Automata and Languages

Transcription

1 Automata and Languages Prof. Mohamed Hamada Software Engineering Lab. The University of Aizu Japan

2 FINITE STATE MACHINES (AUTOMATA)

3 Switch Example Think about the On/Off button

4 Switch Example The corresponding Automaton Off Push Push Input: Push Push Push Push

5 Vending Machine Example 2 Vending machine dispenses Cola for $.45

6 Vending Machine Example 2 Input: D Q Q D 45 is final state Input Accepted

7 Example 3, The machine accepts a string if the process ends in a final state

8 Example 3 states Input Symbols final states (F) q, q q 2 start state (q ) q 3 states

9 Definitions An alphabet Σ is a finite set of symbols (in Ex3, Σ = {,}) A string over Σ is a finite sequence of elements of Σ (e.g. ) For a string s, s is the length of s The unique string of length will be denoted by λ and will be called the empty string The reversal of a string u is denoted by u R. Example: (banana) R = ananab

10 Definitions The concatenation of two strings is the string resulting from putting them together from left to right. Given strings u and v, denote the concatenation by u.v, or just uv. Example: jap. an = japan, QQ. DD = QQDD Q: What s the Java equivalent of concatenation? The + operator on strings Q2: Find a formula for u.v? u.v = u + v

11 Definitions If Σ is an alphabet, Σ * denotes the set of all strings over Σ. A language over Σ is a subset of Σ * i.e. a set of strings each consisting of sequences of symbols in Σ.

12 Example: in our vending machine we have Σ = { D, Q } Σ * = {λ, D, Q, DD, DQ, QD, QQ, DDD, DDQ, DQD, DQQ, QDD, QDQ, QQD, QQQ, DDDD, DDDQ, } L = { u Σ * u successfully vends } Examples Example2: in our switch example we have Σ = { Push} Σ * = {λ, Push, Push Push, Push Push Push, Push Push Push Push, } L = { Push n n is odd }

13 Definitions A finite automaton is a 5-tuple M = (Q, Σ, δ, q, F) Q is the set of states Σ is the alphabet δ is the transition function q Q is the start state F Q is the set of final states L(M) = the language of machine M = set of all strings machine M accepts

14 Definitions State Diagram and Table a b q q a b q 2 a b δ a b q q q q q q 2 Q = { q Σ = { a, b} F = { q 2, q }, q 2 } q 2 q 2 q 2

15 FINITE STATE MACHINES (AUTOMATA) Deterministic Finite Automata (DFA) Non-Deterministic Finite Automata with empty move (λ-nfa) Non-Deterministic Finite Automata (NFA)

16 Deterministic & Nondeterministic Deterministic Non-Deterministic B B E E A H A H C C D F G D F G G I G I E E One choice Multi choice è Backtrack

17 Deterministic & Nondeterministic Deterministic Non-Deterministic a B A a C A a a C D From ONE state machine can go to another ONE state on one input One choice From ONE state machine can go to MANY states on one input Multi choice

18 Deterministic Computation Non-Deterministic Computation reject accept or reject accept

19 DETERMINISTIC FINITE AUTOMATA (DFA)

20 Q is the set of states Σ is the alphabet Definitions A DFA is a 5-tuple M = (Q, Σ, δ, q, F) δ : Q Σ Q is the transition function q Q is the start state F Q is the set of accept states L(M) = the language of machine M = set of all strings machine M accepts

21 Deterministic Finite Automata (DFA) Example is not final state Input Rejected

22 Deterministic Finite Automata (DFA) Example Q: What kinds of bit-strings are accepted? A: Bit-strings that represent binary even numbers.

23 Deterministic Finite Automata (DFA) Example 2 q q 2 q 3 reject accept λ reject accept reject,

24 Exercise q, L(M) = {,}*

25 Exercise q, L(M) =

26 Exercise q q L(M) = { w w has an even number of s}

27 Exercise Build an automaton that accepts all and only those strings that contain, q q q q

28 Exercise Strings over {a,b} containing even number of a s and odd number of b s. q = Even(a).Even(b) q = Even(a).Odd(b) Even(a) Σ* Odd(a) q 2 = Odd(a).Even(b) q 3 = Odd(a).Odd(b) Even(b) Odd(b) Even(b) Odd(b) a q b b a b b a q q 2 q 3 a

29 Exercise Strings over {a,b,c} that has the form (ab)*c q = End(b) q = End(a) valid prefix Σ* invalid prefix q 2 = End(c) q 3 = Error End(a) End(b) End(c) c b a q q q 2 q 3 a,b,c b a,c a,b,c

30 Exercise Design with a friend a machine that tells us when a base- number is divisible by 3. What should your alphabet be? How can you tell when a number is divisible by 3?

31 Answer,3,6,9,4,7 mod 3 mod 3 2,5,8,3,6,9,4,7 2,5,8 2,5,8 2 mod 3,4,7,3,6,9

32 Exercise,3,6,9 mod 3,4,7 Find the formal description of this automaton.,3,6,9,4,7 2,5,8 mod 3 2,5,8 2,5,8,4,7 2 mod 3,3,6,9

33 Answer Q = { mod 3, mod 3, 2 mod 3 } ( rename: {q, q, q 2 } ) Σ = {,, 2, 3, 4, 5, 6, 7, 8, 9 } q = mod 3 F = { mod 3 } δ : Q Σ Q δ( q δ( q,2),7) = = q q 2 2,, δ( q δ( q 2,9) =,3) = q q 2,δ( q,δ( q 2,2),5) = = q q., Question : δ( q δ( i, j) =? q i, j) = q( i + j) mod 3