Definition:Balanced String

Definition

Let $S$ be a string in an alphabet containing brackets.

$S$ is said to be balanced if and only if:

$(1): \quad S$ contains equally many left and right brackets.

$(2): \quad$ Every prefix of $S$ contains at least as many left brackets as it does right brackets.

$S$ is said to be balanced if and only if it satisfies the following:

$(1): \quad$ The null string $\epsilon$ is a balanced string.

$(2): \quad$ If $x$ is a symbol that is specifically not a bracket, then the string consisting just of $x$ is a balanced string.

$(3): \quad$ If $S$ is a balanced string, then $\paren S$ is a balanced string.

$(4): \quad$ If $S$ and $T$ are balanced strings, then $S T$ is a balanced string.

$(5): \quad$ Nothing is a balanced string unless it has been created by one of the rules $(1)$ to $(4)$.