Ordinary interactive small-step algorithms, II

Blass, Andreas; Gurevich, Yuri

doi:10.1145/1243996.1243998

Cited by 16 publications

(82 citation statements)

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“…• In [4,5,6], the analysis of algorithms was extended to the case when an algorithm interacts with the outside environment during a step, and execution waits until all queries of the environment have been responded to.…”

Section: Resultsmentioning

confidence: 99%

“…The three postulates given below (from [25], modified slightly as in [4,5,6,3]) assert that a classical algorithm is a state-transition system operating over first-order structures in a way that is invariant under isomorphisms. An algorithm is a prescription for updating states, that is, for changing some of the interpretations given to symbols by states.…”

Section: Sequential Algorithmsmentioning

confidence: 99%

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The Generic Model of Computation

Dershowitz

2012

Electron. Proc. Theor. Comput. Sci.

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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. BackgroundAbstract state machines (ASMs), invented by Yuri Gurevich [24], constitute a most general model of computation, one that can operate on any desired level of abstraction of data structures and native operations. All (ordinary) models of computation are instances of this one generic paradigm. Here, we give an overview of the foundational considerations underlying the model (cobbled together primarily from [18,3,12]). 1 Programs (of the sequential, non-interactive variety) in this formalism are built from three components:• There are generalized assignments f (s 1 , . . . , s n ) := t, where f is any function symbol (in the vocabulary of the program) and the s i and t are arbitrary terms (in that vocabulary).• Statements may be prefaced by a conditional test, if C then P or if C then P else Q, where C is a propositional combination of equalities between terms.• Program statements may be composed in parallel, following the keyword do, short for do in parallel.An ASM program describes a single transition step; its statements are executed repeatedly, as a unit, until no assignments have their conditions enabled. (Additional constructs beyond these are needed for interaction and large-scale parallelism, which are not dealt with here.) As a simple example, consider the program shown as Algorithm 1, describing a version of selection sort, where F(0), . . . , F(n − 1) contain values to be sorted, F being a unary function symbol. Initially, n ≥ 1 is the quantity of values to be sorted, i is set to 0, and j to 1. The brackets indicate statements that are executed in parallel. The program proceeds by repeatedly modifying the values of i and j, as well as of locations in F, referring to terms F(i) and F( j). When all conditions fail, that is, when j = n and i + 1 = n, the values in F have been sorted vis-à-vis the black-box relation ">". The program halts, as there is nothing left to do. (Declarations and initializations for program constants and variables are not shown.)This sorting program is not partial to any particular representation of the natural numbers 1, 2, etc., which are being used to index F. Whether an implementation uses natural language, or decimal numbers, 1 For a video lecture of Gurevich's on this subject, see http://www.youtube.com/v/7XfA5EhH7Bc. Generic Model of ComputationAlgorithm 1 An abstract-state-machine program for sorting.Algorithm 2 An abstract-state-machine program for bisection search.or binary strings is immaterial, as long as addition behaves as expected (and equality and disequality, too). Furthermore, the program will work regardless of the domain from which...

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Section: Resultsmentioning

confidence: 99%

Section: Sequential Algorithmsmentioning

confidence: 99%

The Generic Model of Computation

Dershowitz

2012

Electron. Proc. Theor. Comput. Sci.

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“…Abstract State Machines (ASMs) have been used since their discovery in the 1990s to model sequential and concurrent systems (see [20,Ch. 6,9] for references), the latter ones based upon various definitions of concurrent ASM runs which eventually were superseded by the definition in [28] of so-called distributed ASM runs as a certain class of partially ordered runs. It turned out, however, that the elegance of this definition has a high price, namely to be too restrictive and thereby impractical.…”

Section: Introductionmentioning

confidence: 99%

“…9] for a survey) that ASMs allow one to faithfully model at any level of abstraction sequential systems found in practice ('ground model' concern [17]) and to provide a basis for the practitioner to rigorously analyze and reliably refine such models to their implementations (correct refinement concern [16]). A considerable research effort has been devoted to extend this sequential ASM thesis to other computational concepts, in particular to parallel machines [7,11], to machines which interact with the environment during a step [8][9][10] (for one more variation see [12,13]), to quantum algorithms [25] and to database systems [56][57][58]60,64]. But up to now no postulates have been found from which a thesis for truly concurrent ASMs can be proven.…”

Section: Introductionmentioning

confidence: 99%

Concurrent abstract state machines

Börger

Schewe

2015

Acta Informatica

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A concurrent Abstract State Machine (ASM) is a family of agents each equipped with a sequential ASM to execute. We define the semantics of concurrent ASMs by concurrent ASM runs which overcome the problems of Gurevich's distributed ASM runs and generalize Lamport's sequentially consistent runs. A postulate characterizing an intuitive understanding of concurrency is formulated. It allows us to state and prove an extension of the sequential ASM thesis to a concurrent ASM thesis.

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“…The Abstract State Machine Thesis, first proposed in [7] and then elaborated in [8,9], asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Beginning in [10] and continuing in [1], [2], [3], and [4], the thesis has been proved for various classes of algorithms. In each case, the class of algorithms under consideration was defined by postulates describing, in very general terms, the nature of the algorithms, and in each case the main theorem was that all algorithms of this class are equivalent, in a strong sense, to ASMs.…”

Section: Introductionmentioning

confidence: 99%

Interactive Small-Step Algorithms II: Abstract State Machines and the Characterization Theorem

Blass¹,

Gurevich²,

Rosenzweig³

et al. 2007

Logical Methods in Computer Science

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Abstract. In earlier work, the Abstract State Machine Thesis -that arbitrary algorithms are behaviorally equivalent to abstract state machines -was established for several classes of algorithms, including ordinary, interactive, small-step algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. In a companion paper [5] the axiomatisation was extended to cover interactive small-step algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment's replies but also the order in which the replies were received. In order to prove the thesis for algorithms of this generality, we extend here the definition of abstract state machines to incorporate explicit attention to the relative timing of replies and to the possible absence of replies. We prove the characterization theorem for extended ASMs with respect to general algorithms as axiomatised in [5].

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Ordinary interactive small-step algorithms, II

Cited by 16 publications

References 6 publications

The Generic Model of Computation

The Generic Model of Computation

Concurrent abstract state machines

Interactive Small-Step Algorithms II: Abstract State Machines and the Characterization Theorem

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