The complexity class P plays an important role in our theory. • Concrete enough to rule out unphysical (exponential) complexity. • Abstract enough to be robust (Extended Church Turing Thesis).

In this paper, we address a collection of state space reachability problems, for linear time-invariant systems, using a minimal number of actuators. In particular, we design a zero-one diagonal...

We begin with a discussion of tasks, machine models, and resource measures, after which we examine serial complexity classes and relationships among them. Complete problems are defined and the P-complete, NP-complete, and PSPACE-complete problems are examined.

In this chapter we de ne many complexity classes and describe natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes. Given a problem, how hard is it?

Let us talk complexity classes. We are interested in classifying func-tions according to their complexity, so it makes sense to lump func-tions into sets of similar complexity: Definition 1. Define DTIME(t(n)) to be the set of functions. DTIME(t(n)) = { f : {0, 1}∗ → {0, 1}| f is computable in time O(t(n))}. Definition 2.

Computational Classes The Euler diagram below illustrates the complexity classes that we discuss in CS 456/656. The diagram shows the widely believed relationshop between the complexity classes. Some of the classes are defined by grammar classes, and others are defined by computational complexity.

To study complexity, we need to answer these questions: •What does “complexity” even mean? •What is an “efficient” solution to a problem?

Dec 16, 2014 · Is there a good way of using the Venn Diagram shown below to tackle these kind of problem? For example, how can I prove that If $A$ is an NP-complete problem, $B$ is reducible to $A$, then $B$ can be NP-hard using the above diagram?

Descriptive Complexity is an attempt to study the complexity of problems and classify them, not on the basis of how difficult it is to compute solutions, but on the basis of how difficult it is to describe the problem. This gives an alternative way to study complexity, independent of particular machine models. Based on definability in logic.

Introduction to Complexity Classes Definition The class P (aka PTIME, aka DTIME) is the class of problems for which there is a number k and an algorithm which solves the problem and whose running time is O(nk) where n is the size of the instance of the problem. These problems are also called polynomial-time problems.