Chapter 12: The Principle of Computational Equivalence

Section 8: Undecidability and Intractability

Lower bounds [on computational complexity]

If one could prove for example that P!=NP then one would immediately have lower bounds on all NP-complete problems. But absent such a result most of the general lower bounds known so far are based on fairly straightforward information content arguments. One cannot for example sort n objects in less than about n steps since one must at least look at each object, and one cannot multiply two n-digit numbers in less than about n steps since one must at least look at each digit. (As it happens the fastest known algorithms for these problems require very close to n steps.) And if the output from a computation can be of size 2n then this will normally take at least 2n steps to generate. Subtleties in defining how big the input to a computation really is can lead to at least apparently exponential lower bounds. An example is testing whether one can match all possible sequences with a regular expression that involves s-fold repetitions. It is fairly clear that this cannot be done in less than about s steps. But this seems exponentially large if s is specified by its digit sequence in the original input regular expression. Similar issues arise in the problem of determining truth or falsity in Presburger arithmetic (see page 1152).

Diagonalization arguments analogous to those on pages 1128 and 1162 show that in principle there must exist functions that can be evaluated only by computations that exceed any given bound. But actually finding examples of such functions that can readily be described as having some useful purpose has in the past seemed essentially impossible.

If one sufficiently restricts the form of the underlying system then it sometimes becomes possible to establish meaningful lower bounds. For example, with deterministic finite automata (see page 957), there are sequences that can be recognized, but only by having exponentially many states. And with DNF Boolean expressions (see page 1096) functions like Xor are known to require exponentially many terms, even—as discovered in the 1980s—if any limited number of levels are allowed (see page 1096).

From Stephen Wolfram: A New Kind of Science [citation]