Minimal History, a theory of plausible explanation

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Minimal History, a theory of plausible
explanation

• John Mayfield
• Department of Genetics, Development, and Cell
  Biology
• Iowa State University

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Slow Growth

• A deep object can only be produced rapidly from
  deep input. Deep output can be created from
  shallow input, but only slowly.

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Off-putting aspects of logical depth

• Strong ties to the notion of discrete state
  computation.
• Logical depth is formally incomputable
  evaluation founders on the halting problem.
• Depth is defined in units of steps rather than
  bits.
• The optimal difference between the minimal input
  length (algorithmic information) and the
  near-minimal input length used to define depth
  (the significance level) is not determined.
• Logical depth provides no explanation of the
  near-minimal input nor of the universal
  computer employed by its definition.
• Logical depth does not relate very well to
  standard physics.

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Five Observations

• Bit strings record sequences of past binary
  decisions.
• Binary decisions can be made in two ways on the
  basis of existing information, or randomly.
  These decision-making modes are not exclusive,
  they can be mixed.
• A computer is the physical manifestation of an
  ordered set of rules.
• Any computation can be defined as discrete input
  manipulated according to specific rules resulting
  in discrete output. This implies that the
  input, the output, and the computer rules
  itself can all be represented as binary strings.
• The elementary steps that characterize the
  actions carried out during a computation can be
  encoded therefore, an entire computation can be
  recorded as a bit string.

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Imaginary process

• randomly generate some rules (bit string r)
• randomly generate some input (bit string p)
• manipulate input p according to rules r and
  record the resulting actions as bit string d.
  The resulting output (if any) is bit string x.

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Definition of Minimal History

• There are 2?r? binary strings of length ?r?, and
  ?r?2?r? characterizes the expected number of
  random choices tosses of an ideal coin, for
  example needed to generate a particular string r
  if there is no a priori knowledge of r.
• It requires of order ?rp?2?rp? steps to randomly
  create a particular pair of strings r and p.

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• It follows that ? ?rpd?2?rp? characterizes a
  creation of x, and that min ? characterizes the
  minimal creation of x in the absence of prior
  information.
• Define ? rpd to be a history of x.

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• Let ?' r'p'd' be a minimal history of x. It is
  a binary string that records all aspects of a
  minimal computation compatible with the minimal
  creation. There may be more than one such
  computation.

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• The length of the minimal history of x is
  formally defined as
• ??'? ?r'p'd'? min?rpd??? ?rpd?2?rp? is
  minimal, and r(p) x (1)
• Min ? characterizes the minimal process for
  creating x from nothing, while ??'? characterizes
  the minimal computation employed by the minimal
  process.

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• Levin (1984) defined a measure of complexity,
  Kt(x) min?p? log t to aid in his proof of
  a time bound on inverting problems. Defining Kd
  logmin ? min?rp? log?rpd? ?r'p'?
  log??'? creates a measure similar to Levin's but
  in which ??'? plays the role of time.

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Comments on ?', d', p', and r'

• ??'?, ?p'?, ?d'? and Kd do not generally
  characterize the shortest computation of x.
• ?', Kd, d', p', and r' are computable (for a
  specified language).
• ?d'? and ??'? are closely related to logical
  depth.
• Bennett's significance factor does not appear.

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more comments

• ?d'? and ??'? have well-defined upper limits.
• ?d'? ? ? r"x d"?2?r"x?/?r'p'?2?r'p'?
• ??'? ?r'p'd'? ? ? r"x d"?2?r"x?/2?r'p'?
• Where, double primes refer to the computation
  print x
• From these relationships we see that ?d'? and
  ??'? are always bounded from above. Recognizing
  that ?d"?is of the same order as ?x? and assuming
  that ?r"? ? ?r'?, we see that the upper bound for
  both ?d'? and ??'? grow approximately as
  ?x?2?x-p'?.

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last comment

• ??'? and ?d'? have the slow growth property.
• Theorem (slow growth law of minimal history) Let
  x and y be two strings with respective minimal
  histories, ?x' and ?y'. If ??y'? ? ??x'?, then
  the amount of minimal history characterizing the
  computation of y from input x must be ? ??y'? -
  ??x'?.

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The information measure hierarchy

• K(x) ? ?x?c and ?x? lt ??'?.
• K(x) is the smallest of all information measures
  of x. It does not include measure of the
  universal computer specified, and it does not
  account for redundancies present in x nor for
  relationships between parts of x that may exist.
• K(x) is most constrained, it cannot increase
  during deterministic computation.

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• ?x? is a direct measure of x. It includes
  measure of redundancy but does not measure
  internal relationships.
• ?x?, is closely tied to thermodynamic concepts of
  entropy and negentropy.
• ?x? cannot increase during a reversible
  computation, and any change in length that occurs
  in a non-reversible deterministic computation
  must be paid for by the use or release of kT?ln2
  of thermodynamic energy per bit.

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• ??'? provides a comprehensive measure of x by
  combining minimized measure of computer, input,
  and the amount of computation required to create
  x. It accounts for all internal computable
  relationships and redundancies that may be
  present in the object. ??'? measures the minimum
  process for producing x from nothing.
• ??'? is subject to the least constraint. It may
  increase during reversible deterministic
  computations without the expenditure of
  thermodynamic energy assuming an ideal machine.
  ??'? is constrained only by time.

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Why do we care?

• Minimal History is computable, but not
  practically so.
• Most important impact is philosophical. This
  exercise shows that within the framework of
  computation theory it is possible to
  conceptualize and define a notion of object
  complexity that incorporates the difficulty of
  creating an object from nothing.
• Minimal History measures all aspects of a
  computable object.
• Minimal History has the property of slow growth.
• Minimal History completes a hierarchical
  arrangement of entropy (information) measures
  that reflect the amount of constraint.

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Questions?
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• Proof
• 1.      Definition The length of the minimal
  history of y is given by ??y'?
• 2.      Definition The length of the minimal
  history of x is given by ??x'?
• 3.      Definition The length of the minimal
  history of the computation of y, with x as the
  only input, is given by ??xy'? ?r"xd"? where r"
  and d" denote the shortest combination of
  computer and computation that outputs y from
  input x that is compatible with ?rxd?2?rx? being
  minimal (condition 1).
• 4.      ??x'? ??xy'? ? ??y'?. This is true
  because an added requirement to produce x (even
  by a separate computation), cannot lessen the
  minimum amount of computation required to create
  y.
• 5.      By rearrangement, ??xy'? ? ??y'? - ??x'?
• Thus, only if ??y'? - ??x'? is small, is it
  possible for any added minimal history
  characterizing a process that produces y from
  input x to be small (a rapid computation). On
  the other hand, if y is deep and x is shallow,
  then ??xy'? must be large (the computation must
  be lengthy).