CS30 - Spring 2015 - Class 23

Lecture notes

  • admin
       - assignment 10
       - lab tomorrow
          - will play NIM a bit
          - then work on assignment

       - NIM tournament in class next Tuesday

  • talk at a high level about theoretical models of computation

  • deterministic finite automata (DFA)
       - defined by 5 components
          1) a set of states Q
          2) the alphabet \sigma
          3) q_0 \in Q, the start state
          4) F subset Q, the accepting (or final) states
          5) the transitions between states
             - (Q x \sigma -> Q)
             - a transition for each state for each letter in the alphabet to some other state

       - a simple (but fairly powerful) model of computation

  • some DFA examples (found in DFA_examples)
       - no_a: Strings of a's and b's with no a's
          - b*
       - even_a: Strings of a's with an even number of a's
          - (aa)*

       - 5a: Strings with a's that are multiples of 5
          - (aaaaa)*

       - one_a: Strings with only one a
          - b*ab*

       - start_a: all strings that start with a
          - a(a|b)*

       - start_end: Strings that start and end with the same letter
          - (a(a|b)*a)|(b(a|b)*b)|a|b

       - odd: odd length strings
          - (a|b)((a|b)(a|b))*

       - two_a: strings with two or more a's
          - (a|b)*a(a|b)*a(a|b)*

       - two_exact_a: strings with exactly two a's
          - b*ab*ab*

       - ab: some number of repetitions of ab
          - (ab)*

  • non-deterministic finite automata (NFA)
       - almost identical definition to DFA except:
          - can have epsilon (or sometimes called lambda) transitions from one state to another
             - doesn't read anything from the input, just transitions
          - for a given state and input, can go to two or more states (rather than just a single one for DFAs)
          - do not require that there is a transition for every alphabet letter for every state
             - if you encounter a state without a transition for a particular letter, it does not accept

       - tend to be a bit easier to work with than DFAs

       - NFA's do not actually have any more expressive power
          - in particular, for any NFA, there is a corresponding DFA that accepts the same language (i.e. same set of strings)

  • some NFA examples (found in NFA_examples)
       - start_end_a: start and end with a
          - a(a|b)*a

       - 2_or_3_a: strings of a's that have lengths divisible by 2 OR 3
          - (aa)*|(aaa)*

       - end_aa: ends in two a's
          - (a|b)*aa

       - aa_bb: has either aa or bb as a substring
          - (a|b)*(aa|bb)(a|b)*

  • regular language
       - any language that can be described by a DFA (or an NFA)
       - any language that can be described by a regular expression!