hi this chapter we change our direc

hi this chapter we change our direction significantly. Until now, we have been interested primarily in simple classes of languages and the ways that they can be used for relatively constrained problems, such as analyzing protocols, search¬ing text, or parsing programs. Now, we shall start looking at the question of what languages can be defined by any computational device whatsoever. This question is tantamount to the question of what computers can do, since recog¬nizing the strings in a language is a formal way of expressing any problem, and solving a problem is a reasonable surrogate for what it is that computers do.
We begin with an informal argument, using an assumed knowledge of C programming, to show that there are specific problems we cannot solve using a computer. These problems are called “undecidable.” We then introduce a venerable formalism for computers, called the Turing machine. While a Turing machine looks nothing like a PC, and would be grossly inefficient should some startup company decide to manufacture and sell them, the Turing machine long has been recognized as an accurate model for what any physical computing device is capable of doing.
In Chapter 9, we use the Turing machine to develop a theory of “undecid¬able” problems, that is, problems that no computer can solve. We show that a number of problems that are easy to express are in fact undecidable. An ex¬ample is telling whether a given grammar is ambiguous, and we shall see many others.
Problems That Computers Cannot Solve
The purpose of this section is to provide an informal, C-programming-based introduction to the proof of a specific problem that computers cannot solve. The particular problem we discuss is whether the first thing a C program prints
is hello, world. Although we might imagine that simulation of the program would allow us to tell what the program does, we must in reality contend with programs that take an unimaginably long time before making any output at all. This problem — not knowing when, if ever, something will occur — is the ultimate cause of our inability to tell what a program does. However, proving formally that there is no program to do a stated task is quite tricky, and we need to develop some formal mechanics. In this section, we give the intuition behind the formal proofs.