Theory of Computation

Transcription

1 Theory of Computation CSRU49-29S-L4 Lecture 4: DFA and NFA Elena Filatova Department of Computer and Information Sciences Fordham University

2 L4-: DFA Deterministic Finite State Automata reads input string one symbol at a time after the string is completely read it is either accepted or rejected NB! Every possible combination of states and input symbols is covered by the δ transition function.

3 L4-: DFA Definition A deterministic finite automaton (DFA) is a 5-tuple M = (K, Σ, δ, s, F) K Set of states Σ Input Alphabet δ : (K Σ) K is a Transition function s K Initial (start) state F K Final (accepting) states

4 L4-2: Connect DFA with a Language DFA is a checker Two main questions asked about automata Given a DFA M, what is L(M) 2 Given a language L, what is a DFA M such that L(M) = L

5 L4-3: DFA Example Give a DFA for recognizing the set of all strings over {, } and a regular expression that generates such strings.

6 L4-4: DFA Example Give a DFA for recognizing the set of all strings over {, } and a regular expression for generating such strings. Regular Expression: ( + ) DFA

7 L4-5: Fun with DFA Create a DFA for: All strings over {, } that end with

8 L4-6: Fun with DFA Create a DFA for: All strings over {, } that end with 2 3

9 L4-7: Fun with DFA Create a DFA for: All strings over {, } that begin with

10 L4-8: Fun with DFA Create a DFA for: All strings over {, } that begin with, 4, 2 3

11 L4-9: Fun with DFA Create a DFA for: All strings over {, } that begin or end with

12 L4-: Fun with DFA Create a DFA for: All strings over {, } that begin or end with,

13 L4-: Fun with DFA Create a DFA for: All strings over {, } that begin and end with

14 L4-2: Fun with DFA Create a DFA for: All strings over {, } that begin and end with,

15 L4-3: Fun with DFA Create a DFA for: All strings over {, } that contain or

16 L4-4: Fun with DFA Create a DFA for: All strings over {, } that contain or ,

17 L4-5: Fun with DFA Create a DFA for: All strings over {a, b} that begin and end with the same letter

18 L4-6: Fun with DFA Create a DFA for: All strings over {a, b} that contain begin and end with the same letter a b a b b 2 a b a a 2 b

19 L4-7: Why DFA? Why are these machines called Deterministic Finite Automata Deterministic: Each transition is completely determined by the current state and next input symbol. That is, for each state / symbol pair, there is exactly one state that is transitioned to Finite: Every DFA has a finite number of states Automata: (singular automaton) means machine (From Merriam-Webster Online Dictionary, definition 2: A machine or control mechanism designed to follow automatically a predetermined sequence of operations or respond to encoded instructions)

20 L4-8: DFA Configuration & M Way to describe the computation of a DFA Configuration: What state the DFA is currently in, and what string is left to process K Σ Binary relation M : What machine M yields in one step M (K Σ ) (K Σ ) M = {((q, aw), (q 2, w)) : q, q 2 K M, w Σ M, a Σ M, ((q, a), q 2 ) δ M }

21 L4-9: DFA Configuration & M Given the following machine M: a,b a 2 b a,b (q, abba), (q 2, bba) M can also be written (q, abba) M (q 2, bba)

22 L4-2: DFA Configuration & M 2 3 (q, ) M (q, ) M (q 2, ) M (q 3, ) M (q, ) M (q, ǫ)

23 L4-2: DFA Configuration & M 2 3 (q, ) M (q, ) M (q, ) M (q, ) M (q 2, ) M (q 3, ǫ)

24 L4-22: DFA Configuration & M M is the reflexive, transitive closure of M Smallest superset of M that is both reflexive and transitive yields in or more steps Machine M accepts string w if: (s M, w) M (f, ǫ) for some f F M

25 L4-23: DFA & Languages Language accepted by a machine M = L[M] {w : (s M, w) M (f, ǫ) for some f F M } DFA Languages, L DFA Set of all languages that can be defined by a DFA L DFA = {L : M, L[M] = L} To think about: How does L DFA relate to L REG

26 L4-24: Non-Determinism A Deterministic Finite Automata s transition function has exactly one transition for each state/symbol pair A Non-Deterministic Finite Automata can have, or more transitions for a single state/symbol pair An NFA is defined in the same way as a DFA except that the following liberalizations are allowed: multiple next states ǫ-transitions

27 L4-25: NFA: Multiple Next Steps DFA: each state has Σ outgoing edges, each alphabet symbol is a label for one of the edges NFA: in a state q and with a symbol a i from the alphabet Σ - there could be more than one next step to take δ(q, a i ) is a subset of the Q set of states δ(q, a i ) = {q, q 2,..., q k }, where any of q, q 2,..., q k could be the next step Special case: δ(q, a i ) = : no next state when the machine is in q and reading an a i machine hangs and the input is rejected same as, going to trap (failure) state in DFA

28 L4-26: ǫ - Transitions In an ǫ-transition, the tape head does not do anything: it does not read and it does not move. However, the state of the machine can be changed. Formally, the transition function δ is given the empty string. More information - next class

29 L4-27: NFA Example Example: L = {w {a, b} : w starts with a} Regular expression?

30 L4-28: NFA Example Example: L = {w {a, b} : w starts with a} a(a+b)* a,b a

31 L4-29: NFA Example Example: L = {w {a, b} : w starts with a} a(a+b)* a,b a What happens if a b is seen in state q? The machine crashes, and does not accept the string

32 L4-3: NFA Example Example: L = {w {a, b} : w contains the substring aa} Regular Expression?

33 L4-3: NFA Example Example: L = {w {a, b} : w contains the substring aa} (a+b)*aa(a+b)* a,b a,b a a 2 What happens if a a is seen in state q?

34 L4-32: NFA Example Example: L = {w {a, b} : w contains the substring aa} (a+b)*aa(a+b)* a,b a,b a a 2 What happens if a a is seen in state q? Stay in state q, or go on to state q Multiple Computational Paths: aab, aaa, abb, abaa

35 L4-33: NFA Example Example: L = {w {a, b} : w contains the substring aa} a,b a,b a a 2 (q, abaa) (q, baa) (q,aa) (q, a) (q, ε) reject crash reject (q, baa) (q, a) (q, ε) (q2, ε) reject accept Does this machine accept abaa? Tree of states for NFA

36 L4-34: NFA Acceptance If there is any computational path that accepts a string, then the machine accepts the string Two ways to think about NFAs: Magic Oracle, which always picks the correct path to take Try all possible paths

37 L4-35: NFA Acceptance NB! nondeterministic has nothing to do with random nondeterministic implies parallelism The difference between a DFA and a NFA is that the state transition table, δ, for a DFA has exactly one target state but for a NFA has a set, possibly empty ( ), of target states.

38 L4-36: NFA Example Example: L = {w {a, b} : w contains the substring aa} a,b a,b a a 2 If a string contains aa, will there be a computational path that accepts it? If a string does not contain aa, will there be a computational path that accepts it?

39 L4-37: NFA Definition Difference between a DFA and an NFA DFA has exactly only transition for each state/symbol pair δ : (K Σ) K NFA has, or more transitions for each state/symbol pair

40 L4-38: NFA Definition Difference between a DFA and an NFA DFA has exactly only transition for each state/symbol pair Transition function: δ : (K Σ) K NFA has, or more transitions for each state/symbol pair Transition relation: ((K Σ) K)

41 L4-39: NFA Definition A NFA is a 5-tuple M = (K, Σ,, s, F) K Set of states Σ Alphabet : (K Σ) K is a Transition relation s K Initial state F K Final states

42 L4-4: Fun with NFA Create an NFA for: All strings over {a, b} that start with a and end with b

43 L4-4: Fun with NFA Create an NFA for: All strings over {a, b} that start with a and end with b a,b a b 3 (example computational paths for ababb, abba, bbab)

44 L4-42: Fun with NFA Create an NFA for: All strings over {, } that contain or

45 L4-43: Fun with NFA Create an NFA for: All strings over {, } that contain or,, 3