Complexity Classes

The purposes of complexity theory are to ascertain the amount of computational resources required to solve important computational problems, and to classify problems according to their difficulty. The resource most often discussed is computational time, although memory (space) and circuitry (or hardware) have also been studied. The main challenge of the theory is to prove lower bounds, i.e., that certain problems cannot be solved without expending large amounts of resources. Although it is easy to prove that inherently difficult problems exist, it has turned out to be much more difficult to prove that any interesting problems are hard to solve. There has been much more success in providing strong evidence of intractability, based on plausible, widely held conjectures. In both cases, the mathematical arguments of intractability rely on the notions of reducibility and completeness—which are the topics of Chapter 23. Before one can understand reducibility and completeness, however, one must grasp the notion of a complexity class—and that is the topic of this chapter. First, however, we want to demonstrate that complexity theory really can prove—to even the most skeptical practitioner—that it is hopeless to try to build hardware or programs that solve certain problems. As our example, we consider the manufacture and testing of logic circuits and communication protocols. Many problems in these domains are solved by building a logical formula over a certain vocabulary, and then determining whether the formula is logically valid, or whether counterexamples (that is, bugs) exist. The choice of vocabulary for the logic is important here, as the next paragraph illustrates. One particular logic that was studied by Stockmeyer and Meyer [42] is called WS1S. (We need not be concerned with any details of this logic.) They showed that any circuit that takes as input a formula with up to 610 symbols and produces as output a correct answer saying whether the formula is valid, requires at least 10125 gates. The 610 symbols can be encoded within 458 bytes, i.e., fewer than 6 lines of ASCII text. According to Stockmeyer [43], “Even if gates were the size of a proton and were connected by infinitely thin wires, the network would densely fill the known universe.” Of course, Stockmeyer’s theorem holds for one particular sort of circuitry, but the awesome size of the lower bound makes it evident that, no matter how innovative the architecture, no matter how clever the software, no computational machinery will enable us to solve the validity problem in this logic. For the practitioner testing validity of logical formulas, the lessons are (1) be careful with the choice of the logic, (2) use small formulas, and often (3) be satisfied with something less than full validity testing. In contrast to this result of Stockmeyer, most lower bounds in complexity theory are stated asymptotically. For example, one might show that a particular problem requires time Ω(t(n)) to solve on a Turing machine, for some rapidly growing function t. For the Turing machine model, no