Deterministic finite automata

1.1. Write a DFA which accepts the language $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of 1s} \} $ Hint:

• in a DFA, delta is total!

1.2 Define a DFA which accepts arithmetic expressions. Consider the following definition for arithmetic expressions:

​
<expr> ::= <var> | <expr> + <expr> | <expr> * <expr>
<var> ::= STRING

Hint:

• how would you define the alphabet for the DFA?
• can <expr> be the empty string?

Consider the following encoding of a DFA:

   <number_of_states>
   <list_of_final_states>
   <state> <symbol> <state>

Example:

4
2 3
0 a 1
1 b 2
2 a 0

2.1. Write a function which takes a DFA encoding as above and returns a DFA representation. Define a class “DFA”.

2.2. Add a method accept which takes a word and returns true if it is accepted by the DFA

2.3. Add a method step with takes a DFA configuration and returns the “next-step” configuration of the DFA. How is a configuration defined?

2.4(*) Write a method which:

• takes a list of DFAs $ a_1, a_2, \ldots, a_n$
• takes a string $ s$. We know the string consists of a sequence of words, each accepted by some dfa in the list.
• returns a list of pairs $ (w_1,a_1), \ldots(w_i,a_i) \ldots (w_n,a_n)$ such that $ w_1w_2\ldots w_n = s$ and the dfa $ a_i$ accepts word $ w_i$, for each i from 1 to n.

Example:

l = [a1, a2, a3]
 ''' 
    a1 accepts sequences of digits [0-9]
    a2 accepts sequences of lowercase symbols [a-z]
    a3 accepts operands (+ and *)
 '''
s = "var+40*2300"

our function returns:
[("var",2), ("+",3), ("40",1), ("*",3), ("2300",1)]