A precise analysis

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A precise analysis

• Determinism, soundness, completeness for
• Determinism for
• Additional properties of
• Comparisons

2

• Analysis for with errors
  
• Observation
• (determinism for ? for
  )
• Theme
• Prove by induction on length of selected path
• For the proof, we need some concepts and results

3

• Applicability of rules
• A rule r is applicable (????) to E, if there is a
  proof tree for E?E, with r associated with the
  root.
• A group of rules is applicable to E if one of
  them is applicable

Why is it an interesting concept?
Note it is a semantic concept!
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• A characterization (if only if ) of
  applicability
• for an application expression E1 E2
• For you Similar characterization for if,
  tuple

5

• Q Are the conditions mutually exclusive?
• Q Have we proved determinism?
• Q Have we proved completeness?
• A we still depend on a semantic condition of
  having a transition

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• A reformulation (with some redundancy) of the
  (iff ) characterization of applicability (for E1
  E2)

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• By removing the transition condition
• ? mutually exclusive, syntactic, only if
  conditions
• But, it is still possible that two rules in a
  group are applicable,
• Or a rule generates more than one transition

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• Corollary (Inversion)
• If E has a transition, then
• Its primary construct
• value tests for (some) direct sub-expressions
• determine the sub-expression of E that has a
  transition
• why inversion ?

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• Observations
• Knowing primary construct of E ? all rules not
  for Es expression kind are excluded
• Value tests on (some) direct sub-expressions ?
  more rules excluded
• rules not excluded are potentially applicable
  (given current knowledge)
• If the remaining rules are proper, they select
  a unique direct sub-expression
• else, E is a candidate for a reduction
• This leads to the notions of
• selected path selected
  sub-expression
• (using the conditions in last table)

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• The selection path in a composite expression E
  (for ?d)
• Contains its root
• If it contains node n labeled by
• Application n E1 E2, then
• It contains E1 (operator) if E1 is not a value
• It contains E2 (operand) if E2 is not a value,
  but E1 is
• Stops at n if both E1 E2 are values
• Tuple ..
• If
• Its end is the (root of) the selected
  sub-expression of E
• As we go down the path, a set of (potentially
  applicable) rules is associated with each node,
  that all select the same child as next node

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• Example
• ,
• The condition v1 in Op (1st expr) selects a
  unique axiom, then we climb up

Application rules
Non-value
tuple rules
with i 1
applic
Application rules
(apply-op) (apply-error) (app-axioms)
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• Soundness (determinism) and completeness
  (progress)
• A composite expression E has a unique proof
  tree for a transition, hence a unique
  transition.
• and
• Es selected sub-expression is a redex, (always
  true for the rules with error)
• If E ? E (regular), then the
  transition for E is E ? E/EE
• Else, E ? ER, and also E ? ER

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• Proof induction on length of selection path
• 0 E is a candidate for a reduction
• always a redex obviously all the
  claims hold
• Ngt0
• Induction hypothesis claim is true for the
  unique child E1 one level down on the
  selected path.
• The (unique) transition of E1 determines
• a unique applicable rule for E
• hence there is a unique proof
  tree/transition for E.

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• Determinism for
• Can prove directly, using same technique
• Or, recall that
• But, for some expressions, there is no ?d
  transition

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Additional properties

• ? is not a functional relation!
• Possibly many successors to an expression, many
  computations
• Claim (confluence)
• All ? computations from an expression E
  terminate in same final (extended) value
• (proof in chapter)

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• Functionality, Soundness completeness for
  natural
• For , for each E there is at most
  one
• applicable rule,
• proof tree,
• value V, s.t.
• Proved by induction on E (we did that already)
• For , replace
• at most by unique
• Value V by XV (V or ER)
• Here, uniqueness of proof tree fails for
  derivations of ER, but there is never an
  evaluation to a value, another to ER

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Comparison of transition, natural

• 
  (w/o ER)
• 
  (with ER)
  
• The semantics express the
• same intuitive semantics (for
  values of expressions)
• in different styles
• (which one is easier to write transition or
  natural?)