number-theory

8C The search algorithm1238C6.1 Example (cf. Example 8B3.) Applying the search algorithm to the typeproduces the following sets:SI(T, 0) = {VTj,T - (a-*b--*c)-+(a--+b)->a--+c{1x1-b-cxz-bx3.xa-b-cVl Vzd(T,2) = l 1 2 3 1 3( 2a-.b-.c a-.b a, a-b-c a a-b a(T, 3) = 1x2 x3 x1 x3(x2 x3)}8C6.2 Example (cf. Example 8B4.) Applying the search algorithm to the typeT =_ a-+ ... --+a-+a(with m+1 a's) produces the following sets:sd(T,0) = {VT},(T,1) = {(2xi ... xm'xi), ... , (Axj ... xmxm)}.8C6.3 Example (cf. Example 8B6.) Applying the search algorithm to the typeT((a-+b)-+a)->aproduces the following sets:,%/(T,0) = {VT},,V(T 1)d(T, 2) = 0.= {Ax(a-.b)-.a,x(a-+b)-ava_.b}1 1 18C6.4 Exercise* List members of the sets sad(T,0),szl(T,1).... for all the types inTable 8B7a.8C6.5 Comments (i) Besides Ben-Yelles' 1979 search algorithm there are othersin Zaionc 1985 and 1988 and in Takahashi, Akama and Hirokawa 1994. Alsoprocedures essentially equivalent to the main parts of 8C6 are used in some decisionalgorithmsfor provability in intuitionist logic, cf. 6B7.3, and in algorithms that seekto unify pairs of typed A-terms, for example the original one in Huet 1975 §3.(ii) The algorithm in 8C6 can be viewed as a context-free grammar for generatinga language whose expressions are exactly the inhabitants of T. In the standardnotation for context-free grammars (see for example Hopcroft and Ullman 1979Ch. 4) the meta-variables in 8C6 would be called non-terminal variables and eachsuitable replacement defined in Subsubstep Ila of 8C6 would determine a productionin the grammar. The set of all such productions need not be finite, however, so theoriginal definition of "grammar" must be relaxed if this view is to be made precise.This view was taken in Huet 1975 §3, Zaionc 1985 §3 and 1988 §5, and was usedand analysed in detail in Takahashi, Akama and Hirokawa 1994.(iii) A very smooth description of a search algorithm in terms of solving sets ofpolynomial equations over the domain {0, 1, 2, ... , co} is in Zaionc 199-.