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In computer science, an operator-precedence parser is a bottom-up parser that interprets an operator-precedence grammar.For example, most calculators use operator-precedence parsers to convert from the human-readable infix notation relying on order of operations to a format that is optimized for evaluation such as Reverse Polish notation (RPN).
Calculators generally perform operations with the same precedence from left to right, [1] but some programming languages and calculators adopt different conventions. For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.
An operator precedence grammar is a kind of grammar for formal languages. Technically, an operator precedence grammar is a context-free grammar that has the property (among others) [ 1 ] that no production has either an empty right-hand side or two adjacent nonterminals in its right-hand side.
The operator precedence is a number (from high to low or vice versa) that defines which operator takes an operand that is surrounded by two operators of different precedence (or priority). Multiplication normally has higher precedence than addition, [ 1 ] for example, so 3+4×5 = 3+(4×5) ≠ (3+4)×5.
The + examples have been given twice. The first version is for simple calculators, showing how it is necessary to rearrange operands in order to get the correct result. The second version is for scientific calculators, where operator precedence is observed. Different forms of operator precedence schemes exist.
For example, "1 2 +" is not a valid infix expression, but would be parsed as "1 + 2". The algorithm can however reject expressions with mismatched parentheses. The shunting yard algorithm was later generalized into operator-precedence parsing.
The associativity and precedence of an operator is a part of the definition of the programming language; different programming languages may have different associativity and precedence for the same type of operator. Consider the expression a ~ b ~ c. If the operator ~ has left associativity, this expression would be interpreted as (a ~ b) ~ c.
The above properties of L, R, and k are actually shared by all shift-reduce parsers, including precedence parsers. But by convention, the LR name stands for the form of parsing invented by Donald Knuth, and excludes the earlier, less powerful precedence methods (for example Operator-precedence parser). [1]