Chapter 2. Finite Automata. (part A) (2015/10/20) Hokkaido, Japan

Transcription

1 Chapter 2 Finite utomata (part ) (25//2) Hokkaido, Japan

2 Outline 2 Introduction 2 n Informal Picture of Finite utomata 22 Deterministic Finite automata 23 Nondeterministic Finite utomata 24 n pplication: Text Search (in part b) 25 Finite utomata with psilon-transitions (in part b) 2

3 2 Introduction Two types of finite automata (F): deterministic F (DF) nondeterministic F (NF) Both types are of the same power: the former is easier for hardware and software implementations; and the latter is more efficient for descriptions of applications of F 2 n Informal Picture of Finite utomata complete application example of finite automata for protocol design and verification Involving a concept of product of two automata (read details in the textbook by yourself) 22 Deterministic Finite utomata 22 Review (supplemental) Recall the example of designing a vending machine selling 2-dolllar food packs in Chapter (see Fig 2) What abstract concepts are involved in the design? See the definition next $5 $5 $ $5 start $ $5 $ $5 $5 $ $ $ $2 t he transition diagram Fig 2 Transition diagram of a finite automaton for a vending machine (from Chapter ) 22 Definition of DF Definition --- deterministic finite automata (DF) consists of 5-tuples (/2): a finite (nonempty) set of states Q; a finite (nonempty) set of input symbols ; a (state) transition function such that (q, a) = p Means automaton, in state q, takes input a and enters state p a start state q ; a set of (nonempty) final or accepting states F 3

4 DF may be written as a 5-tuple as follows: = (Q,,, q, F) graphic model for a DF --- see Fig 22 tape 磁帶 讀取頭 tape reader finite 有限 control 控制器 Fig 圖 一部有限自動機的圖形模式 graphic model for a DF 222 How a DF processes strings Given an input string x = a a 2 a n, if (q, a ) =, (, a 2 ) =,, (q i, a i ) = q i,, (q n, a n ) = q n, and q n F, then x is accepted ; otherwise, rejected very transition is deterministic xample Design an F to accept the language L = {xy x and y are any strings of s and s} xamples of strings in L:,, Transitions: (q, ) = q, (q, ) =, (, ) =, (, ) =, (, ) =, (, ) = 5-Tuple: = ({q,, }, {, },, q, { }) 223 Simpler Notations for DF s For the last example, we may have other ways of descriptions as follows Transition diagram --- see Fig 23 Start q, Fig 23 The transition diagram for the DF of xample 2 4

5 Transition table --- see Table 2 Table 2 The transition table for xample 2 q q * (Notations: : initial state; *: final state) 224 xtending transition function to strings xtended transition function ˆ --- If x = a a 2 a n and is such that (p, a ) =, (, a 2 ) =,, (q i, a i ) = q i,, (q n, a n ) = q, then we define ˆ to be ˆ (p, x) = q ˆ may also be defined recursively as follows: Basis: ˆ (q, ) = q Induction: if w = xa (a is the last symbol of w), then ˆ (q, w) = ( ˆ (q, x), a) graphic diagram for the concept of the above induction step --- see Fig 24 x a ˆ(q, x) ˆ(q, w=xa), Fig 24 n illustration for the induction step in the definition of ˆ Difference between and ˆ --- the former accepts single symbols as input while the latter may accept strings as input 225 The Language of a DF The language of a DF is defined as L() = {w ˆ (q, w) is in F} If L is L() for some DF, then we say L is a regular language 5

6 23 Nondeterministic Finite utomata 23 n informal view of NF s Review of a previous example of DF (of xample 2) --- Original version --- see Fig 25 (the same as Fig 23) Start q, Fig 25 The transition diagram for the DF of xample 2 (Fig 23 repeated here for comparison) nondeterministic finite automaton (NF) version of the above DF --- see Fig 26 More intuitive! How to design an NF will be described later, Start q $2, Fig 26 nother transition diagram for the DF of xample 2 using the extended transition function ˆ Some properties of NF s (see Fig 26 for the illustration) --- Some transitions may die, like ˆ (, ) Some transitions have multiple choices, like ˆ (q, ) = q and xample Design an NF accepting the following language L = {w w{, }* and ends in } By intuition, we can draw the transition diagram for an NF to accept L as shown in Fig 27 Nondeterminism may create many transition paths, but if there is one path leading to a final state, then the input is accepted, Start q $2 Fig 27 design of the NF of xample 22 by intuition When input x =, the NF processes x in a way as shown in Fig 28 6

7 q q q q q q Stuck! q2 Stuck! ccept! Fig 28 The process of the NF of xample 22 to accept the input x = (Fig 2 in the textbook) 232 Definition of NF n NF is a 5-tuple = (Q,,, q, F) where Q = a finite (nonempty) set of states; = a finite (nonempty) set of input symbols; q = a start state; F = a set of (nonempty) final or accepting states; = a (state) transition function such that (q, a) = {p, p 2,, p m } which means automaton, in state q, takes input a and enters one of the states p, p 2,, p m xample The transition table of the NF of the last example (xample 26) is as shown in Table 22 Table 22 Transition table of NF of xample 22 q {q, } {q } { } * 233 xtended Transition Function Recursive definition of ˆ (with strings, not symbols, as input) --- Basis: ˆ (q, ) = {q} Induction: if w = xa (with a as the last symbol of w), ˆ (q, x) = {p, p 2,, p k }, and (p i, a) = {r, r 2,, r m }, then ˆ (q, w) = {r, r 2,, r m } graphic diagram for the concept involved in the induction step above is as shown in Fig 29(supplemental): 7

8 x ˆ (q, x) ={ p p 2 p k } a a a ˆ (q, w=xa) = {r r 2 r m } Fig 2 graphic model for explaining the induction step of the extended transition function xample 28 (continued from xample 26) --- With the input w = for the NF of xample 26, we have ˆ (q,) = {q }; 2 ˆ (q, ) = (q, ) = ; 3 ˆ (q, ) = (q, ) (, ) = = 234 The Language of an NF Definition --- If = (Q,,, q, F) is an NF, then the language accepted by is L() = {w ˆ (q, w) F } That is, as long as a final state is reached, which is among possibly many, the input is accepted 235 quivalence of DF and NF comparison of DF s and NF s is as follows NF s are more intuitive to construct for many languages; DF s are easier for use in software and hardware implementations Definition: Two models are said to be equivalent if they have identical languages Theorem: very NF N has an equivalent DF D, ie, L(D) = L(N) The proof of the theorem is based on the technique of subset construction Concept of subset construction: each state of DF is a subset of NF Detail of proof: Given an NF N = (Q N,, N, q, F N ), we construct a DF D = (Q D,, D, q ', F D ) such that: the two alphabets are identical; Q D is the power set of Q N (ie, set of all subsets of Q N ); inaccessible states in Q D are deleted; q ' = {q }; each subset S of Q N is put in the final state set F D if S includes at least one accepting 8

9 state in F N (ie, S F N ); for each subset S of Q N and for each input symbol a, D is defined as D (S, a) = N (p, a) ps xample For the NF of xample 26 (see Fig 27), we have The power set of Q N = {q,, } is Q D = {, {q }, { }, { },, {q, }, {, }, {q,, }} The transition table is as shown Table 23 Table 23 The transition table for the NF of xample 26 {q } { } *{ } *{q, } *{, } *{q,, } {q } { } {q, } {q } { } {q, } limination of inaccessible states --- Changing the names of the states in Table 23 according to the following rules, we get a result as shown in Table 24 (which is Fig 23 in the textbook): {q } = B, { } = C, { } = D, =, {q, } = F, {, } = G, {q,, } = H Table 24 transition table transformed from Table 23 for the NF of xample 26 B C *D *F *G *H B D F B D F Checking accessible states from the start state (marked by, ), we get a result shown as the red ones in Table 24 (only B,, F left); the other states are inaccessible transition diagram for the above is shown in Fig 2 9

10 Start B F t he Fig 2 The transition diagram for the simplified NF of xample 2 t nother way to eliminate inaccessible states by lazy evaluation --- starting from the start state, only accessible states are evaluated in order to avoid generation of inaccessible states (see Fig 27 again during the evaluation process) xample 2 (continued) --- The result according to lazy evaluation for xample 2 using Table 23 is shown by the red parts in Table 25 which are the same as those red states shown in Table 24 The corresponding transition diagram is shown in Fig 22 Table 25 The transition table for the NF of xample 26 using lazy evaluation {q } { } *{ } *{q, } *{, } *{q,, } {q } { } {q, } {q } { } {q, } Start {q } t {q, } n Fig 22 The transition diagram of xample 2 resulting from lazy evaluation

11 Theorem 2 If DF D = (Q D,, D, {q }, F D ) is constructed from NF N = (Q N,, N, q, F N ) by subset construction, then L(D) = L(N) Proof See the textbook Theorem 22 (equivalence of DF and NF): language L is accepted by some DF if and only if L is accepted by some NF Proof See the textbook 236 Bad Case of Subset Construction The DF constructed from an NF usually has the same number of accessible states of the NF But the worse case is m = 2 n where m is the number of states in the DF n is the number of states in the NF Regarding NF s as DF s --- In each state of a DF, for each input there must be a next state In an NF, for a certain input there might be no next state So the automaton shown in Fig 23 is an NF But it is deterministic in nature --- either getting into a next state or becoming dead (getting into a dead state) Such a kind of NF has at most one transition out of any state on any symbol, and may be regarded as a DF start t t h th e the n t Fig 23 n NF which may be regarded as a DF (to be continued in part II of this chapter)