George Boole and Claude Shannon

This chapter presents brief biographical sketches of George Boole and Claude Shannon. George was born in Lincoln, a town in the north of England, on November 2, 1815. His father John, while simple tradesman (a cobbler), taught George geometry and trigonometry, subjects John had found of great aid in his optical studies. Boole was essentially self-taught, with a formal education that stopped at what today would be a junior in high school. Eventually he became a master mathematician (who succeeded in merging algebra with logic), one held in the highest esteem by talented, highly educated men who had graduated from Cambridge and Oxford. Claude was born on April 30, 1916, in Petoskey, Michigan. He enrolled at the University of Michigan, from which he graduated in 1936 with double bachelor's degrees in mathematics and electrical engineering. It was in a class there that he was introduced to Boole's algebra of logic.

Introduction

This introductory chapter considers the work of mathematician George Boole (1815–1864), whose book An Investigation of the Laws of Thought (1854) would have a huge impact on humanity. Boole's mathematics, the basis for what is now called Boolean algebra, is the subject of this book. It is also called mathematical logic, and today it is a routine analytical tool of the logic-design engineers who create the electronic circuitry that we now cannot live without, from computers to automobiles to home appliances. Boolean algebra is not traditional or classical Aristotelian logic, a subject generally taught in college by the philosophy department. Boolean algebra, by contrast, is generally in the hands of electrical engineering professors and/or the mathematics faculty.

Epilogue

This chapter present the author's vision of the sort of logical problem that may soon be one that even a quantum computer would find a struggle to deal with—the decipherment of entangled legalese, the sort of monstrous gobbledegook one finds, for example, in the increasingly convoluted IRS tax code. In the form of a short story, that vision is “The Language Clarifier,” which first appeared in the May 1979 issue of Omni Magazine. The original title was “The Anti-Amphibological Machine,” but Omni's fiction editor thought that a tad too mystifying for readers and “suggested” the change.

What You Need to Know to Read This Book

This chapter details the background knowledge needed to read this book. Specifically, it assumes some knowledge of mathematics and electrical physics and an appreciation for the value of analytical reasoning—but no more than a technically minded college-prep high school junior or senior would have. In particular, the math level is that of algebra, including knowing how matrices multiply. The electrical background is simple: knowing (1) that electricity comes in two polarities (positive and negative) and that electrical charges of like polarity repel and of opposite polarity attract; and (2) understanding Ohm's law for resistors (that the voltage drop across a resistor in volts is the current through the resistor in amperes times the resistance in ohms) and the circuit laws of Kirchhoff (that the sum of the voltage drops around any closed loop is zero, which is an expression of the conservation of energy; the sum of all the currents into any node is zero).

Beyond Boole and Shannon

Boole and Shannon never studied the physics of computation. Obviously Boole simply could not have, as none of the required physics was even known in his day, and Shannon was nearing the end of his career when such considerations were just beginning. And yet, both Boole's algebra and Shannon's information concepts to make many of our calculations. This chapter touches on how fundamental physics—the uncertainty principle from quantum mechanics, and thermodynamics, for example—constrain what is possible, in principle, for the computers of the far future. It argues that while there are indeed finite limitations, present-day technology falls so far short of those limits that there will be good employment for computer technologists for a very long time to come.

Turing Machines

This chapter discusses Turing machines. A Turing machine is the combination of a sequential, finite-state machine plus an external read/write memory storage medium called the tape (think of a ribbon of magnetic tape). The tape is a linear sequence of squares, with each square holding one of several possible symbols. The Turing machine's power to compute comes from its tape, for two reasons. First, Turing was the first to conceive of the idea of a stored program that could be changed by the operation of the machine itself. The program, and its input data, exist together on the tape as sequences of symbols. Second, because of the arbitrarily long length of the tape, a Turing machine has the ability to “remember” what has happened in the arbitrarily distant past.

Some Combinatorial Logic Examples

The entire point of Shannon's 1948 “A Mathematical Theory of Communication” was to study the theoretical limits on the transmission of information from point A (the source) to point B (the receiver) through an intervening medium (the channel). The information is imagined first to be encoded in some manner before being sent through the channel. Shannon considers two distinct types of channels: the so-called continuous channel that would carry, for example, a continuous signal like the human voice; and the so-called discrete channel that would carry, again for example, a keyboard's output in the form of a digital stream of bits. This chapter focuses on this second case. In a perfect world the digital stream would arrive at the receiver exactly as it was sent, but in the real world the channel is noisy and so, occasionally, a bit will arrive in error.

Boole, Shannon, and Probability

George Boole and Claude Shannon shared a deep interest in the mathematics of probability. Boole's interest was, of course, not related to the theory of computation—he was a century too early for that—while Shannon's mathematical theory of communication and information processing is replete with probabilistic analyses. There is, nevertheless, an important intersection between what the two men did, which is shown in this chapter. The aim is to provide a flavor of how they reasoned and of the sort of probabilistic problem that caught their attention. Once we have finished with Boole's problem, the reader will see that it uses mathematics that will play a crucial role in answering Shannon's concern about “crummy” relays.

Boolean Algebra

This chapter discusses how Boole cast logic into algebraic form. Boole was interested in symbolic analysis years before he wrote his Laws of Thought. In fact, others before him—in particular, the German mathematician Gottfried Wilhelm Leibniz (1646–1716)—had pursued a similar goal of reducing logic to algebra, but it was Boole who finally succeeded. What Boole described in his books is not exactly what modern users call Boolean algebra, but nevertheless it is from Boole that the modern presentation springs. The chapter first describes the essence of what Boole did using the language of sets. It then discusses Boole's algebra of sets, examples of Boolean analysis, and visualizing Boolean functions.

Sequential-State Digital Circuits

This chapter begins by discussing two sequential state problems. The first example drives home the point that the concept of a physical system changing state with time predates Shannon and Boole by centuries. The second example uses a more modern situation that just about everyone of driving age has encountered: the automated parking garage. It then argues that the design of a sequential state digital machine is simply that of determining how to interconnect these individual latches so as to transition from state to state with the goal of accomplishing a desired task. The remainder of the chapter shows how the latch can be modified in several important ways to arrive at the practical logic element called the clocked, edge-triggered flip-flop, and concludes with the design of a specific machine.