Let n be the number of states in a DFA. Show that A_{DFA} ∈ P.
We have to show that all the states (n) in the DFA given, exist in P.
How to re-term it in order to make it simple to prove? We can say we have to show that-
All the states in a DFA that can be reached from initial state 'i' and get accepted would ∈ P. But, how to show it?
This can be proved by using DFS (Depth first search) or BFS(breadth first search).
Please note that DFS would take more time to complete whereas BFS would take up more space, so it depends on you which method you want to choose if you implement it.
So, using DFS/BFS on the deterministic finite automata, starting from initial state 'i', keep on traversing to find accepting states.
If you ever find a non-accepting state which is reachable, deny it. This means that the DFA doesn't accept a certain pattern.
Once you reach a particular final state which is reachable, backtrack and search for other accepting states. Thus, all the 'n' states get covered and for all reachable states from initial state 'i', we have successfully proved that A_{DFA} ∈ P.
Using DFS/BFS, we were able to show this in polynomial time.
Let PALINDROME DFA = { <M> | M is a DFA, and for all s E L(M), s is a palindrome }. Show that PALINDROME DFA E P by providing an algorithm for it that runs in polynomial time.
Let M be a DFA that recognizes a finite language A, and suppose M has n states. Determine if the following statement is true or false: if w Element of A, then |w| < = n. Prove your answer.
5. Let CONTAINPDA DFA L(M1) C (M2)}. Show that CONTAIN PDA DFA is decidable. {{M1, M2) M1 is a PDA and M2 is a DFA such that = 5. Let CONTAINPDA DFA L(M1) C (M2)}. Show that CONTAIN PDA DFA is decidable. {{M1, M2) M1 is a PDA and M2 is a DFA such that =
If a_(1)=0 and a_(n+1)=2a_(n)-3, find a_(4)
Let PALINDROMEDFA = { | M is a DFA, and for all s L(M), s is a palindrome }. Show that PALINDROMEDFA P by providing an algorithm for it that runs in polynomial time. Let PALINDROMEDFA = {<M> Mis a DFA, and for alls e L(M), s is a palindrome }. Show that PALINDROMEDFA E P by providing an algorithm for it that runs in polynomial time.
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative. 12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
Let A be an n x p matrix with n p. (a) Show that r(AA) = r(A). (b) Show that I - A(ATA) AT is idempotent. (c) Show that r(1-A(ATAYA") = n-r(A) Let A be an n x p matrix with n p. (a) Show that r(AA) = r(A). (b) Show that I - A(ATA) AT is idempotent. (c) Show that r(1-A(ATAYA") = n-r(A)
Problem 1. (15 points) Apply the DFA minimization algorithm to the DFA shown below. Show the matrix of distinguishable pairs of states after each iteration of the loop.
7.(15) Let PALINDROMEDFA = { <M> Mis a DFA, and for all s E L(M), s is a palindrome } Show that PALINDROMEDFA E P by providing an algorithm for it that runs in polynomial time.
Linear Algebra: Show that for each i = 1, ..., n there is a natural number p. j- 1v1, . . . , Vnf is a canonical Let be a linear operator on V and Jordan basis, ie. ΤΊβ is a canonical Jordan form. Show that for each i-1, . . . ,n there is some p є N such that (T-ÀI)" (vi-0, where is the diagonal entry of the matrix [T]β on the ith column. j- 1v1, . ....