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Abstract State Machines Meet Computational Models

Abstract State Machines Meet Computational Models. Udi Boker. Abstract State Machines. Mathematically define algorithms Simulate every machine step for step. Computational Models. Define a computing mechanism Its machines are defined on top of the common framework

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Abstract State Machines Meet Computational Models

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  1. Abstract State MachinesMeetComputational Models Udi Boker Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  2. Abstract State Machines • Mathematically define algorithms • Simulate every machine step for step Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  3. Computational Models • Define a computing mechanism • Its machines are defined on top of the common framework • Examples: Turing machines, finite automata, counter machines, … Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  4. The Motivation • We want ASMs to define computational models • Not a single machine or algorithm, but the general model’s mechanism • The individual machines of the model can then be defined on top of the model’s configuration Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  5. The Gap • A single machine vs. a framework • ASMs simulate single machines • Extensionality (implemented function) • ASMs have no extensionality • Domain of computation • ASMs have no domains (they have all domains) due to the isomorphism closure Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  6. In The Literature • There are many examples in the literature of ASM defining computational models. • There is no general framework, nor analysis: As a side effect of this – epistemologically significant – generality of the postulates, the application of the Blass and Gurevich proof scheme to established models of computation may yield “abstract” machine models which are more involved than necessary and may blur features which really distinguish different concrete systems. ASM book, 2003 Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  7. Filling the Gap Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  8. Extensionality - Demands • There is a domain of computation – input and output domains • The machine has no control over the possible inputs Input Ouput Computation Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  9. Extensionality - Solution • The signature contains two designated nullary functions named In, Out • There is a specific domain over which the machine is defined • All initial states are the same, except for the value of In • There is exactly one initial state for each domain element e, in which In  e Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  10. Comp. Models - Demands • Every computational model has its own concepts and building blocks, on top of which its machines are defined. Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  11. Comp. Models - Solution • All model’s machines share the same signature • All machines share the same ASM program • An ASM state has the following configuration: Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  12. Examples • Most examples in the literature fit the framework • Some (improper) exceptions, e.g. with a different ASM program for each simulated machine • Turing machines: ctl_state := Nxtctl(ctl_state,tape(head)) tape(head) := Write(ctl_state,tape(head)) head := head + Move(ctl_state,tape(head)) Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  13. Analysis Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  14. Conflicts? • Does the suggested framework conflict with the ASM concepts? • There are four potential issues: • In, Out – two specific nullary functions • Fixed state configuration • Specific domain for the structures • Fixed initial state, up to changes in In Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  15. Conflict Resolution – In, Out • Designated functions are not a problem, and are often used in the ASM approach, e.g. with distributed ASMs (Mod, Self) Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  16. Resolution – State Configuration • The separation of static and dynamic functions is well established in ASMs • The finitary and infinitary properties are preserved under isomorphism, hence well defined in the ASM approach • The separation of the machine’s part and model’s part is a higher level view, not concerning ASM internals Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  17. Resolution – Specific Domain • ASM defines an algorithm, operating over all domains. • Every specific run operates over a specific domain • This domain is the interpretation given to the abstract machine’s entities by the user, who views the machine as performing a specific computation. • The ASM defining a computational model is a general (all domains) ASM, in which we relate only to those runs that apply to the chosen domain Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  18. Fixed Domain View Strings Integers ∙ ∙ ∙ Initial states Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007 18

  19. Resolution – Fixed Initial State • As with the domain, the ASM is general • Relating to a fixed view of the elements Isomorphic Views Fixed View ∙ ∙ ∙ Initial states

  20. Summary • We provide a framework for representing computational models by ASMs • The framework provides a specific configuration, which goes along with the ASM approach • This representation scheme is intended for analyzing properties of computational models, for example effectivenes Perspectives on the ASM Theorem Berlin, 26-27 Feb 2007

  21. Thanks

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