Deterministic Finite Automata (DFAs)

1. Deterministic Finite Automata (DFAs)

2. Reminder: Functions vs Relations Let P = {p: p is a person} M = {m: m is a male} S1 = {(m,p): m is in M, p is in P and m is the father of m} S2 = {(m,p): m is in M, p is in P and m is an ancestor of m} • True or false: S1 M  P • True or false: S2  M  P • Is either S1 or S2 a relation in M  P? • Is either S1 or S2 a function f:M  P?

3. Deterministic Automata (Informal) “current state” • Key questions: if a automaton is confronted with a certain state where a choice must be made, • 1. are all the alternatives transitions known?, and • 2. given some input data, is it known which transition the machine will make? “new state” If the answer to both of these questions is “yes”, the automaton is said to be deterministic “transition”

4. Nondeterministic Automata (Informal) If the answer to any of these questions is “no”, the automaton is said to be nondeterministic • That is, either • some transitions are unknown, or • given some input data, the machine can make more than one transition

5. Deterministic Automata (Informal) of the computation d s’ s Transition: ((s,d), s’) We are going to define automata indicating for a state s and some input data d, which is the state that will be reached Let Q be the set of all states and  be the set of all input data. Then, the set of transitions is a subset of (Q  )  Q

6. Determinism, Nondeterminism, Relations and Functions The set of transitions defining an automaton is a subset of (Q  )  Q If the automaton is deterministic, should the set of transitions be a relation or a function? • Deterministic automata: Since for each pair (s,d) there should be one and only one s’, the set of transitions must be a function • Nondeterministic automata: Since for each pair (s,d) there might not be any s’ or there might be more than one s’, the set of transitions must be a relation

7. Finite Automata Finite automata use a constant amount of memory • Problem 1: Design a computer program that given a sequence of numbers a1, a2, …, an returns their sum a1 + a2 +… + an • Problem 2: Design a computer program that given a sequence of numbers a1, a2, …, an returns the list in the inverted order: an, …, a2, a1 • How many memory units are needed for a program to execute: • problem 1: • Problem 2: 1 n

8. Deterministic Finite Automaton (DFA) no! (or yes!) • A deterministic finite automaton (DFA) is a 5-tuple (Q,,,s,F) where: • Q is a finite set of elements called states •  is a finite input alphabet •  is a transition function, (Q × ) × Q (or : (Q × )  Q) • s Q called the initial state • F  Q called the favorable states Constant! The fact that  is a function makes the automaton deterministic … a1 a2

9. Finite State Diagram a ((q,a),q’) is a transition in  Indicates: q’ q > The initial state is marked with: The favorable states are marked with: • A finite state diagram is a graphic representation for a DFA • A finite state diagram is a directed graph, where nodes represent elements in Q (i.e., states) and arrows are characters in  such that:

10. Example # 1 Trap/ Dead State a b b b a a r r s q q > a a b Formally, this automaton (Q,,,s,F) is defined as: • : 6 transitions: • ((s,b), a) • ((s,a), q) • ((q,a), r) • ((q,b), s) • ((r,a), r) • ((r,b), r) Q = {s,q,r} Is this automaton deterministic?  = {a,b} s: initial state F = {r}

11. Real-Life Example: Finite Automaton Controlling a computer-Generated character E S inspect ~S D S • States • Attack • Chase • Spawn • Wander • Events • E: see an enemy • S: hear a sound • D: die Attack ~E D E Wander E ~E Spawn D Thief movie

12. Preliminary Definitions • Given a set B, B* denote the set of all strings made of elements in B. Strings in B* are also called words. • For example, if  = {a,b}, then * = {a, b, aa, ab, bb, …, baaaba, …} • Given an DFA A = (Q,,,s,F) , A configuration is any element in Q × * • In the example in the previous slide possible configurations include (s,abbba), (s, aaa), (q,bab) and (r, bbbb) • The symbol e denotes the empty string

13. Configuration Yields Configuration  q’ q • Given an DFA A= (Q,,,s,F), the configuration (q,w) yields the configuration (q’,w’) in one step, if w = w’ with    and q’ = (q, ) • (q,w) yields (q’,w’) if there is a sequence of configurations: • (q1,w1), (q2,w2), …, (qk,wk) • such that (qi,wi) yields (qi+1,wi+1) in one step

14. Example Trap/ Dead State a b • What is the configuration yield after 3 yield-steps for (s,aab)? • What is the configuration yield after 4 yield-steps for (s,baab)? b b a a r r s q q > a a b

15. String Accepted by Automaton “such that” • Given an automaton A = (Q,,,s,F), and a string w *, we say that w is accepted by A if the configuration (s,w) yields the configuration (f,e), where f is a favorable state, and e is the empty string • In Example # 1, the automaton accepts aaabbbb but not b • Given an automaton A, the language accepted by A, written L(A), is defined by: • L(A) = {w * : w is accepted by A} • The language accepted in Example # 1 is all the words that contains two consecutive a’s

16. Example Trap/ Dead State a b • What is the language accepted by this finite automaton? b b a a r r s q q > a a b

17. Example # 2 a b b b a a r r s q q > a a b What is the language accepted by this automaton?