Perfect Supercompilation Master’s Thesis

up By the induction hypothesis P fulfil the requirements. Let be a node in P’s frontier that such that P( ) = ht;Ri is non-trivial, and let P( ) = ht 0;R 0i be a non-trivial ancestor such that t 0 t. Let t 0 u t = (s; fx1:=s1; : : : ; xn :=sng; ). Since t 0 6$ t, s is not a variable. Also, since t 0 6 lt, then sfx1:= s1; : : : ; xn := sng 6 : = s. Hence the abstract operation is welldefined and results in that the subtree at is replaces by a let-term and the restriction set R 0. Since is now placed in the frontier, Q fulfil the requirements. 2 We will now see show that our supercompiler indeed terminates on one of the canonical examples. Example 87 (Accumulating Parameter Terminates) data List(a) = CONS(a; List(a)) j NIL reverse(xs) = rev(xs; NIL) rev(NIL; ys) = ys rev(CONS(x; xs); ys) = rev(xs; CONS(x; ys)) Supercompiling the term reverse(vs)will proceed as follows. Let nodes in the frontier be marked by double nodes and locally unfolded nodes be indicated as dashed nodes. Before the first generalisation occurs, the process will look like this: _ _ _ _ _ M M _ _ _ _ _ reverse(vs) G F E D @ A B C rev(vs; NIL) vs=[ ] yy vs=u:us G F E D @ A B C NIL G F E D @ A B C ? > = < 8 9 : ; rev(us; CONS(u; NIL)) us=[ ] xx us=u 0:us 0 G F E D @ A B C CONS(u; NIL) G F E D @ A B C rev(us 0; CONS(u 0; CONS(u; NIL))) When the frontier to the right is selected, it is discovered that it embeds the parent. Since it is not an instance of the parent and they share common structure, an upwards abstraction is made. _ _ _ _ _ M M _ _ _ _ _ reverse(vs) G F E D @ A B C ? > = < 8 9 : ; let ys=NIL in rev(vs; ys)