The tightest lower bound on the number of comparisons, in the worst case, for comparison-based sorting is of the order of

Consider the grammar rule $$E \to {E_1} - {E_2}$$ for arithmetic expressions. The code generated is targeted to a CPU having a single user register. The subtraction operation requires the first operand to be in the register. If E₁ and E₂ do not have any common sub-expression, in order to get the shortest possible code.

Consider the grammar with the following translation rules and E as the start symbol.

$$\eqalign{ & E \to {E_1}\# T\,\,\left\{ {E.value = {E_1}.value*T.value} \right\} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,|T\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {E.value = T.value} \right\} \cr & T \to {T_1}\& F\,\,\,\left\{ {T.value = {T_1}.value*F.value} \right\} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,|\,F\,\,\,\,\,\,\,\,\,\,\,\left\{ {T.value = F.value} \right\} \cr & F \to num\,\,\,\,\,\,\,\left\{ {F.value = num.value} \right\} \cr} $$

Compute E.value for the root of the parse tree for the expression:
2 # 3 & 5 # 6 & 4.

Which of the following grammar rules violate the requirements of an operator grammar? P, Q, R are nonterminals, and r, s, t are terminals.

$$\eqalign{ & 1.\,P \to QR \cr & 2.\,P \to QsR \cr & 3.\,P \to \varepsilon \cr & 4.\,P \to QtRr \cr} $$