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First Steps Into Metapredicativity in Explicit Mathematics
, 1999
"... The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarc ..."
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The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarchies is bounded by # 0 . 1 Introduction Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose prooftheoretic strength is beyond the FefermanSchutte ordinal # 0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories # ID # whose detailed prooftheoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman's ATR that can be measured against transfinitely iterated fixed point ...
Design Choices in Specification Languages and Verification Systems
, 1991
"... We describe some of the design choices that should be considered in the development and application of specification languages and verification systems. A principal issue is the need to reconcile the desire for expressiveness in the specification language with the ability to provide effective mechan ..."
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We describe some of the design choices that should be considered in the development and application of specification languages and verification systems. A principal issue is the need to reconcile the desire for expressiveness in the specification language with the ability to provide effective mechanical support. We argue that this reconciliation is assisted by a novel approach to specification language design that requires theorem proving to be used during typechecking.
On the Proof Theory of Applicative Theories
 PHD THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT
, 1996
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Formalizing NonTermination of Recursive Programs
 J. of Logic and Algebraic Programming
, 2001
"... In applicative theories the recursion theorem provides a term rec which solves recursive equations. However, it is not provable that a solution obtained by rec is minimal. In the present paper we introduce an applicative theory in which it is possible to dene a least xed point operator. Still, o ..."
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In applicative theories the recursion theorem provides a term rec which solves recursive equations. However, it is not provable that a solution obtained by rec is minimal. In the present paper we introduce an applicative theory in which it is possible to dene a least xed point operator. Still, our theory has a standard recursion theoretic interpretation. 1
Impredicative Overloading in Explicit Mathematics
, 2000
"... In this article we introduce the system OTN of explicit mathematics based on elementary separation, product, join and weak power types. We present a settheoretical model for OTN, and we develop in OTN a theory of impredicative overloading. Together this yields a solution to the problem of impredica ..."
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In this article we introduce the system OTN of explicit mathematics based on elementary separation, product, join and weak power types. We present a settheoretical model for OTN, and we develop in OTN a theory of impredicative overloading. Together this yields a solution to the problem of impredicativity encountered in denotational semantics for overloading and latebinding. Further, our work provides a first example of an application of power types in explicit mathematics. Keywords: Objectoriented constructs, type structure, proof theory. 1 Introduction Overloading is an important concept in objectoriented programming. For example, it occurs when a method is redefined in a subclass or when a class provides several methods with the same name but with di#erent argument types. Theoretically speaking, overloading denotes the possibility that several functions f i with respective types S i # T i may be combined to a new overloaded function f of type {S i # T i } i#I . We then ...
Formal Specification and Verification for Critical Systems: Tools, Achievements, and Prospects
"... Abstract Formal specification and verification use mathematical techniques to help document, specify, design, analyze, or certify computer software and hardware. Mathematicallybased notation can provide specifications that are precise and unambiguous and that can be checked mechanically for certain ..."
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Abstract Formal specification and verification use mathematical techniques to help document, specify, design, analyze, or certify computer software and hardware. Mathematicallybased notation can provide specifications that are precise and unambiguous and that can be checked mechanically for certain types of error. Formal verification uses theorem proving techniques to establish consistency between one level of formal specification and another. This paper describes some of the issues in the design and use of formal specification languages and verification systems, outlines some examples of the application of formal methods to critical systems, and identifies the benefits that may be obtained from this technology.
A Semantics for ...: A Calculus With Overloading and LateBinding
, 2000
"... Up to now there was no interpretation available for #calculi featuring overloading and latebinding, although these are two of the main principles of any objectoriented programming language. In this paper we provide a new semantics for a stratified version of Castagna's # {} , a #calculus co ..."
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Up to now there was no interpretation available for #calculi featuring overloading and latebinding, although these are two of the main principles of any objectoriented programming language. In this paper we provide a new semantics for a stratified version of Castagna's # {} , a #calculus combining overloading with latebinding. The modelconstruction is carried out in EETJ + (Tot) + (FI N ), a system of explicit mathematics. We will prove the soundness of our model with respect to subtyping, typechecking and reductions. Furthermore, we show that our semantics yields a solution to the problem of loss of information in the context of type dependent computations.
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"... Traditionally the view has been that direct expression of control and store mechanisms and clear mathematical semantics are incompatible requirements. This paper shows that adding objects with memory to the callbyvalue lambda calculus results in a language with a rich equational theory, satisfying ..."
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Traditionally the view has been that direct expression of control and store mechanisms and clear mathematical semantics are incompatible requirements. This paper shows that adding objects with memory to the callbyvalue lambda calculus results in a language with a rich equational theory, satisfying many of the usual laws. Combined with other recent work this provides evidence that expressive, mathematically clean programming languages are indeed possible. 1. Overview Real programs have effectscreating new structures, examining and modifying existing structures, altering flow of control, etc. Such facilities are important not only for optimization, but also for communication, clarity, and simplicity in programming. Thus it is important to be able to reason both informally and formally about programs with effects, and not to sweep effects either to the side or under the store parameter rug. Recent work of Talcott, Mason, Felleisen, and Moggi establishes a mathematical foundation for...