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Axiomatizing the Quote
"... We study reflection in the Lambda Calculus from an axiomatic point of view. Specifically, we consider various properties that the quote �· � must satisfy as a function from Λ to Λ. The most important of these is the existence of a definable left inverse: a term E, called the evaluator for �·�, that ..."
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We study reflection in the Lambda Calculus from an axiomatic point of view. Specifically, we consider various properties that the quote �· � must satisfy as a function from Λ to Λ. The most important of these is the existence of a definable left inverse: a term E, called the evaluator for �·�, that satisfies E�M � = M for all M ∈ Λ. Usually the quote �M � encodes the syntax of a given term, and the evaluator proceeds by analyzing the syntax and reifying all constructors by their actual meaning in the calculus. Working in Combinatory Logic, Raymond Smullyan [12] investigated which elements of the syntax must be accessible via the quote in order for an evaluator to exist. He asked three specific questions, to which we provide negative answers. On the positive side, we give a characterization of quotes which possess all of the desired properties, equivalently defined as being equitranslatable with a standard quote. As an application, we show that Scott’s coding is not complete in this sense, but can be slightly modified to be such. This results in a minimal definition of a complete quoting for Combinatory Logic.
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.
Global semantic typing for inductive and coinductive computing
 Logical Methods in Computer Science
"... ..."
The ChurchTuring Thesis as an Immature Form of the ZuseFredkin Thesis (More Arguments in Support of the “Universe as a Cellular Automaton” Idea)
"... In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational ..."
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In [1] we have shown a strong argument in support of the &quot;Universe as a computer &quot; idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only &quot;some kind of computer&quot;, but also a concrete computational model known as a &quot;cellular automaton&quot;.
A Staging Calculus and its Application to the Verification of Translators
, 1993
"... We develop a calculus in which the computation steps required to execute a computer program can be separated into discrete stages. The calculus, denoted 2 , is embedded within the pure untyped calculus. The main result of the paper is a characterization of sucient conditions for conuence for terms ..."
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We develop a calculus in which the computation steps required to execute a computer program can be separated into discrete stages. The calculus, denoted 2 , is embedded within the pure untyped calculus. The main result of the paper is a characterization of sucient conditions for conuence for terms in the calculus. The condition can be taken as a correctness criterion for translators that perform reductions in one stage leaving residual redexes over for subsequent computation stages. As an application of the theory, we verify the correctness of a macro expansion algorithm. The expansion algorithm is of some interest in its own right since it solves the problem of desired variable capture using only the familiar capture avoiding substitutions. 1 Introduction The calculus is widely used as a metalanguage for programming language semantics because most of the complexities of real programming languages can be modeled by relatively simple and wellunderstood aspects of the calculus. ...
A Rulebased Approach to the Implementation of Evaluation Strategies Mircea Marin
 PETCU D., NEGRU V., ZAHARIE D., JEBELEAN T., Eds., Proceedings of SYNASC 2004
"... We describe a new methodology to program evaluation strategies, which relies on some advanced features of the rulebased programming language #Log. We illustrate how our approach works for a number of important evaluation strategies. 1 ..."
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We describe a new methodology to program evaluation strategies, which relies on some advanced features of the rulebased programming language #Log. We illustrate how our approach works for a number of important evaluation strategies. 1
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
 J. Complexity
"... 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. ..."
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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 an according list for f −1 [V]. Analogously, effective openness requires the mapping U↦ → f[U] on open real subsets to be effective. The present work reveals important and rich classes of functions to be effectively open and thus applicable in the foundations of solid modeling to computations on regular sets. We also address the general relation between effective openness and computability of functions. 1
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
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