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A Hypercomputational Alien
, 2005
"... Is there a physical constant with the value of the halting function? An answer to this question, as holds true for other discussions of hypercomputation, assumes a fixed interpretation of nature by mathematical entities. We discuss the subjectiveness of viewing the mathematical properties of nature, ..."
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Is there a physical constant with the value of the halting function? An answer to this question, as holds true for other discussions of hypercomputation, assumes a fixed interpretation of nature by mathematical entities. We discuss the subjectiveness of viewing the mathematical properties of nature, and the possibility of comparing computational models having different views of the world. For that purpose, we propose a conceptual framework for power comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing for the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful ” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete ” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is more powerful than Turing machines.
Effectively Open Real Functions
, 2005
"... Abstract. A function f is continuous iff the preimage f −1 [V] of any open set V is open again. Dual to this topological property, f is called open iff the image f[U] of any open set U is open again. Several classical Open Mapping Theorems in Analysis provide a variety of sufficient conditions for ..."
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Abstract. A function f is continuous iff the preimage f −1 [V] of any open set V is open again. Dual to this topological property, f is called open iff the image f[U] of any open set U is open again. Several classical Open Mapping Theorems in Analysis provide a variety of sufficient conditions for openness. By the Main Theorem of Recursive Analysis, computable real functions are necessarily continuous. In fact they admit a wellknown characterization in terms of the mapping V ↦ → f −1 [V] being effective: Given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f −1 [V]. Analogously, effective openness requires the mapping U ↦ → f[U] on open real subsets to be effective. By effectivizing classical Open Mapping Theorems as well as from application of Tarski’s Quantifier Elimination, the present work reveals several rich classes of functions to be effectively open. 1
Theory for Software Verification
, 2009
"... Semantic models are the basis for specification and verification of software. Operational, denotational, and axiomatic or algebraic methods offer complementary insights and reasoning techniques which are surveyed here. Unifying theories are needed to link models. Also considered are selected program ..."
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Semantic models are the basis for specification and verification of software. Operational, denotational, and axiomatic or algebraic methods offer complementary insights and reasoning techniques which are surveyed here. Unifying theories are needed to link models. Also considered are selected programming features for which new models are needed.
Theory and Implementation of a Functional Programming Language
, 2000
"... The goal of this research is to design and implement a small functional programming language that incorporates some of the features that arise from the theoretical study of programming language semantics. We begin with the study of the λcalculus, an idealized mathematical language. We present the l ..."
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The goal of this research is to design and implement a small functional programming language that incorporates some of the features that arise from the theoretical study of programming language semantics. We begin with the study of the λcalculus, an idealized mathematical language. We present the language PPL, a strongly typed, callbyname language which supports recursion and polymorphism. We describe a compiler of PPL into a lowlevel assembly language. The compilation is based on an abstract machine interpretation and includes a type inference algorithm. 1
The NASA STI Program Office provides
, 2000
"... Since its founding, NASA has been dedicated to the advancement of aeronautics and space ..."
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Since its founding, NASA has been dedicated to the advancement of aeronautics and space
LambdaCalculus and Functional Programming
"... This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, bo ..."
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This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, booleans, and strings. It is clearly desirable to have a method of writing a piece of code that can accept the specific type as an argument. Milner developed his ideas in terms of type assignment to lambdaterms. It is based on a result due originally to Curry (Curry 1969) and Hindley (Hindley 1969) known as the principal typescheme theorem, which says that (assuming that the typing assumptions are sufficiently wellbehaved) every term has a principal typescheme, which is a typescheme such that every other typescheme which can be proved for the given term is obtained by a substitution of types for type variables. This use of type schemes allows a kind of generality over all types, which is known as polymorphism.
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
Types in Programming Languages
"... Studies about types have influenced, in a significant way, the design and definition of programming languages. This survey presents an introductory overview of concepts related to types and type systems for modern programming languages. We introduce by identifying why types are useful, and go on ..."
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Studies about types have influenced, in a significant way, the design and definition of programming languages. This survey presents an introductory overview of concepts related to types and type systems for modern programming languages. We introduce by identifying why types are useful, and go on to discuss the formalization of the syntax of programming languages by pointing out which properties should type systems satisfy, in particular with respect to denotational and operational semantic definitions. We provide an overview of simple type systems, polymorphic type systems, type inference, constrained polymorphism, subtyping and abstract types.
Church’s Thesis
"... In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively ..."
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In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively calculable functions, and the ideas behind Alonzo Church’s (1903–1995) proposal to identify the
On Fixed point and Looping Combinators in Type Theory
"... Abstract. The type theories λU and λU − are known to be logically inconsistent. For λU, this is known as Girard’s paradox [Gir72]; for λU − the inconsistency was proved by Coquand [Coq94]. It is also known that the inconsistency gives rise to a so called ”looping combinator”: a family of terms Ln su ..."
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Abstract. The type theories λU and λU − are known to be logically inconsistent. For λU, this is known as Girard’s paradox [Gir72]; for λU − the inconsistency was proved by Coquand [Coq94]. It is also known that the inconsistency gives rise to a so called ”looping combinator”: a family of terms Ln such that Lnf is convertible with f(Ln+1f). It was unclear whether a fixed point combinator exists in these systems. Later, Hurkens [Hur95] has given a simpler version of the paradox in λU − , giving rise to an actual proof term that can be analyzed. In the present paper we analyze the proof of Hurkens and we study the looping combinator that arises from it: it is a real looping combinator (not a fixed point combinator) but in the Curry version of λU − it is a fixedpoint combinator. We also analyze the possibility of typing a fixed point combinator in λU − and we prove that the Church and Turing fixed point combinators cannot be typed in λU −. 1