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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally e ..."
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Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
In Some Curved Spaces, One Can Solve NPHard Problems in Polynomial Time
"... In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved s ..."
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Cited by 6 (6 self)
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In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. 1 Introduction and Formulation of the Problem Many practical problems are NPhard. It is well known that many important practical problems are NPhard; see, e.g., [11, 14, 27]. Under the usual hypothesis that P̸=NP, NPhardness has the following intuitive meaning: every algorithm which solves all instances of the corresponding problem requires, for
An ASMcharacterization of a class of distributed algorithms, Proceedings of the Dagstuhl seminar on rigorous methods for software construction and analysis
 May 2006, available at http://www2.informatik.huberlin.de/top/download/publications/ GlauschR2007 dagstuhl.pdf (viewed Mar. 26, 2008). Longer version: Distributed abstract state machines and their expressive power, InformatikBerichte 196, HumboldtUnive
, 2006
"... Abstract. Conventional computation models restrict to particular data structures to represent states of a computation, e.g. natural numbers, sequences, stacks, etc. Gurevich’s Abstract State Machines (ASMs) take a more liberal position: any firstorder structure may serve as a state. In [7] Gurevich ..."
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Abstract. Conventional computation models restrict to particular data structures to represent states of a computation, e.g. natural numbers, sequences, stacks, etc. Gurevich’s Abstract State Machines (ASMs) take a more liberal position: any firstorder structure may serve as a state. In [7] Gurevich characterizes the expressive power of sequential ASMs: he defines the class of sequential algorithms by means of only a few, amazingly general requirements and proves this class to be equivalent to sequential ASMs. In this paper we generalize Gurevich’s result to distributed algorithms: we define a class of distributed algorithms by likewise general requirements and show that this class is covered by a distributed computation model based on sequential ASMs. 1
The Expressive Power Of AbstractState Machines
, 2003
"... STATE MACHINES Wolfgang Reisig Institut fur Informatik Humboldt Universitat zu Berlin Unter den Linden 6 10099 Berlin, Deutschland email: reisig@informatik.huberlin.de Abstract. Conventional computation models assume symbolic representations of states and actions. Gurevich's "AbstractSt ..."
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STATE MACHINES Wolfgang Reisig Institut fur Informatik Humboldt Universitat zu Berlin Unter den Linden 6 10099 Berlin, Deutschland email: reisig@informatik.huberlin.de Abstract. Conventional computation models assume symbolic representations of states and actions. Gurevich's "AbstractState Machine" model takes a more liberal position: Any mathematical structure may serve as a state. This results in "a computational model that is more powerful and more universal than standard computation models" [5].
The Generic Model of Computation
"... Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a ..."
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Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein. 1
Semantic Blueprints of Discrete Dynamic Systems: Challenges and Needs in Computational Modeling of Complex Behavior
"... How can one cope with the notorious problem of establishing the correctness and completeness of abstract functional requirements in the design of controlintensive software systems prior to actually building the system? The answer given here explores abstract state machines (ASMs): a universal math ..."
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How can one cope with the notorious problem of establishing the correctness and completeness of abstract functional requirements in the design of controlintensive software systems prior to actually building the system? The answer given here explores abstract state machines (ASMs): a universal mathematical framework for semantic modeling of discrete dynamic systems. Combining common abstraction principles from computational logic and discrete mathematics, ASMs provide a universal model of computation and an effective instrument for analyzing and reasoning about complex semantic properties of realworld systems. Widely recognized applications include semantic foundations of virtually all kinds of architectures, languages and protocols. In this paper we focus on empirical aspects in modeling concurrent and reactive behavior.
How Expressive are Petri Net Schemata?
"... Abstract. Petri net schemata are an intuitive and expressive approach to describe highlevel Petri nets. A Petri net schema is a Petri net with edges and transitions inscribed by terms and Boolean expressions, respectively. A concrete highlevel net is gained by interpreting the symbols in the inscr ..."
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Abstract. Petri net schemata are an intuitive and expressive approach to describe highlevel Petri nets. A Petri net schema is a Petri net with edges and transitions inscribed by terms and Boolean expressions, respectively. A concrete highlevel net is gained by interpreting the symbols in the inscriptions by a structure. Its semantics can then be described in terms of a transition system. Therefore, the semantics of a Petri net schema can be conceived as a family of transition systems indexed by structures. In this paper we characterize the expressive power of a general version of Petri net schemata. For that purpose we examine families of transition systems in general and characterize the families as generated by Petri net schemata. It turns out that these families of transition systems can be characterized by simple and intuitive requirements. 1
On the Expressive Power of UnboundedNondeterministic Abstract State Machines
"... Abstract. Conventional computational models assume a symbolical representation of states. Gurevich’s Abstract State Machines (ASMs) take a more liberal position: any mathematical structure may serve as a state. In [7] Gurevich characterizes the expressive power of sequential ASMs: he defines the cla ..."
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Abstract. Conventional computational models assume a symbolical representation of states. Gurevich’s Abstract State Machines (ASMs) take a more liberal position: any mathematical structure may serve as a state. In [7] Gurevich characterizes the expressive power of sequential ASMs: he defines the class of sequential algorithms by help of only a few, amazingly general requirements and proves this class to be equivalent to sequential ASMs. In [8] this result is extended to the class of boundednondeterministic ASMs. This paper considers the general case of unboundednondeterministic ASMs: in each step, a nondeterministic ASM may choose among infinitely many alternatives. We define the class of nondeterministic algorithms by a conservative extension of Gurevich’s original requirements to sequential algorithms. We show that this class is equivalent to unboundednondeterministic ASMs. 1