Consider the language L = {w |w ∈ {0, 1} ∗ , 2 × #0 (w) = #1 (w)} , here #0(w) and #1(w) means the numbers of 0s and 1s in w respectively. That is, this is the language consisting of all strings with exactly twice as many 1s as 0s: for example, 101 ∈ L, 010 ∈/ L. In this problem we will show how to construct a context-free grammar for it. (a) (2 points) For a length n word x, we define a function f (i) = (2 × number of 0s in locations [1 . . . i]) − (number of 1s in locations [1 . . . i]). Show that we have f(i) = f(j) for some i < j if and only if the substring in indices [i + 1, . . . j] has exactly twice as many 1s as 0s (i. e. xi+1,i+2,...,j ∈ L).

We as humans write math expression in infix notation, e.g. 5 + 2 (the operators are written in-between the operands). in a computer’s language, however, it is preferred to have the operators on the right side of the operands, i.e. 5 2 +. for more complex expressions that include parenthesis and multiple operators, a compiler has to convert the expression into postfix first and then evaluate the resulting postfix.write a program that takes an “infix” expression as input, uses stacks to convert it into postfix expression, and finally evaluates it. it must support the following operations: + - / * ^ % (example infix expression: (7 - 3) / (2 + 2)postfix expression: 7 3 - 2 2 + /result: 1guidelines: 1. you will need to use stacks in three placesa. one for the parenthesis check [char stack]b. one during infix to postfix [char stack]c. one during evaluation [int stack]for a and b above, you can use same array and same push, pop method as both ofthem are char. but for evaluation you have int stack and you might consider to createanother push pop method to handle it. maybe push_int, pop_int, etc. or find otherstrategy to utilize existing push pop method2. you can create a function for obtaining operator priority. that function should take anoperator as input and return its priority as an integer. this function will you a lot andreduce repeated code3. during evaluation you will need to convert char into integer. example for single digit: char c = '5'; int x = c - '0';

Discuss the ways in which electronic information associated with payments is addressed in terms of security. include encryption, secure sockets layers, and secure electronic transactions in your discussion. are there any other ways that consumers and businesses can keep their payment information secure in an electronic commerce environment? do you feel that your information is safe when conducting electronic business? why or why not?

Martha is designing a single-player game. her manager suggests that she plan the design to incorporate future modifications. which principle of game design relates to planning for future modifications?