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INTRODUCTION AND PRELIMINARIES use visual studio .net uss code 39 implement todevelop 3 of 9 with .net Microsoft Office Development. Microsoft Office 2000/2003/2007/2010 De nition 1.2 (solving visual .net barcode 39 a decision problem): Let S {0, 1} .

A function f : {0, 1} {0, 1} solves the decision problem of S (or decides membership in S) if for every x it holds that f (x) = 1 if and only if x S. We often identify the decision problem of S with S itself, and identify S with its characteristic function (i.e.

, with the function S : {0, 1} {0, 1} de ned such that S (x) = 1 if and only if x S). Note that if f solves the search problem of R then the Boolean def function f : {0, 1} {0, 1} de ned by f (x) = 1 if and only if f (x) = solves the decision problem of {x : R(x) = }. Re ection.

Most people would consider search problems to be more natural than decision problems: Typically, people seek solutions more than they stop to wonder whether or not solutions exist. De nitely, search problems are not less important than decision problems; it is merely that their study tends to require more cumbersome formulations. This is the main reason that most expositions choose to focus on decision problems.

The current book attempts to devote at least a signi cant amount of attention also to search problems.. 1.2.2.

3. Promise Probl ems (an Advanced Comment) Many natural search and decision problems are captured more naturally by the terminology of promise problems, in which the domain of possible instances is a subset of {0, 1} rather than {0, 1} itself. In particular, note that the natural formulation of many search and decision problems refers to instances of a certain type (e.

g., a system of equations, a pair of numbers, a graph), whereas the natural representation of these objects uses only a strict subset of {0, 1} . For the time being, we ignore this issue, but we shall revisit it in Section 2.

4.1. Here we just note that, in typical cases, the issue can be ignored by postulating that every string represents some legitimate object (e.

g., each string that is not used in the natural representation of these objects is postulated as a representation of some xed object). 1.

2.3. Uniform Models (Algorithms).

Science is One. Laci Lov sz (according to Silvio Micali, ca. 1990) a We nally reach the he art of the current section (Section 1.2), which is the de nition of uniform models of computation. We are all familiar with computers and with the ability of computer programs to manipulate data.

This familiarity seems to be rooted in the positive side of computing; that is, we have some experience regarding some things that computers can do. In contrast, Complexity Theory is focused at what computers cannot do, or rather with drawing the line between what can be done and what cannot be done. Drawing such a line requires a precise formulation of all possible computational processes; that is, we should have a clear model of all possible computational processes (rather than some familiarity with some computational processes).

. 1.2.3.

1. Overview and ANSI/AIM Code 39 for .NET General Principles Before being formal, let we offer a general and abstract description, which is aimed at capturing any arti cial as well as natural process.

Indeed, arti cial processes will be associated with computers, whereas by natural processes we mean (attempts to. 1.2. COMPUTATIONAL TASKS AND MODELS model) the mechanical barcode 39 for .NET aspects of the natural reality (be it physical, biological, or even social). A computation is a process that modi es an environment via repeated applications of a predetermined rule.

The key restriction is that this rule is simple: In each application it depends on and affects only a (small) portion of the environment, called the active zone. We contrast the a priori bounded size of the active zone (and of the modi cation rule) with the a priori unbounded size of the entire environment. We note that, although each application of the rule has a very limited effect, the effect of many applications of the rule may be very complex.

Put in other words, a computation may modify the relevant environment in a very complex way, although it is merely a process of repeatedly applying a simple rule. As hinted, the notion of computation can be used to model the mechanical aspects of the natural reality, that is, the rules that determine the evolution of the reality (rather than the speci c state of the reality at a speci c time). In this case, the starting point of the study is the actual evolution process that takes place in the natural reality, and the goal of the study is nding the (computation) rule that underlies this natural process.

In a sense, the goal of science at large can be phrased as nding (simple) rules that govern various aspects of reality (or rather one s abstraction of these aspects of reality). Our focus, however, is on arti cial computation rules designed by humans in order to achieve speci c desired effects on a corresponding arti cial environment. Thus, our starting point is a desired functionality, and our aim is to design computation rules that effect it.

Such a computation rule is referred to as an algorithm. Loosely speaking, an algorithm corresponds to a computer program written in a high-level (abstract) programming language. Let us elaborate.

We are interested in the transformation of the environment as affected by the computational process (or the algorithm). Throughout (most of) this book, we will assume that, when invoked on any nite initial environment, the computation halts after a nite number of steps. Typically, the initial environment to which the computation is applied encodes an input string, and the end environment (i.

e., at the termination of the computation) encodes an output string. We consider the mapping from inputs to outputs induced by the computation; that is, for each possible input x, we consider the output y obtained at the end of a computation initiated with input x, and say that the computation maps input x to output y.

Thus, a computation rule (or an algorithm) determines a function (computed by it): This function is exactly the aforementioned mapping of inputs to outputs. In the rest of this book (i.e.

, outside the current chapter), we will also consider the number of steps (i.e., applications of the rule) taken by the computation on each possible input.

The latter function is called the time complexity of the computational process (or algorithm). While time complexity is de ned per input, we will often consider it per input length, taking the maximum over all inputs of the same length. In order to de ne computation (and computation time) rigorously, one needs to specify some model of computation, that is, provide a concrete de nition of environments and a class of rules that may be applied to them.

Such a model corresponds to an abstraction of a real computer (be it a PC, mainframe, or network of computers). One simple abstract model that is commonly used is that of Turing machines (see, 1.2.

3.2). Thus, speci c algorithms are typically formalized by corresponding Turing machines (and their time complexity is represented by the time complexity of the corresponding Turing machines).

We stress, however, that most results in the theory of computation hold regardless of the speci c computational model used, as long as it is reasonable .
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