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As a trade off we have to learn and get famiiiax with the new notation. The main points of our notation can be summarized as follows. Function application is written as juxtaposition as in f y. The semicolon is used to break the normal precedence of function application as in f x; g z. Special parentheses ~... are introduced to distinguish syntactic from semantics objects. Attempts are made to have variables named by single letters; we use of roman, greek, and script fonts in upper and lower case.

Xn} is a sequence with elements x l , . •, x~. " is used to denote concatenation. w. If 8 is a sequence ! s I denotes the length of s. *(di.... 4. F i x e d p o i n t s [Fixed points] Let f • D ~ D, then f i x f denotes the least fixed point of f , that is the least element of {gi g == f(g)}. For monotonic f the least fixed point exists; for continuous f the least fixed point is given by fiz f = U { f i } where f0 = A_, f i + l = f(fO. 1. F i x e d p o i n t i n d u c t i o n Suppose we are given an object which is defined as the least fixed point of a function, say f = f i x h.

Given the above theory, the implementation of the semantic analysis phase is straightforward. 6. 1. N e c e s s a r y t h e o r y We choose a fairly low level definition of the source language. For example, addressing is done relative to base addresses. For new procedure invocations new base addresses are set up. A display mechanism describes how memory of outer scopes is to be accessed. These concepts are familiar to the compiler writer and are well suited as our specification. For some statements of LS we give a modified definition which is better suited for the verification of the code generation.