Perspectives on the ASM Theorem

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Abstract State MachinesMeetComputational Models
Udi Boker
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Abstract State Machines

• Mathematically define algorithms
• Simulate every machine step for step

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Computational Models

• Define a computing mechanism
• Its machines are defined on top of the common
  framework
• Examples Turing machines, finite automata,
  counter machines,

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The Motivation

• We want ASMs to define computational models
• Not a single machine or algorithm, but the
  general models mechanism
• The individual machines of the model can then be
  defined on top of the models configuration

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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

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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
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Filling the Gap
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Extensionality - Demands

• There is a domain of computation input and
  output domains
• The machine has no control over the possible
  inputs

Input
Ouput
Computation
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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

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Comp. Models - Demands

• Every computational model has its own concepts
  and building blocks, on top of which its machines
  are defined.

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Comp. Models - Solution

• All models machines share the same signature
• All machines share the same ASM program
• An ASM state has the following configuration

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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(he
ad) Write(ctl_state,tape(head)) head head
Move(ctl_state,tape(head))
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Analysis
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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

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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)

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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 machines part and models
  part is a higher level view, not concerning ASM
  internals

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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 machines 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

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Fixed Domain View
Strings
Integers

Initial states
Perspectives on the ASM Theorem Berlin, 26-27
Feb 2007
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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
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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

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Thanks