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**Sample text**

The deﬁnition of atoms above can be translated directly into an OCaml function, for example using union for ‘∪’ and [x] for ‘{x}’. However, we prefer to deﬁne it in terms of the existing iterator atom union: let atoms fm = atom_union (fun a -> [a]) fm;; For example: # atoms <

~p \/ (r <=> s)>>;; - : prop list = [P "p"; P "q"; P "r"; P "s"] Because the interpretation of a propositional formula p depends only on the valuation’s action on the ﬁnite (say n-element) set atoms(p), and it can only make two choices for each, the ﬁnal truth-value is completely determined by all 2n choices for those atoms.

Write a simpliﬁer that uses a more intelligent traversal strategy to avoid wasteful evaluation of subterms such as E in 0 · E or E − E. Write a function to generate huge expressions in order to test how much more eﬃcient it is. g. transform (x+1)3 −3·(x+1)2 +3·(2·x−x) into x3 −2. We will eventually develop similar functions in Chapter 5. Many concrete strings with slightly diﬀerent bracketing or spacing correspond to the same abstract syntax tree, so we can’t expect print(parse(s)) = s in general.

As planned, we will implement a parser and prettyprinter to translate between abstract syntax trees and concrete strings (‘x + 0’), setting them up to be invoked automatically by OCaml for input and output of expressions. We model our concrete syntax on ordinary algebraic notation, except that in a couple of respects we will follow the example of computer languages rather than traditional mathematics. We allow arbitrarily long ‘words’ as variables, whereas mathematicians traditionally use mostly single letters with superscripts and subscripts; this is especially important given the limited stock of ASCII characters.