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. Here we extend the axiomatization and, in a companion paper, the proof, 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.
The characterization theorem of @cite_12 establishes the ASM thesis for the class A of algorithms defined by Sequential Time, Abstract State, and Bounded Exploration and the class M of machines defined by the basic ASM language of update rules, parallel rules and conditional rules @cite_3 @cite_15 @cite_12 .
@cite_15 @cite_3 @cite_12
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We consider parallel algorithms working in sequential global time, for example, circuits or parallel random access machines (PRAMs). Parallel abstract state machines (parallel ASMs) are such parallel algorithms, and the parallel ASM thesis asserts that every parallel algorithm is behaviorally equivalent to a parallel ASM. In an earlier article, we axiomatized parallel algorithms, proved the ASM thesis, and proved that every parallel ASM satisfies the axioms. It turned out that we were too timid in formulating the axioms; they did not allow a parallel algorithm to create components on the fly. This restriction did not hinder us from proving that the usual parallel models, like circuits or PRAMs or even alternating Turing machines, satisfy the postulates. But it resulted in an error in our attempt to prove that parallel ASMs always satisfy the postulates. To correct the error, we liberalize our axioms and allow on-the-fly creation of new parallel components. We believe that the improved axioms accurately express what parallel algorithms ought to be. We prove the parallel thesis for the new, corrected notion of parallel algorithms, and we check that parallel ASMs satisfy the new axioms. We present an idea how to simplify Gurevich's parallel ASM thesis. The key idea is to modify only the bounded exploration postulate from the sequential ASM thesis by allowing also non-ground comprehension terms. The idea arises from comparison with work on ASM foundations of database transformations. We examine sequential algorithms and formulate a sequential-time postulate, an abstract-state postulate, and a bounded-exploration postulate . Analysis of the postulates leads us to the notion of sequential abstract-state machine and to the theorem in the title. First we treat sequential algorithms that are deterministic and noninteractive. Then we consider sequential algorithms that may be nondeterministic and that may interact with their environments. This is the second in a series of three articles extending the proof of the Abstract State Machine Thesis---that arbitrary algorithms are behaviorally equivalent to abstract state machines---to algorithms that can interact with their environments during a step, rather than only between steps. As in the first article of the series, we are concerned here with ordinary, small-step, interactive algorithms. This means that the algorithms: (1) proceed in discrete, global steps, (2) perform only a bounded amount of work in each step, (3) use only such information from the environment as can be regarded as answers to queries, and (4) never complete a step until all queries from that step have been answered. After reviewing the previous article's formal description of such algorithms and the definition of behavioral equivalence, we define ordinary, interactive, small-step abstract state machines (ASMs). Except for very minor modifications, these are the machines commonly used in the ASM literature. We define their semantics in the framework of ordinary algorithms and show that they satisfy the postulates for these algorithms. This material lays the groundwork for the final article in the series, in which we shall prove the Abstract State Machine thesis for ordinary, intractive, small-step algorithms: All such algorithms are equivalent to ASMs.