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27
Typability and Type Checking in the SecondOrder lambdaCalculus Are Equivalent and Undecidable
, 1993
"... We consider the problems of typability and type checking in the Girard/Reynolds secondorder polymorphic typedcalculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pureterms. These problems have been considere ..."
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Cited by 13 (1 self)
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We consider the problems of typability and type checking in the Girard/Reynolds secondorder polymorphic typedcalculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pureterms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lowerbound complexity results for System F have been achieved, but they have remained "embarrassing open problems" 3 for System F itself. We first prove that type checking in System F is undecidable by a reduction from semiunification. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructingterms such that in all type derivations, specific bound variables must always be assigned a specific type. Using this technique, we can require that specif subterms must be typable using a specific, fixed type assignment in order for the entire term to be typable at all. Any desired type assignment maybe simulated. We develop this method, which we call \constants for free", for both the K and I calculi.
Raising ML to the Power of System F
 In ICFP ’03: Proceedings of the eighth ACM SIGPLAN international conference on Functional programming
, 2003
"... We propose a type system MLF that generalizes ML with firstclass polymorphism as in System F. We perform partial type reconstruction. As in ML and in opposition to System F, each typable expression admits a principal type, which can be inferred. Furthermore, all expressions of ML are welltyped, wi ..."
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Cited by 10 (0 self)
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We propose a type system MLF that generalizes ML with firstclass polymorphism as in System F. We perform partial type reconstruction. As in ML and in opposition to System F, each typable expression admits a principal type, which can be inferred. Furthermore, all expressions of ML are welltyped, with a possibly more general type than in ML, without any need for type annotation. Only arguments of functions that are used polymorphically must be annotated, which allows to type all expressions of System F as well.
A Formalization of the Strong Normalization Proof for System F in LEGO
, 1993
"... We describe a complete formalization of a strong normalization proof for the Curry style presentation of System F in LEGO. The underlying type theory is the Calculus of Constructions enriched by inductive types. The proof follows Girard et al [GLT89], i.e. we use the notion of candidates of reducibi ..."
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Cited by 7 (0 self)
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We describe a complete formalization of a strong normalization proof for the Curry style presentation of System F in LEGO. The underlying type theory is the Calculus of Constructions enriched by inductive types. The proof follows Girard et al [GLT89], i.e. we use the notion of candidates of reducibility, but we make essential use of general inductive types to simplify the presentation. We discuss extensions and variations of the proof: the extraction of a normalization function, the use of saturated sets instead of candidates, and the extension to a Church Style presentation. We conclude with some general observations about Computer Aided Formal Reasoning.
Representing Proof Transformations for Program Optimization
 IN PROCEEDINGS OF THE 12TH INTERNATIONAL CONFERENCE ON AUTOMATED DEDUCTION
, 1994
"... In the proofs as programs methodology a program is derived from a formal constructive proof. Because of the close relationship between proof and program structure, transformations can be applied to proofs rather than to programs in order to improve performance. We describe a method for implementing ..."
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Cited by 6 (1 self)
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In the proofs as programs methodology a program is derived from a formal constructive proof. Because of the close relationship between proof and program structure, transformations can be applied to proofs rather than to programs in order to improve performance. We describe a method for implementing transformations of formal proofs and show that it is applicable to the optimization of extracted programs. The method is based on the representation of derived logical rules in Elf, a logic programming language that gives an operational interpretation to the Edinburgh Logical Framework. It results in declarative implementations with a general correctness property that is verified automatically by the Elf type checking algorithm. We illustrate the technique by applying it to the problem of transforming a recursive function definition to obtain a tailrecursive form.
Inferable ObjectOriented Typed Assembly Language
"... A certifying compiler preserves type information through compilation to assembly language programs, producing typed assembly language (TAL) programs that can be verified for safety independently so that the compiler does not need to be trusted. There are two challenges for adopting certifying compil ..."
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Cited by 4 (2 self)
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A certifying compiler preserves type information through compilation to assembly language programs, producing typed assembly language (TAL) programs that can be verified for safety independently so that the compiler does not need to be trusted. There are two challenges for adopting certifying compilation in practice. First, requiring every compiler transformation and optimization to preserve types is a large burden on compilers, especially when adopting certifying compilation into existing optimizing noncertifying compilers. Second, type annotations significantly increase the size of assembly language programs. This paper proposes an alternative to traditional certifying compilers. It presents iTalX, the first inferable TAL type system that supports existential types, arrays, interfaces, and stacks. We have proved our inference algorithm is complete, meaning if an assembly
Program Extraction in a Logical Framework Setting
 IN PROCEEDINGS OF THE 5TH INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING AND AUTOMATED REASONING
, 1994
"... This paper demonstrates a method of extracting programs from formal deductions represented in the Edinburgh Logical Framework, using the Elf programming language. Deductive systems are given for the extraction of simple types from formulas of firstorder arithmetic and of calculus terms from n ..."
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Cited by 3 (1 self)
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This paper demonstrates a method of extracting programs from formal deductions represented in the Edinburgh Logical Framework, using the Elf programming language. Deductive systems are given for the extraction of simple types from formulas of firstorder arithmetic and of calculus terms from natural deduction proofs. These systems are easily encoded in Elf, yielding an implementation of extraction that corresponds to modified realizability. Because extraction is itself implemented as a set of formal deductive systems, some of its correctness properties can be partially represented and mechanically checked in the Elf language.
EXISTENTIAL TYPE SYSTEMS BETWEEN CHURCH AND CURRY STYLE
"... Abstract. We study type checking, typability, and type inference problems for typefree style and Curry style secondorder existential systems where the typefree style differs from the Curry style in that the terms of the former contain information on where the existential quantifier elimination an ..."
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Abstract. We study type checking, typability, and type inference problems for typefree style and Curry style secondorder existential systems where the typefree style differs from the Curry style in that the terms of the former contain information on where the existential quantifier elimination and introduction take place but omit the information on which types are involved. We show that all the problems are undecidable employing reduction of secondorder unification in case of the typefree system and semiunification in case of the Curry style system. This provides a fine border between problems yielding to a reduction of secondorder unification problem and the semiunification problem. In addition, we investigate the subject reduction property of the system in the Currystyle. 5 10
Theoretical Foundations for Practical ‘Totally Functional Programming’
, 2007
"... Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt ..."
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Interpretation is an implicit part of today’s programming; it has great power but is overused and has
significant costs. For example, interpreters are typically significantly hard to understand and hard
to reason about. The methodology of “Totally Functional Programming” (TFP) is a reasoned
attempt to redress the problem of interpretation. It incorporates an awareness of the undesirability
of interpretation with observations that definitions and a certain style of programming appear to
offer alternatives to it. Application of TFP is expected to lead to a number of significant outcomes,
theoretical as well as practical. Primary among these are novel programming languages to lessen or
eliminate the use of interpretation in programming, leading to betterquality software. However,
TFP contains a number of lacunae in its current formulation, which hinder development of these
outcomes. Among others, formal semantics and typesystems for TFP languages are yet to be
discovered, the means to reduce interpretation in programs is to be determined, and a detailed
explication is needed of interpretation, definition, and the differences between the two. Most
important of all however is the need to develop a complete understanding of the nature of
interpretation. In this work, suitable typesystems for TFP languages are identified, and guidance
given regarding the construction of appropriate formal semantics. Techniques, based around the
‘fold’ operator, are identified and developed for modifying programs so as to reduce the amount of
interpretation they contain. Interpretation as a means of languageextension is also investigated.
v
Finally, the nature of interpretation is considered. Numerous hypotheses relating to it considered in
detail. Combining the results of those analyses with discoveries from elsewhere in this work leads
to the proposal that interpretation is not, in fact, symbolbased computation, but is in fact something
more fundamental: computation that varies with input. We discuss in detail various implications of
this characterisation, including its practical application. An often moreuseful property, ‘inherent
interpretiveness’, is also motivated and discussed in depth. Overall, our inquiries act to give
conceptual and theoretical foundations for practical TFP.
THE UNDECIDABILITY OF TYPE RELATED PROBLEMS IN TYPEFREE STYLE SYSTEM F
"... Abstract. We consider here a number of variations on the System F, that are predicative secondorder systems whose terms are intermediate between the Curry style and Church style. The terms here contain the information on where the universal quantifier elimination and introduction in the type infere ..."
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Abstract. We consider here a number of variations on the System F, that are predicative secondorder systems whose terms are intermediate between the Curry style and Church style. The terms here contain the information on where the universal quantifier elimination and introduction in the type inference process must take place, which is similar to Church forms. However, they omit the information on which types are involved in the rules, which is similar to Curry forms. In this paper we prove the undecidability of the typechecking, type inference and typability problems for the system. Moreover, the proof works for the predicative version of the system with finitely stratified polymorphic types. The result includes the bounds on the Leivant’s level numbers for types used in the instances leading to the undecidability. 1.