By J. Roger Hindley
Style thought is likely one of the most crucial instruments within the layout of higher-level programming languages, comparable to ML. This booklet introduces and teaches its suggestions via concentrating on one rather neat method and learning it intimately. via focusing on the foundations that make the speculation paintings in perform, the writer covers all of the key principles with out becoming concerned within the issues of extra complex structures. This booklet takes a type-assignment method of variety thought, and the procedure thought of is the best polymorphic one. the writer covers the entire uncomplicated rules, together with the system's relation to propositional common sense, and offers a cautious remedy of the type-checking set of rules that lies on the center of each such method. additionally featured are different fascinating algorithms that previously were buried in inaccessible technical literature. The mathematical presentation is rigorous yet transparent, making it the 1st publication at this point that may be used as an advent to variety idea for machine scientists.
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Additional info for Basic Simple Type Theory
The first informal use of a PT algorithm in combinatory logic or A-calculus was probably due to Haskell Curry in the 1950's. 284-293) and show very clearly the equation-solving method on which several later formal algorithms were based. But, as mentioned earlier, Curry was mainly interested in stronger systems at that time and only afterwards began to look at simple type-assignment on its own. When he did, he wrote up his earlier methods as a formal equation-solving PT algorithm for TAc and added a proof of its correctness; these survive in his notes Curry 1966 and were published in Curry 1969.
U Fn- To prove (i): by part (i) of the induction hypothesis every type in a Ai occurs in p; or F, and hence in F; also the type of y occurs in F. Hence (i) holds. To prove (ii): the argument above shows that A' must use the same rules at the same positions as in A. ,Pn are determined by the type of y. 1 Note (Subformula property) Part (i) of 2B3 corresponds to what is usually called in logic the subformula property; this says that in a Natural Deduction system every formula in an irreducible deduction occurs in either the conclusion or an undischarged assumption.
Further, sp can be shown to be the leftmost symbol of a unique subtype p" of p. ) Similarly to is the leftmost symbol of a unique subtype v" of v. Choose one of p", v" that is a variable and call it "a". tn = SI ... Smtm+l ... tn with it > m. This is left as a (rather dull) exercise for the reader. 2 Note (History) As mentioned earlier, the above unification algorithm is due to J. A. Robinson, see Robinson 1965 §5; it was the first to be published complete with a correctness-proof. 2 Another early unification algorithm is implicit in Maslov 1964.
Basic Simple Type Theory by J. Roger Hindley