1. Show that P is closed under *. Recall that ** ("crane") is the operation on sets defined by The Class P *Definition: P the set of all languages that can be decided by a TM in polynomial time in the input size. * Formally: PUDTIME) * The class P the class of efficiently decidable languages *Properties: *Model independent: can replace TM with any model *Composition: if an efficient program M calls an efficient subroutine M' then the whole procedure is efficient. * Definition: Efficient Polynomial Time in the input size 1. Show that P is closed under *. Recall that ** ("crane") is the operation on sets defined by The Class P *Definition: P the set of all languages that can be decided by a TM in polynomial time in the input size. * Formally: PUDTIME) * The class P the class of efficiently decidable languages *Properties: *Model independent: can replace TM with any model *Composition: if an efficient program M calls an efficient subroutine M' then the whole procedure is efficient. * Definition: Efficient Polynomial Time in the input size