the x's by means of many recursive calls to X, and then a b, and returns to S and finally recognizes a d. The key concept here is inherent in the phrase again descends into X. The process of looking forward, failing, backing up, and then retrying the next alternative is known in parsing as backtracking, and it is primarily backtracking that presents opportunities for memoization in parsing. Consider a function RuleAcceptsSomeInput(Rule, Position, Input), where the parameters are as follows: Rule is the name of the rule under consideration. Position is the offset currently under consideration in the input. Input is the input under consideration. Let the return value of the function RuleAcceptsSomeInput be the length of the input accepted by Rule, or 0 if that rule does not accept any input at that offset in the string. In a backtracking scenario with such memoization, the parsing process is as follows: When the rule A descends into X at offset 0, it memoizes the length 5 against that position and the rule X. After having failed at d, B then, rather than descending again into X, queries the position 0 against rule X in the memoization engine, and is returned a length of 5, thus saving having to actually descend again into X, and carries on as if it had descended into X as many times as before. In the above example, one or many descents into X may occur, allowing for strings such as xxxxxxxxxxxxxxxxbd. In fact, there may be any number of x's before the b. While the call to S must recursively descend into X as many times as there are x's, B will never have to descend into X at all, since the return value of RuleAcceptsSomeInput(X, 0, xxxxxxxxxxxxxxxxbd) will be 16 (in this particular case). Those parsers that make use of syntactic predicates are also able to memoize the results of predicate parses, as well, thereby reducing such constructions as: S → (A)? A A → /* some rule */ to one descent into A. If a parser builds a parse tree during a parse, it must memoize not only the length of the input that matches at some offset against a given rule, but also must store the sub-tree that is generated by that rule at that offset in the input, since subsequent calls to the rule by the parser will not actually descend