From … to Induction CS 270 Math Foundations of CS Jeremy Johnson

Objective To illustrate the shortcomings to the proof techniques seen so far and to introduce the principle of induction which allows the proof of infinitely many cases at once. With practice, students should be able to carry out simple inductive proofs following the lecture.

Outline 1.Motivating Example 2.Induction Principle 3.Inductive Proofs 1.Distributive Law 2.DeMorgan’s Law 3.Sums 4.Counting Clauses

Atmost One Recall the condition for at most one of the variables P 1,…,P t to be true P 1  (  P 2     P t ) … P t-2  (  P t-1   P t ) P t-1   P t We see a pattern and fill in the dots

Atmost One When converting to CNF we used a generalized version of the distributive law P 1  (  P 2     P t )  P 1  (  P 2     P t ) (  P 1   P 2 )    (  P 1   P t ) Again we fill in the dots and assume the distributive law generalizes appropriately

Generalized Distributive Law A  (B  C)  (A  B)  (A  C) What about A  (B  C  D)  (A  B)  (A  C)  (A  D) What does this mean? (B  C  D) and (A  B)  (A  C)  (A  D)

Generalized Distributive Law A  (B  C  D)  A  ((B  C)  D)  (A  (B  C))  (A  D)  ((A  B)  (A  C))  (A  D)  (A  B)  (A  C)  (A  D)

Generalized Distributive Law A  (B  C  D  E)  A  ((B  C  D)  E)  (A  (B  C  D))  (A  E)  ((A  B)  (A  C)  (A  D))  (A  E)  (A  B)  (A  C)  (A  D)  (A  E)

Generalized Distributive Law A  (B  C  D  E  F)  A  ((B  C  D  E)  F)  (A  (B  C  D  E))  (A  F)  ((A  B)  (A  C)  (A  D)  (A  D))  (A  F)  (A  B)  (A  C)  (A  D)  (A  E)  (A  F) …

Generalized Distributive Law

Induction Principle Let S(n) be a statement paramterized by a non-negative integer n If S(0) is true and S(n)  S(n+1) then S(n) holds for all non-negative integers. S(0), S(0)  S(1)  S(1) S(1), S(1)  S(2)  S(2) S(2), S(2)  S(3)  S(3) … This allows a proof of infinitely many cases

Inductive Proofs Let S(n) be a statement paramterized by n a nonnegative integer. To prove S(n) holds for all non-negative integers. 1.Prove S(0) [Base case] 2.Assume S(n) [inductive hypothesis] and prove S(n+1). This proves S(n)  S(n+1) Can start with a positive integer k and show S(n) holds for all integers  k.

Generalized Distributive Law

Exercise Generalized DeMorgan’s Law

Solution

Counting Clauses We would like a formula for the number of clauses in the N-queens problem. Count the number of clauses for atmost_one, atleast_one and exactly_one Sum over all rows, columns and diagonals

Number of Clauses in atmost_one

Summation Formula

Counting Clauses M(n) number of clauses for atmost_one L(n) number of clauses for atleast_one E(n) number of clauses for exactly_one M(n) = n(n-1)/2 L(n) = 1 E(n) = M(n)+E(n)

Clauses in N-Queens SAT Problem