CSCE 355 Lecture 4: Induction Proofs and DFA Examples - Prof. M. Matthews, Study notes of Computer Science

Download CSCE 355 Lecture 4: Induction Proofs and DFA Examples - Prof. M. Matthews and more Study notes Computer Science in PDF only on Docsity! CSCE 355 Lecture 4 - Outline September 8, 2008 1 • Induction Proofs format review • Induction Proofs in HW – ∑i=n i=0 i ∗ 2i = 2 + 2n−1 ∗ (n− 1) – commonly made mistakes – A full complete tree with n leaves has 2n-1 nodes. • Induction Proof Pseudo Pop Quiz • Review - – For M = (Q,Σ, δ, q0, F ): path determined by input, δ̂ – string accepted by a DFA, Language accepted by a DFA – Examples: given M what is L(M)? – Examples: given L what is M with L(M) = L? – Pigeon Hole Principle: If take a string w with as many characters as states in DFA then the path specified must loop. – Notes on Website http://www.cse.sc.edu/ matthews/Courses/355/Lectures.html • Homework from email • Ruby Simulation of DFA Handouts/dfa0.rb • more Examples: Given M find L(M) (Example 2.4) • Example: exercise 2.2.4: a ending in 00 b with three consecutive 0’s c strings with 011 as a substring • Given L1 and L2 and Machines M1 and M2 with L1 = L(M1) and L2 = L(M2) construct the DFA that accepts L1 ∪ L2 • Nondeterministic Finte Automata (NFA): δ : Q × Σ → 2Q • Homework: – page 54 2.2.5 b The set of all strings from {0, 1}∗ such that the tenth symbol from the right is a 1. – page 54 2.2.5 c The set of all strings from {0, 1}∗ that either begin with 01 or end with 01 (or both). – page 54 2.2.5 d The set of all strings from {0, 1}∗ such that the number of )’s is divisible by five and the number of one’s is divisible by 3. – Given L1 and L2 and Machines M1 and M2 with L1 = L(M1) and L2 = L(M2) construct the DFA that accepts L1 ∩ L2 – page 54 exercise 2.2.7 Let A be a DFA and q a particular state of A, such that δ(q, a) = q for all input Symbols a. Show by induction of the length of the input that for all strings w, δ̂(q, w) = q – Practice - not to be submitted, i.e., Solved ones on website 1. 2.2.2 Induction 2.