What Are They? Who Needs ‘em? An Example: Scoring in Tennis

What Are They? Who Needs ‘em? An Example: Scoring in Tennis
Finite Automata What Are They? Who Needs ‘em? An Example: Scoring in Tennis

What is a Finite Automaton?
A formal system. Remembers only a finite amount of information. Information represented by its state. State changes in response to inputs. Rules that tell how the state changes in response to inputs are called transitions.

Why Study Finite Automata?
Used for both design and verification of circuits and communication protocols. Used for many text-processing applications. An important component of compilers. Describes simple patterns of events, etc.

Tennis Like ping-pong, except you are very tiny and stand on the table. Match = 3-5 sets. Set = 6 or more games.

Scoring a Game One person serves throughout.
To win, you must score at least 4 points. You also must win by at least 2 points. Inputs are s = “server wins point” and o = “opponent wins point.”

Server Wins Opp’nt s o s o Love-40 15-30 30-15 40-Love s o deuce s o 30-40 40-30 s o Ad-out Ad-in s o 40-15 15-40 30-all s o Love-30 15-all 30-Love s o Start Love-15 15-Love s o Love

Acceptance of Inputs Given a sequence of inputs (input string ), start in the start state and follow the transition from each symbol in turn. Input is accepted if you wind up in a final (accepting) state after all inputs have been read.

Example: Processing a String
s o s o s o s o s o s s Love Start Love-15 15-Love s o Love-30 15-all 30-Love Love-40 15-30 30-15 40-Love Server Wins Opp’nt 40-15 15-40 30-all 30-40 40-30 deuce Ad-out Ad-in *

Example: Processing a String
* s o s o s o s o s o s s Love Start Love-15 15-Love s o Love-30 15-all 30-Love Love-40 15-30 30-15 40-Love Server Wins Opp’nt 40-15 15-40 30-all 30-40 40-30 deuce Ad-out Ad-in Good going server.

Language of an Automaton
The set of strings accepted by an automaton A is the language of A. Denoted L(A). Different sets of final states -> different languages. Example: As designed, L(Tennis) = strings that determine the winner.

Deterministic Finite Automata
Alphabets, Strings, and Languages Transition Graphs and Tables Some Proof Techniques

Alphabets An alphabet is any finite set of symbols. Examples:
ASCII, Unicode, {0,1} (binary alphabet ), {a,b,c}, {s,o}, set of signals used by a protocol.

Strings A string over an alphabet Σ is a list, each element of which is a member of Σ. Strings shown with no commas or quotes, e.g., abc or Σ* = set of all strings over alphabet Σ. The length of a string is its number of positions. ε stands for the empty string (string of length 0).

Example: Strings {0,1}* = {ε, 0, 1, 00, 01, 10, 11, 000, 001, . . . }
Subtlety: 0 as a string, 0 as a symbol look the same. Context determines the type.

Languages A language is a subset of Σ* for some alphabet Σ.
Example: The set of strings of 0’s and 1’s with no two consecutive 1’s. L = {ε, 0, 1, 00, 01, 10, 000, 001, 010, 100, 101, 0000, 0001, 0010, 0100, 0101, 1000, 1001, 1010, }

Deterministic Finite Automata
A formalism for defining languages, consisting of: A finite set of states (Q, typically). An input alphabet (Σ, typically). A transition function (δ, typically). A start state (q0, in Q, typically). A set of final states (F ⊆ Q, typically). “Final” and “accepting” are synonyms.

The Transition Function
Takes two arguments: a state and an input symbol. δ(q, a) = the state that the DFA goes to when it is in state q and input a is received. Note: δ is a total function: always a next state – add a dead state if no transition (Example on next slide).

Server Wins Opp’nt s o s o Love-40 15-30 30-15 40-Love s o deuce s o Dead s, o 30-40 40-30 s o Ad-out Ad-in s o 40-15 15-40 30-all s o Love-30 15-all 30-Love s o Start 15-Love s Love o Love-15

Graph Representation of DFA’s
Nodes = states. Arcs represent transition function. Arc from state p to state q labeled by all those input symbols that have transitions from p to q. Arrow labeled “Start” to the start state. Final states indicated by double circles.

Example: Recognizing Strings Ending in “ing”
Not i or g Not i Not i or n i nothing Saw i Saw in Saw ing i n g i Start i Not i

Example: Protocol for Sending Data
timeout data in Ready Sending Start ack

Example: Strings With No 11
Start 1 A C B 0,1 Consecutive 1’s have been seen. String so far has no 11, does not end in 1. String so far has no 11, but ends in a single 1.

Alternative Representation: Transition Table
Final states starred * Columns = input symbols 1 A A B B A C C C C Arrow for start state Each entry is δ of the row and column. Rows = states Start 1 A C B 0,1

Convention: Strings and Symbols
… w, x, y, z are strings. a, b, c,… are single input symbols.

Extended Transition Function
We describe the effect of a string of inputs on a DFA by extending δ to a state and a string. Intuition: Extended δ is computed for state q and inputs a1a2…an by following a path in the transition graph, starting at q and selecting the arcs with labels a1, a2,…, an in turn.

Inductive Definition of Extended δ
Induction on length of string. Basis: δ(q, ε) = q Induction: δ(q,wa) = δ(δ(q,w),a) Remember: w is a string; a is an input symbol, by convention.

Example: Extended Delta
1 A A B B A C C C C δ(B,011) = δ(δ(B,01),1) = δ(δ(δ(B,0),1),1) = δ(δ(A,1),1) = δ(B,1) = C

Delta-hat We don’t distinguish between the given delta and the extended delta or delta-hat. The reason: δ(q, a) = δ(δ(q, ε), a) = δ(q, a) ˄ ˄ Extended deltas

Language of a DFA Automata of all kinds define languages.
If A is an automaton, L(A) is its language. For a DFA A, L(A) is the set of strings labeling paths from the start state to a final state. Formally: L(A) = the set of strings w such that δ(q0, w) is in F.

Example: String in a Language
String 101 is in the language of the DFA below. Start at A. Start 1 A C B 0,1

String 101 is in the language of the DFA below. Follow arc labeled 1. 0,1 1 1 A B C Start

String 101 is in the language of the DFA below. Then arc labeled 0 from current state B. 0,1 1 1 A B C Start

String 101 is in the language of the DFA below. Finally arc labeled 1 from current state A. Result is an accepting state, so 101 is in the language. 0,1 1 1 A B C Start

Example – Concluded The language of our example DFA is:
{w | w is in {0,1}* and w does not have two consecutive 1’s} Read a set former as “The set of strings w… Such that… These conditions about w are true.

Proofs of Set Equivalence
Often, we need to prove that two descriptions of sets are in fact the same set. Here, one set is “the language of this DFA,” and the other is “the set of strings of 0’s and 1’s with no consecutive 1’s.”

Proofs – (2) In general, to prove S = T, we need to prove two parts: S ⊆ T and T ⊆ S. That is: If w is in S, then w is in T. If w is in T, then w is in S. Here, S = the language of our running DFA, and T = “no consecutive 1’s.”

Part 1: S ⊆ T To prove: if w is accepted by
then w has no consecutive 1’s. Proof is an induction on length of w. Important trick: Expand the inductive hypothesis to be more detailed than the statement you are trying to prove. Start 1 A C B 0,1

The Inductive Hypothesis
If δ(A, w) = A, then w has no consecutive 1’s and does not end in 1. If δ(A, w) = B, then w has no consecutive 1’s and ends in a single 1. Basis: |w| = 0; i.e., w = ε. (1) holds since ε has no 1’s at all. (2) holds vacuously, since δ(A, ε) is not B. “length of” Important concept: If the “if” part of “if..then” is false, the statement is true.

Inductive Step Assume (1) and (2) are true for strings shorter than w, where |w| is at least 1. Because w is not empty, we can write w = xa, where a is the last symbol of w, and x is the string that precedes. IH is true for x. Start 1 A C B 0,1

Inductive Step – (2) Need to prove (1) and (2) for w = xa.
Start 1 A C B 0,1 Need to prove (1) and (2) for w = xa. (1) for w is: If δ(A, w) = A, then w has no consecutive 1’s and does not end in 1. Since δ(A, w) = A, δ(A, x) must be A or B, and a must be 0 (look at the DFA). By the IH, x has no 11’s. Thus, w has no 11’s and does not end in 1.

Inductive Step – (3) Now, prove (2) for w = xa: If δ(A, w) = B, then w has no 11’s and ends in 1. Since δ(A, w) = B, δ(A, x) must be A, and a must be 1 (look at the DFA). By the IH, x has no 11’s and does not end in 1. Thus, w has no 11’s and ends in 1. Start 1 A C B 0,1

Part 2: T ⊆ S X Y Now, we must prove: if w has no 11’s, then
w is accepted by Contrapositive : If w is not accepted by then w has 11. Y Start 1 A C B 0,1 Start 1 A C B 0,1 Key idea: contrapositive of “if X then Y” is the equivalent statement “if not Y then not X.”

Using the Contrapositive
Start 1 A C B 0,1 Because there is a unique transition from every state on every input symbol, each w gets the DFA to exactly one state. The only way w is not accepted is if it gets to C.

Using the Contrapositive – (2)
The only way to get to C [formally: δ(A,w) = C] is if w = x1y, x gets to B, and y is the tail of w that follows what gets to C for the first time. If δ(A,x) = B then surely x = z1 for some z. Thus, w = z11y and has 11. Start 1 A C B 0,1

Regular Languages A language L is regular if it is the language accepted by some DFA. Note: the DFA must accept only the strings in L, no others. Some languages are not regular. Intuitively, regular languages “cannot count” to arbitrarily high integers.

Example: A Nonregular Language
L1 = {0n1n | n ≥ 1} Note: ai is conventional for i a’s. Thus, 04 = 0000, e.g. Read: “The set of strings consisting of n 0’s followed by n 1’s, such that n is at least 1. Thus, L1 = {01, 0011, ,…} Proof？

Suppose there is a DFA with m states
S00m1m …S10m-11m… S2m-11S2m For the first m moves, there are m+1 states PHP! At least one state happened more than once Suppose the state is q Si = Sj = q S00m1m …q0m-i1m… q0m-j1m… S2m-11S2m How about S00m-j+i1m

Another Example L2 = {w | w in {(, )}* and w is balanced }
Balanced parentheses are those sequences of parentheses that can appear in an arithmetic expression. E.g.: (), ()(), (()), (()()),…

But Many Languages are Regular
They appear in many contexts and have many useful properties. Example: the strings that represent floating point numbers in your favorite language is a regular language.

Example: A Regular Language
L3 = { w | w in {0,1}* and w, viewed as a binary integer is divisible by 23} The DFA: 23 states, named 0, 1,…,22. Correspond to the 23 remainders of an integer divided by 23. Start and only final state is 0.

Transitions of the DFA for L3
If string w represents integer i, then assume δ(0, w) = i%23. Then w0 represents integer 2i, so we want δ(i%23, 0) = (2i)%23. Similarly: w1 represents 2i+1, so we want δ(i%23, 1) = (2i+1)%23. Example: δ(15,0) = 30%23 = 7; δ(11,1) = 23%23 = 0.

Another Example L4 = { w | w in {0,1}* and w, viewed as the reverse of a binary integer is divisible by 23} Example: is in L4, because its reverse, is 46 in binary. Hard to construct the DFA. But there is a theorem that says the reverse of a regular language is also regular.

Nondeterministic Finite Automata
Nondeterminism Subset Construction ε-Transitions

Nondeterminism A nondeterministic finite automaton has the ability to be in several states at once. Transitions from a state on an input symbol can be to any set of states.

Nondeterminism – (2) Start in one start state.
Accept if any sequence of choices leads to a final state. Intuitively: the NFA always “guesses right.”

Example: Moves on a Chessboard
States = squares. Inputs = r (move to an adjacent red square) and b (move to an adjacent black square). Start state, final state are in opposite corners.

Example: Chessboard – (2)
r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, * 1 2 5 7 9 3 4 8 6 1 r b 4 2 1 5 3 5 7 1 3 9 7 Accept, since final state reached

Formal NFA A finite set of states, typically Q.
An input alphabet, typically Σ. A transition function, typically δ. A start state in Q, typically q0. A set of final states F ⊆ Q.

Transition Function of an NFA
δ(q, a) is a set of states. Extend to strings as follows: Basis: δ(q, ε) = {q} Induction: δ(q, wa) = the union over all states p in δ(q, w) of δ(p, a)

Language of an NFA A string w is accepted by an NFA if δ(q0, w) contains at least one final state. The language of the NFA is the set of strings it accepts.

Example: Language of an NFA
For our chessboard NFA we saw that rbb is accepted. If the input consists of only b’s, the set of accessible states alternates between {5} and {1,3,7,9}, so only even-length, nonempty strings of b’s are accepted. What about strings with at least one r? 1 2 5 7 9 3 4 8 6

Equivalence of DFA’s, NFA’s
A DFA can be turned into an NFA that accepts the same language. If δD(q, a) = p, let the NFA have δN(q, a) = {p}. Then the NFA is always in a set containing exactly one state – the state the DFA is in after reading the same input.

Equivalence – (2) Surprisingly, for any NFA there is a DFA that accepts the same language. Proof is the subset construction. The number of states of the DFA can be exponential in the number of states of the NFA. Thus, NFA’s accept exactly the regular languages.

Subset Construction Given an NFA with states Q, inputs Σ, transition function δN, state state q0, and final states F, construct equivalent DFA with: States 2Q (Set of subsets of Q). Inputs Σ. Start state {q0}. Final states = all those with a member of F.

Critical Point The DFA states have names that are sets of NFA states.
But as a DFA state, an expression like {p,q} must be understood to be a single symbol, not as a set. Analogy: a class of objects whose values are sets of objects of another class.

Subset Construction – (2)
The transition function δD is defined by: δD({q1,…,qk}, a) is the union over all i = 1,…,k of δN(qi, a). Example: We’ll construct the DFA equivalent of our “chessboard” NFA.

Example: Subset Construction
r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, r b {1} {2,4} {5} {2,4} {5} Alert: What we’re doing here is the lazy form of DFA construction, where we only construct a state if we are forced to. *

r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, * r b {1} {2,4} {5} {2,4} {2,4,6,8} {1,3,5,7} {5} {2,4,6,8} {1,3,5,7}

r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, * r b {1} {2,4} {5} {2,4} {2,4,6,8} {1,3,5,7} {5} {2,4,6,8} {1,3,7,9} {2,4,6,8} {1,3,5,7} * {1,3,7,9}

r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, * r b {1} {2,4} {5} {2,4} {2,4,6,8} {1,3,5,7} {5} {2,4,6,8} {1,3,7,9} {2,4,6,8} {2,4,6,8} {1,3,5,7,9} {1,3,5,7} * {1,3,7,9} * {1,3,5,7,9}

r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, * r b {1} {2,4} {5} {2,4} {2,4,6,8} {1,3,5,7} {5} {2,4,6,8} {1,3,7,9} {2,4,6,8} {2,4,6,8} {1,3,5,7,9} {1,3,5,7} {2,4,6,8} {1,3,5,7,9} * {1,3,7,9} * {1,3,5,7,9}

r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, * r b {1} {2,4} {5} {2,4} {2,4,6,8} {1,3,5,7} {5} {2,4,6,8} {1,3,7,9} {2,4,6,8} {2,4,6,8} {1,3,5,7,9} {1,3,5,7} {2,4,6,8} {1,3,5,7,9} * {1,3,7,9} {2,4,6,8} {5} * {1,3,5,7,9}

r b 2, 4, ,3,5 2, 2, ,5,7 2,4,6,8 1,3,7,9 2, ,5,9 4, 4, ,7,9 6, * r b {1} * {1,3,5,7,9} {2,4,6,8} {1,3,5,7,9} * {1,3,7,9} {2,4,6,8} {5} {2,4,6,8} {2,4,6,8} {1,3,7,9} {5} {2,4} {2,4,6,8} {1,3,5,7} {1,3,5,7} {2,4} {5}

Proof of Equivalence: Subset Construction
The proof is almost a pun. Show by induction on |w| that δN(q0, w) = δD({q0}, w) Basis: w = ε: δN(q0, ε) = δD({q0}, ε) = {q0}.

Induction Assume IH for strings shorter than w.
Let w = xa; IH holds for x. Let δN(q0, x) = δD({q0}, x) = S. Let T = the union over all states p in S of δN(p, a). Then δN(q0, w) = δD({q0}, w) = T.

NFA’s With ε-Transitions
We can allow state-to-state transitions on ε input. These transitions are done spontaneously, without looking at the input string. A convenience at times, but still only regular languages are accepted.

Example: ε-NFA ε 0 1 ε A {E} {B} ∅ B ∅ {C} {D} C ∅ {D} ∅ D ∅ ∅ ∅
ε A {E} {B} ∅ B ∅ {C} {D} C ∅ {D} ∅ D ∅ ∅ ∅ E {F} ∅ {B, C} F {D} ∅ ∅ * C E F A B D 1 ε

Closure of States CL(q) = set of states you can reach from state q following only arcs labeled ε. Example: CL(A) = {A}; CL(E) = {B, C, D, E}. Closure of a set of states = union of the closure of each state. C E F A B D 1 ε

Extended Delta ˄ δ Intuition: (q, w) is the set of states you can reach from q following a path labeled w. Basis: (q, ε) = CL(q). Induction: (q, xa) is computed by: Start with (q, x) = S. Take the union of CL(δ(p, a)) for all p in S. ˄ δ ˄ δ ˄ δ

Example: Extended Delta
C E F A B D 1 ε Example: Extended Delta ˄ δ (A, ε) = CL(A) = {A}. (A, 0) = CL({E}) = {B, C, D, E}. (A, 01) = CL({C, D}) = {C, D}. Language of an ε-NFA is the set of strings w such that (q0, w) contains a final state. ˄ δ ˄ δ ˄ δ

Equivalence of NFA, ε-NFA
Every NFA is an ε-NFA. It just has no transitions on ε. Converse requires us to take an ε-NFA and construct an NFA that accepts the same language. We do so by combining ε–transitions with the next transition on a real input.

Picture of ε-Transition Removal
To here a We’ll go from here a a Transitions on ε Transitions on ε

Equivalence – (2) Start with an ε-NFA with states Q, inputs Σ, start state q0, final states F, and transition function δE. Construct an “ordinary” NFA with states Q, inputs Σ, start state q0, final states F’, and transition function δN.

Equivalence – (3) Compute δN(q, a) as follows:
Let S = CL(q). δN(q, a) is the union over all p in S of δE(p, a). F’ = the set of states q such that CL(q) contains a state of F.

Equivalence – (4) δ Prove by induction on |w| that
CL(δN(q0, w)) = E(q0, w). Thus, the ε-NFA accepts w if and only if the “ordinary” NFA does. ˄ δ

Example: ε-NFA-to-NFA
Interesting closures: CL(B) = {B,D}; CL(E) = {B,C,D,E} Example: ε-NFA-to-NFA Doesn’t change, since B, C, D have no transitions on 0. ε A {E} {B} ∅ B ∅ {C} {D} C ∅ {D} ∅ D ∅ ∅ ∅ E {F} ∅ {B, C} F {D} ∅ ∅ * A {E} {B} B ∅ {C} C ∅ {D} D ∅ ∅ E {F} {C, D} F {D} ∅ * Since closures of B and E include final state D. * * Since closure of E includes B and C; which have transitions on 1 to C and D. ε-NFA

Summary DFA’s, NFA’s, and ε–NFA’s all accept exactly the same set of languages: the regular languages. The NFA types are easier to design and may have exponentially fewer states than a DFA. But only a DFA can be implemented!

What Are They? Who Needs ‘em? An Example: Scoring in Tennis

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