 Operator associativity

For the mathematical concept of associativity, see Associativity.
In programming languages and mathematical notation, the associativity (or fixity) of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators (for example, "^ 4 ^"), and those operators have equal precedence, then the operand may be used as input to two different operations (i.e. the two operations indicated by the two operators). The choice of which operations to apply the operand to, is determined by the "associativity" of the operators. Operators may be leftassociative (meaning the operations are grouped from the left), rightassociative (meaning the operations are grouped from the right) or nonassociative (meaning there is no defined grouping). 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 operator symbol.
Consider the expression a ~ b ~ c. If the operator ~ has left associativity, this expression would be interpreted as (a ~ b) ~ c and evaluated lefttoright. If the operator has right associativity, the expression would be interpreted as a ~ (b ~ c) and evaluated righttoleft. If the operator is nonassociative, the expression might be a syntax error, or it might have some special meaning.
Many programming language manuals provide a table of operator precedence and associativity; see, for example, the table for C and C++.
Contents
Examples
Associativity is only needed when the operators in an expression have the same precedence. Usually + and  have the same precedence. Consider the expression 7 − 4 + 2. The result could be either (7 − 4) + 2 = 5 or 7 − (4 + 2) = 1. The former result corresponds to the case when + and − are leftassociative, the latter to when + and  are rightassociative.
Usually the addition, subtraction, multiplication, and division operators are leftassociative, while the exponentiation, assignment and conditional operators are rightassociative. To prevent cases where operands would be associated with two operators, or no operator at all, operators with the same precedence must have the same associativity.
A detailed example
Consider the expression 5^4^3^2. A parser reading the tokens from left to right would apply the associativity rule to a branch, because of the rightassociativity of ^, in the following way:
 Term 5 is read.
 Nonterminal ^ is read. Node: "5^".
 Term 4 is read. Node: "5^4".
 Nonterminal ^ is read, triggering the rightassociativity rule. Associativity decides node: "5^(4^".
 Term 3 is read. Node: "5^(4^3".
 Nonterminal ^ is read, triggering the reapplication of the rightassociativity rule. Node "5^(4^(3^".
 Term 2 is read. Node "5^(4^(3^2".
 No tokens to read. Apply associativity to produce parse tree "5^(4^(3^2))".
This can then be evaluated depthfirst, starting at the top node (the first ^):
 The evaluator walks down the tree, from the first, over the second, to the third ^ expression.
 It evaluates as: 3^{2} = 9. The result replaces the expression branch as the second operand of the second ^.
 Evaluation continues one level up the parse tree as: 4^{9} = 262144. Again, the result replaces the expression branch as the second operand of the first ^.
 Again, the evaluator steps up the tree to the root expression and evaluates as: 5^{262144} ≈ 6.2060699 × 10^{183230}. The last remaining branch collapses and the result becomes the overall result, therefore completing overall evaluation.
A leftassociative evaluation would have resulted in the parse tree ((5^4)^3)^2 and the completely different results 625, 244140625 and finally ~5.9604645 × 10^{16}.
Rightassociativity of assignment operators
Assignment operators in imperative programming languages are usually defined to be rightassociative. For example, in C, the assignment a = b is an expression that returns a value (namely, b converted to the type of a) with the side effect of setting a to this value. An assignment can be performed in the middle of an expression. (An expression can be made into a statement by following it with a semicolon; i.e. a = b is an expression but a = b; is a statement). The rightassociativity of the = operator allows expressions such as a = b = c to be interpreted as a = (b = c), thereby setting both a and b to the value of c. The alternative (a = b) = c does not make sense because a = b is not an lvalue.
Nonassociative operators
Nonassociative operators are operators that have no defined behavior when used in sequence in an expression. In Prolog, the infix operator : is nonassociative because constructs such as "a : b : c" constitute syntax errors.
Another possibility distinct from left or rightassociativity is that the expression is legal but has different semantics. An example is the comparison operators (such as >, ==, and <=) in Python: a < b < c is shorthand for (a < b) and (b < c), not equivalent to either (a < b) < c or a < (b < c).^{[1]}
See also
 Order of operations (in arithmetic and algebra)
 Common operator notation (in programming languages)
 Associativity (the mathematical property of associativity)
References
Categories: Parsing
 Programming language topics
 Operators (programming)
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