Overview of Chapters 7.2, 7.3 and 8.2

Abstract:

We haven't covered that much material since your last hour exam, so look for detailed problems using the techniques we've considered from these chapters:

• translating between logic networks, truth functions, and boolean expressions;
• creating logic networks to solve tasks (e.g. the stairway light problem)
• minimizing logic networks using
  • Karnough maps
  • Quine-McCluskey
• creating finite state machines to solve tasks (especially recognition of regular sets)
• create regular expressions which describe regular sets
• minimize a finite state machine

Section 7.2: Logic Networks

• Equivalent representations of a Logic Network:
  • Truth Functions
  • Boolean Expressions
  • Logic Network
• Converting between the three forms
• Rudimentary simplification of Boolean expressions
• Adding binary numbers (as Boolean expressions, half-adder, full-adder)

Section 7.3

Section 7.3 included two methods of simplification:

• Karnough maps
• Quine-McCluskey

Remember how the Karnough map uses idempotence to produce multiple copies of elements for the matches (pairs, quads, etc.), which it then can simplify.

Remember that you may use the techniques of the Karnough map to produce different simplified Boolean expressions: make sure that you've made things as simple as possible, but not simpler! (You can always check your Boolean expressions to make sure that they are equal by using the original ``tuples'' in the simplified expressions, and you should get 1!

In Quine-McCluskey, make sure that you've found all the ``reduced'' expressions possible, comparing every pair that may differ in only a single place.

Section 8.2

Definition: A finite-state machine M is a structure where

  
Table: Elements of a finite-state machine.

Construction of finite-state machines to perform tasks

• the binary adder, for example
• set recognition.

Given a machine, determine whether a set is recognized by it or not. Or determine what set is recognized by it. Remember that machine recognition is defined as follows, with emphasis on the word ``ends'' below:

Definition: Finite-State Machine Recognition A finite-state machine M with input alphabet I recognizes a subset S of (the set of finite-length strings over the input alphabet I) if M, beginning in state and processing an input string , ends in a final state (a state with output 1) if and only if .

Regular expressions define sets of input strings which are the ones that finite-state machines can actually recognize. The existence of reasonable sets, which one should reasonably be able to detect (e.g. , where stands for n copies of a), finite-state machines are obviously not sufficient to understand all of computation.