R. Harper and R. Pollack. Type Checking with Universes. Theoretical COmputer Science, 89: 107-136, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is that that all types in ECC are a member of some type universe in the hierarchy, which is not true for UTT because the objects in universes are just codes for types in the kind Type, and not the types themselves. We are also considering how to incorporate the flexibility of universe polymorphism [10] into the system. 4 Implementing the translations Plastic implements some simple syntax translations already. They are very useful for reducing the amount of information in type expressions so that they are more readable. For example, FA A [x:A]B can be entered as, and shown as, fx:AgB (but note ....

R. Harper and R. Pollack. Type checking, universe polymorphism, and typical ambiguity in the calculus of constructions. Theoretical Computer Science, 89(1), 1991.

....hold (see [CPM90, Ore90] where extensions of ECC by inductive types are discussed; also see [Luo90c] for a recent proposal using Martin Lof s W types) Concerning about the predicative universes, it is often tedious to always write the level subscripts. A technique has been developed [Hue87, HP89, Pol90] to ease the tension of worrying about universe levels so that, in practice, one can omit the universe levels to write Type instead of Type i . This is nicely implemented in the proof development system Lego [Pol89, LPT89] With such a facility, to assume X :T ype in a context is in some ....

R. Harper and R. Pollack. Type checking, universe polymorphism, and typical ambiguity in the calculus of constructions. Theoretical Computer Science, 1989. to appear.

.... (see for example, page 84 of [26] Also, 31] provides a treatment of trans nite orders as universes, 25] discusses predicative universes in the Calculus of Constructions [7] 8] introduces the generalised Calculus of Constructions CC which includes a cumulative hierarchy of universes, [13] studies type checking and well typedness in CC and in an extended version of it with an anonymous universe Type which is intended to model Russell and Whitehead s typical ambiguity convention, and [37] uses orders in proof theory. Moreover, orders are closely related to the degree of ....

....ed formalization of rtt, which is based on a more extensive formalization given in [22] 3. We give the rst account of embedding rtt in a relevant modern type theory. This is done in Section 4, where we present an embedding of rtt in Nuprl s type system. Note that this is very di erent from [13] which did not give a presentation of rtt, but instead, extended CC with an anonymous universe Type and intended this extension to model Russell and Whitehead s typical ambiguity convention. 4. Our study is the rst to connect rtt to the modern way of writing type theory as a PTS. As we ....

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R. Harper and R. Pollack. Type Checking with Universes. Theoretical COmputer Science, 89: 107-136, 1991.

....necessary to perform Chapter 3 s embedding into the predicative type theory of Nuprl. Since these annotations are tedious to provide, I discuss some alternatives in Section 3.5. In a practical system based on this annotation mechanism, the system would have to infer level annotations for the user [47]. The kind of functions mapping constructors of kind 1 to constructors of kind 2 may be denoted 1 2 , but since constructors may appear within kinds, it is desirable to allow a more precise characterization of such functions: The kind Piff: 1 : 2 includes all functions mapping 1 ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107--136, 1991.

....CC may give several types to a given term, it does not de ne any subtyping relation and there is no notion of principal type in this system. For instance, a term t with type Type 0 Type 0 may have also the type Type 0 Type 1 or not, depending on t. In fact, typechecking CC is complex [vBJMP93,HP91] and relies on universe inference. Therefore we choose to replace CC by Zhaohui Luo s Extended Calculus of Constructions [Luo89,Luo90] ECC extends CC with a real subtyping relation, called the cumulativity relation (thus Type 0 Type 0 is included in Type 0 Type 1 in ECC) It enjoys the ....

R. Harper and R. Pollack. Type checking, universe polymorphism, and typical ambiguity in the calculus of constructions. Theoretical Computer Science, 89(2), 1991.

....the de nition of a reduction relation between generalized universes independent from the concept of reduction. The typing under universes is captured in the system by an independent additional rule. These systems are a natural generalization of PTS and other theories, as Ecc [7] and CC [6], for a proper initial relation . Generic PTS systems and their properties have already been studied in [10 12] In the next section we will see the principal properties of PTS that we will use in through this paper. An important property to be studied in PTS systems is the Expansion ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107-136, 1991.

....of for this result compute the normal forms of types, hence have bad worst case behavior. For many TS of interest there are natural algorithms that do only necessary reduction, and work quite well in practice; for example P [Coquand and Huet, 1988] ECC [Luo, 1990b] and its subsystem GCC [Harper and Pollack, 1991]. The system P above, of first order dependent types is also decideable. All of these systems are implemented by LEGO [Luo and Pollack, 1992] Coq [Dowek et al. 1991] also implements a strong TS with inductive types. 3 Typical Ambiguity Consider a TS P = Sort; Ax; Rule; with a ....

.... is, LEGO solves what we might call the implicit type synthesis problem: given Gamma and M , find A, Gamma 0 , M 0 and A 0 such that Gamma; M;A ) Gamma 0 M 0 : A 0 ) In fact, there is an algorithm that correctly and completely solves this problem over several TS of interest [Harper and Pollack, 1991] ; e.g. ECC, GCC, and PU . The idea for how to do this, suggested in [Huet, 1987] is to view the explicit universe levels, i, j, appearing in the translation system as metavariables that are constrained by the side conditions. For example in PU , given input ffl and , the algorithm returns a ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107--136, 1991.

.... be described mathematically and proven correct (This involves not just a type theory, but a represenation of a type theory; for example the Constructive Engine s translation into nameless representation [Pol94] Could the Constructive Engine be extended to a type theory with universes [Hue87,HP91] Type checking is only the start of the problem. For example, LEGO supports definitions, assigning a name to a (typed) term; do global definitions preserve normalizability of CC Do local definitions also preserve normalizability [SP94] LEGO also uses meta variables to implement refinement ....

....argument synthesis, typical ambiguity, and universe polymorphism; these features have to be explained too. I ve still only begun to outline the the questions that actually come up in implementing a proofchecker. In 1988, Bob Harper and I addressed the problem of typechecking cumulative universes [HP91] we were far from formalizing our definitions and theorems, let al..one their proofs, but we reasoned about complex algorithmic issues in terms of formal systems (i.e. inductively defined relations) and I began to see that it might be possible to formally reason about the kinds of questions ....

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Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107--136, 1991.

....number, we add the notation Type(ID) where ID is a LEGO identifier. All instances of Type(nam) are the same universe, but what absolute universe that is is constrained only by how Type(nam) is used. The oerational meaning of this notation, and the algorithm for implementing it, is explained in [HP91] although we knew of no interesting application until the implementation of inductive types. Consider the inductive type of lists declared as Inductive [list:Type] Parameters [A Type] Constructors [nil:list] cons:A list list] This is inconsistent, as the type of the parameter may be ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107--136, 1991.

....several axioms for c to choose, or which of several rules for s 1 , s 2 . These problems are reduced to a decision problem about the sorts, axioms and rules of the PTS by making these decisions schematically and collecting the conditions constraining the schematic choices. This subsection follows [HP91], where more details of a related application of the same ideas can be found. The same technique can be used to derive algorithms from syntax directed derivation systems for other classes of type theory, such as the systems of Jutting. Notation: we write Gamma ds M ) X; C for Gamma ds M ) ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107--136, 1991.

.... semifull PTS; the other uses in this paper could be replaced by more difficult argument, but to have a suitably deterministic syntax directed system for non functional systems requires the technique of sort variables, schematic terms, and constraints, which is not discussed in this paper (see [HP91, vBJMP94] Functionality plays a more important role in type checking for PTS that are not semi full, as in this case the failure of subject expansion for non functional PTS becomes problematic, and we must normalize some terms to be sure we have all of their types. In fact the general ....

....characterizes all the types of a term. In this setting lemma 4 is replaced by a lemma saying that sdsf computes the principal type of a term. ECC, with principal types, is well behaved; for an (informal) development of type checking for a system similar to ECC, but not having principal types, see [HP91] Remark. Lemma 4 is surprisingly weak. sdsf is called syntax directed , but it has two sources of non determinism which explain why sdsf types are unique only up to conversion. 1. I am being informal about variable names, but in detail there is a need to choose fresh variables in the rules ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107--136, 1991.

....Similarly Pi sfnsd, Lda1 sfnsd and Lda2 sfnsd may be non syntax directed due to non functionality of R. For functional PTS sfnsd is already syntax directed, and may be used directly for typechecking. For non functional semi full PTS we will remove this non determinism by a method suggested in [Hue87, HP91, Pol92], and for this purpose we first introduce some technical machinery. Schematic Terms Introduce a new set Sigma , disjoint from V and S . Elements of Sigma will be called sort variables and will be denoted by oe; oe 0 ; oe 1 ; The symbols ff; fi; fl; will range over Sigma [S . ....

....of variables and constants) by checking acyclicity of a directed graph whose nodes are variables and constants, and whose edges are the relations and . This result is due to Chan [Cha77] and its application for solving constraints of typechecking was suggested by Huet [Hue87] and detailed in [HP91]. ut Our purpose will be to compute for any pair Gamma , a a term X 2 T [ T Sigma and a constraint C , we denote this by Gamma sfsd a : X j C ) such that i If Gamma sfsd a : X j C and OE j= C then Gamma a : OE X ii If Gamma a : A then Gamma sfsd a : X j C and 9 OE [OE j= C ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107-- 136, 1991.

....occurrences of M in N with the variable x. If definitions are treated formally, they are introduced as a new concept that extends the type theory causing additional complication. With singleton types we get a form of definition in the system for free, and derive similar rules to those given in [HP91, SP94]. The typed definitions of Severi and Poll [SP94] have the form: x = M : A in N This is similar to a abstraction over a singleton type in fg , applied to the trivially appropriate argument: x: fMgA : N)M which in turn can be compared with the usual trick for writing definitions in systems ....

Robert Harper and Robert Pollack. Type checking with universes. Theoretical Computer Science, 89:107--136, 1991.

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R. Harper and R. Pollack. Type Checking with Universes. Theoretical COmputer Science, 89: 107-136, 1991.

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