Y.C. Fuh and P. Mishra, "Type inference with subtypes", Proc. ESOP'88, LNCS 300, pp.94-114, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of subtype # # . Subtype constraints are systems of inequations t#t # that talk about the subtype relation. Terms t and t # in subtype constraints are built from type variables and type constructors. Two logical operations on subtype constraints were investigated: satisfiability and entailment [6, 12, 1, 4, 18, 20]. Subtype satisfiability can be checked in cubic time for many type languages [9, 16] A quadratic time algorithm for the variable free case is presented in [10] Interest in subtype entailment was first raised by practical questions on type inference engines with subtyping [5, 21, 19] The ....

Y. Fuh and P. Mishra. Type inference with subtypes. Theo. Comp. Science, 73, 1990.

....A with DV (A with [ FV (E with ) x2Dom(E) FV (E(x) with ) The typing rules are given in gure 1. They are very similar to the rules for ML, except for the additional handling of constraints, reminiscent of the treatment of subtyping hypotheses in type inference systems with subtypes [12, 6]. The rules de ne the proposition term a has type A under assumptions E and constraints , written E a : A with . The typing environment E is a partial mapping from term variables to type schemas. We write E[x A] for the environment identical to E, except that x is mapped to A. We assume the ....

Y.-C. Fuh and P. Mishra. Type inference with subtypes. In ESOP '88, volume 300 of Lecture Notes in Computer Science, pages 94-114. Springer Verlag, 1988.

....since such annotations would generally be verbose. Second, a variety of program analyzes may be described as type inference processes. The rst algorithm for type inference with atomic subtyping was proposed by Mitchell [6, 11] and improved for handling structural subtyping by Fuh and Mishra [12, 13]. However, quoting Hoang and Mitchell [10] it has seen little if any practical use , mostly because it is inecient and the output, even for relatively simple input expressions, appears excessively long and cumbersome to read . The theoretical complexity of the problem has been largely studied. ....

Fuh, Y.C., Mishra, P.: Type inference with subtypes. In Ganzinger, H., ed.: Proceedings of the European Symposium on Programming. Volume 300 of Lecture Notes in Computer Science., Springer Verlag (1988) 94-114

....since such annotations would generally be verbose. Second, a variety of program analyzes may be described as type inference processes. The rst algorithm for type inference with atomic subtyping was proposed by Mitchell [6, 11] and improved for handling structural subtyping by Fuh and Mishra [12, 13]. However, quoting Hoang and Mitchell [10] it has seen little if any practical use , mostly because it is inecient and the output, even for relatively simple input expressions, appears excessively long and cumbersome to read . The theoretical complexity of the problem has been largely studied. ....

Fuh, Y.C., Mishra, P.: Type inference with subtypes. In: European Symposium on Programming. Volume 300 of LNCS. (1988) 94-114

.... is computable in polynomial time [23] Moreover, if ( Sigma; is itree likej, then type inference is computable in polynomial time [6] In general, the type inference problem is PSPACE hard [14, 23] Type inference with atomic subtyping has also been studied in combination with ML polymorphism [10, 11]. Objects do not have base types, so type inference with atomic subtyping does not apply to Abadi and Cardelli s calculus. The object types considered in this paper are rather like record types. Type inference for calculi with records but no subtyping has been studied by Wand [24] and Remy [22] ....

You-Chin Fuh and Prateek Mishra. Type inference with subtypes. In Proc. ESOP'88, European Symposium on Programming, pages 94114. SpringerVerlag (LNCS 300), 1988.

....done by uni cation. With subtyping, type inference is equivalent to solving a system of inequalities between type expressions. Several algorithms for solving subtyping inequalities have been proposed in the litterature. They can be classi ed along several axes: the domain of types: nite types [10, 9, 7], recursive types [4, 15, 11] or in nite types [14] the structure of types: lattices [4, 13, 11] quasi lattices [12] posets with suprema [6] partial orders [7] the subtyping relation: structural extension of a subtyping relation on basic types, subtyping relation between di erent type ....

Y.C. Fuh and P. Mishra. Type inference with subtypes. In Proc. ESOP'88, number 300 in LNCS, pages 94-114, 1988.

....type system. Concluding remarks are given in Section 5. Proofs are omitted from this paper due to space limitations. The full paper will be available as [23] 1. 1 Related Work Constraint based type inference systems have been designed for many languages, including ones that support subtyping [6, 11], objects [5] overloading [8, 22] and linear types [21] The inference issues in [21] are especially close to ours, and we received considerable inspiration from this work. One novel aspect of our system which does not arise in these other works is the need to associate constraints with each ....

Y.-C. Fuh and P. Mishra. Type inference with subtypes. Theoretical Comput. Sci., 73:155--175, 1990.

....its supertypes. This approach also supports certain aspects of method inheritance and provides a typetheoretical basis for object oriented programming. Cardelli and Wegner [1985] extended this approach to a second order type system. Type inference systems with subtyping have also been developed [Fuh and Mishra 1988; Mitchell 1984; Stansifer 1988] It is, however, not clear whether or not the mechanism for record polymorphism should be coupled with a strong mechanism of subtyping. In the presence of subtyping, a static type no longer represents the exact record structure of a runtime value. For example, a ....

Fuh, Y.-C. and Mishra, P. 1988. Type inference with subtypes. In Proceedings of ESOP '88.

....memory cells containing elements of type set(t 0 ) Pointer dereferencing is then the operational analogue to invoking the conversion rule for a value of type t to get a value of type set(t 0 ) We refine the simple type inference system for SQ2 to a subtype inference system (c.f. Mit84, FM88] by adding propositional formulas of the form 0 and the rule schemes presented in Appendix C. The rules specify that type containments are reflexive, transitive and closed with respect to type constructors set( and ( The rule (COERCE) connects the type containments to SQ2 ....

....applications of the (COERCE) rule into the type inference system of Appendix B. This is done by systematically replacing every assumption of the form e : set( or e : 0 ; 00 ) by e : oe; oe set( respectively e : oe; oe ( 0 ; 00 ) where oe is a new meta variable. 8 See [Wan87, FM88, Hen88] for a sample of reductions of type inference to type constraint systems and [Hen91b, Hen91a] for reductions very similar to the one necessary to prove this theorem. can be extended in a canonical fashion to a mapping from type expressions to subsets of U by defining T [ fv : 9e) e ....

Y. Fuh and P. Mishra. Type inference with subtypes. In Proc. 2nd European Symp. on Programming, pages 94--114. Springer-Verlag, 1988. Lecture Notes in Computer Science 300.

....the research and the application of this kind of languages. One of the traditional topics of research in declarative paradigms has been the incorporation of type systems as an attempt to provide more structured readable compile time type safe programs and run time optimizations. In the functional [8, 18] and the logic paradigm [20, 12, 17, 15, 4, 5] the combination of parametric types and order sorted hierarchies, so called subtyping, has been adopted as a simple type system allowing to subdivide and parametrize the universe of discourse. In particular, the logic paradigm has benefited from the ....

.... L( C ffl ff = K(ff) ff 1 1 ; ff n n ; C[ff=K(ff) where K L 5) ff fi; C ffl ff = fi; C[ff=fi] 6) ff fi;Cffl ff =K(ff) fi =L( fi) ff 1 fi 1 ; ff n fi n ;C[ff=K(ff) fi=L( fi) where K L These rules for solving subtype conditions are similar to those presented in [4, 18, 8]. Rules obtaining a supremum 0 3 and binding lower bound type variables 0 4 , are defined in a similar way to their corresponding 3 and 4 . There is don t 7 3 needs occur check of X in c(e) Occur check in lazy unification differs from syntactic occur check. Occur check in 3 checks that X ....

Y. Fuh, P. Mishra. Type Inference with Subtypes, TCS 73, pp. 155-175, 1990.

....including limited union and intersection types, a type complement, and a notion of conditional type. Versions of constrained types which either disallow recursive constraints [Smi94, Kae92, BM96] or use a notion of subtyping which relates types with the same structure only [Mit84, Mit91, MR85, FM88] have also been studied but are of less interest for our approach to typing classes and objects (discussed in Section 6 below) which requires recursion and record subtyping. 1 One consequence of such a type representation is the possibility for over constraining a type, for instance t Int n ....

Y.-C. Fuh and P. Mishra. Type inference with subtypes. In European Symposium on Programming, 1988. 8

....these analyses are typically exponential and use an arbitrary limit on the number of contexts being distinguished (e.g. JW95, NN97] 2.4. 5 Constraint Simpli cation Program analyses based on polymorphic subtyping depend crucially on constraint simpli cation in order to be at all practical [FM88, FA96, Pot96, Mos96, FF97, Pot98]. In the particular case of Mossin s algorithm for inferring derivations of POLYFLOW copy , the constraint set C is simpli ed before forming the quanti ed type 8 :C ) Without this simpli cation, the constraint sets can grow exponentially in n due to the copying. In the case of polymorphic ....

....for polymorphic recursive functions is avoided altogether. It appears indirectly in the CFL algorithm where the xpoint is computed for the set of non terminal edges that can be added to the graph. CFL based approaches do not obviate all constraint simpli cation. For example, Ssimpli cation [FM88] may be useful in obtaining smaller polymorphic types (i.e. fewer distinct labels) and thus fewer instantiation constraints. Furthermore, the computation of the CFL closure may bene t from techniques such as online cycle elimination on M edges [FFSA98] 2.4.6 Other Related Work Henglein s ....

Y. Fuh and P. Mishra. Type inference with subtypes. In Proceedings of the 1988 European Symposium on Programming, pages 94-114, 1988.

....elements. Therefore we can write one function len with the functionality List ff Nat where ff is a sort variable. If we combine the subsorting facility with the parametric polymorphism we get a richer sort system which is decidable by static program analysis if a few restrictions are considered [3]. However, the principal sort of an expression, representing all valid sorts of the expression, cannot be expressed by a sort expression alone. We need a pair consisting of a sort expression and a set of subsort relations between sort expressions, called coercions. The expression f.x.f(f x) for ....

Y.-C. Fuh and P. Mishra. Type inference with subtypes. In ESOP 88, pages 94--114. Springer Verlag, 1988.

....therefore also constraint implication) is PSPACE hard. In particular, conversion to the normal form described above may in general entail an exponential growth in the size of the constraint. Nevertheless, in [3] we give an algorithm (inspired by an incomplete algorithm given by Fuh and Mishra in [9]) that we expect to be polynomial for real world programs. We can now define the ordering on types in terms of constraint implication. In practice, we work relative to a constraint context Delta = # : The use of the constraint context will become clear when we explain type checking in ....

....theorem. This semantics performs dynamic dispatch on run time types, as opposed to ML style pattern matching, which is performed on run time values. We have not addressed type inference in this paper. Nevertheless, we conjecture that standard type inference techniques such as described in [2, 8, 9, 10, 13, 18, 20] can be used to infer the types of untyped ML programs except for method declarations. It is an open question whether or not methods have principal types in this context. The report [3] discusses several extensions to the base language presented here. In particular, we propose a module system, ....

Y.-C. Fuh, P. Mishra. Type inference with Subtypes. 2nd European Symp. on Programming, LNCS 300 (1988) 94--114

....for each primitive type) and we can assume that all fields have type u, a type with just one element. Only the presence or absence of a field will be important in this paper; we are not actually interested in records as data structures. Unfortunately type reconstruction in this system is difficult [2, 4, 8, 10]. The type system r, inspired by the work of R emy [6, 7] is now displayed. This is the system in which we will perform type reconstruction. Type system r has two syntactic categories, or sorts the usual one for types, and an additional sort for fields of records. ff j fl 1 : F 1 ; ....

Y.-C. Fuh and P. Mishra. Type inference with subtypes. In H. Ganzinger, editor, ESOP '88: 2nd European Symposium on Programming, Nancy, France, March 21--24, pages 94--130, Berlin, 1988. Springer-Verlag.

....is a set of variables, is a constraint, and is a monotype. Constrained polymorphic types have been proposed by some authors, in particular to accommodate overloading in various ways [19, 20, 29, 39] or to perform type inference for object oriented languages or languages with primitive subtyping [5, 21, 22, 23, 26, 29, 33, 34, 35, 41, 42]. Our system is original in that it merges the two approaches: methods are, in a sense, cleanly overloaded functions , and their polymorphic constrained types have constraints which are conjunctions of inequalities between monotypes. As we shall see, this modification leads to a clean notion of ....

....possible algorithm is to try to arbitrarily fix G(v 2 ) fv 1 g for some v 1 2 G(v 2 ) to reapply the greatest fixpoint algorithm starting from G instead of the maximum element of the lattice, backtracking if necessary. This 4 This property, as well as the functional Phi, are already given in [22], but the authors conjecture incorrectly that the existence of a pre solution is sufficient to ensure that the problem has a solution. 52 procedure Solve(F ) is G = T i 0 Phi i (F ) if 9v 2 2 # 2 : jG(v 2 )j = 0 then fail ; if 8v 2 2 # 2 : jG(v 2 )j = 1 then succeed ; for v 2 2 # 2 ....

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Y.-C. Fuh, P. Mishra. Type inference with Subtypes. 2nd European Symp. on Programming, LNCS 300 (1988) 94--114

....such that an expression of a trusted type can automatically be coerced to an untrusted type. The ordering between base types as determined by their trustworthiness is extended to higher types using the usual contra co variant structural subtyping idea of Mitchell (Mitchell, 1984) Fuh and Mishra (Fuh Mishra, 1990), Cardelli (Cardelli, 1984) and others. The trust type system differs from many other type systems in that given a bare value (e.g. 7) it is not possible to see by just examining the value whether it is trustworthy or not. This is where our three extensions to the basic calculus comes in. They ....

....have: t u s v if and only if u v and t s; and for bare arrow types we define: oe 0 oe 0 if and only if 0 and oe oe 0 ; so argument types are ordered contravariantly. This is inspired by the work on structural subtyping by Mitchell (Mitchell, 1984) Fuh and Mishra (Fuh Mishra, 1990), Cardelli (Cardelli, 1984) and others. As in the denotational semantics we denote by t the least upper bound operation on the trust lattice according to the ordering. In Section 5 we discuss several extensions of the type system to cope with recursion, modules, polymorphism and more general ....

Fuh, You-Chin, & Mishra, Prateek. (1990). Type Inference With Subtypes. Theoretical computer science, 73(1), 155--175.

....only the features we are interested in so as to understand them independently of their possible interactions with other language features. This is not to say that other features are irrelevant or of less interest. In fact, operator overloading [52,118] implicit and explicit type coercions [103,78,27], abstract and dependent types [82,69,34] recursive types [110,70,77] and especially inclusion polymorphism [10,11,112,50,98,121,122] a type theoretic view of the behavior of object oriented programming languages, are significant in the typing disciplines of modern strongly typed programming ....

Y. Fuh and P. Mishra. Type inference with subtypes. In Proc. 2nd European Symp. on Programming, pages 94--114, Springer-Verlag, 1988. Lecture Notes in Computer Science 300.

....issues are, of course, important, but (solely due to reasons of lack of space) we refer the interested reader to the last two articles mentioned above for a full treatment of the semantical part of our theory. For readers interested in other issues regarding subtyping in type theory we refer to [CaWe85,FuMi88,KiLa89,Ohor90,Stan88,Wand87]. 3.1 Cardelli type theory In this section we give a brief summary of the system described in [Card84,Card88] Full details of syntactical and proof theoretical aspects can be found in the technical report. 3.1.1 Expressions and Types Types in our version of the Cardelli system are either 1. ....

Y. C. Fuh & P. Mishra, "Type inference with subtypes," in Proceedings Second European Symposium on Programming (ESOP88), H. Ganzinger, ed., 1988, 94--114, Lecture Notes in Computer Science #300.

.... as the front end of our analysis and the algorithm(s) 5 Similar considerations can be applied to other type inference problems to derive systematically reductions to solvability of type constraint systems; in particular, the reductions of Mitchell [Mit84] Wand [Wan87] Fuh and Mishra [FM88] Stansifer [Sta88] Henglein [Hen88b] can be derived in this fashion. for solving these constraints as its back end . We hope that, possibly after some suitable generalization, our constraints are both expressive enough to capture many interesting program analyses and still constrained enough ....

Y. Fuh and P. Mishra. Type inference with subtypes. In Proc. 2nd European Symp. on Programming, pages 94--114. Springer-Verlag, 1988. Lecture Notes in Computer Science 300.

....function in the simple type system. This is not a coincidence and we show that it is implied by a semantic relation between types and terms stronger than soundness and that we call strong soundness. The constraints of the form ff v fi are familiar in the context of constrained type systems, [FM88], AW93] PS96] We define some obvious constrained type systems to capture a general semantic relation between terms and types defined only by the descriptive character of types. Then we define proper type systems which correspond to different ways of solving the set of constraints derived by ....

Y. Fuh and P. Mishra. Type inference with subtypes. In Proceedings of the 1988 European Symposium on Programming, 1988.

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Y.C. Fuh and P. Mishra, "Type inference with subtypes", Proc. ESOP'88, LNCS 300, pp.94-114, 1988.

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Fuh, Y.C., Mishra, P.: Type inference with subtypes. In: European Symposium on Programming. Volume 300. (1988)

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You-Chin Fuh and Prateek Mishra. Type inference with subtypes. In H. Ganzinger, editor, Proceedings of the European Symposium on Programming, volume 300 of Lecture Notes in Computer Science, pages 94--114. Springer Verlag, 1988.

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Y.C. Fuh and P. Mishra. Type inference with subtypes. In Proc. ESOP'88, number 300 in LNCS, pages 94-114, 1988.

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