Consider the following Nondeterministic Finite Automaton (NFA). 1 a b a 3 4 2 (a) What are the possible states that this NFA could be in, after reading the input string aba? (b) Prove, by induction on n, that for all positive integers n, the string (aba)" is accepted by this NFA. (This string is obtained by n repetitions of aba.)

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Given two strings X and Y, where X consists of the sequence of symbols X1, X2, Xm and Y consists of the sequence of symbols y₁,Y2, ***,yn. Consider the sets {1, 2, ..., m} and {1, 2, ..., n} as representing the different positions in the strings X and Y, and consider a matching of these sets, where a matching is a set of ordered pairs with the property that each item occurs in at most one pair. A matching M of these two sets is an alignment if there are no "crossing" pairs: if (i,j). (i*,j') = Mandi 0 that defines a gap penalty. For each position of X or Y that is not matched in M (it is a gap), we incur a cost of 8. Second, for each pair of letters p and q (p + q) in our alphabet, there is a mismatch cost of app for lining up p with q. Thus, for each (i,j) = M, we pay the appropriate mismatch cost axiy for lining up x; with y;. The cost of M is the sum of its gap and mismatch costs, and the problem seeks an alignment of minimum cost. Define the minimum alignment cost OPT(i,j) (0 ≤ i ≤…

create a Deterministic Finite Automaton (DFA) for the regular expression below: A ( (AB) | (AC) )* C

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Let LPrime = {I the number of strings accepted by M is prime}. Classify this language as either (i) decidable, (ii) recognizable but not decidable, or (iii) not recognizable. Prove your answer. You may give a high-level description of any TM programs used in your proof. ( You should not use Rice's theorem.)

1. Examine the following DFA: B 0 1 A D 1 Ο 0 Identify in the list below the string that this automaton accepts. a) ε b) 0110 c) 0111010 d) 10000 2. The binary string 1000000 is a member of which of the following problems? Remember, a "problem" is a language whose strings represent the cases of a problem that have the answer "yes." In this question, you should assume that all languages are sets of binary strings interpreted as base-2 integers. The exception is the problem of finding palindromes, which are strings that are identical when reversed, like 0110110, regardless of their numerical value. a) Is the given string not a perfect square? b) Is the given string less than or equal to 50? Is the given string not a multiple of 3? d) Is the given string a prime? 3. What is the concatenation of abc and cda? a) abccda b) abc||cda c) cdaabc d) acbdca 4. Convert the following nondeterministic finite automaton: 1 A B C 1 to a DFA, including the dead state, if necessary. Which of the following…

Build a deterministic FA Ma such that L= L(Ma) = a((a+b)(a+b))*b For example, baba e La and aab e L4, but ab e La and abaaab e La.

es remaining 8. Consider the function f:NxN-N defined recursively by: 1) Base case: Let meN and define (0,m) = 0 2) Recursive case: For any x,meN, x>0, define f(x,m) = (x-1,m) + (m+m) Prove the following theorem holds using proof by induction: Thereom: For any n,meN, m>0 we have (n.m) I m = n+n Fill in your answer here 9 Help BIU X, x L - ɔE =N E X Format

Convert the non-deterministic finite automaton (NFA) M below into a deterministic finite automaton (DFA) M'. 90 (91 a a, b 92