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A Theory of Explicit Mathematics Equivalent to ID_1
"... We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_1. ..."
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We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_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 ...
Weak theories of truth and explicit mathematics. Submitted for publication. 19
"... We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able t ..."
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We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able to interpret expressive feasible subsystems of explicit mathematics. 1
Axioms for Strict and Lazy Functional Programs
 Annals of Pure and Applied Logic
"... We show the adequacy of axioms and proof rules for strict and lazy functional programs. Our basic logic comprises a huge part of what is common to both styles of functional programming. The logic for callby value is obtained by adding the axiom that says that all variables are defined, whereas ..."
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We show the adequacy of axioms and proof rules for strict and lazy functional programs. Our basic logic comprises a huge part of what is common to both styles of functional programming. The logic for callby value is obtained by adding the axiom that says that all variables are defined, whereas the logic for callbyname is obtained by adding the axiom that postulates the existence of undefined object for each type. To show the correctness of the axiomatization we do not use denotational semantics and the adequacy of the evaluation of programs with respect to the semantics. Instead we use the standard term models based on callbyvalue and callbyname evaluation. We introduce a new method to prove on the syntactical level the monotonicity of the evaluation of functional programs with unbounded recursion. The direct method yields result about the prooftheoretic strength of the axiomatization. As a side result we obtain a syntactical proof of the context lemma for simply typed lambda terms with recursion.
On the Proof Theory of Applicative Theories
 PHD THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT
, 1996
"... ..."
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
A Semantics for ...: A Calculus With Overloading and LateBinding
, 1999
"... 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 stratied version of Castagna's fg , a calculus combini ..."
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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 stratied version of Castagna's fg , 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. Keywords: Explicit mathematics, typed calculus, overloading, latebinding, loss of information. 1 Introduction Polymorphism is one of the concepts to which the objectoriented paradigm owes its power. The distinction is made between parametric (or universal) and \ad hoc" polymorphism. Using parametric polymorp...
ProofTheoretic Notions for Software Maintenance
, 2000
"... In this report we give an outline how prooftheoretic notions can be useful for questions related to software maintenance. ..."
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In this report we give an outline how prooftheoretic notions can be useful for questions related to software maintenance.