Write the syntax definitions of the following objects:

(a) Natural number (1, 2, 3, ...)
(b) Unsigned integer (0, 1, 2, 3, ...)
(c) Integer (..., -2, -1,0, 1, 2, ...)
(d) Odd number (..., -3, -1, 1, 3, ...)
(e) Even number (..., -4, -2,0, 2, 4, ...)
(f) Integer divisible by 5 (..., -10, -5,0, 5, 10, ...)

Solve each problem from (a) to (f) using BNF, EBNF, syntax diagrams, and the parenthetic notation. At the end compare all solutions and write what is in your opinion the most suitable metalanguage for solving this type of problems.

An external consultant recommends that a plant installs a bank of capacitors for power factor correction. this will reduce the peak electrical demand charges by an average of 93 kw every month. the plant current pays $13 per kw in peak demand charges. the capacitor bank will include 223 kw of fixed capacitors, and 183 of variable capacitors. the fixed capacitors cost $59 per kw, and the variable capacitors will cost $65 per kw. the consultant charges 21% of the equipment costs to install the capacitors. because this project will reduce the demand for the electric utility, they are prepared to provide a one-time rebate of $42 per kw of reduced demand. what is the simple payback period for this project (in years)?