constraints have received much attention in constraint based type inference and program analysis for different programming languages [3, 12, 15, 17, 21, 28]. Other applications of set constraints include order sorted unification [29] and constraint logic programming [20] Expressiveness and Complexity. Expressiveness and complexity have been widely studied for various classes of set constraint [1, 2, 7, 9, 11, 14, 27] The complexity of their ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In 6 ACM Conf. on Functional Programming and Computer Architecture, pages 31-- 41, 1993.

.... many type inference problems have a natural and simple formulation as the satisfiability problem of an appropriate constraint system (e.g. 20, 31] Constraints were also instrumental in generalizing the ML type system towards record polymorphism [19, 22, 32] overloading [6, 18] and subtyping [1, 8] (see also [28] Along this line, we adopt feature trees [3] as the semantic domain of the constraint system underlying our type system. A feature tree is a possibly infinite tree with unordered marked edges (called features) and with marked nodes (called labels) where the features at the ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In 6 ACM Conf. on Functional Programming and Computer Architecture, pp. 31--41. ACM Press, June 1993.

....type t with sub typing constraints C in an environment H. The second one means that the recursive de nition D produces an environment H 0 and a set of constraints C. The de nition of these two predicates is given in gure 1. This is a classical type system with sub typing constraints [1,8,9]. We adopt a simpler (and less general) presentation adapted to the initialization problem. constant receive type h0i, saying that they are always de ned. primitive operators need their arguments to have the same type. when typing a node declaration, we rst type its body, generalize its ....

Alexander Aiken and Edward Wimmers. Type inclusion constraints and type inference. In Seventh Conference on Functional Programming and Computer Architecture, pages 31-41, Copenhagen, Denmark, June 1993. ACM.

....kept in the log matrices of the analysis framework. Regular trees, on the contrary, naturally express deep invariants of the concrete values. A sound mathematical basis comes along with them. Analyses using regular trees should be considered. They have been used by Aiken (and collaborators) in [5, 4, 6] and presented by Courcelle in [18] The results by Aiken showed an impressive representation power but did not seem to be efficient enough. 7.3.3 Extensions Other Languages Although we explicitly aim at analyzing dynamically typed languages, we believe that the type analysis could be useful in ....

Alexander Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In Proceedings of the Conference on Functional Programming Languages and Computer Architec- ture, pages 31-41, jun 1993.

....approaches for defining subtyping: the syntactic approach and the semantic one. The syntactic approach (by far the most widespread) consists in defining the subtyping relation by axiomatizing it in a formal system (a set of inductive or coinductive rules) in the semantic approach (for instance, [1, 3]) instead, one starts with a model of the language and an interpretation of types as subsets of the model, then defines the subtyping relation as the inclusion of denoted sets, and, finally, when the relation is decidable, derives a subtyping algorithm from the semantic definition. The semantic ....

A. Aiken and E. L. Wimmers. Type inclusion constraints and type inference. In Proceedings of the Seventh ACM Conference on Functional Programming and Computer Architecture, pages 31--41, Copenhagen, Denmark, June 93.

....size in lines of code after macro expansion, the number of additionally used primitives, the number of birthplaces, the number of birthplaces recognized as dynamic binding, the number of iterations the worklist algorithm used, and the analysis time. least solution of the original constraints [2, 14, 32]. The worklist algorithm always terminates. Every program induces only a finite set of abstract values (Abs#) and there is only a finite number of potential nodes since there is only a finite number of program points, variables, and birthplace environments. Hence, the analysis propagates a finite ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In Proceedings of the FPCA 1993.

....other. Their type inference algorithm is always based on some form of first order unification. Similar in motivation to these is Pfenning s work on type inference for F2 [Pfe88] which uses higher order unification. Extensions of the Hindley Milner system with subtyping have also been studied [AW93, TS96, EST95, Pot96, Nor98, Pot98] They are usually based on constrained types [OSW99] which include a set of subtype constraints as part of a type. A problem in practice is that constraint sets can become very large. Trifonov and Smith [TS96] as well as Pottier [Pot98] have proposed schemes to ....

Alexander Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In FPCA '93: Conference on Functional Programming Languages and Computer Architecture, Copenhagen, Denmark, pages 31--41, New York, June 1993. ACM Press.

....other. Their type inference algorithm is always based on some form of first order unification. Similar in motivation to these is Pfenning s work on type inference for F2 [Pfe88] which uses higher order unification. Extensions of the Hindley Milner system with subtyping have also been studied [AW93, TS96, EST95, Pot96, Nor98, Pot98] They are usually based on constrained types [OSW99] which include a set of subtype constraints as part of a type. A problem in practice is that constraint sets can become very large. Trifonov and Smith [TS96] as well as Pottier [Pot98] have proposed schemes to ....

Alexander Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In FPCA '93: Conference on Functional Programming Languages and Computer Architecture, Copenhagen, Denmark, pages 31--41, New York, June 1993. ACM Press.

....extend Curry types to Hindley Milner types. An extension of our type inference algorithm remains to be found. 3.5. EXTENSIONS 57 Finding a good type inference algorithm for a type system with both structural subtyping and polymorphism is a nontrivial task although the work by Aiken and Wimmers [AW93] and by Eifrig, Smith and Trifonov [EST95] is promising. A Trust case Construction. One could imagine the usefulness of a trust case construction that would allow dynamic dispatch on the trustworthiness of a value. The reduction rules added for such a construction could be: trust case E F G ....

....type system that is able to handle both polymorphism and sub typing together with the extra complication of tuple concatenation. Type inference for systems combining sub typing with polymorphism and higher order languages has proven to be quite hard, and only recently some inroads have been made [AW93, EST95, OL96] We eschew this hard problem by integrating a control flow analysis (along the lines of Shivers s 0 CFA [Shi91] into the analysis. Parametric polymorphism allows recording of dependences between inputs and outputs. In the polymorphic type: ff; fi) fi; ff) we can see that the ....

Alexander Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In FPCA'93: Proc. Conference on Functional Programming Languages and Computer Architecture, pages 31--41, 1993.

....studied by Wand [24] and Remy [22] In the presence of subtyping, unication based approaches to type inference seem not to apply. Type inference for calculi with records and subtyping has been studied by Eifrig, Smith, and Trifonov [9] using the approach to type inference of Aiken and Wimmers [5]. Their algorithm does not immediately apply to Abadi and Cardelli s calculus because of the following dioeerence between the subtyping relations. For records, the conventional subtyping relations makes every record type constructor covariant in all arguments. For example, l : A; m : C] is a ....

Alexander Aiken and Edward Wimmers. Type inclusion constraints and type inference. In Proc. Conference on Functional Programming Languages and Computer Architecture, pages 3141, 1993.

....from type variables to types and this is the usual result of type inference. 1.6.1 Comparison with Others Work Mitchell and others have given inference algorithms for idealised subtyped languages [11] 10] his language does not cover all the complexities of MLj. This work is extended by [7] and [1]. Rather than looking at general subtyped languages, Eiefig, Smith and Trifonov look at type inference for objects in [5] Constraints are also used in program analysis. Nielson, Nielson and Hankin describe this in [12] Francois Pottier s work [13] is closest to our own, in particular in the ....

A. Aiken and E.L. Wimmers. Type Inclusion Constraints and Type Inference. In Conference on Functional Programming Languages and Computer Architecture, pages 31--41. ACM, June 1993.

....type variables to types and this is the usual result of type inference. Previous work on constraints has included John Mitchell s inference algorithms for idealised subtyped languages [Mit96, Mit91] His language does not include all the complexities of MLj. This work is extended by [JM93] and [AW93] Rather than looking at general subtyped languages, Eiefig, Smith and Trifonov look at type inference for objects in [EST95] Constraints are also used in program analysis. Nielson, Nielson and Hankin describe this in [NNH98] Francois Pottier s work [Pot96] is closest to MLj s, in particular in ....

A. Aiken and E.L. Wimmers. Type Inclusion Constraints and Type Inference. In Conference on Functional Programming Languages and Computer Architecture, pages 31--41. ACM, June 1993.

....is more natural for compilation. General research into intersection types that has influenced our thinking includes the work of Van Bakel [29] and Jim [15] Research on both intersection and union types that we have consulted includes the work by Pierce [22] Aiken, Wimmers, and Lakshman [3, 4], Barbanera and Dezani Ciancaglini [7] and Trifonov and Smith [28] Of the above, only Pierce considers intersection and union types in an explicitly typed language. Even that is somewhat distant from our work because Pierce includes a general subtyping relation on intersection and union types ....

A. S. Aiken and E. L. Wimmers. Type inclusion constraints and type inference. In FPCA '93, Conf. Funct. Program. Lang. Comput. Arch., pp. 31--41. ACM, 1993.

....and rejecting only provably ill typed programs we can use more powerful type languages without restricting the set of accepted programs due to inaccuracies of the type checker and without enforcing explicit type annotations. e.g. subtyping as a step towards powerful type languages as found in e.g. [1], 8] or partial types as presented in [9, 10] can be processed by our type checker. Our type checker is described as an abstract interpretation and yields detailed support in detecting type errors. Furthermore, warnings of an additional sound soft typing system [2] 16] or a system with output ....

A. S. Aiken and E. L. Wimmers. Type inclusion constraints and type inference. In Functional Programming and Computer Architecture, pages 31-41. ACM Press, June 1993.

....Source #a Var, Var #a Var, or Var #a Sink. If the constraint system contains a non atomic constraint, the resolution rules from Figure 3 are used to generate new atomic constraints, as described in Section 4.3. We use annotated constraint graphs based on the inductive form representation [3]. Inductive form is an e#cient sparse representation that does not explicitly represent the transitive closure of the constraint graph. The graphs are represented with adjacency lists pred (n) and succ(n) stored at each node n. Edge (n1 , n2 , a) where a is an annotation, is represented either as ....

....into several atomic constraints and their corresponding edges are added to the graph. The closure of a constraint graph under the Trans rule is the solved inductive form of the corresponding constraint system. The least solution of the system is not explicit in the solved inductive form [3], but is easy to compute by examining all predecessors of each variable. For constraint graphs without annotations, the least solution LS(v) for a variable v is LS(v) c( c( # pred(v) # # u#pred(v) LS(u) In this case, LS(v) can be computed by transitive acyclic ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In Conference on Functional Programming Languages and Computer Architecture, pages 31--41, June 1993.

....one unit, and then performing the analysis directly on the abstract syntax trees (AST) of the entire code base would require substantial computing resources especially memory. A better approach would be to convert the ASTs into a more compact representation such as some form of set constraints [1, 2, 8, 11, 9, 10]. However, even this representation is problematic when the number of initial constraints approaches one million. Worse, the usual indexing schemes used to speed up analysis (e.g. dependency lists for each set variable so that when a set variable is changed the change can be efficiently propagated ....

A. Aiken and E. Wimmers, "Type Inclusion Constraints and Type Inference", Proc. of the

....system and must be given explicitly in the expression. In their paper, this decoration were automatically generated when compiling a higher level language in their typed formalism. To compute types, we proceed by solving set constraints. This approach has been defined by Aiken and Wimmers in [4] and used in the context a functional object language in [13, 14] The resulting system is very soft and allows to type more programs than [10] or [15] The difference is clear in the example of a server and two clients, the following program uses the same convention as in the previous example: ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In Proc. of the ACM Symp. on FPCA, June 1993.

....of multisets instead of sets. This article describes the multiset based type abstraction and gives insights on the associated constraints solver. 1 Introduction A powerful set constraint solver and its application to type inference for functional languages was proposed by Aiken et al. in [AW92, AW93, AWL94] This work has been extended by Aiken et al. in the BANE project [AFFS98] in order to provide a generic framework for developping ecient set based analysis. As advocated by Smith et al. in [EST95a, EST95b, TS96] set based type inference can be applied to sequential object oriented ....

....the work of Kobayashi et al. in [KPT96] this constraint is introduced in the type system using unlimited types (hm i ( i ) i2I i) and unlimited environments (in which all types are unlimited) 4. 3 Step 3: Multiset constraints solver Aiken and Wimmers work on set constraint resolution [AW93, AW92] was adapted to the concurrent objects speci c context in [CPS97] This algorithm principle consists in using operator properties to put the system in a resolved form (also called inductive system) 6 : hm i i ( S M k2K i ok ) i2I i j t j t S [ hm i (hi) i2I) i ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In Proc. of the ACM Symp. on FPCA, June 1993.

....[TMC 96, PJM97] While type information has been tightly integrated into modern intermediate languages, flow information has not. The only implemented or partially implemented languages of which we are aware which merge type and flow information are those based on constrained types [Cur90, AW93, AWL94, EST95] It is not clear whether any work with constrained types has used a typed intermediate language. Also, support in constrained type systems for type polymorphism has been via let style polymorphism, which is di#cult to use in a typed intermediate language without losing the ability ....

..... Imperative features: It is important to show that our techniques are still applicable in the context of imperative features like references and exceptions. Connections with constrained types: We are currently exploring connections between flow types and constrained types [Cur90, EST95, AW93, AWL94] Our goal is to incorporate the polyvariant power of intersections and unions into the constrained type framework. ....

Alexander S. Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In FPCA '93, Conf. Funct. Program. Lang. Comput. Arch., pages 31--41. ACM, 1993.

....sets using Boolean operations and function applications. Program behavior can be described by set constraints on the sets of values that variables assume during execution. Then, solving set constraints gives a useful information about the program. For example, it can be used for type inference[11, 2] or control flow analysis. The obtained information can be used to optimize the program or to point out potential errors to the programmer. This paper starts with a formal definition of set constraints (Section 2) Then, it is shown how to construct a system of set constraints from a program ....

A. Aiken and E. Wimmers. Type inclusion constraints and type inference. Proceedings of Conference on Functional Programming Languages and Computer Architecture, pp. 31-41, 1993. 9

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Alexander Aiken and Edward L. Wimmers. Type inclusion constraints and type inference. In Proceedings of the 1993 Conference on Functional Programming Languages and Computer Architecture, pages 31--41, Copenhagen, June 1993.

.... for consistency of the given constraint set according to a set of syntactic rules [20, 30, 33] Corresponding to polymorphic type schemes in Hindley Milner style type systems, polymorphic subtype systems have so called constrained types, in which a type is restricted by a system of constraints [1, 3, 42]. An ML style polymorphic type can be viewed as a constrained type with no constraints. For example, # # int int, int # is a constrained type. Let # C be a constrained type, and let # be a satisfying valuation for C. The ground type #(#) is called a instance of # C. There are a ....

A. Aiken and E. Wimmers. Type Inclusion Constraints and Type Inference. In Proceedings of the 1993.

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A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In FPCA, pages 31--41, 1993.

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