Boolean logic

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Boolean logic

(mathematics)

Boolean logic

The "mathematics of logic," developed by English mathematician George Boole in the mid-19th century. Its rules govern logical functions (true/false) and are the foundation of all electronic circuits in the computer. As add, subtract, multiply and divide are the primary operations of arithmetic, AND, OR and NOT are the primary operations of Boolean logic. Boolean logic is turned into logic gates on the chip, and the logic gates make up logic circuits that perform functions such as how to add two numbers together.

Various permutations of AND, OR and NOT are used, including NAND, NOR, XOR and XNOR. The rules, or truth tables, for AND, OR and NOT follow. See Boolean search, binary, logic gate and Bebop to the Boolean Boogie.


An AND Gate (Wired in Series)
AND requires both inputs to be present in order to provide output. When both inputs pulse both switches closed, current flows from the source to the output.



Curious About the Chip?


Wired in patterns of Boolean logic and in less space than a postage stamp, transistors in one of today's high-speed chips collectively open and close quadrillions of times every second. If you are curious about how it really works down deep in the layers of the silicon, read the rest of "Boolean logic," then "chip" and, finally, "transistor." It is a fascinating venture into a microscopic world.

The following AND, OR and NOT examples use mechanical switches to show open and closed transistors. The switching part of an actual transistor is solid state (see transistor).


An AND Gate (Wired in Series)
AND requires both inputs to be present in order to provide output. When both inputs pulse both switches closed, current flows from the source to the output.







An OR Gate (Wired in Parallel)
OR requires only one of the two inputs to be present in order for current to flow from the source to the output.







A NOT Gate (Input Is Reversed)
No pulse in puts current out (as shown). A pulse in puts no current out, as follows: an input pulse closes switch #1 and the current goes to #2. Switch #2 is normally closed, and a pulse from #1 opens it and stops the flow.







The Hierarchy
The gates make up circuits, and circuits make up logical devices, such as a CPU. We're going to look at a circuit that is present in every computer. It adds one bit to another.







Adding Two Bits Together
The half-adder circuit adds one bit to another and yields a one-bit result with one carry bit. This circuit in combination with a shift register, which moves over to the next bit, is how a string of binary numbers are added. This diagram shows the four possible binary additions for two bits.







The Half-Adder Circuit
Trace the current through the example above. See how AND, OR and NOT react to their inputs. The 1 is represented in red (flow of current), and the 0 in blue (no current). Try it yourself below.









Try It Yourself
Print this diagram and try your Boolean skill. Review the combinations of 0 and 1 above and pick any pair. With a pen or pencil, draw a line to represent a 1. Draw nothing for 0, and see if you can get the right answer.




Try It Yourself
Print this diagram and try your Boolean skill. Review the combinations of 0 and 1 above and pick any pair. With a pen or pencil, draw a line to represent a 1. Draw nothing for 0, and see if you can get the right answer.
References in periodicals archive ?
It is represented in the boolean circuits formalism by a circuit Ch, for some h, equivalent to ?
A quite related topic, which we plan to consider, is the use of boolean circuits for the computation of queries to distributed relational databases.
Boolean circuits are a suitable theoretical model for the study of the computability and parallel complexity of queries to relational databases.
Given an r-ary query and a natural number n that represents the size of the domain of a given database, we showed how to build a finite subfamily of boolean circuits which preserves the property of uniformity, and which has a much better relation between size and depth, thus improving the time needed for the parallel evaluation of the query, as well as the appreciation of the parallelizability of the query.
Then any FO formula can be expressed as a subfamily of boolean circuits in the class Size-Depth(nO(1),log n).
Translation of FO to Boolean Circuits When the atomic formula is interpreted for a given valuation, it is satisfied or no, that is, true or false.
Our boolean circuits belong to the complexity class NC, which makes formal the notion of well parallelizability of a function.