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Comparative Analysis of Hypercomputational Systems Submitted in partial fulfilment
"... In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosoph ..."
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In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosophers and scientists are suggesting models of hypothetical computing devices which can outperform the Turing machine, performing some calculations the latter is unable to. The ChurchTuring Thesis, which the Turing machine model embodies, has raised discussions on whether it could be possible to solve undecidable problems which Turing’s model is unable to. Models which could solve these problems, have gone further to claim abilities relating to quantum computing, relativity theory, even the modeling of natural biological laws themselves. These so called ‘hypermachines ’ use hypercomputational abilities to make the impossible possible. Various models belonging to different disciplines of physics, mathematics and philosophy, have been suggested for these theories. My (primarily researchoriented) project is based on the study and review of these different hypercomputational models and attempts to compare the different models in terms of computational power. The project focuses on the ability to compare these models of different disciplines on similar grounds and
Kleene Automata And Recursion Theory
, 1992
"... . Following the introduction of a theoretical computational model on infinite objects, the !input Turing machine, we present a new type of infinite automata, the Kleene automata. We show it recognizes exactly the class of arithmetical !languages. Essentially, it is a propositional automaton for w ..."
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. Following the introduction of a theoretical computational model on infinite objects, the !input Turing machine, we present a new type of infinite automata, the Kleene automata. We show it recognizes exactly the class of arithmetical !languages. Essentially, it is a propositional automaton for which the transition relation is recursive and the interpretation of atomic formulas associated with each state is recursive. The acceptance conditions are built up hierarchically by adding to each level, the recursive disjunctions of negations of the previous level's formulas. The first level is a propositional temporal logic restricted to the only one temporal operator next. We show the expressive power of this logic to be the class of recursive !languages. 1. Background and introduction. Concurrent systems are often modeled as having infinite behavior and their specification or verification should be based on a theory of computation for infinite objects. Vardi [Var87] proposed an infinita...
A Note on First Order Unification
, 2000
"... this article a language L for the predicate calculus whose only predicate symbol = is the 2ary one for the equality and whose function symbols are a 2ary one ap for application together with a set of 0ary symbols divided into ground symbols and unknowns. The unknowns correspond to the function le ..."
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this article a language L for the predicate calculus whose only predicate symbol = is the 2ary one for the equality and whose function symbols are a 2ary one ap for application together with a set of 0ary symbols divided into ground symbols and unknowns. The unknowns correspond to the function letters of [2], page 263, while the ground symbols to its function symbols. We remark: unknowns are function symbols of the language L that should not be confused with variables of L. Usually, we use capital letters X , Y , Z, perhaps indexed, for denoting unknowns; the letters u, v, w for variables; other small letters and numbers for ground symbols.
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
INTERMEDIATE LOGICS AND THE DE JONGH PROPERTY
, 2009
"... Abstract. We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property. Dedicated to Petr Hájek, on the occasion of his 70th Birthday 1. Preface The three authors of this paper have enjoyed Petr Hájek’s acquaintance since the late e ..."
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Abstract. We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property. Dedicated to Petr Hájek, on the occasion of his 70th Birthday 1. Preface The three authors of this paper have enjoyed Petr Hájek’s acquaintance since the late eighties, when a lively community interested in the metamathematics of arithmetic shared ideas and traveled among the beautiful cities of Prague, Moscow, Amsterdam, Utrecht, Siena, Oxford and Manchester. At that time, Petr Hájek and Pavel Pudlák were writing their landmark book Metamathematics of FirstOrder Arithmetic [HP91], which Petr Hájek tried out on a small group of eager graduate students in Siena in the months of February and March 1989. Since then, Petr Hájek has been a role model to us in many ways. First of all, we have always been impressed by Petr’s meticulous and clear use of correct notation, witness all his different types of dots and corners, for example in the Tarskian ‘snowing’snowing lemmas [HP91]. But also as a human being, Petr has been a role model by his example of living in truth, even in averse circumstances [Hav89]. The tragic story of the Logic Colloquium 1980, which was planned to be held in Prague and of which Petr Hájek was the driving force, springs to mind [DvDLS82]. Finally, we were moved by Petr’s openmindedness when coming to terms with a situation that turned out to look disconcertingly unlike the ‘standard model ’ 1. Therefore, in this paper, we would like to pay homage to Petr Hájek. Unfortunately, we cannot hope to emulate his correct use of dots and corners. Instead, we do our best to provide some pleasing nonstandard models and nonclassical arithmetics. 2.
Can't decide? Undecide!
"... In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs ..."
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In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs into the formalisms, this area can seem remote from most areas of mathematics and irrelevant to the efforts of most workaday mathematicians. But that’s just not so! Undecidable problems surround us, everywhere, even in recreational mathematics!
Who Can Name the Bigger Number?
, 1999
"... In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces ”Eightythree! ” The second, mightily impressed, replies ”You win.” A biggest number contest is clearly pointless when the contestants take turns. But what if the contestants wri ..."
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In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces ”Eightythree! ” The second, mightily impressed, replies ”You win.” A biggest number contest is clearly pointless when the contestants take turns. But what if the contestants write down their numbers simultaneously, neither aware of the other’s? To introduce a talk on ”Big Numbers, ” I invite two audience volunteers to try exactly this. I tell them the rules: You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number–not an infinity–on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature. So contestants can’t say ”the number of sand grains in the Sahara, ” because sand drifts in and out of the Sahara regularly. Nor can they say ”my opponent’s number plus one, ” or ”the biggest number anyone’s ever thought of plus one”–again, these are illdefined, given what our reasonable mathematician has available. Within the rules, the contestant who names the bigger number wins. Are you ready? Get set. Go. The contest’s results are never quite what I’d hope. Once, a seventhgrade boy filled his card with a string of successive 9’s. Like many other bignumber tyros, he sought to maximize his number by stuffing a 9 into every place value. Had he chosen easytowrite 1’s rather than curvaceous 9’s, his number could have been millions of times bigger. He still would been decimated, though, by the girl he was up against, who wrote a string of 9’s followed by the superscript 999. Aha! An exponential: a number multiplied by itself 999 times. Noticing this innovation, I declared the girl’s victory without bothering to count the 9’s on the cards. And yet the girl’s number could have been much bigger still, had she stacked the mighty exponential more than once. Take 999, for example. This behemoth, equal to 9387,420,489, has 369,693,100 digits. By comparison, the number of elementary particles in the observable universe has a meager 85 digits, give or take. Three 9’s, when stacked exponentially, already lift us incomprehensibly ∗ Revised by Florian Mayer 1 beyond all the matter we can observe–by a factor of about 10369,693,015. And we’ve said nothing of 9 999
TuringCompleteness of Polymorphic Stream Equation Systems
"... Polymorphic stream functions operate on the structure of streams, infinite sequences of elements, without inspection of the contained data, having to work on all streams over all signatures uniformly. A natural, yet restrictive class of polymorphic stream functions comprises those definable by a sys ..."
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Polymorphic stream functions operate on the structure of streams, infinite sequences of elements, without inspection of the contained data, having to work on all streams over all signatures uniformly. A natural, yet restrictive class of polymorphic stream functions comprises those definable by a system of equations using only stream constructors and destructors and recursive calls. Using methods reminiscent of prior results in the field, we first show this class consists of exactly the computable polymorphic stream functions. Using much more intricate techniques, our main result states this holds true even for unary equations free of mutual recursion, yielding an elegant model of Turingcompleteness in a severely restricted environment and allowing us to recover previous complexity results in a much more restricted setting.