Schema induction

Gick, M., 8c Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1-38. 

Lemme VI 1.2.1. Soient S un schema, X un S-schema localement de presentation finie, B un S-schema localement de presentation finie qui est le graphe d une S-relation d dquivalence sur X (i.e. pour tout S-schema T, B(T) est le graphe d une relation d dquivalence sur X(T), dependant fonctoriellement de T). Alors le faisceau fppf quotient X/B = X commute aux limites inductives filtrantes d anneaux et par suite, si T/R est un schema, X est localement de presentation finie sur S (EGA IV 8.14.2) ... 

A est limite inductive de ses soua-Z-algAbres A de type fini. Le "proc ld brevete" d jA utilise (explicit en long et en large dans EGA IV 8) montre qu il exists un i, un schema affine X sur = Spec(A ) et un ouvert dans X, dont X,U se deduisent par changement de base S —. Soit E la partie... 

Let s now relax the requirement that predicate r be binary. But we keep the (so far implicit) constraint that the induction parameter be simple. Supposing predicate r is -ary (where n is a schema-variable), this new setting implies that Y becomes a vector Y ofn-1 variables Yj. and that vector TY becomes a vector TY of n-l vectors TKy, each of which is a vector of t variables TYji (where j, I are notation-variables). Similarly, HY becomes a vector HY of n-l vectors HYj, each of which is a vector of h j) variables HYji, where h /l is a schema function-variable. Thus ... 

Third, we only aim at the synthesis of single-loop logic algorithms. In other words, we assume that the only loop is the one that is achieved in the schema by the recursion on the induction parameter, and that none of the instances of the predicate-variables is defined recursively (possibly as a divide-and-conquer logic algorithm). 

How to discover compound induction parameters Due to our restriction to version 3 of the divide-and-conquer schema. Task A only considers simple induction parameters. Meeting this challenge is thus considered future research. According to what well-founded relation to decompose the induction parameter Step 3 (Synthesis of Decompose) does this non-deterministically by considering all predefined decomposition operators (which each reflect some well-founded relation) of a typed database, and possibly by listening to the specifier s hints. 

The support of any number of minimal or non-minimal forms and of compound induction parameters is considered future research, as considerable extensions to the already defined tasks need to be developed. Note however that Section 5.2.2 shows that single minimal forms and non-minimal forms are more frequent than one might believe at first sight. As outlined in Section 8.4, there is still a lot of space for designing even more sophisticated divide-and-conquer schemas. 

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