Function Lfp Applies a Function F to an Argument 5.6 Implementation 5.2 Finding Function K 4.4 Implementation 4.2 Generating Trees Fixed Points Catamorphisms on Rose-trees 2.1 Implementation Lattices and Cpo's Parse Trees Bottom-up Grammar Analysis | a Functional Formulation |

x until it reaches a xed point. In order to determine whether an argument is a xed point, f x is compared with x. In the case of grammar analysis problems, the rst components of the elements of the grammar analysis problems are always the nonterminals of the given grammar. It follows that the rst components of the elements of f x and x are always equal, and that equality depends just on the second components of the problems. The rst condition of function lfp may be replaced by map snd x = map snd (f x). 6 Conclusions This paper discusses bottom-up grammar analysis problems. We give a very general specii-cation of bottom-up grammar analysis problems, and from this speciication we derive, by means of program transformation applying laws to the components of the intermediate expressions, an algorithm for performing bottom-up grammar analysis. The driving force in the derivation of the algorithm is the construction of a xed point. To obtain such a xed point a number of conditions have to be imposed upon the components of the bottom-up grammar analysis problem. Thus we derive both the algorithm and the conditions under which the xed point exists in one go. The derivation is an example of a derivation of a real-world program, which would have been diicult to obtain without a derivation. The research reported on in this paper is still in progress: in the next version we want to split the calculation in two parts. The rst part of the derivation assumes that the function that computes the property of a parse tree is a Rosetree catamorphism and the second part of the derivation adds, if necessary , the extra information (for example in the case of rsts, where we use information about the empties). This simpliies the derivation. Future research will be directed towards the derivation of an algorithm for top-down grammar analysis. =) sen)) x = deenition of qn (see below), and p qn nt (p x) Function qn is deened by qn nt x = and x Furthermore, we have to show that ^, the operator of the reduction for and, distributes over _, that is, a ^ (b _ c) = (a ^ b) _ (a ^ c) (a _ b) ^ c = (a ^ c) _ (b ^ c) and that false is a zero of ^. These equalities hold for false, …