Abstract

Practice designing and working with NFAs.

1 JFLAP

1. Return to the Finite Automata_ section of the Tutorial. Read the subsection devoted to Nondeterministic Finite Automata, specifically the sections titled “Construct and Run”, “Manipulating Transitions”, and "“Add a Trap State to DFA”. Also read “Convert to DFA”.

    
   

2. The NFA shown to the right is intended to accept the language $\{aa,aab\}^{*}\{b\}$. Try creating it in JFlap and running it on each of the following inputs to see if it works as expected:

   • b
   • aa
   • aab
   • aabb
   • aaab
   • aaaab
   • aaaabb
   • aaaabbb

   Save your NFA.

    
   

3. The NFA on the right uses a lambda transition, to help recognize the language $\{aab\}^*\{a,aba\}^*$. Look at the figure and compare to the structure of the set expression. Can you see how each element of the set expression is reflect in the arrangement of the states and transitions?

   Now build it in JFLAP and try running it on each of the following inputs to see if it works as expected:

   • a
   • aa
   • aab
   • aba
   • aaba
   • aabaab
   • aabaaba
   • aababa
   • abaaab

   Save your NFA.

4. JFLAP can convert NFAs to DFAs. Try converting each of your saved NFAs. Notice that the process used is directly comparable to the process described in section 2.3.5 of your text.

5. Look back at the example I gave of an NFA for the union of two languages.

   Here is the NFA for the union.

   Note: By default, JFLAP marks instantaneous transitions with a $\lambda$ instead of the $\epsilon$ used in your textbook. Both conventions are quite common. But for the sake of consistency, change the option in the JFLAP settings to use $\epsilon$.

   Run this NFA on the following inputs, “Step” by step, observing how transitions take us to multiple states at once. Take note of which inputs are accepted.

   • a
   • b
   • ba
   • aba
   • babba
   • aabba

6. Use JFlap to convert this to a DFA. Compare the results and the steps with what we went through in the lecture notes.

7. Run the same inputs as the previous step through this new DFA. Convince yourself that it does accept the inputs that the NFA did.

8. Revisit some of the DFA problems that you worked in the final step of the earlier JFLAP exercise. Try rewriting some of them as NFAs.

   • In many cases, an NFA will use fewer states and transitions than the corresponding DFA. Certainly, it should never need more states or transitions than a DFA (because every DFA is also an NFA that just happens to not use non-determinism).

   • You can think of this simplification as the pay-off for the slightly more complicated conceptual effort involved in understanding non-determinism.

9. Exercise 2.3.4 at the end of section 2.3 2 of your text suggests drawing a number of NFAs. Pick one or two of these and create them in JFLAP. Run a handful of inputs through each one to convince yourself that you have done so correctly.

1. the string 101

   Reveal

   

   (JFLAP)

   (video)

   Same as DFA.

2. all strings beginning with 101

   Reveal

   

   (JFLAP)

   (video)

3. all strings ending with 101

   Reveal

   

   (JFLAP)

   (video)

4. all strings containing 101

   Reveal

   

   (JFLAP)

   (video)

1. all strings beginning with and ending with 101
   Reveal

   In essence, we need to merge these:

    

   

    

   

    

   and we get:

   

   (JFLAP)

   (video)

1. strings beginning with 101 or 110
   Reveal

   

   (JFLAP)

   (video)

2.2.3 Repetition

Design an FSA to recognize the language consisting of…

1. All strings containing one or more instances of 101

   Reveal

   Same as saying “all strings that contain 101”

    

   

2. All strings containing zero or more instances of 101

   Reveal

   Same as saying “all strings”

    

   

3. all strings composed of 1 or more (non-overlapping) occurrences of 101

   Reveal

   

   (JFLAP)

   (video)

4. all strings composed of zero or more occurrences of 101 only

   Reveal

   

   (JFLAP)

   (video)

5. all strings that do not contain 101

   Reveal

   

   (JFLAP)

   (video)

• Converting a Non-deterministic automaton to deterministic