A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 17.3 Impossible Problems 587another function combine that takes a program P and an input string I asparameters. Function combine modifies P to store I as a static variable Sand further modifies all calls to input functions within P to instead get theirinput from S. Call the resulting program P ′ . It should take no stretch of theimagination to believe that any decent compiler could be modified to takecomputer programs and input strings and produce a new computer programthat has been modified in this way. Now, take P ′ and feed it to Ehalt. IfEhalt says that P ′ will halt, then we know that P would halt on input I.In other words, we now have a procedure for solving the original HaltingProblem. The only assumption that we made was the existence of Ehalt.Thus, the problem of determining if a program will halt on no input mustbe unsolvable.✷Example 17.5 For arbitrary program P, does there exist any input forwhich P halts?Proof: This problem is also uncomputable. Assume that we had a functionAhalt that, when given program P as input would determine if there issome input for which P halts. We could modify our compiler (or writea function as part of a program) to take P and some input string w, andmodify it so that w is hardcoded inside P, with P reading no input. Call thismodified program P ′ . Now, P ′ always behaves the same way regardless ofits input, because it ignores all input. However, because w is now hardwiredinside of P ′ , the behavior we get is that of P when given w as input. So, P ′will halt on any arbitrary input if and only if P would halt on input w. Wenow feed P ′ to function Ahalt. If Ahalt could determine that P ′ haltson some input, then that is the same as determining that P halts on input w.But we know that that is impossible. Therefore, Ahalt cannot exist. ✷There are many things that we would like to have a computer do that are unsolvable.Many of these have to do with program behavior. For example, provingthat an arbitrary program is “correct,” that is, proving that a program computes aparticular function, is a proof regarding program behavior. As such, what can beaccomplished is severely limited. Some other unsolvable problems include:• Does a program halt on every input?• Does a program compute a particular function?• Do two programs compute the same function?• Does a particular line in a program get executed?