Loyola University Chicago Loyola eCommons Computer Science: Faculty Publications and Other Works Faculty Publications 1-1996 Putting Type Annotations to Work Martin Odersky Universität Karlsruhe Konstantin Laufer Loyola University Chicago, [email protected] Follow this and additional works at: https://ecommons.luc.edu/cs_facpubs Part of the Computer Sciences Commons Recommended Citation M. Odersky and K. Läufer, Putting type annotations to work, in Proceedings of the 23rd ACM SIGPLAN- SIGACT symposium on Principles of programming languages, ser. POPL '96. New York, NY, USA: ACM, 1996, pp. 54-67. [Online]. Available: http://webpages.cs.luc.edu/~laufer/papers/popl96.pdf This Conference Proceeding is brought to you for free and open access by the Faculty Publications at Loyola eCommons. It has been accepted for inclusion in Computer Science: Faculty Publications and Other Works by an authorized administrator of Loyola eCommons. For more information, please contact [email protected]. This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License. Copyright © 1996 Martin Odersky and Konstantin Läufer Putting Typ e Annotations to Work Martin Odersky Konstantin Läufer Department of Computer Science Department of Mathematical Sciences University of Karlsruhe Loyola University Chicago 76128 Karlsruhe, Germany Chicago, Illinois 60626, USA o [email protected] [email protected] Abstract egories. Curry-style reconstruction lls in p olymorphic abstractions and applications together with typ e an- We study an extension of the Hindley/Milner system notations. This style of reconstruction is complicated with explicit typ e scheme annotations and typ e decla- by the lack of principal typ es in F . The prop osed 2 rations. The system can express p olymorphic function schemes all have rather complex inference rules with arguments, user-dened data typ es with abstract com- cumb ersome conversions b etween declared and inferred p onents, and structure typ es with p olymorphic elds. typ es [McC84, OG89]. By contrast, Church-style re- More generally, all programs of the p olymorphic lambda construction requires the p osition of typ e abstractions calculus can b e enco ded by a translation b etween typ- and applications to b e indicated in the original source. ing derivations. We show that typ e reconstruction in This style of reconstruction also called partial typ e this system can b e reduced to the decidable problem of reconstruction [Bo e89] was shown to b e reducible to rst-order unication under a mixed prex. higher-order unication [Pfe88]. Even though Church- style reconstruction is thus undecidable in general, this result op ens up the p ossibility for semi-decision pro ce- 1 Intro duction dures that work well in practice. On the other hand, the p osition of a p olymorphic application has to b e in- Two of the most imp ortant cornerstones of typ e theory dicated explicitly in the source, which leads to a rather for programming languages are the Hindley/Milner sys- unfamiliar co ding style, at least for programmers used tem and the second-order p olymorphic -calculus. This to the Hindley/Milner system. pap er tries to explore some of the design space b etween Recently there havebeenseveral approaches towards them. extending the Hindley/Milner system with some form The Hindley/Milner system [Mil78] extends the of emb edded quantiers without going all the wayto simply-typ ed -calculus with p olymorphic let-b ound the p olymorphic -calculus. For instance, Launch- identiers. It thus adds considerable expressivepower bury and Peyton Jones have presented an elegant yet retains the prop erty thatnotyp e annotations in typ e system for syntactic control of interference [LPar] programs are needed, since most general typ es can b e that uses second-order universal quantication. Perry inferred [DM82]. This prop erty has made the Hind- [Per90] and Läufer and Odersky [LO94]have studied ley/Milner system very app ealing as a basis of typ e existential quantication in algebraic datatyp es, which systems for programming languages. yields a Hindley/Milner style version of Mitchell and By contrast, the second-order p olymorphic - Plotkin's abstract typ es [MP88]. This style of existen- calculus F [Gir71, Rey74] allows p olymorphic typ es 2 tial quantication has b een implemented in compilers everywhere, but requires explicit annotations of b oth for Hop e [Per90], Haskell [Aug94] and CAML [MP93]. argumenttyp es and typ e instantiations. The general Rémy [Rém94] has extended Läufer and Odersky's sys- problem of typ echecking without typ e annotations is tem with universal quantication in datatyp es, so that undecidable [Wel94], but there havebeenseveral ap- ob jects with p olymorphic metho ds can b e expressed. proaches towards typ e reconstruction where some typ e Jones [Jon95] has investigated record typ es with p oly- information is given. These generally fall into two cat- morphic elements as a way to capture essential asp ects of mo dule systems. A prop osal along these lines has b een accepted for inclusion in Haskell 1.3. It seems that a combination of all of the ab ove sys- tems, while feasible, would b e rather unwieldy. Fortu- nately, it turns out that it is go o d enough to consider as a generalization a far simpler typ e system that captures only to typ es, never to typ e schemes. We get back the extensions' commonalities and expresses their dif- the full p ower of F in an indirect way,by allowing 2 ferences via enco dings. The extensions all have in com- typ e schemes as comp onents of explicitly declared mon that some form of explicit typ e information is re- data typ es. We show that we can enco de all of F 2 quired. For instance, Läufer and Odersky's and Rémy's by providing typ e declarations for all p olymorphic systems restrict existential quantication to the com- typ es in a given F program. This shows that our 2 p onents of explicitly declared datatyp es, while Jones typing discipline provides essentially the same ca- restricts universal quantication to elds of explicitly pabilities as F ,even though the enco ding in F 2 2 declared record typ es. do es not supp ort a formal comparison of expres- Here we study a typ e system that allows but do es sivepower in the sense of Felleisen [Fel90] since it not require explicit typ e scheme annotations for func- fails to b e comp ositional. tion arguments. The idea is that a formal function pa- Our typing discipline is a conservative extension of rameter is p olymorphic only if annotated with a typ e the Hindley/Milner system. Every typable program in scheme; otherwise the parameter is monomorphic, i.e. it that system continues to b e typable. This holds also if has a typ e, not a typ e scheme. As an imp ortant sp ecial typ e annotations in the style of ML or Haskell are added case we admit a rudimentary form of user-dened data to Hindley/Milner. Wewere able to show principal typ e declaration that intro duces a value constructor typ e prop erties and soundness and completeness of typ e with a single, p ossibly p olymorphic argument. Finally, inference fully analogous to the results stated by Damas we also allowtyp e scheme annotations for expressions. and Milner [DM82]. Since the engineering issues of Note that this is roughly the kind of typ e annota- ML-like programming languages and typ e checkers are tions that most programming languages oer or require. bynowwell understo o d, we b elieve that this makes The crucial extension of this pap er is that annotations our system promising as a practical kernel language on and declarations can refer to p olymorphic typ e schemes whichtyp e-systematic extensions of ML or Haskell can instead of just typ es. The ramications of this simple b e based. idea are quite substantial. The rest of this pap er is organized as follows. Sec- tion 2 presents our typ e system. Section 3 shows how We can express p olymorphic function arguments previous p olymorphic extensions of ML can b e em- by annotating the argument with a typ e scheme. b edded in it. Section 4 discusses an enco ding of the We can express data typ es and record typ es by p olymorphic -calculus. Section 5 states the most gen- their usual Church enco dings in a typ e-correct eral instantiation problem and presents an algorithm to way. solve it. Section 6 presents a typ e inference algorithm. Section 7 concludes. By slightly mo difying these Church enco dings, we can also express existentially or universally quan- tied comp onenttyp es of records and data typ es, 2 The Typ e System thereby subsuming the typ e systems of Perry, Figure 1 presents the abstract syntax of our kernel lan- Läufer and Odersky,Rémy, and Jones. The en- guage, Exp: . As in the Hindley/Milner system, we co dings give us principal typ e prop erties and typ e distinguish b etween typ es, which cannot contain quan- inference algorithms for these systems for free . tication over typ e variables, and typ e schemes, which Unlike the situation in the simply typ ed -calculus can. Compared to Milner's language Exp there are two [Mor68] or ML [Mil78], it is no longer p ossible to extensions that can b e considered indep endently, but reduce typ e inference to a simple Herbrand uni- that are most useful in combination. One extension cation problem. We need to consider instead considers typ e annotations for formal arguments and the problem of nding a most general substitution expressions; the other considers typ e declarations.