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Structural Induction

Structural Induction. CS 611. Structural Induction. Useful for proving properties about languages. Useful for proving properties about algorithms defined on languages. A generalization of the induction you know and love. Induction on the Naturals. 0. n. n+1. Example.

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Structural Induction

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  1. Structural Induction CS 611

  2. Structural Induction • Useful for proving properties about languages. • Useful for proving properties about algorithms defined on languages. • A generalization of the induction you know and love.

  3. Induction on the Naturals 0 n n+1

  4. Example Prove: every natural number is odd or even. Base: 0 is even. Induction: if n is odd or even then n+1 is odd or even. Suppose n is odd, then n+1 is even Suppose n is even then n+1 is odd. 0 n n+1

  5. Induction on Aexp Suppose I want to prove a property of all arithmetic expressions. Like “eval (aexp) is finite” for some evaluation function eval. Aexp has 2 atoms and 2 constructors. So induction on Aexp has ___ base cases and ___ inductive steps? What is the induction theorem for arithmetic expressions? var const + Aexp Aexp - Aexp Aexp

  6. Induction on Aexp var const + Aexp Aexp - Aexp Aexp

  7. Example Eval (aexp a) if a = var then val(a) if a = const then a if a = (b+c) then eval(b)*eval(c) if a = (b-c) then eval(b)/eval(c) Prove: forall a:aexp. eval(a) terminates Base 1: eval(var) terminates Base 2: eval(const) terminates Ind1: suppose eval(b) and eval(c) terminate prove eval (b+c) termiantes Ind2: suppose eval(b) and eval(c) terminate prove eval(b-c) terminates

  8. Example Eval (aexp a) if a = var then val(a) if a = const then a if a = (b+c) then eval(b)*eval(c) if a = (b-c) then eval(b)/eval(c) Prove: forall a:aexp. eval(a) = correct_val(a) Base 1: eval(var) = correct_val(var) Base 2: eval(const) = correct_val(const) Ind1: suppose eval(b) = correct_eval(b) and eval(c) = correct_eval(c) prove eval(b+c) = correct_eval(b+c) Ind2: suppose eval(b) = correct_eval(b) and eval(c) = correct_eval(c) prove eval(b-c) = correct_eval(b-c)

  9. Example Eval (aexp a) if a = var then val(a) if a = const then a if a = (b+c) then eval(b)*eval(c) if a = (b-c) then eval(b)/eval(c) Prove: forall a:aexp. eval(a) is finite Base 1: eval(var) is finite Base 2: eval(const) is finite Ind1: suppose eval(b) is finite and eval(c) is finite prove eval(b+c) is finite Ind2: suppose eval(b) is finite and eval(c) is finite prove eval(b-c) is finite

  10. Reminders • Reverse engineer proof of Thm 3.1 to scratch work for Monday. • Start on proofs of Thms 3.2, 3.3, probably for Tuesday. • Thm 3.4 and Cor 3.2 likely for Friday • Paper 1 on Friday • Cancel Godel encoding part of hwk.

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