# Computing Patterns in Strings by William Smyth

By William Smyth

The computation of styles in strings is a basic requirement in lots of components of technology and knowledge processing. The operation of a textual content editor, the lexical research of a working laptop or computer software, the functioning of a finite automaton, the retrieval of data from a database - those are all actions that may require that styles be positioned and computed. In different parts of technology, the algorithms that compute styles have purposes in such varied fields as information compression, cryptography, speech acceptance, laptop imaginative and prescient, computational geometry and molecular biology.

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In the present section we will apply these results to study the question of whether there exist inherently sequential problems. Although this question is still (as of 1991) open, we can study it in a manner reminiscent of the theory of NP-complete problems. As is done there, we restrict our attention to a particular class of problems known as decision problems. These are computations that produce a boolean 0 or 1 as their result. It will turn out that there are reasonable candidates for inherently sequential problems even among this restricted subset.

18. When performing the computations on nodes of depth k, the order of the computations is irrelevant. This is due to the definition of depth — it implies that the output of any vertex of depth k is input to a vertex of strictly higher depth (since depth is the length of the longest path from an input vertex to the vertex in question). The simulation of computations at depth k proceeds as follows: 1. Processors read the data from the output areas of the data-structures for vertices at depth k − 1.

15. The circuit value problem, as defined above, is logspace-complete for P. P ROOF. Let CVP denote the Circuit Value Problem. We must show, that if Z is any problem in P, then Z ∝logspace CVP. We will assume that Z is computed by a Turing machine T that always halts in a number of steps that is bounded by a polynomial of the complexity parameter of the input. We will construct a (fairly large) circuit whose final value is always the same as that computed by T. We will also show that the construction can be done in logspace.