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35
What Are Principal Typings and What Are They Good For?
, 1995
"... We demonstrate the pragmatic value of the principal typing property, a property more general than ML's principal type property, by studying a type system with principal typings. The type system is based on rank 2 intersection types and is closely related to ML. Its principal typing property prov ..."
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Cited by 93 (0 self)
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We demonstrate the pragmatic value of the principal typing property, a property more general than ML's principal type property, by studying a type system with principal typings. The type system is based on rank 2 intersection types and is closely related to ML. Its principal typing property provides elegant support for separate compilation, including "smartest recompilation" and incremental type inference, and for accurate type error messages. Moreover, it motivates a novel rule for typing recursive definitions that can type many examples of polymorphic recursion.
A direct algorithm for type inference in the rank2 fragment of the secondorder λcalculus
, 1993
"... We study the problem of type inference for a family of polymorphic type disciplines containing the power of CoreML. This family comprises all levels of the stratification of the secondorder lambdacalculus by "rank" of types. We show that typability is an undecidable problem at every rank k >= 3 o ..."
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Cited by 78 (14 self)
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We study the problem of type inference for a family of polymorphic type disciplines containing the power of CoreML. This family comprises all levels of the stratification of the secondorder lambdacalculus by "rank" of types. We show that typability is an undecidable problem at every rank k >= 3 of this stratification. While it was already known that typability is decidable at rank 2, no direct and easytoimplement algorithm was available. To design such an algorithm, we develop a new notion of reduction and show howto use it to reduce the problem of typability at rank 2 to the problem of acyclic semiunification. A byproduct of our analysis is the publication of a simple solution procedure for acyclic semiunification.
Type Error Slicing in Implicitly Typed HigherOrder Languages
, 2004
"... Previous methods have generally identified the location of a type error as a particular program point or the program subtree rooted at that point. We present a new approach that identifies the location of a type error as a set of program points (a slice) all of which are necessary for the type error ..."
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Cited by 46 (3 self)
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Previous methods have generally identified the location of a type error as a particular program point or the program subtree rooted at that point. We present a new approach that identifies the location of a type error as a set of program points (a slice) all of which are necessary for the type error. We identify the criteria of completeness and minimality for type error slices. We discuss the advantages of complete and minimal type error slices over previous methods of presenting type errors. We present and prove the correctness of algorithms for finding complete and minimal type error slices for implicitly typed higherorder languages like Standard ML.
The Barendregt Cube with Definitions and Generalised Reduction
, 1997
"... In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, ..."
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Cited by 37 (17 self)
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In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, Definitions, Barendregt Cube, Church Rosser, Subject Reduction, Strong Normalisation. Contents 1 Introduction 3 1.1 Why generalised reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why definition mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The item notation for definitions and generalised reduction . . . . . . . . . . 4 2 The item notation 7 3 The ordinary typing relation and its properties 10 3.1 The typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Properties of the ordinary typing relation . . . . . . . . . . . . . . . . . . . . 13 4 Generalising reduction in the Cube 15 4.1 The generalised...
Union Types for Semistructured Data
 University of Pennsylvania Dept. of CIS
, 1999
"... Semistructured databases are treated as dynamically typed: they come equipped with no independent schema or type system to constrain the data. Query languages that are designed for semistructured data, even when used with structured data, typically ignore any type information that may be present. ..."
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Cited by 35 (4 self)
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Semistructured databases are treated as dynamically typed: they come equipped with no independent schema or type system to constrain the data. Query languages that are designed for semistructured data, even when used with structured data, typically ignore any type information that may be present. The consequences of this are what one would expect from using a dynamic type system with complex data: fewer guarantees on the correctness of applications. For example, a query that would cause a type error in a statically typed query language will return the empty set when applied to a semistructured representation of the same data. Much semistructured data originates in structured data. A semistructured representation is useful when one wants to add data that does not conform to the original type or when one wants to combine sources of different types. However, the deviations from the prescribed types are often minor, and we believe that a better strategy than throwing away all typ...
Rank 2 Type Systems and Recursive Definitions
, 1995
"... We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We int ..."
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Cited by 26 (1 self)
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We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We introduce a rank 2 system combining intersections and polymorphism, and prove that it types exactly the same terms as the other rank 2 systems. The combined system suggests a new rule for typing recursive definitions. The result is a rank 2 type system with decidable type inference that can type some interesting examples of polymorphic recursion. Finally,we discuss some applications of the type system in data representation optimizations such as unboxing and overloading.
Calculi of Generalised βReduction and Explicit Substitutions: The TypeFree and Simply Typed Versions
, 1998
"... Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substit ..."
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Cited by 14 (7 self)
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Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the  calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fireduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs, and show that welltyped terms are strongly normalizing and that other properties,...
Functional Database Query Languages as Typed Lambda Calculi of Fixed Order (Extended Abstract)
 In Proceedings 13th PODS
, 1994
"... We present a functional framework for database query languages, which is analogous to the conventional logical framework of firstorder and fixpoint formulas over finite structures. We use atomic constants of order 0, equality among these constants, variables, application, lambda abstraction, and le ..."
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Cited by 12 (5 self)
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We present a functional framework for database query languages, which is analogous to the conventional logical framework of firstorder and fixpoint formulas over finite structures. We use atomic constants of order 0, equality among these constants, variables, application, lambda abstraction, and let abstraction; all typed using fixed order ( 5) functionalities. In this framework, proposed in [21] for arbitrary order functionalities, queries and databases are both typed lambda terms, evaluation is by reduction, and the main programming technique is list iteration. We define two families of languages: TLI = i or simplytyped list iteration of order i +3 with equality, and MLI = i or MLtyped list iteration of order i+3 with equality; we use i+3 since our list representation of databases requires at least order 3. We show that: FOqueries ` TLI = 0 ` MLI = 0 ` LOGSPACEqueries ` TLI = 1 = MLI = 1 = PTIMEqueries ` TLI = 2 , where equality is no longer a primitive in TLI = 2 . We also show that ML type inference, restricted to fixed order, is polynomial in the size of the program typed. Since programming by using low order functionalities and type inference is common in functional languages, our results indicate that such programs suffice for expressing efficient computations and that their MLtypes can be efficiently inferred.
Finite Model Theory In The Simply Typed Lambda Calculus
, 1994
"... Church's simply typed calculus is a very basic framework for functional programming language research. However, it is common to augment this framework with additional programming constructs, because its expressive power for functions over the domain of Church numerals is very limited. In this thesi ..."
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Cited by 8 (5 self)
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Church's simply typed calculus is a very basic framework for functional programming language research. However, it is common to augment this framework with additional programming constructs, because its expressive power for functions over the domain of Church numerals is very limited. In this thesis: (1) We reexamine the expressive power of the "pure" simply typed calculus, but over encodings of finite relational structures, i. e., finite models or databases . In this novel framework the simply typed calculus expresses all elementary functions from finite models to finite models. In addition, many common database query languages, e. g., relational algebra, Datalog : , and the Abiteboul/Beeri complex object algebra, can be embedded into it. The embeddings are feasible in the sense that the terms corresponding to PTIME queries can be evaluated in polynomial time. (2) We examine fixedorder fragments of the simply typed calculus to determine machine independent characterizations of complexity classes. For this we augment the calculus with atomic constants and equality among atomic constants. We show that over ordered structures, the order 3, 4, 5, and 6 fragments express exactly the firstorder, PTIME, PSPACE, and EXPTIME queries, respectively, and we conjecture that for general k 1, order 2 k + 4 expresses exactly the kEXPTIME queries and order 2 k + 5 expresses exactly the kEXPSPACE queries. (3) We also reexamine other functional characterizations of PTIME and we show that method schemas with ordered objects express exactly PTIME. This is a firstorder framework proposed for objectoriented databasesas opposed to the above higherorder frameworks. In summary, this research provides a link between finite model theory (and thus computational complexity), dat...