csc501-ln007

csc501-ln007 - Induction Up to this point we have...

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Induction Up to this point we have considered proofs of the following type: Given a selected expression a Aexp and some state σ Σ compute the semantic value k, ( a ) 7→ k . Or we have considered equivalence proofs of the form, Given two selected programs c , c 0 Com , show that they are semantically equivalent, c c 0 if σ Σ , σ 0 Σ . ( c ) 7→ σ 0 ( c 0 ) 7→ σ 0 Question: How would we prove that a certain property holds for all expressions or for all commands ? The syntax sets Aexp , Bexp , and Com are inFnite and therefore we need a proof technique that can deal with the inFnite nature of these sets Induction.
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Induction Induction is a proof principle that allows us to prove properties of possibly inFnite sets. 1 The power of this technique derives from the fact that we can test a particular property on a few select elements of the set and, if the tests hold, we can conclude that the property holds over the whole set. In order for this to work the set has to have a certain structure and the tests have to be selected in a manner to provide us with maximum information. 1 Sidebar: It is interesting to note that there is evidence that inductive proofs were already in use in Aristotle and Plato’s time.
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Mathematical Induction We start with mathematical induction over the natural numbers N . Consider the inductive defnition of the natural numbers, 0 N , If m N ,then m +1 N . The inductive structure of the natural numbers allows us to position the values along the number line without conflict, 012345 ... a precise ordering, 0 < 1 < 2 < 3 < 4 < 5 <... <
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Mathematical Induction Now, if we want to show that a predicate P holds for all n N ,thenwe can structure this as an inductive proof in two parts: 1 Base Case – show that P (0) holds, that is, show that the property of interest holds for the least element of the natural numbers. 2 Inductive Step – show that if P ( m ) holds for any m N ,then P ( m +1)alsoholds. If both parts can be shown to be true then we can conclude that the statement holds for all n N . This is sometimes also called the domino eFect ;herestep2aboveimp l ies P (0) P (1) and P (1) P (2) and P (2) P (3), etc. Later on we will prove that this domino eFect has to hold for inductively de±ned structures and therefore the proof principle is sound.
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Mathematical Induction Proposition: (Mathematical Induction) Let P be a predicate over the natural numbers N ,then n N . P ( n )if P (0) ∧∀ n N . P ( n ) P (
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csc501-ln007 - Induction Up to this point we have...

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