J. C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211--249, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ul. Banacha 2, 02 097 Warsaw, Poland essentially allows to treat an object with a certain set of functionalities also as an object that has a subset of such functionalities [9] The relation of subtyping for (parametric) polymorphic types ( has been axiomatized by Mitchell in 1988 [12]. In that work, Mitchell introduced the concept of retyping functions , which are better known in the literature as coercions. The word coercion , as remarked in [10, 11] occurs often in object oriented contexts. For example, in languages such as Java and C , coercions are connected to the ....

....of the variable s content, in which case the conversion is done at run time when the content is known. More semantic interpretations characterize coercions with respect to identity functions, which do not imply a change to the underlying representation (and this is the way they are seen in [12]) In practice, conversions are performed statically at compile time, and used during type checking. In this case, we talk about subtyping instead of type casting . There is, of course, a trade off between an actual dynamic conversion, which implies a less safe use of types and some loss of ....

[Article contains additional citation context not shown here]

Mitchell, J. C.: Polymorphic type inference and containment, Information and Computation, 76(2/3), 1988.

...., as choose id, but not with s 2 ; otherwise auto could be applied, for instance, to the successor function, which would lead to a runtime error. Hence, s 1 cannot be safely coerced to s 2 . Conversely, however, there is a retyping function a function whose type erasure h reduces to the identity [19] from type s 2 to type s 1 , namely, l(g : s 2 ) l(x : s id ) l(a) g a (x a) Actually, s 2 is a principal type for choose id in F h (System F closed by h expansion) 19] While the argument of auto must be at least as polymorphic as s id , the argument of the function choose id need not be ....

....to s 2 . Conversely, however, there is a retyping function a function whose type erasure h reduces to the identity [19] from type s 2 to type s 1 , namely, l(g : s 2 ) l(x : s id ) l(a) g a (x a) Actually, s 2 is a principal type for choose id in F h (System F closed by h expansion) [19]. While the argument of auto must be at least as polymorphic as s id , the argument of the function choose id need not be polymorphic: it may be any instance t of s id and the type of the return value is then t. We could summarize these constraints by saying that: auto : 8(a=s id ) a a choose ....

[Article contains additional citation context not shown here]

J. C. Mitchell. Polymorphic type inference and containment. Information and Computation, 2/3(76):211--249, 1988.

.... 1 , as choose id, but not with 2 ; otherwise auto could be applied, for instance, to the successor function, which would produce an error. Hence, 1 can not be safely coerced to 2 . Conversely, however, there is a retyping function a function whose type erasure reduces to the identity [Mit88] from type 2 to type 1 , that is, fun (g : 2 ) fun (x : 8 ( id ) fun ( g (x ) Actually, 2 is a principal type for choose id in F (System F closed by expansion) Mit88] In fact, the argument of the function choose id does not have to be polymorphic: the function ....

.... however, there is a retyping function a function whose type erasure reduces to the identity [Mit88] from type 2 to type 1 , that is, fun (g : 2 ) fun (x : 8 ( id ) fun ( g (x ) Actually, 2 is a principal type for choose id in F (System F closed by expansion) Mit88] In fact, the argument of the function choose id does not have to be polymorphic: the function simply returns a value that is at best as polymorphic as both its argument and the identity. Conversely, the argument to the function auto must be at least as polymorphic as 8 ( id . We could ....

[Article contains additional citation context not shown here]

John C. Mitchell. Polymorphic type inference and containment. Information and Computation, 2/3(76):211--249, 1988.

....chosen in di#erent semantic theories. Examples are the set theoretic semantics of type systems for the # calculus, such as the F semantics, the simple semantics or, more generally, the so called inference semantics; they all assume this definition, each with a di#erent specification of [10] [12]. Observe that if one also abstracts from the semantics of the typing judgement t : # (which associates a type with a term) the above definition simply becomes the condition that f : # # i# x : # f(x) # (3) which is the minimal condition that must be satisfied by whatever notion of ....

J. Mitchell. Polymorphic Type Inference and Containment. Information and Computation 76:211--249, 1988.

....explored in ML [15, 28, 29] This style of polymorphism, called implicit polymorphism, is based on the idea that programs are type free, with types interpreted as predicates expressing properties of programs under evaluation. Numerous extensions of these ideas have been explored in the literature ([7, 25, 31, 45], to name just a few) Although implicit polymorphism is appealingly simple and natural, it does not scale well to more sophisticated language features such as modularity and abstract types [20] Recent languages, notably Quest [4] and LEAP [35] are based instead on the notion of explicit ....

John C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211--249, 1988. (Reprinted in [22], pp. 153--194.).

....Meseguer [25] considered categorical, preorder and partial order interpretations of a rewriting logic for conditional rewriting modulo a set of equations; he proves soundness and completeness results relative to all three classes of interpretations. In section 4, following an idea of Mitchell [28], we consider a generalisation of the simple notion of type interpretation, for the i We may informally understand a rule to be admissible in a formal system if, for any instantiation of the schematic variables that occur in the rule, the conclusion of the rule is provable in the formal system ....

.... system by expansion or equality rules) Formal evidence to this effect can be provided by theorems such as the Subject Construction Theorem for simple types (see Theorem 9B1 in [10] and Theorem 14D2 in [11] Mitchell s characterisation of his pure typing system for the polymorphic type discipline [28] and Lemma 2.8 for the intersection type discipline of Barendregt, Coppo, and Dezani [3] These show that the systems are syntax directed, in that the types inferred for a term are a function of the types inferred for their immediate subterms. Note that there is, unfortunately, no guarantee for ....

[Article contains additional citation context not shown here]

Mitchell, J.C. Polymorphic type inference and containment. Information and Computation 76, pp. 211-249 (1988).

....from a chain of arrows in Cocont 6.2 Coherence So far we have proved results concerning #Cat categories. In order to make these results directly applicable to Prof we should extend them to hold for bicategories with the #Cat property [9] A way of doing this is by means of coherence results [106, 107, 109, 127, 42]. Roughly speaking these are results that state when an up to isomorphism situation can be replaced with a strict one without losing any property of interest. The category theory literature abounds with examples of such results. The first notable one is Mac Lane s coherence result for monoidal ....

....to arise more naturally than 2 categories. They are often more di#cult to work with because of the extra coherence conditions that one has to carry along. Fortunately there are coherence results that permit us to strictify a bicategory into a 2 category without losing its relevant properties [107, 106, 109]. 232 APPENDIX A. BASIC DEFINITIONS OF ENRICHED CATEGORY THEORY Appendix B Some proofs for Chapter 6 B.1 Theorem 6.4.1 be two pseudo #Cat algebraically complete 2 categories and be a pseudo #Cat functor. For any A write TA for the object part of a chosen pseudo initial algebra for ....

A. John Power. Why tricategories? Information and Computation, 120(2):251-- 262, 1995.

....the rule. z: v is an expression having the same sort than the variable x. Special implication rules: P t : A t : P ) A t : P ) A P t : A ; P A B A P ) B A B P P ) A B 4 Permutation axioms: We add immediately two axioms: the rst one is Mitchell s axiom [11] for quanti cation. It is needed to give the completeness of subtyping. The second one is a similar axiom concerning our special implication. Axiom 6. 8A;B (8x (A(x) B(x) 8x A(x) 8x B(x) 8P;A;B (P ) A ) B A ) P ) B) Fact 7. We prove immediately 8A 8B (8x (A(x) B) 8x A(x) ....

J.C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76:211-249, 1988.

.... as illustrated by the following examples: f( ff ff Boolg: ff ff Bool ( f( ff ff ffg: ff ff ff sort : f( ff ff Boolg: ff] ff] Another form of constrained polymorphism, called subtype bounded polymorphism (or simply bounded polymorphism) CW85, Mit88, CG92, FM96, FM98] uses subtyping constraints. 18 Suppose, for example, a (recursively defined) record type Ord: f( Ord Ord Boolg A sorting function may be defined and, in a type system with subtype bounded polymorphism, be given the type: 8ff : Ord: ff] ff] However, this ....

J.C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211--249, 1988.

.... M : 8 :A Figure 13. Decorating the sequent calculus with terms . Observe that ( rule can have at most one assumption. Observe also that the two rules for the second order formulas are not encoded by any term. Namely, we introduce a system analogous to Mitchell s language Pure Typing Theory [Mit88]. In this case, the logical system of reference is II order ILAL, in place of System F [GLT89] 8. The Dynamics for the Concrete Syntax Figure 14 de nes the basic rewriting relations on . The rst relation is the trivial generalization of the rule of Calculus to abstractions that bind ....

J.C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76:211 - 249, 1988.

....undecidable. This is already the case for ML which is theoretically DEXPTIME [11] but these worst cases never appear in real programs. Contribution of this paper The main contribution of this paper is a new semi algorithm that may reach this goal. We choose to study Mitchell s version of system F [10], because it is based on a subtyping relation and subtyping is really needed to have a powerful module system. We give a new version of the subtyping for this system which has the property of being deterministic: there is only one applicable rule (in fact there may be two applicable rules, but in ....

....terms. 1 Ax ; x : A F x : A ; x : A F t : B i F x t : A B F t : A B F u : A e F (t u) B F t : A X 62 V( 8 i F t : 8X A F t : 8X A 8 e F t : A[X ( C] Table 1: The rules for system F Related work Our presentation of subtyping di ers from Mitchell s [10] because it can handle the introduction of the quanti cation. This means that we do need to keep this rule in the type system. Moreover, in our system, the transitivity is only an admissible rule. This is also the case in Longo, Milsted and Soloviev work [9] However, in their presentation there ....

J.C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76:211 249, 1988.

.... reduces to the identity function can be given a type which is not admissible for the identity function itself. The system F introduced in this paper xes this problem. This system is therefore equivalent to system F with the reduction rule and so it is also equivalent to Mitchell system [13] (see the section on related work for a more detailed comparison) The system F uses a new notion of sub typing to deal with the quanti er rules. It allows more freedom in the manipulation of quanti ers. One of the consequences is that types are more general. For instance 8X(X X) is a ....

....2 but is neither complete nor terminating. It seems that these problems have little consequences for the practical use of our algorithm. Both the theoretical results and the experiments give hope that our system could by implemented in a full scale programming language 1 . The work by Mitchell [13] on system F with containment presents another system using sub typing which is equivalent to our presentation of system F . The main di erence is that Mitchells system is much more non deterministic. It is always possible to use the transitivity rule (which is not present in our system) and ....

J.C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76:211{ 249, 1988.

....to observe that P (f.g) P (f) P (g) and id P (id) for f, g, and id in the due types. This proves that P is a quasi functor. Page 31 We claim that the interpretation of subtyping we are using, faithfully corresponds to the intuitive semantics of subtyping (or is compelling , as suggested in [Mitchell 88] with reference to [Bruce Longo 89] Note first that the coercion c A,B in general is not a mono (or injective map) in PER. It happens to be so only when Q(A) Q(B) that is, when one also has In(A) In(B) as w sets. Indeed, the topos theoretic notion of subobject as mono from A to B, given ....

J.Mitchell: Polymorphic Type Inference and Containment, Information and Computation, Vol. 76, Numbers 2/3, (211-249).

....term c:A B when A B, with motivation of giving semantics to calculi with subtyping and inheritance (see, e.g. BCGS91] where no equational theory was studied for the calculus with coercions) Others have also considered coercions in different frameworks of subtyping. See, for example, [MCMS94, PS94, LMS95, Tiu95, Che96, Che98]. These studies had in common that coercion terms were supposed to possess a priori some special properties distinguishing them among all definable terms. For example, in [LMS95, Tiu95, Che96] coercions are almost identical maps. They are characterized formally as terms that become j equal in ....

J.C. Mitchell, L. Cardelli, S. Martini, and A. Schedrov. An extension of system f with subtyping. Information and Computation, 1994.

....also easy to observe that P (f.g) P (f) P (g) and id P (id) for f, g, and id in the due types. This proves that P is a quasi functor. We claim that the interpretation of subtyping we are using, faithfully corresponds to the intuitive semantics of subtyping (or is compelling , as suggested in [Mitchell 88] with reference to [Bruce Longo 89] Note first that the coercion c A,B in general is not a mono (or injective map) in PER. It happens to be so only when Q(A) Q(B) that is, when one also has In(A) In(B) as w sets. Indeed, the topos theoretic notion of subobject as mono from A to B, given by ....

J.Mitchell: Polymorphic Type Inference and Containment, Information and Computation, Vol. 76, Numbers 2/3, (211-249).

....rest of the paper is organized as follows. Section 3 introduces the type rules of our system. Section 4 presents the type inference algorithm and proves theorems of soundness and computation of principal types. Section 5 concludes. 3 Type System We use a language similar to core ML[Mil78,DM82,Mit88,Mit96] but new overloadings can be introduced in let bindings. Recursive let bindings are not considered (since they can be replaced by a fix point operator) For simplicity, letbindings do not introduce nested scopes. Thus: let x = e 1 in let y = e 2 in e is viewed as in a form let fx = e 1 ; y ....

John Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211--249, 1988.

.... rule since it leads to unsoundness when functions involve effects: ff not free in A) 8ff: A B) A 8ff: B If we were to include this rule, our subtyping relation for the fragment containing functions and parametric polymorphism would be equivalent to that proposed by Mitchell [11]. Mitchell s subtyping relation has been shown to be undecidable [18, 16] but the techniques used in these proofs do not seem to apply directly in the absence of distributivity. We therefore do not know at present whether our subtyping relation is decidable. As before, we present an more ....

J. C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211--249, 1988.

....difference between the BCD system and the strict one is that the former is closed for j reduction, whereas the latter is not. The main result proved for the strict system in [1] is that of completeness of type assignment without the relation, by using inference type semantics as defined in [20] instead of the simple type semantics as defined in [16] As stated above, if for the construction of a type inference system for an untyped functional programming language instead of Curry s system a type assignment system with intersection types is to be used, then such a system should at least ....

J.C. Mitchell. Polymorphic Type Inference and Containment. Information and Computation, 76:211--249, 1988.

....another combinations of ideas of type control in presence of polymorphism. During last decade a substantial progress has been achieved both on elucidating theoretical foundations of the underlying type systems and practical implementation of type checkers for programming systems and environments [CW85, Car88, Mit88, Car89, BL90, Mit90, BTCCS91, CL91, Bar92, CG92, Pie92, KS92, CP94, CMMS94]. Among various existing notions of polymorphism the most important and influential are the following two: The parametric (or horizontal) polymorphism characterizes a function uniformly applicable to objects of any type. This is best captured by universal quantification. For example, the ....

J. C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76:211--249, 1988.

No context found.

J. C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211--249, 1988.

No context found.

J.C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211 -- 249, 1988.

No context found.

John Power. Why tricategories? Information and Computation, 120(2):251--262, 1995.

No context found.

John Power. Why tricategories? Information and Computation, 120(2):251--262, 1995.

No context found.

John C. Mitchell. Polymorphic type inference and containment. Information and Computation, 2/3(76):211--249, 1988.

No context found.

John C. Mitchell. Polymorphic type inference and containment. Information and Computation, 76(2/3):211--249, 1988. (Reprinted in [22], pp. 153--194.).

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC