COMP 455: Models of Languages and Computations Homework 1 Fall 2024 Matlab

Java Python Homework Assignment 1:

COMP 455: Models of Languages and Computations

Fall 2024

DEADLINE: 11:59 P.M., FRIDAY, Sep 6, 2023

Problem 1 (20 points ; 4 parts)   Design DFAs to accept the following languages over the alphabet {a, b}. Show the DFAs in terms of their transition diagrams. Clearly indicate the initial state, the accepting states and the transitions following the conventions used in the lectures.

1.1 (5 points) L1 is the language of all strings in which every ‘b’ is immediately preceded by a ’a’.

1.2 (5 points) L2 is the language of all strings that begin with ‘ba’ and end with ‘ab’.

1.3 (5 points) L3 is the language of all strings that have an even number of a’s and an even number of b’s.

1.4 (5 points) L4 is the language of all strings that contain neither ‘aa’ nor ’bb’ as substrings.

fig1

Problem 2: (12 points; 3 parts)

Consider the two DFAs M1 (on the left) and M2 (on the right) over the alphabet {0, 1} shown in fig1. Let L1 and L2 be the languages accepted by M1 and M2 respectively.

2.1 (4 points) Describe succinctly in English the languages L1 and L2

2.2 (6 points) Using the product construction derive the transition diagram COMP 455: Models of Languages and Computations Homework Assignment 1 Fall 2024 Matlab of the automaton accepting the language L1 ∩ L2

2.3 (2 points) Using the same product, derive the transition diagram of the automaton accepting the language L1 ∪ L2

Problem 3 (10 points; 3 parts)

3.1 (5 points) Construct an NFA M over the alphabet {a, b} such that the language accepted by M consists of strings in which the last but one symbol is ‘a’. Make sure M exhibits the two features that a DFA can not exhibit. Show the transition diagram of your NFA.

3.2 (2 points) Show a successful execution sequence for the input string babaa. You can show your result by writing an alternating sequence of states and input symbols so that it represents a sequence of transitions of M.

3.3 (3 points) Show three unsuccessful executions of M on the input aababab.

fig2

Problem 4 (10 points; 3 parts) Consider the NFA M shown in fig2.

4.1 (2 points) Show the formal specification of this NFA in terms of its 5 components. Use the notations developed in the class.

4.2 (2 points) Show the transition table of M

4.2 (6 points) Derive an equivalent DFA using the subset construction method. Show the transition diagram of the DFA. Show only the states that are reachable from the initial state of the DFA. You may use the breadth-first-method that optimizes the subset construction but you don’t have to