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Constraints on Hypercomputation, in
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
"... “To infinity, and beyond!”, Buzz Lightyear, Toy Story, Pixar, 1995. Many attempts to transcend the fundamental limitations to computability implied by the Halting Problem for Turing Machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous, as opposed to ..."
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“To infinity, and beyond!”, Buzz Lightyear, Toy Story, Pixar, 1995. Many attempts to transcend the fundamental limitations to computability implied by the Halting Problem for Turing Machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous, as opposed to bounded or discrete, computation. Thus, such schemes may include the deployment of actualised rather than potential infinities of physical resources, or of physical representations of real numbers to arbitrary precision. Here, we argue that such bases for hypercomputation are not materially realisable and so cannot constitute new forms of effective calculability. 1
Universal Computation with WatsonCrick D0L Systems
, 2000
"... WatsonCrick D0L systems, introduced in 1997 by V. Mihalache and A. Salomaa, arise from two major principles: the Lindenmayer rewriting and the WatsonCrick complementarity principle. Complementarity can be viewed as a purely languagetheoretic operation. Majority of a certain type of symbols in a s ..."
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WatsonCrick D0L systems, introduced in 1997 by V. Mihalache and A. Salomaa, arise from two major principles: the Lindenmayer rewriting and the WatsonCrick complementarity principle. Complementarity can be viewed as a purely languagetheoretic operation. Majority of a certain type of symbols in a string (purines vs. pyrimidines) triggers a transition to the complementary string. The paper deals with an expressive power of deterministic interactionless WatsonCrick Lindenmayer systems. A rather surprising result is obtained: these systems, consisting of iterated morphism and a basic DNA operation, are alone able to express any Turing computable function. 1
Computability in an Introductory Course on Programming
 Bulletin of the European Association for Theoretical Computer Science 73 (2001
"... The programming approach to computability presented in the textbook by Kfoury, Moll, and Arbib in 1982 has been embedded into a programming course following the textbook by Abelson and Sussman. This leads to a course concept teaching good programming practice and clear theoretical concepts simult ..."
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The programming approach to computability presented in the textbook by Kfoury, Moll, and Arbib in 1982 has been embedded into a programming course following the textbook by Abelson and Sussman. This leads to a course concept teaching good programming practice and clear theoretical concepts simultaneously. Here, we explain some of the main points of this approach: the halting problem, primitive and recursive functions and the operational counterpart of these functions, i.e., the Loop and the While programs. 1
Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
Length Of Polynomial Ascending Chains And Primitive Recursiveness
, 1992
"... In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \Delta \Delta \Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s ..."
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In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \Delta \Delta \Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s of such a chain depending only on n and f , which is recursive in f for every n and primitive recursive in f for n = 2. In this paper we give a better bound, expressed in a rather simple way in terms of f , which is attained when f is an increasing function. We prove that it is primitive recursive in f for all n. We also show that, on the contrary, there is no bound which is primitive recursive in n in general.
On Properties of WatsonCrick D0L Systems
"... WatsonCrick D0L systems, introduced in 1998 by Arto Salomaa, arise from two major principles: the Lindenmayer rewriting and the WatsonCrick complementarity principle. Complementarity can be viewed as a purely languagetheoretic operation. Majority of a certain type of symbols in a string (purines v ..."
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WatsonCrick D0L systems, introduced in 1998 by Arto Salomaa, arise from two major principles: the Lindenmayer rewriting and the WatsonCrick complementarity principle. Complementarity can be viewed as a purely languagetheoretic operation. Majority of a certain type of symbols in a string (purines vs. pyrimidines) triggers a transition to the complementary string. This paper deals with an expressive power of deterministic interactionless WatsonCrick Lindenmayer systems. A surprising result is obtained: these systems, consisting of iterated morphism and a basic DNA operation, are alone able to express any Turing computable function. 1 WatsonCrick D0L systems For elements of formal language theory we refer to [6, 7]. Here we only briey x some notation. Let (; ) be a free monoid with the catenation operation and let the empty word be denoted : We denote jwj a the number of occurrences of a symbol a in a string w for a 2 ; w 2 : Further we denote jwj = P a2 jwj a for an a...
Generality’s price: Inescapable deficiencies in machinelearned programs
 Annals of Pure and Applied Logic
, 2006
"... This paper investigates some delicate tradeoffs between the generality of an algorithmic learning device and the quality of the programs it learns successfully. There are results to the effect that, thanks to small increases in generality of a learning device, the computational complexity of some su ..."
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This paper investigates some delicate tradeoffs between the generality of an algorithmic learning device and the quality of the programs it learns successfully. There are results to the effect that, thanks to small increases in generality of a learning device, the computational complexity of some successfully learned programs is provably unalterably suboptimal. There are also results in which the complexity of successfully learned programs is asymptotically optimal and the learning device is general, but, still thanks to the generality, some of those optimal, learned programs are provably unalterably information deficient — in some cases, deficient as to safe, Preprint submitted to Elsevier Science 11 March 2007 algorithmic extractability/provability of the fact that they are even approximately optimal. For these results, the safe, algorithmic methods of information extraction will be by proofs in arbitrary, true, computably axiomatizable extensions of Peano Arithmetic. Key words: Computational learning theory; Applications of computability theory
The WHILE Hierarchy of Program Schemes is Infinite
, 1998
"... . We exhibit a sequence Sn (n # 0) of while program schemes, i. e., while programs without interpretation, with the property that the while nesting depth of Sn is n, and prove that any while program scheme which is scheme equivalent to Sn , i. e., equivalent for all interpretations over arbitrary ..."
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. We exhibit a sequence Sn (n # 0) of while program schemes, i. e., while programs without interpretation, with the property that the while nesting depth of Sn is n, and prove that any while program scheme which is scheme equivalent to Sn , i. e., equivalent for all interpretations over arbitrary domains, has while nesting depth at least n. This shows that the while nesting depth imposes a strict hierarchy (the while hierarchy) when programs are compared with respect to scheme equivalence and contrasts with Kleene's classical result that every program is equivalent to a program of while nesting depth 1 (when interpreted over a fixed domain with arithmetic on nonnegative integers). Our proof is based on results from formal language theory; in particular, we make use of the notion of star height of regular languages. 1 Introduction When comparing programming languages, one often has a vague impression of one language being more powerful than another. However, a basic result of the ...
Computational Aspects of FirstOrder Logic on Finite Structures
, 1999
"... Descriptive complexity aims to classify properties of finite structures according to the logical resources that are needed to express them. The ImmermanVardi Theorem states that a property of finite ordered structures is computable in polynomialtime if and only if it can be expressed in least fixe ..."
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Descriptive complexity aims to classify properties of finite structures according to the logical resources that are needed to express them. The ImmermanVardi Theorem states that a property of finite ordered structures is computable in polynomialtime if and only if it can be expressed in least fixedpoint logic. Here, least fixedpoint logic is the extension of firstorder logic with the ability to buildin inductive definitions. It is known that firstorder logic is a fairly weak expressive language in the context of descriptive complexity. For example, least fixedpoint logic is strictly more expressive than firstorder logic on the class of all finite ordered structures. The ordered conjecture, formulated by Kolaitis and Vardi in 1992, asks whether this is also the case for any infinite class of finite ordered structures. Although some significant progress has been made since its formulation, the ordered conjecture remains open. In fact, it has been shown that any way of resolving ...
SCHEMATA: THE CONCEPT OF SCHEMA IN THE HISTORY OF LOGIC
"... Abstract. Schemata have played important roles in logic since Aristotle’s Prior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 19 ..."
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Abstract. Schemata have played important roles in logic since Aristotle’s Prior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in firstorder number theory where Peano’s secondorder Induction Axiom is approximated by Herbrand’s InductionAxiom Schema [23]. Similarly, in firstorder set theory, Zermelo’s secondorder Separation Axiom is approximated by Fraenkel’s firstorder Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a templatetext or schemetemplate, a syntactic string composed of one or more “blanks ” and also possibly significant words and/or symbols. In accordance with a side condition the templatetext of a schema is used as a “template ” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argumenttexts, called instances of the schema. The side condition is a second component. The collection of instances may