## Interactive small-step algorithms I: Axiomatization, (2006)

Citations: | 4 - 2 self |

### BibTeX

@MISC{Blass06interactivesmall-step,

author = {Andreas Blass and Yuri Gurevich and Dean Rosenzweig and Benjamin Rossman},

title = { Interactive small-step algorithms I: Axiomatization, },

year = {2006}

}

### OpenURL

### 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. 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.

### Citations

2446 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...) that are computable in principle, disregarding limitations of time and memory. The simplifications, however, typically lower the level of abstraction in a drastic way; just compare any algorithm in =-=[1]-=- or [29] with a Turing machine implementation of that algorithm. The latter involves details — the data representation on the tape, indications how the read/write head should find the right place for ... |

1541 |
Distributed Algorithms
- Lynch
- 1996
(Show Context)
Citation Context ... of distributed algorithms, by programming languages attempting to provide means for communication between computing agents, and by computing technology. A few references that seem relevant to us are =-=[4, 28, 31, 32, 33]-=-. 3. Postulates for Algorithms This section is devoted to the description of interactive, small-step algorithms by means of suitable definitions and postulates. Some parts of this material are essenti... |

1167 |
On computable numbers, with an application to the entscheidungsproblem
- Turing
- 1937
(Show Context)
Citation Context ...n [7] and the class M of machines defined in [8] is proved in [9]. 2.2. Other Related Work. Efforts to characterize algorithms mathematically began with the work of Alonzo Church [14] and Alan Turing =-=[34]-=- in the 1930’s. Turing gave a careful analysis and a gradual simplification of what can happen during an algorithmic computation performed by a person. The result was the Turing machine model of compu... |

440 |
The ⇡-calculus: a Theory of Mobile Processes
- Sangiorgi, Walker
- 2001
(Show Context)
Citation Context ... of distributed algorithms, by programming languages attempting to provide means for communication between computing agents, and by computing technology. A few references that seem relevant to us are =-=[4, 28, 31, 32, 33]-=-. 3. Postulates for Algorithms This section is devoted to the description of interactive, small-step algorithms by means of suitable definitions and postulates. Some parts of this material are essenti... |

405 | Evolving algebras 1993: Lipari guide
- Gurevich
- 1995
(Show Context)
Citation Context ...(2) can use not only the environment’s replies but also the order in which the replies were received. 1. Introduction The Abstract State Machine (ASM) Thesis, first proposed in [20] and elaborated in =-=[21, 22]-=-, asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Beginning in [23] and continuing in [5], [7], [8], and [9], the thesis has been proved ... |

267 |
An unsolvable problem of elementary number theory
- Church
- 1936
(Show Context)
Citation Context ... algorithms defined in [7] and the class M of machines defined in [8] is proved in [9]. 2.2. Other Related Work. Efforts to characterize algorithms mathematically began with the work of Alonzo Church =-=[14]-=- and Alan Turing [34] in the 1930’s. Turing gave a careful analysis and a gradual simplification of what can happen during an algorithmic computation performed by a person. The result was the Turing m... |

216 | Abstract state machines: A method for high-level system design and analysis
- Börger
- 2003
(Show Context)
Citation Context ...n. The ASM Thesis of [22] asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Subsequent experimentation provided confirmation of the thesis =-=[2, 3, 13]-=-. Paper [23] was the first of a series of papers offering speculative justification for the thesis, for particular classes of algorithms. They all follow the same general pattern; (1) Describe axiomat... |

151 |
Introduction to Algorithms: A Creative Approach
- Manber
- 1989
(Show Context)
Citation Context ...are computable in principle, disregarding limitations of time and memory. The simplifications, however, typically lower the level of abstraction in a drastic way; just compare any algorithm in [1] or =-=[29]-=- with a Turing machine implementation of that algorithm. The latter involves details — the data representation on the tape, indications how the read/write head should find the right place for the next... |

114 | Sequential abstract state machines capture sequential algorithms
- Gurevich
(Show Context)
Citation Context ...t looks like creation is then regarded as taking an element from out of the reserve and into the active part of the state. (The nondeterministic choice of the element is made by the environment.) See =-=[22, 23, 7, 8]-=- and the next section for discussion. The idea behind the third requirement is that all relevant state information is reflected in the vocabulary: if your algorithm can distinguish red integers from g... |

64 | The Undecidable - Davis - 1965 |

58 | Abstract State Machines Capture Parallel Algorithms
- Blass, Gurevich
(Show Context)
Citation Context ..., first proposed in [20] and elaborated in [21, 22], asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Beginning in [23] and continuing in =-=[5]-=-, [7], [8], and [9], 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... |

53 |
On the definition of an algorithm
- Kolmogorov, Uspensky
- 1958
(Show Context)
Citation Context ...description of computation, presumably motivated by the physics of computation rather than by an analysis of the actions of a human computer. For a detailed presentation of Kolmogorov’s approach, see =-=[27]-=-. Also see [6] and the references there for information about research on pointer machines. Like Turing’s model, these computation models also lower the abstraction level of algorithms. Yiannis Moscho... |

31 | Ordinary interactive small-step algorithms, Part I
- Blass, Gurevich
- 2006
(Show Context)
Citation Context ...st proposed in [20] and elaborated in [21, 22], asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Beginning in [23] and continuing in [5], =-=[7]-=-, [8], and [9], 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... |

26 |
Cédric Fournet. Modern concurrency abstractions for C
- Benton, Cardelli
(Show Context)
Citation Context ... of distributed algorithms, by programming languages attempting to provide means for communication between computing agents, and by computing technology. A few references that seem relevant to us are =-=[4, 28, 31, 32, 33]-=-. 3. Postulates for Algorithms This section is devoted to the description of interactive, small-step algorithms by means of suitable definitions and postulates. Some parts of this material are essenti... |

23 | What is an algorithm
- Moschovakis
- 2001
(Show Context)
Citation Context ... see [6] and the references there for information about research on pointer machines. Like Turing’s model, these computation models also lower the abstraction level of algorithms. Yiannis Moschovakis =-=[30]-=- proposed that the informal notion of algorithm be identified with the formal notion of recursor. A recursor is a monotone operator over partial functions whose least fixed point includes (as one comp... |

19 | Algorithms: a quest for absolute definitions
- Blass, Gurevich
- 2003
(Show Context)
Citation Context ...computation, presumably motivated by the physics of computation rather than by an analysis of the actions of a human computer. For a detailed presentation of Kolmogorov’s approach, see [27]. Also see =-=[6]-=- and the references there for information about research on pointer machines. Like Turing’s model, these computation models also lower the abstraction level of algorithms. Yiannis Moschovakis [30] pro... |

15 | Kleene, Introduction to Metamathematics - Cole - 1980 |

10 |
Evolving Algebras: An Introductory Tutorial
- Gurevich
- 1991
(Show Context)
Citation Context ...(2) can use not only the environment’s replies but also the order in which the replies were received. 1. Introduction The Abstract State Machine (ASM) Thesis, first proposed in [20] and elaborated in =-=[21, 22]-=-, asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Beginning in [23] and continuing in [5], [7], [8], and [9], the thesis has been proved ... |

8 |
On the concept of algorithm”, Uspekhi Mat. Nauk 8:4
- Kolmogorov
- 1953
(Show Context)
Citation Context ...TERACTIVE SMALL-STEP ALGORITHMS I 5 The Bounded Exploration Postulate expresses the idea that a sequential algorithm (in the traditional meaning of the term) “computes in steps of bounded complexity” =-=[26]-=-. More explicitly, it asserts that the values of a finite set W of terms (also called expressions), that depends only on the algorithm and not on the input or state, determine the state change (more e... |

7 |
webpage, http://www.eecs.umich.edu/gasm/, maintained by James K
- Michigan
(Show Context)
Citation Context ...n. The ASM Thesis of [22] asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Subsequent experimentation provided confirmation of the thesis =-=[2, 3, 13]-=-. Paper [23] was the first of a series of papers offering speculative justification for the thesis, for particular classes of algorithms. They all follow the same general pattern; (1) Describe axiomat... |

7 | W.,: Distributed Abstract State Machines and Their Expressive Power
- Glausch, Reisig
- 2008
(Show Context)
Citation Context ...ernal intra-step interaction, but with communication between subprocesses in the parallel case), and ordinary interactive small-step algorithms in [7, 8, 9]. Also, Andreas Glausch and Wolfgang Reisig =-=[16, 17]-=- adapted the postulates of [23] to describe a restricted but important class of “small-step” distributed algorithms. In this section, we briefly overview the work in [23, 7, 8, 9] directly leading to ... |

5 |
Composite interactive algorithms” (tentative title), in preparation
- Blass, Gurevich, et al.
(Show Context)
Citation Context ...t C, working in the presence of other components, then it would be possible for queries to be answered without having been asked by C, simply because another component could have asked the query. See =-=[11]-=- for a study of components. It follows immediately from the definitions that any initial segment of a coherent history is coherent. Definition 3.15. A history ξ for a state X is complete if Pending X(... |

5 |
A new thesis,” Abstract 85T-68-203
- Gurevich
- 1985
(Show Context)
Citation Context ...ies from that step and (2) can use not only the environment’s replies but also the order in which the replies were received. 1. Introduction The Abstract State Machine (ASM) Thesis, first proposed in =-=[20]-=- and elaborated in [21, 22], asserts that every algorithm is equivalent, on its natural level of abstraction, to an abstract state machine. Beginning in [23] and continuing in [5], [7], [8], and [9], ... |

5 |
Saharon Shelah, “On polynomial time computation over unordered structures
- Blass, Gurevich
(Show Context)
Citation Context ...science, the ASM computation model is the most generalINTERACTIVE SMALL-STEP ALGORITHMS I 3 computation model that we know. It supports for example computing with structures rather than strings; see =-=[12]-=- for particular examples leading to interesting theoretical results and questions. Finally it may be disappointing that, unlike the Church-Turing Thesis, the ASM thesis has not been used for negative ... |

4 | Interactive small-step algorithms II: Abstract state machines and the characterization theorem” (tentative title), in preparation
- Blass, Gurevich, et al.
(Show Context)
Citation Context ...ause ξ, being attainable, is coherent, which implies Dom( ˙ ξ) ⊆ IssuedX(ξ).INTERACTIVE SMALL-STEP ALGORITHMS I 25 4. Equivalence of Algorithms One of our principal aims in this paper and its sequel =-=[10]-=- is to show that every algorithm, in the sense defined above, is behaviorally equivalent, in a strong sense, to an ASM. Of course, this goal presupposes a precise definition of the notion of behaviora... |

3 |
Interactive Algorithms 2005 with Added Appendix
- Gurevich
(Show Context)
Citation Context ...istent Turing machine” [19]. But this is not so if one intends to preserve the abstraction level of the given interactive algorithm. In particular, PTMs cannot step-for step simulate interactive ASMs =-=[24]-=-. The topic of the present paper is intra-step interaction between a single sequential-time algorithm and its environment (that is, the rest of the world). We are not aware of any previous literature ... |

2 |
Varianten des ASM-Theorems, Diploma thesis
- Glausch
- 2005
(Show Context)
Citation Context ...ernal intra-step interaction, but with communication between subprocesses in the parallel case), and ordinary interactive small-step algorithms in [7, 8, 9]. Also, Andreas Glausch and Wolfgang Reisig =-=[16, 17]-=- adapted the postulates of [23] to describe a restricted but important class of “small-step” distributed algorithms. In this section, we briefly overview the work in [23, 7, 8, 9] directly leading to ... |