Structural induction

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Structural induction

• Richard Connor

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Infinite and unbounded

• Keep the difference clear in mind!
• Infinite
• an entity greater than n for any n
• Unbounded
• An entity less than n for some n
• but n can only be determined based on the entity
• The set a
• Defines an infinite number of strings
• A member chosen at random has unbounded length

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Defining infinite sets

• Iterative
• L 0-9
• Inductive
• ltdigitsgt ltdigitgtltdigitsgt
• ltdigitsgt ?
• ltdigitgt 0 1 2 3
• Inductive
• Gives a way to construct terms in the language

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Properties of the infinite set

• We can consider the infinite set by
• consideration based on the construction mechanism
• The infinite set is defined by a finite number of
  rules
• If we can cover all the rules, then we cover the
  whole set
• This is known as structural induction
• (the term structural recursion is also used)
• Core mathematical technique for
• Defining relations, eg semantics
• Proving properties, eg soundness

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Example defining semantics

• ltdigitsgt ltdigitsgtltdigitgt
• ltdigitsgt ?
• ltdigitgt 0 1 2 3
• ?ltdigitgt ltdigitgt
• ltdigitsgtltdigitgt ltdigitsgt 10
  lt digitgt
• 0 0
• 1 1

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Example proving a property

• Property every sentence has a meaning
• Nb this is the soundness property easier for
  this language than most!
• Rewrite termination every rhs involves a
  decomposition of the lhs, so every obligation is
  smaller than the previous
• Also if m, n have meaning, so does m x 10 n
• Property the meaning of any defined string is a
  natural number
• Proof by structural induction over semantic
  relation
• 0 is a natural number
• 1 is a natural number
• If m,n are natural numbers, so is m x 10 n