Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Operator associativity
open-in-new
<h2 id="examples">Examples</h2> <p>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 left-associative, the latter to when + and - are right-associative. </p><p>In order to reflect normal usage, <a href="/facts/Addition/apFWJBFB">addition</a>, <a href="/facts/Subtraction/dEUrp6k0">subtraction</a>, <a href="/facts/Multiplication/VtcUjpNv">multiplication</a>, and <a href="/facts/Division_(mathematics)/FqLKNkmq">division</a> operators are usually left-associative,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> while for an <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a> operator (if present)<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>[<i>better source needed</i>] there is no general agreement. Any <a href="/facts/Assignment_(computer_science)/kLevAoyJ">assignment</a> operators are typically right-associative. 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. </p> <h3>A detailed example</h3> <p>Consider the expression 5^4^3^2, in which ^ is taken to be a right-associative exponentiation operator. A parser reading the tokens from left to right would apply the associativity rule to a branch, because of the right-associativity of ^, in the following way: </p> <ol><li>Term 5 is read.</li> <li>Nonterminal ^ is read. Node: "5^".</li> <li>Term 4 is read. Node: "5^4".</li> <li>Nonterminal ^ is read, triggering the right-associativity rule. Associativity decides node: "5^(4^".</li> <li>Term 3 is read. Node: "5^(4^3".</li> <li>Nonterminal ^ is read, triggering the re-application of the right-associativity rule. Node "5^(4^(3^".</li> <li>Term 2 is read. Node "5^(4^(3^2".</li> <li>No tokens to read. Apply associativity to produce parse tree "5^(4^(3^2))".</li></ol> <p>This can then be evaluated depth-first, starting at the top node (the first ^): </p> <ol><li>The evaluator walks down the tree, from the first, over the second, to the third ^ expression.</li> <li>It evaluates as: 32 = 9. The result replaces the expression branch as the second operand of the second ^.</li> <li>Evaluation continues one level up the <a href="/facts/Parse_tree/FeBEtXcz">parse tree</a> as: 49 = 262,144. Again, the result replaces the expression branch as the second operand of the first ^.</li> <li>Again, the evaluator steps up the tree to the root expression and evaluates as: 5262144 ≈ 6.2060699×10183230. The last remaining branch collapses and the result becomes the overall result, therefore completing overall evaluation.</li></ol> <p>A left-associative evaluation would have resulted in the parse tree ((5^4)^3)^2 and the completely different result (6253)2 = 244,140,6252 ≈ 5.9604645×1016. </p> <h2 id="right-associativity-of-assignment-operators">Right-associativity of assignment operators</h2> <p>In many <a href="/facts/Imperative_programming_language/6b5dUnE1">imperative programming languages</a>, the <a href="/facts/Assignment_operator/kLevAoyJ">assignment operator</a> is defined to be right-associative, and assignment is defined to be an expression (which evaluates to a value), not just a statement. This allows <a href="/facts/Assignment_(computer_science)/kLevAoyJ">chained assignment</a> by using the value of one assignment expression as the right operand of the next assignment expression. </p><p>In <a href="/facts/C_(programming_language)/Ky2No763">C</a>, the assignment a = b is an expression that evaluates to the same value as the expression b converted to the type of a, with the <a href="/facts/Side_effect_(computer_science)/AU3GlLzG">side effect</a> of storing the <a href="/facts/Value_(computer_science)/QgQPbewf">R-value</a> of b into the <a href="/facts/Value_(computer_science)/QgQPbewf">L-value</a> of a.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Therefore the expression a = (b = c) can be interpreted as b = c; a = b;. The alternative expression (a = b) = c raises an error because a = b is not an L-value expression, i.e. it has an R-value but not an L-value where to store the R-value of c. The right-associativity of the = operator allows expressions such as a = b = c to be interpreted as a = (b = c). </p><p>In <a href="/facts/C%2B%2B/Et0F2qn9">C++</a>, the assignment a = b is an expression that evaluates to the same value as the expression a, with the side effect of storing the R-value of b into the L-value of a. Therefore the expression a = (b = c) can still be interpreted as b = c; a = b;. And the alternative expression (a = b) = c can be interpreted as a = b; a = c; instead of raising an error. The right-associativity of the = operator allows expressions such as a = b = c to be interpreted as a = (b = c). </p> <h2 id="non-associative-operators">Non-associative operators</h2> <p>Non-associative operators are operators that have no defined behavior when used in sequence in an expression. In <a href="/facts/Prolog/r5h66Qh1">Prolog</a> the infix operator :- is non-associative because constructs such as "a :- b :- c" constitute syntax errors. </p><p>Another possibility is that sequences of certain operators are interpreted in some other way, which cannot be expressed as associativity. This generally means that syntactically, there is a special rule for sequences of these operations, and semantically the behavior is different. A good example is in <a href="/facts/Python_(programming_language)/YbuGqofa">Python</a>, which has several such constructs.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> Since assignments are statements, not operations, the assignment operator does not have a value and is not associative. <a href="/facts/Assignment_(computer_science)/kLevAoyJ">Chained assignment</a> is instead implemented by having a grammar rule for sequences of assignments a = b = c, which are then assigned left-to-right. Further, combinations of assignment and augmented assignment, like a = b += c are not legal in Python, though they are legal in C. Another example are comparison operators, such as >, ==, and <=. A chained comparison like a < b < c is interpreted as (a < b) and (b < c), not equivalent to either (a < b) < c or a < (b < c).<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Order_of_operations/sWsoTzbH">Order of operations</a> (in arithmetic and algebra)</li> <li><a href="/facts/Common_operator_notation/UWQe0RSI">Common operator notation</a> (in programming languages)</li> <li><a href="/facts/Associativity/EQX8lV6r">Associativity</a> (the mathematical property of associativity)</li></ul> <h2 id="notes">Notes</h2> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Education Place: The Order of Operations <a href="http://eduplace.com/math/mathsteps/4/a/index.html" target="_blank">http://eduplace.com/math/mathsteps/4/a/index.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Khan Academy: The Order of Operations, timestamp 5m40s <a href="/wiki/Khan_Academy" target="_blank">/wiki/Khan_Academy</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Virginia Department of Education: Using Order of Operations and Exploring Properties, section 9 <a href="http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3" target="_blank">http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Exponentiation Associativity and Standard Math Notation Codeplea. 23 Aug 2016. Retrieved 20 Sep 2016. <a href="https://codeplea.com/exponentiation-associativity-options" target="_blank">https://codeplea.com/exponentiation-associativity-options</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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. <a href="/wiki/Statement_(programming)" target="_blank">/wiki/Statement_(programming)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>The Python Language Reference, "6. Expressions" <a href="https://docs.python.org/3/reference/" target="_blank">https://docs.python.org/3/reference/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>The Python Language Reference, "6. Expressions": 6.9. Comparisons <a href="https://docs.python.org/3/reference/" target="_blank">https://docs.python.org/3/reference/</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>