to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus. While no longer used much in logic, Polish notation has since found a place in computer science. Explanation The expression for adding the numbers 1 and 2 is written in Polish notation as (prefix), rather than as (infix). In more complex expressions, the operators still precede their operands, but the operands may themselves be expressions including again operators and their operands. For instance, the expression that would be written in conventional infix notation as can be written in Polish notation as Assuming a given arity of all involved operators (here the "−" denotes the binary operation of subtraction, not the unary function of sign-change), any well-formed prefix representation is unambiguous, and brackets within the prefix expression are unnecessary. As such, the above expression can be further simplified to The processing of the product is deferred until its two operands are available (i.e., 5 minus 6, and 7). As with any notation, the innermost expressions are evaluated first, but in Polish notation this "innermost-ness" can be conveyed by the sequence of operators and operands rather than by bracketing. In the conventional infix notation, parentheses are required to override the standard precedence rules, since, referring to the above example, moving them or removing them changes the meaning and the result of the expression. This version is written in Polish notation as When dealing with non-commutative operations, like division or subtraction, it is necessary to coordinate the sequential arrangement of the operands with the definition of how the operator takes its arguments, i.e., from left to right. For example, , with 10 to the left of 5, has the meaning of 10 ÷ 5 (read as "divide 10 by 5"), or , with 7 left to 6, has the meaning of 7 − 6 (read as "subtract from 7 the operand 6"). Evaluation algorithm Prefix/postfix notation is especially popular for its innate ability to express the intended order of operations without the need for parentheses and other precedence rules, as are usually employed with infix notation. Instead, the notation uniquely indicates which operator to evaluate first. The operators are assumed to have a fixed arity each, and all necessary operands are assumed to be explicitly given. A valid prefix expression always starts with an operator and ends with an operand. Evaluation can either proceed from left