Although the automatic sequences form a large and interesting class, one drawback is that they need to take their values in a finite set. But many interesting sequences, such as (s2(n))n.0 (counting the sum of the bits in the base-2 representation of n), take their values in N (or Z, or any semiring). We would like to find a generalization that allows this.

Up to now our logical formulas have allowed us to state formulas concerning a single automatic sequence, or perhaps two at the same time (as in Section 8.9.3). In some cases, however, it’s possible to use our approach to prove results about infinitely many sequences (even uncountably many sequences) at once!

A typical modern computational system is structured like a tower, with eachlayer’s proper behavior contingent on the correctness of the onebelow. The website that you use to send money to a friend relies on both astack of networking protocols (HTTP relying on TCP, which is relying on IP,etc.), as well as a stack of applications on your computer or your phone(your browser relying on your operating system, which is relying on thehardware itself). A key theme in computer science is this idea ofabstraction: that, so long as it’s workingproperly, you can rely on the next layer in one of these towers (or afunction in a large program, or . ..) without worryingabout how exactly it works. You just have to trustthat it works.

Imagine converting a color photograph to grayscale (as in Figure 2.1).Implementing this conversion requires interacting with a slew offoundational data types (the basic “kinds of things”) thatshow up throughout CS. A pixel is a sequence of three colorvalues, red, green, and blue. (And an image is a two-dimensional sequence ofpixels.)

This book is designed for an undergraduate student who has taken a computerscience class or three. Most likely, you are a sophomore or juniorprospective or current computer science major taking your firstnon-programming-based CS class. If you are a student in this position, youmay be wondering why you’re taking this class (or why youhave to take this class!).

In which our heroes encounter many choices, some of which may lead them tolive more happily than others, and a precise count of their number ofoptions is calculated.

Imagine writing a program to implement a student registration system at acollege or university. When a student is registering for classes,you’ll need to be able to answer questions of the form “isAlice eligible to be added to the roster for Price Theory?” to decidewhether to allow her to click to add that particular course. To do so,you’ll need to know Price Theory’s prerequisites: what classesmust you have already passed before you can take Price Theory?

I often say that when you can measure what you are speaking about, andexpress it in numbers, you know something about it; but when you cannotmeasure it, when you cannot express it in numbers, your knowledge is of ameagre and unsatisfactory kind.

This chapter introduces probability, the study ofrandomness. Our focus, as will be no surprise by this point of the book, ison building a formal mathematical framework for analyzing random processes.We’ll begin with a definition of the basics of probability: defininga random process that chooses one particular outcome from aset of possibilities (any one of which occurs some fraction of the time).We’ll then analyze the likelihood that a particularevent occurs—in other words, asking whether thechosen outcome has some particular property that we care about. We thenconsider independence and dependence ofevents, and conditional probability: how, if at all, doesknowing that the randomly chosen outcome has one particular property changeour calculation of the probability that it has a different property?