Conceptual Foundations © 2008 Pearson Education Australia Lecture slides for this course are based on teaching materials provided/referred by: (1) Statistics for Managers using Excel by Levine (2) Computer Algorithms: Introduction to Design & Analysis by Baase and Gelder (slides by Ben Choi to accompany the Sara Baase’s text). (3) Discrete Mathematics by Richard Johnsonbaugh 1 Conceptual Foundations (MATH21001) Lecture Week 8 Reading: Textbook, Chapter 9: Automata, Grammars and Languages

2 Learning Objectives In this lecture, you will learn:  Finite-state machines  Finite-state automata  Languages and grammars  Nondeterministic finite-state automata  Relationships between languages and automata

3 Finite-state Machines  A finite-state machine is an abstract model of a machine with a primitive internal memory.  A finite-state machine M consists of (a) A finite set I of input symbols. (b) A finite set O of output symbols. (c) A finite set S of states. (d) A next-state function f from S x I into S. (e) An output function g from S x I into O (f) An initial state σ ε S.  We write M = (I, O, S, f, g, σ)

4 Example  Let I={a, b}, O={0, 1}and S={σ0, σ1}. Define the pair of functions f and g by the rules given in the table below. Table 1 fg S Ia b σ0σ1σ0σ1 σ0 σ1 σ

5 Example Then M = (I, O, S, f, g, σ0) is a finite state machine. Table 1 is interpreted as follows: f(σ0, a) = σ0 g(σ0, a) = 0 f(σ0, b) = σ1 g(σ0, b) = 1 f(σ1, a) = σ1 g(σ1, a) = 1 f(σ1, b) = σ1 g(σ1, b) = 0 The next state and output functions can also be defined by a transition diagram.

6 What is a Transition Diagram?  Let M = (I, O, S, f, g, σ) be a finite state machine. The transition diagram of M is a diagraph G whose vertices are the members of S.  Transition diagram is a digraph. The vertices are the states. The initial state is indicated by an arrow.  If we are in state σ and inputting i causes output o and moves us to state σ1, we draw a directed edge from vertex σ to vertex σ1 and level it i/o.

7 Example – Transition Diagram  Transition diagram of example on slide 4 is shown below. Table 1 shows that if we are in state σ0 and we input a, then we will output 0 and will remain in state σ0. Thus we draw a directed loop on vertex σ0 and label it a/0. By considering all such possibilities we obtain the transition diagram as follows: σ0σ0σ1σ1 a/0 a/1 b/1 b/0

8 Finite-state Automata  A finite state automaton is a special kind of finite state machine.  A finite state automaton A = (I, O, S, f, g, σ) is a finite state machine in which the set of output symbols is {0,1} and where the current state determines the last output.

9 Example fg S Ia b σ0 σ 1 σ2 σ1 σ0 σ2 σ0 1 0 Draw the transition diagram of the finite state machine A defined by the table. The initial state is σ0. Show that A is a finite state automaton.

10 Example  The transition diagram is shown below. If we are in state σ0, the last output was 0. If we are in either state σ1 or σ2, the last output was 1; thus A is a finite state automaton. Example – Transition Diagram σ0σ0σ1σ1 b/0 a/1 b/0 σ2σ2 a/1

Conceptual Foundations © 2008 Pearson Education Australia Lecture slides for this course are based on teaching materials provided/referred by: (1) Statistics for Managers using Excel by Levine (2) Computer Algorithms: Introduction to Design & Analysis by Baase and Gelder (slides by Ben Choi to accompany the Sara Baase’s text). (3) Discrete Mathematics by Richard Johnsonbaugh 11

12 Languages & Grammars  Definition  Let A be a finite set. A formal language L over A is a subset of A*, the set of all strings over A.  One way to define a language is to give a list of rules that the language is assumed to obey.

13 Grammar A grammar G =(N, T, P, S) consists of  A finite set N of non-terminal symbols  A finite set T of terminal symbols  A finite subset P  A starting symbol S which is a non-terminal.

14 Example – A grammar for Integers  The following grammar generates all integers. ::=0|1|2|3|4|5|6|7|8|9 ::= | ::=+ |- ::= |

15 Summary  Introduced finite-state machines (FSM) and finite-state automata (FSA).  Reviewed various examples of FSM and FSA.  Discussed transition diagram.  Introduced language and grammar. In this lecture, we have