Some Assumptions about Problem Solving Representation in Turing's Model of Intelligence

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Some Assumptions aboutProblem Solving
Representation inTuring's Model of Intelligence

• Raymundo Morado
• Institute for Philosophical Research, UNAM
• morado_at_minerva.filosoficas.unam.mx
• Francisco Hernández Quiroz
• Faculty of Science, UNAM
• fhq_at_fciencias.unam.mx

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Abstract

• Turing machines as a model of intelligence can be
  motivated by some assumptions, both mathematical
  and philosophical
• Some of these are about the possibility, the
  necessity, and the limits of representing problem
  solving by mechanical means
• The assumptions about representation are related
  to information representability and availability,
  processing as solving, non-essentiality of
  complexity issues, and finiteness, discreteness
  and sequentiality of the representation

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• What happens if these assumptions are rejected or
  weakened?
• Tinkering with these assumptions sheds light on
  the import of alternative computational models

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Introduction

• The modeling of methodical intelligence with
  Turing Machines (TM's) implies some
  idealizations
• The specification of TM's leaves unexplained the
  intuitions that lead to posit those
  characteristics and no others
• Some principles can be removed without affecting
  the computational power of the model, but not all

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General Assumptions about Representation

• Problem Representability
• Information Processing
• Solving as syntactic transformation
• Stimulus and representation
• Information acquisition

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Problem Representability

• Any problem that can be solved mechanically can
  be represented.
• Any problem can be syntactically presented.
• Therefore we need a system of formulae both to
  present the problem and to present the answer

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Information Processing

• To solve a problem is akin to process
  information
• To solve a problem is to modify a representation
  until it becomes a representation which is the
  solution
• However an action from the agent is needed for a
  solution to become reality!

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Solving as syntactic transformation

• To solve a problem is to change its
  representation to "read" the problem and to
  "write" the solution
• The actions of the computer are specified by
  telling how to modify the current symbol or how
  to move on to read another one

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Stimulus and representation

• The machine perceives stimuli and can be affected
  by different situations
• This is necessary if we accept that to solve a
  problem is like processing information
• Both assumptions stand or fall together

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Information acquisition

• The machine must be capable of some actions to
  gather information
• In order to grasp more symbols the machine could
  do something such as moving its reading head or
  moving itself

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Specific assumptions about representation

• Finiteness of vocabulary
• Finiteness of the stimulus
• Sequentiality of the representation
• Two kinds of memory
• Access to memory
• Spatial complexity
• Discrete representation

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Finiteness of vocabulary

• In Turing's model, the number of symbols is
  finite
• Which agrees with the thesis that to solve
  effectively a problem requires processing a
  finite amount of information
• Also agrees with the assumption that effectively
  calculable solutions demand only a finite number
  of operations

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Finiteness of the stimulus

• A finite number of symbols does not force a
  finite input, but to solve effectively a problem
  the initial input must be finite
• But this is not always true!
• For instance, we can ask whether an infinite data
  stream contains more than two digits

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Sequentiality of the representation

• This assumption is not surprising if the input
  must be represented finitely
• In this restricted case there are trivial
  encodings for non-sequential information
• But this not always true if the input cannot be
  represented finitely

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Two kinds of memory

• A fixed finite set of states
• An infinite tape
• But the tape does not record every possible event

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Access to memory

• Access to the second kind of memory should be
  unlimited
• Restricting this access would lead to a weaker
  model

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Spatial complexity

• There is not a pre-established limit to a
  problem's spatial complexity
• To reject this assumption produces a strictly
  weaker mechanism than Turing Machines
• Then, this assumption is necessary, but
  unrealistic

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Discrete representation

• The transit between m-configurations is discrete
• It is mathematically conceivable to think
  otherwise?
• Some people argue in favor of explicitly
  analogical methods and data

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Final Remarks

• We have tried to make explicit principles that
  support the TM's model of mechanical computation
  as a form of intelligence
• Some principles are essential and others
  non-essential
• This distinction is important if we do not
  endorse the TM's as the model of mechanical
  problem solving

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Two tasks for alternative models

• To single out what essential principles should be
  omitted
• Alternatively, to state what features are missing
  in Turings model
• The rejection or addition of any essential
  principle leads to alternative models with
  different computational power