SMITH, G. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming 23, 2-3 (1994), 197--226.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 ....

G. S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23(2-3):197-226, Dec. 1994.

....may grow exponentially in the number of required iterations. We can drastically reduce the number of constraints if we can simplify the constraints to fewer but equivalent constraints. It is therefore no surprise that lots of work has been put into techniques for how to simplify constraints [FM89,Cur90,Kae92,Smi94,EST95,Pot96,TS96,FA96,AWP97,Reh97,FF97]. Another approach is to make the constraint language more powerful so that constraints can be generated by a simple linear time traversal over the program. This can be achieved by making substitution instantiation a syntactic construct in the constraint language. But when we make the constraint ....

....a b is in the constraint becomes just set membership in the appropriate set. 4 Related Work The motivation for this paper is to reduce the cost of the combination of subtyping and polymorphism and in this respect it is related to numerous papers on constraint simplification techniques [FM89,Cur90,Kae92,Smi94,EST95,Pot96,TS96,FA96,AWP97,Reh97,FF97]. Our work is particularly related to the work by Dussart, Henglein and Mossin on binding time analysis with binding time polymorphic recursion [DHM95] where they use constraint simplification techniques in combination with a clever fixedpoint iteration to obtain a polynomial time algorithm. In ....

G. S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, 1994.

....OBJ3 use instead inclusion polymorphism [9] so called dynamic type systems) increasing the representation facilities by allowing subdivide the universe of discourse flexibly. The combination of both kinds of polymorphism has also been treated in literature. As a matter of fact, we have the works [8, 20] referring to functional programming and [21] 13] 15] 4, 5] related to logic programming. The type systems of these languages offer advantages from the traditional programming (static consistence at compiletime, avoidance of non meaningful expressions, explicit data structures and better ....

G.S. Smith. Principal Type Schemes for Functional Programs with Overloading and Subtyping, Science of Computer Programming 23, pp. 197-226, 1994.

....#, # # = P e : # succeeds, fv(#) #, and # #P # # # . Detailed proofs of this and other results stated in this section can be found in the author s dissertation [Nor99] 28 7 Related work Algorithms for polymorphic subtype inference are described in [Mit84, FM90, FM89, Mit91, Kae92, Smi94, AW93, EST95, FA96, Hen96, Pot96, TS96, Reh97, MW97, Seq98, Pot98, MOW99, Pot00] Strategies for simplification of the inferred constraint sets are discussed in [FM89, AW93, EST95, Reh97, MW97, Pot98, Pot00] in particular. The choice to settle for a deliberately incomplete inference algorithm ....

....subtype related if and only if they have the same structure (a function, a pair, a list, etc) and their respective constituent parts are subtype related. This leads to a subtyping theory where the only 29 primitive subtype relationships are between nullary base types. The systems described in [Smi94, EST95, Seq98] follow [CW85] and allow two records to be subtype related if their set of selectors are in a superset relationship. Still, it is the structure of a record that defines its position in the subtype hierarchy, not any intentions explicitly declared by the programmer. Henglein s type ....

G. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, (23):197--226, 1994.

....ow labels of the form 0 , qualifying the type . Since C may contain quanti ed labels from , such an approach gives rise to copies of the captured constraints at all instantiation sites for that type, as we will illustrate next. A standard 1 polymorphic constrained type (see, e.g. [Smi94, TS96, Mos96]) for our example function id of Figure 3 is 8 1 2 :f 1 2g ) int 1 int 2 In this typing, the constraint set f 1 2g captures the fact that any value (represented by 1) passed as argument to id ows to the result of the function (represented by 2 ) In a copy based framework, ....

....site has been identi ed as a major problem, making it very dicult to scale polymorphic subtyping to large programs. 2 In particular, the problem has generated a signi cant amount of research on constraint simpli cation, which aims at compacting constraint sets before they are copied [FM89, Cur90, Kae92, Smi94, EST95, Mos96, Pot96, TS96, FA96, AWP97, Reh97, FF97]. It is unlikely that constraint simpli cation techniques alone will solve this problem, and complete simpli cation is a hard problem itself [Reh98, FF97] 2.2 Problem 2: Demand driven ow computation Constraint copying methods for polymorphic subtyping systems are not demand driven [Mos96] For ....

G. S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197-226, 1994.

....becomes intractable in practice. If we can simplify the constraints for the polymorphic functions to fewer but equivalent constraints that can help to reduce the cost of instantiations. It is therefore no surprise that lots of work has been put into techniques for how to simplify constraints [FM89,Cur90,Kae92,Smi94,EST95,Pot96,TS96,FA96,AWP97,Reh97,FF97]. Another approach is to make the constraint language more powerful so that constraints can be generated by a simple linear time traversal over the program. This can be achieved by making substitution instantiation a syntactic construct in the constraint language. But when we make the constraint ....

....a b is in the constraint becomes just set membership in the appropriate set. 4 Related Work The motivation for this paper is to reduce the cost of the combination of subtyping and polymorphism and in this respect it is related to numerous papers on constraint simplification techniques [FM89,Cur90,Kae92,Smi94,EST95,Pot96,TS96,FA96,AWP97,Reh97,FF97]. To have substitution instantiation in the constraint language is not a new idea. It goes back at least to Henglein s work on type polymorphic recursion [Hen93] where he uses semiunifictaion constraints to represent instantiation. We have previously used constraint abstraction to formulate ....

G. S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, (23):197--226, 1994.

....type systems with inclusion polymorphism are dynamic type systems, that is type information must be checked at run time. The combination of both kind of polymorphism has been studied for logic programming (cfr. Smo89] Han91] HiTo92] Bei95] and for functional programming (cfr. FuMi90] [Smi94]) These approaches investigate the problem of type inference (cfr. Smo89] FuMi90] Smi94] Bei95] or operational semantics based on typed unification for logic programming (cfr. Smo89] Han91] HiTo92] TEL is a language which combines both kinds of polymorphism in a logic language. ....

....must be checked at run time. The combination of both kind of polymorphism has been studied for logic programming (cfr. Smo89] Han91] HiTo92] Bei95] and for functional programming (cfr. FuMi90] Smi94] These approaches investigate the problem of type inference (cfr. Smo89] FuMi90] [Smi94], Bei95] or operational semantics based on typed unification for logic programming (cfr. Smo89] Han91] HiTo92] TEL is a language which combines both kinds of polymorphism in a logic language. The integration of logic and functional programming has been studied in the last years (see ....

G.S. Smith. Principal Type Schemes for Functional Programs with Overloading and Subtyping, Science of Computer Programming, 23, pp. 197-226, 1994.

....systems give rise to a principal typing property for object expressions, using subtype qualified type schemes with relatively few subtype qualifications. Principal typings and their simplification by both syntactic and semantic entailment are the subject of a substantial number of papers; see e.g. [9, 5, 15, 29, 28, 6, 13, 23, 31, 24, 8]. All of these, however, operate in a context where co or contravariant subtyping is admitted and exploited during simplification. Since these simplifications are, in principle, unsound for invariant subtyping, principal typings with invariant object subtyping and their simplification is another ....

Geoffrey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, 1994.

....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 ....

G.S. Smith. Principal Type Schemes for FunctionalPrograms with Overloading and Subtyping, Science of Computer Programming 23, pp. 197-226, 1994.

....expressions that may be well typedness but which cannot be checked at compile time and attaches a tag to take them into account at run time. The combination of both parametric and inclusion polymorphism has been already studied in both paradigms (for instance, in the functional paradigm in [17] [53] and in the logic paradigm in [55] 29, 30] 35] 10] The quoted works in functional programming deal with type inference, showing algorithms for ML like programs, and addressing the subtype constraints solving [17] and the introduction of overloading [53] On the side of logic programming, the ....

.... in the functional paradigm in [17] 53] and in the logic paradigm in [55] 29, 30] 35] 10] The quoted works in functional programming deal with type inference, showing algorithms for ML like programs, and addressing the subtype constraints solving [17] and the introduction of overloading [53]. On the side of logic programming, the cited works take interest in the use of order sorted types with the aim of replacing syntactic uni cation by typed uni cation in order to improve and prune Prolog style (or SLDNF) computations. Such works di er in the form of the type speci cations, for ....

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G. S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23(2-3):197{ 226, 1994.

....and Wimmers [2] uses ML s let polymorphism, and, therefore, we believe it does not have principal typings. The subsystem without let polymorphism, though, is still of interest, and may have principal typings (but this is not clear) The constraint based systems of Jones [12] Kaes [14] and Smith [28] are also based on ML. Constraint satisfaction, including subtype satisfaction, is an important component of each of these systems. Our method for solving constraints involving quantifiers ( 82;1 satisfaction) is a significant advance over these systems. Along with intersections, this is the ....

Geoffrey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197-- 226, 1994.

....size, which makes typing information difficult to read and may seriously slow down type inference in polymorphic frameworks. The problem is now widely recognized and has generated a substantial amount of work that aims at simplifying typings generated by subtype inference algorithms (see, e.g. [9, 5, 12, 21, 6, 11, 18, 24, 20, 1, 7]. Recently, a number of researchers have independently suggested that the problem should be attacked by introducing stronger systems, based on more powerful, model theoretic notions of entailment (j= rather than syntactic proof relations such as P . Such works include [18, 24, 7, 1, 8] The ....

G. S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, 1994.

.... :C ) where C is a set of captured subtyping constraints qualifying the type . Since C may contain quanti ed labels from , such an approach gives rise to copies of the captured constraints at all instantiation sites for that type. A standard 1 polymorphic constrained type (see, e.g. [Smi94, TS96, Mos96]) for id is 8 1 2 :f 1 2 g ) int 1 int 2 . In a copybased framework, program e is typed by copying the constraint set f 1 2 g associated with id at each of the instantiation sites id i and id j , yielding the standard polymorphic typing judgment f 3 4 ; 5 6 g; ....

....site has been identi ed as a major problem, making it very dicult to scale polymorphic subtyping to large programs. 2 The problem has generated a signi cant amount of research, including work on constraint simpli cation, which aims at compacting constraint sets before they are copied [FM89, Cur90, Kae92, Smi94, EST95, Pot96, TS96, FA96, AWP97, Reh97, FF97]. It is unlikely that constraint simpli cation techniques alone will solve this problem, and complete simpli cation is a hard problem itself [Reh98, FF97] 1.2 A new method based on instantiation constraints In this paper, we tackle the constraint copying problem in a new way. Our starting point ....

G. S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197-226, 1994.

.... :C ) where C is a set of captured subtyping constraints qualifying the type . Since C may contain quanti ed labels from , such an approach gives rise to copies of the captured constraints at all instantiation sites for that type. A standard 1 polymorphic constrained type (see, e.g. [Smi94, TS96, Mos96]) for id is 8 1 2 :f 1 2 g ) int 1 int 2 . In a copy based framework, program e is typed by copying the constraint set f 1 2 g associated with id at each of the instantiation sites id i and id j , yielding the standard polymorphic typing judgment f 3 4 ; 5 6 g; ....

....site has been identi ed as a major problem, making it very dicult to scale polymorphic subtyping to large programs. 2 The problem has generated a signi cant amount of research, including work on constraint simpli cation, which aims at compacting constraint sets before they are copied [FM89, Cur90, Kae92, Smi94, EST95, Pot96, TS96, FA96, AWP97, Reh97, FF97]. It is unlikely that constraint simpli cation techniques alone will solve this problem, and complete simpli cation is a hard problem itself [Reh98, FF97] 1.2 A New Method Based on Instantiation Constraints In this article, we tackle the constraint copying problem in a new way. Our starting ....

Georey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23(2-3):197-226, December 1994.

....similar to, but not identical to, the one in Section 7. Pottier gives a method that can eliminate redundant variables from constraint sets [16] Both of these methods are heuristic; i.e. they are sound but not complete. Constraint simplification is also a component of the systems described in [11, 17]. It is not claimed that either system performs complete constraint simplification. Our focus in this paper is quite different. The question of variable elimination arises in any type system with polymorphism and subtyping, not just in systems with constrained types. Our purpose is to explore the ....

Geoffrey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, 1994.

....complete; however, we provide a completeness result w. r. t. the Hindley Milner type system as a form of characterizing lower bound. 1 Introduction The combination of subtyping with polymorphic type inference has been under intensive study for more than a decade [Mit84, FM90, FM89, Mit91, Kae92, Smi94, AW93, EST95, Hen96, Reh97, MW97] From the outset, this line of research has focused on complete inference algorithms, and the related notion of principal types. This direction is not hard to justify considering the evident merits of polymorphic languages like ML or Haskell: program fragments ....

....introduced in the next section. 3 Type system The starting point of our work is the let polymorphic type system of ML (a.k.a the Hindley Milner system) extended with subtype constraints, a subtype relation, and a subsumption rule. Such extensions are well represented in the literature [Kae92, Smi94, AW93, EST95] in this paper we will adopt a formulation by Henglein [Hen96] which has the interesting merit of being generic in its subtyping theory. Since our system is obtained by instantiation with a particular form of subtyping theory, the results of [Hen96] directly carry over to our ....

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G. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, (23):197--226, 1994.

.... 8ff:D ) No satisfiability check [Jon92] C [ D; Gamma e : ff 62 tv(C) tv( Gamma) C; Gamma e : 8 ff:D ) 8 Intro 1) Weak satisfiability check [AW93] C [ D; Gamma e : 9D ff 62 tv(C) tv( Gamma) C; Gamma e : 8 ff:D ) 8 Intro 2) Strong satisfiability check [Smi94] C [ D; Gamma e : C [ ff]D ff 62 tv(C) tv( Gamma) C; Gamma e : 8 ff:D ) 8 Intro 3) Duplication [EST95] C [ D; Gamma e : ff 62 tv(C) tv( Gamma) C [ D; Gamma e : 8 ff:D ) 8 Intro 4) Figure 1: Versions of the quantifier introduction rule While there is ....

....Hence, the system with (8 Intro 2) does not enjoy the property of subject reduction, which says that if a term is typable then its reduction instances are typable as well. Where Aiken and Wimmers require only a weak form of satisfiability for traded constraints, G. Smith requires a strong one [Smi94] In rule (8 Intro 3) the traded constraint D must be solvable by instantiation of only the quantified variables ff. Hence, all three previous examples would be untypable under his system. However, 8 Intro 3) rule might seem overly restrictive, depending on the constraint system used. In a ....

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Geoffrey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, 1994.

....written between the assumptions and : ff : 00 ; x : g; f 00 = 0 g f x : 0 . As a generalization, one can use inclusion constraints rather than equality constraints; then we have f 00 0 g in place of the above constraint. See Aiken et al. 1] and Smith [21] for type systems defined in this way. In the flow type system, our example inference looks as follows: ff : 00 ; x : g; f 00 hj; 0 ig f x : 0 As can be seen from the inference rules in Figure 6, typing judgements in our system are of the form A; S e : where A is a ....

....generalize over Vars. This restriction is the mechanism with which we combine flow information from the various uses of a let bound variable. Note also the tag constraints in the ABS, ALT and LET rules. Another thing to note about the LET rule is that, in contrast to systems like those in [1] and [21], we do not have the restriction that Sol(S) be nonempty instead we have the restriction that x must occur in e 0 (the body of the let expression) Studying the THUNK and EVAL rules, we see that the only thing that happens, type wise, when we suspend or restart computation is the addition or ....

G.S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, December 1994.

....the constraint set; correspond to allowing constructs of the form, say, nat s, nat nat s, where s is a type variable; and corresponds to allowing constraints of the form, say, s nat, s nat nat. Constrained types without cycles in the constraint set were studied by Kaes [17] and Smith [31]. Type inference for constrained types extended with let polymorphism was studied for functional languages by Aiken and Wimmers [2] and later for object oriented languages by Eifrig, Smith, and Trifonov [10, 9] An algorithm for deciding a subtyping relation for constrained types, including type ....

Geoffrey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, 1994.

....In [EST95] Eifrig, Smith, and Trifonov and separately Pottier [Pot96] describe methods for eliminating unnecessary variables from constraint sets. Both of these methods are heuristic; i.e. they are sound but not complete. Constraint simplification is also a component of the systems described in [Kae92, Smi94]. Our focus in this paper is quite different. The question of variable elimination arises in any type system with polymorphism and subtyping, not just in systems with constrained types. Our purpose is to explore the basic structure of this problem in the simplest settings. To the best of our ....

Geoffrey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23:197--226, 1994. This article was processed using the L A T E X macro package with LLNCS style

....order sorted logic programs have considered this combination, obtaining the type of the data variables in every program rule (cfr. Smo89] HiTo92] and [Bei95] For these type systems, the problem of inference of the most general type has been treated for functional programming in [FuMi90] and [Smi94]. The integration of logic and functional programming has been studied in the last years (see [Han94] for a survey) considering the lazy narrowing as a sound and complete operational semantics [GHLR96] This combination is adequate in order to include lazy evaluation allowing partial non strict ....

.... [ n i=1 tvar( i ) tvar( 0 ) 2 ffl FUN is a finite set of function type declarations of the form f : 1 : n 0 , where i 2 T S (X ) i = 0; n (n is called arity of f ) We admit a unique type declaration for every data constructor and function, excluding overloading [GoMe92] [Smi94]. We will write 2 CONS [ FUN if : oe 2 CONS [ FUN . The set [ defined by f 1 ae: n ae 0 ae j ae 2 TSUST (S; X )g denotes the set of instances of the type declaration of : 1 : n 0 2 CONS [ FUN . In the following we suppose a fixed polymorphic signature Sigma. Let V be a ....

G.S. Smith. Principal Type Schemes for Functional Programs with Overloading and Subtyping, Science of Computer Programming, 23, pp. 197-226, 1994.

....formulated in [Cur90] AW93, AWL94] formulate a similar version with a somewhat richer collection of type operations 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 ....

....it is open whether 8 sem is decidable, and we develop a powerful decidable approximation 8 dec . These results improve on previous work which defined weaker relations [EST95a, Pot96, Jim96] Subtyping relations for constrained types have also been studied for non recursive constraint sets [Smi94, BM96] We have yet to find a realistic subtyping example which is semantically sound but is not derivable using 8 dec , but most convincing would be testing the performance of a system that uses it for signature matching and simplifications on real code. 1 i.e. on the left of an even ....

Geoffrey S. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23, 1994. 9

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SMITH, G. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming 23, 2-3 (1994), 197--226.

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G. Smith. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming, 23(2-3):197--226, 1994.

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SMITH, G. Principal type schemes for functional programs with overloading and subtyping. Science of Computer Programming 23, 2-3 (1994), 197--226.

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