Characterizing unambiguous precedence systems in expressions without superfluous parentheses

When infix notation is used, parentheses are sometimes omitted according to a precedence relation between the operators as well as a classification of (binary) operators as being left or right associative. We analyze these concepts by first giving a definition of a general precedence system, that declares the superfluous parenthesis pairs for any given expression. We give a characterization of unambiguity in this general setting, and study the complexity of parsing expressions without superfluous parentheses. Also, we study the two notions of maximal unambiguous and complete precedence systems and give a characterization for each one of these notions. Finally, we show that complete precedence systems can be equivalently described by a chain of left associative and right associative classes of operators, with some extra restrictions on the relative positions and the associativity of unary operators.