If "'L "'is finitely axiomatizable ( and has a recursive set of recursive rules ) and has the fmp, then it is decidable.
12.
A formula is either a term or defined by the recursive rule : if ? and ? are formulas, then ?? and ~ ( ? ) are likewise formulas.
13.
At each step in the derivation except the first and last, we used the self-recursive rules A \ to aA, B \ to bB, C \ to cC.
14.
If we had not use the recursive rules, instead using, say, A \ to a, B \ to bB, C \ to cC, where one of the rules is not self-recursive, the number of non-terminals would have decreased to 2, thus making the string unable to be derived further because it would have too few non-terminals to be rewritten.
15.
The first workshop on Approaches and Applications of Inductive Programming ( AAIP ) held in conjunction with ICML 2005 identified all applications where " learning of programs or recursive rules are called for, [ . . . ] first in the domain of software engineering where structural learning, software assistants and software agents can help to relieve programmers from routine tasks, give programming support for end users, or support of novice programmers and programming tutor systems.
