1.3.3: From Circuits to Propositions
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So, every compound proposition is computed by a logic circuit with one output wire. Is the reverse true? That is, given a logic circuit with one output, is there a proposition that expresses the value of the output in terms of the values of the inputs? Not quite. When you wire together some logic gates to make a circuit, there is nothing to stop you from introducing feedback loops. A feedback loop occurs when the output from a gate is connected—possibly through one or more intermediate gates—back to an input of the same gate. Figure 2.5 shows an example of a circuit with a feedback loop. Feedback loops
Figure 2.5: This circuit contains a feedback loop, so it is not a combinatorial logic circuit. The feedback loop includes the and gate and the or gate on the right. This circuit does not compute the value of a compound proposition. This circuit does, however, play an important role in computer memories, since it can be used to store a logical value.
cannot be described by compound propositions, basically because there is no place to start, no input to associate with a propositional variable. But feedback loops are the only thing that can go wrong. A logic circuit that does not contain any feedback loops is called a combinatorial logic circuit. Every combinatorial logic circuit with just one output computes the value of some compound proposition. The propositional variables in the compound proposition are just names associated with the input wires of the circuit. (Of course, if the circuit has more than one output, you can simply use a different proposition for each output.)
The key to understanding why this is true is to note that each wire in the circuit—not just the final output wire—represents the value of some proposition. Furthermore, once you know which proposition is represented by each input wire to a gate, it’s obvious what proposition is represented by the output: You just combine the input propositions with the appropriate ∧, ∨, or ¬ operator, depending on what type of gate it is. To find the proposition associated with the final output, you just have to start from the inputs and move through the circuit, labeling the output wire of each gate with the proposition that it represents. Figure 2.6 illustrates this process.