Traversal of Object Structures

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Traversal of Object Structures

• Karl Lieberherr

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Definition of traversals

• Traversals of object graphs.
• Informally, a traversal is a (possibly infinite)
  set of concrete class graph paths when used in
  conjunction with an object graph, it results in a
  sequence of objects, called the traversal
  history.
• The traversal history is a depth-first traversal
  of the object graph along object paths agreeing
  with the given concrete path set.

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Definition of Traversals

• For a set of sequences R subset of S for an
  alphabet S, define
• head(R) x in S exists a.(xa in R)
• tail(R,x) a xa in R for some x in S .
• Intuitively, head(R) is the set of all first
  elements of R, and tail(R,x) is the set of all
  tails'' of sequences of R that start with x
  (where a tail of a sequence is the whole sequence
  except its first element).

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Definition of Traversals

• Fix a class graph G. If O is an acyclic object
  graph which is an instance of G, o an object in
  O, R a set of concrete paths corresponding to
  paths of G, and H a sequence of objects, then the
  judgment Judge(O,o,R,H) means that when
  traversing the object graph O starting with o,
  and guided by the concrete path set R, then H is
  the traversal history. This judgment holds when
  it is derivable using the following rules

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Definition of Traversals

• -----------------------------------------------
• Judge(O,o,R,epsilon)
• if tail(R,Class(o)) where epsilon denotes the
  empty history, and
• Judge(O, o_i, tail(tail(R,Class(o)),l_i),H_i)
  forall i in 1..n
• --------------------------------------------------
  -------------------------------
• O \vdash_s o R \rhd o H_1 ... H_n
• Judge(O, o, R, H_1 ... H_n
• if head(tail(R,Class(o))) l_i i in 1..n,
• edge o l_i o_i is in O, i in 1..n, and
• l_j lt l_k for 1 lt j lt k lt n.

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Discussion of semantics

• R may be an infinite set, so semantics does not
  immediately provide an interpreter.
• What we need for an interpreter is a finite
  representation of R