FORMAL LANGUAGES AND AUTOMATA THEORY

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1 1 FORMAL LANGUAGES AND AUTOMATA THEORY Course Summary This course covers foundational theory and practice of finite state machines, regular expression matching, and context-free grammars. The following are the course learning objectives: design of finite state automata and regular expressions conversion amongst DFA's, NFA's, and regular expressions prove that a language is not regular design of push-down automata and context-free grammars conversion amongst push-down automata and context-free grammars prove that a language is not context-free. Overview: Unit 1 This chapter gives the concepts of Finite State Automata, DFA, NFA, Regular languages and Regular Expressions, Applications of finite automata, Pumping Lemma for regular languages. Objectives: This course introduces the fundamental mathematical models of computation. We study both the inherent capabilities and limitations of these computational models as well as their relationships with formal languages. Topics to be covered include: Finite automata and regular languages Deterministic and nondeterministic computations Context-free grammars, languages, and pushdown automata Turing machines Undecidable problems Computational complexity, NP-completeness

2 2 Introduction: what is automata theory? Study of abstract computing devices, or machines Automaton = an abstract computing device Automata theory is a further step in abstracting your attention away from any particular kind of computer or particular programming language. In automata theory we consider a of computing. Such a model strips the computational machinery the programming language down to the bare minimum, so that it s easy to manipulate these theoretical machines (there are several such models, for different purposes, as you ll soon see) mathematically to prove things about their capabilities. For the most part, these mathematical models are not used for practical programming problems. Real programming languages are much more convenient to use. But the very flexibility that makes real languages easier to use also makes them harder to talk about in a formal way. The stripped-down theoretical machines are designed to be examined mathematically. What s a mathematical model? You ll see one shortly, called a finite-state machine. The point of this study is that the mathematical models are, in some important ways, to real computers and real programming languages. What this means is that any problem that can be solved on a real computer can be solved using these models, and vice versa. Anything we can prove about the models sheds light on the real problems of computer programming as well. Theory of Computation: A Historical Perspective Alan Turing ( )_Father of Modern ComputerScience, _ English mathematician Studied abstract machines calledturing machines even beforereal computers existed 1936 Alan Turing invented the Turing machine, and proved that there exists an unsolvable problem s Stored-program computers were built McCulloch and Pitts invented finite automata Kleene invented regular expressions and proved the equivalence of regular expression and finite automata 1956 Chomsky defined Chomsky hierarchy, which organized languages recognized by different automata into hierarchical classes.

3 Rabin and Scott introduced nondeterministic finite automata and proved its equivalence to (deterministic) finite automata s-1960 s More works on languages, grammars, and compilers 1965 Hartmantis and Stearns defined time complexity, and Lewis, Hartmantis and Stearns defined space complexity Cook showed the first NP-complete problem, the satisfiability prooblem Karp Showed many other NP-complete problems Diffie and Helllman defined Modern Cryptography based on NP-complete problems Rivest, Shamir and Adelman proposed a public-key encryption scheme, RSA.

4 4 Mathematical preliminaries and notations Symbol An atomic unit, such as a digit, character, lower-case letter, etc. Sometimes a word. [Formal language does not deal with the meaning of the symbols.] Alphabet A finite set of symbols, usually denoted by Σ. Σ = {0, 1} Σ = {0, a,, 4} Σ = {a, b, c, d} String A finite length sequence of symbols, presumably from some alphabet. u=ε w = 0110 y = 0aa x = aabcaa z = 111 special string: ε (also denoted by λ) concatenation: wz = length: w = 4 x = 6 but u = 0 reversal: y R = aa0 Some special sets of strings: Σ * Σ + All strings of symbols from Σ Σ * - {ε} Example: Σ = {0, 1} Σ * = {ε, 0, 1, 00, 01, 10, 11, 000, 001, } Σ + = {0, 1, 00, 01, 10, 11, 000, 001, } A (formal) language is: 1) A set of strings from some alphabet (finite or infinite), in other words 2) any subset L of Σ * Some special languages: {} The empty set/language, containing no strings {ε} A language containing one string, the empty string. Examples:

5 5 Σ = {0, 1} L = {x x is in Σ * and x contains an even number of 0 s} Σ = {0, 1, 2,, 9,.} L = {x x is in Σ * and x forms a finite length real number} = {0, 1.5, 9.326, } Σ = {a, b, c,, z, A, B,, Z} L = {x x is in Σ * and x is a Pascal reserved word} = {BEGIN, END, IF, } Σ = {Pascal reserved words} U { (, ),., :, ;, } U {Legal Pascal identifiers} L = {x x is in Σ * and x is a syntactically correct Pascal program} Σ = {English words} L = {x x Σ * and x is a syntactically correct English sentence} Deterministic Finite State Automata (DFA) One-way, infinite tape, broken into cells One-way, read-only tape head. Finite control, I.e., a program, containing the position of the read head, current symbol being scanned, and the current state.

6 6 A string is placed on the tape, read head is positioned at the left end, and the DFA will read the string one symbol at a time until all symbols have been read. The DFA will then either accept or reject. The finite control can be described by a transition diagram: Example #1: q 0 q 0 q 1 q 0 q 0 q 0 One state is final/accepting, all others are rejecting. The above DFA accepts those strings that contain an even number of 0 s Example #2:

7 7 a c c c b accepted q 0 q 0 q 1 q 2 q 2 q 2 a a c rejected q 0 q 0 q 0 q 1 Accepts those strings that contain at least two c s Formal Definition of a DFA A DFA is a five-tuple: M = (Q, Σ, δ, q 0, F) Q Σ q 0 F δ A finite set of states A finite input alphabet The initial/starting state, q 0 is in Q A set of final/accepting states, which is a subset of Q A transition function, which is a total function from Q x Σ to Q δ: (Q x Σ) > Q δ is defined for any q in Q and s in Σ, and δ(q,s) = q is equal to some state q in Q, could be q =q Intuitively, δ(q,s) is the state entered by M after reading symbol s while in state q. For example #1: Q = {q 0, q 1 } Σ = {0, 1} Start state is q 0

8 8 F = {q 0 } δ: 0 1 q 0 q 1 q 0 q 1 q 0 q 1 For example #2: Q = {q 0, q 1, q 2 } Σ = {a, b, c} Start state is q 0 F = {q 2 } δ: a b c q 0 q 0 q 0 q 1 q 1 q 1 q 1 q 2 q 2 q 2 q 2 q 2 Since δ is a function, at each step M has exactly one option. It follows that for a given string, there is exactly one computation.

9 9 Extension of δ to Strings δ^ : (Q x Σ * ) > Q δ^(q,w) The state entered after reading string w having started in state q. Formally: 1) δ^(q, ε) = q, and 2) For all w in Σ * and a in Σ δ^(q,wa) = δ (δ^(q,w), a) Recall Example #1: What is δ^(q 0, 011)? Informally, it is the state entered by M after processing 011 having started in state q 0. Formally: δ^(q 0, 011) = δ (δ^(q 0,01), 1) by rule #2 = δ (δ ( δ^(q 0,0), 1), 1) by rule #2 = δ (δ (δ (δ^(q 0, λ), 0), 1), 1) by rule #2 = δ (δ (δ(q 0,0), 1), 1) by rule #1 = δ (δ (q 1, 1), 1) by definition of δ = δ (q 1, 1) by definition of δ = q 1 by definition of δ

10 10 Is 011 accepted? No, since δ^(q 0, 011) = q 1 is not a final state. Note that: δ^ (q,a) = δ(δ^(q, ε), a) by definition of δ^, rule #2 = δ(q, a) by definition of δ^, rule #1 Therefore: δ^ (q, a 1 a 2 a n ) = δ(δ( δ(δ(q, a1), a2) ), a n ) Hence, we can use δ in place of δ^: δ^(q, a 1 a 2 a n ) = δ(q, a 1 a 2 a n ) Recall Example #2: What is δ(q 0, 011)? Informally, it is the state entered by M after processing 011 having started in state q 0. Formally: δ(q 0, 011) = δ (δ(q 0,01), 1) by rule #2 = δ (δ (δ(q 0,0), 1), 1) by rule #2 = δ (δ (q 1, 1), 1) by definition of δ = δ (q 1, 1) by definition of δ = q 1 by definition of δ Is 011 accepted? No, since δ(q 0, 011) = q 1 is not a final state. What is δ(q 1, 10)? δ(q 1, 10) = δ (δ(q 1,1), 0) by rule #2

11 11 = δ (q 1, 0) by definition of δ = q 2 by definition of δ Is 10 accepted? No, since δ(q 0, 10) = q 1 is not a final state. The fact that δ(q 1, 10) = q 2 is irrelevant!

12 12 Definitions for DFAs Let M = (Q, Σ, δ,q 0,F) be a DFA and let w be in Σ *. Then w is acceptedby Miff δ(q 0,w) = p for some state p in F. Let M = (Q, Σ, δ,q 0,F) be a DFA. Then the language accepted by M is the set: L(M) = {w w is in Σ * and δ(q 0,w) is in F} Another equivalent definition: L(M) = {w w is in Σ * and w is accepted by M} Let L be a language. Then L is a regular languageiff there exists a DFA M such that L = L(M). Let M 1 = (Q 1, Σ 1, δ 1, q 0, F 1 ) and M 2 = (Q 2, Σ 2, δ 2, p 0, F 2 ) be DFAs. Then M 1 and M 2 are equivalentiff L(M 1 ) = L(M 2 ). Notes: A DFA M = (Q, Σ, δ,q 0,F) partitions the set Σ * into two sets: L(M) and Σ * - L(M). If L = L(M) then L is a subset of L(M) and L(M) is a subset of L. Similarly, if L(M 1 ) = L(M 2 ) then L(M 1 ) is a subset of L(M 2 ) and L(M 2 ) is a subset of L(M 1 ). Some languages are regular, others are not. For example, if L 1 = {x x is a string of 0's and 1's containing an even number of 1's} and Questions: L 2 = {x x = 0 n 1 n for some n >= 0} then L 1 is regular but L 2 is not. How do we determine whether or not a given language is regular? How could a program simulate a DFA?

13 13 Give a DFA M such that: L(M) = {x x is a string of 0 s and 1 s and x >= 2} Give a DFA M such that: L(M) = {x x is a string of (zero or more) a s, b s and c s such that x does not contain the substring aa} Give a DFA M such that: L(M) = {x x is a string of a s, b s and c s such that x contains the substring aba}

14 14 Give a DFA M such that: L(M) = {x x is a string of a s and b s such that x contains both aa and bb} Nondeterministic Finite State Automata (NFA) An NFA is a five-tuple: M = (Q, Σ, δ, q 0, F) Q Σ q 0 F δ A finite set of states A finite input alphabet The initial/starting state, q 0 is in Q A set of final/accepting states, which is a subset of Q A transition function, which is a total function from Q x Σ to 2 Q δ: (Q x Σ) > 2 Q -2 Q is the power set of Q, the set of all subsets of Q δ(q,s) -The set of all states p such that there is a transition labeled s from q to p δ(q,s) is a function from Q x S to 2 Q (but not to Q)

15 15 Example #1: some 0 s followed by some 1 s Q = {q 0, q 1, q 2 } Σ = {0, 1} Start state is q 0 F = {q 2 } Example #2: pair os 0 s or pair of 1 s Q = {q 0, q 1, q 2, q 3, q 4 } Σ = {0, 1} Start state is q 0 F = {q 2, q 4 }

16 16 Notes: δ(q,s) may not be defined for some q and s (why?). Informally, a string is said to be accepted if there exists a path to some state in F. The language accepted by an NFA is the set of all accepted strings. Question: How does an NFA find the correct/accepting path for a given string? NFAs are a non-intuitive computing model.

17 17 We are primarily interested in NFAs as language defining devices, i.e., do NFAs accept languages that DFAs do not? Other questions are secondary, including practical questions such as whether or not there is an algorithm for finding an accepting path through an NFA for a given string, Determining if a given NFA (example #2) accepts a given string (001) can be done algorithmically Each level will have at most n states Another example (010): All paths have been explored, and none lead to an accepting state.

18 18 Extension of δ to Strings and Sets of States Step #1: Given δ: (Q x Σ) > 2 Q define δ # : (2 Q x Σ) > 2 Q as follows: 1) δ # (R, a) = δ(q, a) for all subsets R of Q, and symbols a in Σ Note that: δ # ({p},a) = δ(q, a) by definition of δ #, rule #1 above Hence, we can use δ for δ # = δ(p, a) δ({q 0, q 2 }, 0) δ({q 0, q 1, q 2 }, 0) These now make sense, but previously they did not. Example: Step #2: δ({q 0, q 2 }, 0) = δ(q 0, 0) U δ(q 2, 0) = {q 1, q 3 } U {q 3, q 4 } = {q 1, q 3, q 4 } δ({q 0, q 1, q 2 }, 1) = δ(q 0, 1) U δ(q 1, 1) U δ(q 2, 1) = {} U {q 2, q 3 } U {} = {q 2, q 3 } Given δ: (2 Q x Σ) > 2 Q define δ^: (2 Q x Σ*) > 2 Q as follows: δ^(r,w) The set of states M could be in after processing string w, having starting from any state in R. Formally: 2) δ^(r, ε) = R for any subset R of Q 3) δ^(r,wa) = δ (δ^(r,w), a) for any w in Σ *, a in Σ, and subset R of Q

19 19 Note that: Hence, we can use δ for δ^ δ^(r, a) = δ(δ^(r, ε), a) by definition of δ^, rule #3 above = δ(r, a) by definition of δ^, rule #2 above δ({q 0, q 2 }, 0110) δ({q 0, q 1, q 2 }, ) These now make sense, but previously they did not. Example: What is δ({q 0 }, 10)? Informally: The set of states the NFA could be in after processing 10, having started in state q 0, i.e., {q 1, q 2, q 3 }. Formally: δ({q 0 }, 10) = δ(δ({q 0 }, 1), 0) = δ({q 0 }, 0) = {q 1, q 2, q 3 } Is 10 accepted? Yes!

20 20 Example: What is δ({q 0, q 1 }, 1)? δ({q 0, q 1 }, 1) = δ({q 0 }, 1) U δ({q 1 }, 1) = {q 0 } U {q 2, q 3 } = {q 0, q 2, q 3 } What is δ({q 0, q 2 }, 10)? δ({q 0, q 2 }, 10) = δ(δ({q 0, q 2 }, 1), 0) = δ(δ({q 0 }, 1) U δ({q 2 }, 1), 0) = δ({q 0 } U {q 3 }, 0) = δ({q 0,q 3 }, 0) = δ({q 0 }, 0) U δ({q 3 }, 0) = {q 1, q 2, q 3 } U {} = {q 1, q 2, q 3 } Example: δ({q 0 }, 101) = δ(δ({q 0 }, 10), 1) = δ(δ(δ({q 0 }, 1), 0), 1) = δ(δ({q 0 }, 0), 1) = δ({q 1, q 2, q 3 }, 1) = δ({q 1 }, 1) U δ({q 2 }, 1) U δ({q 3 }, 1) = {q 2, q 3 } U {q 3 } U {} = {q 2, q 3 } Is 101 accepted? Yes!

21 21 Definitions for NFAs Let M = (Q, Σ, δ,q 0,F) be an NFA and let w be in Σ *. Then w is accepted by M iffδ({q 0 }, w) contains at least one state in F. Let M = (Q, Σ, δ,q 0,F) be an NFA. Then the language accepted by M is the set: L(M) = {w w is in Σ * and δ({q 0 },w) contains at least one state in F} Another equivalent definition: L(M) = {w w is in Σ * and w is accepted by M} Equivalence of DFAs and NFAs Do DFAs and NFAs accept the same class of languages? Is there a language L that is accepted by a DFA, but not by any NFA? Is there a language L that is accepted by an NFA, but not by any DFA? Observation: Every DFA is an NFA. Therefore, if L is a regular language then there exists an NFA M such that L = L(M). It follows that NFAs accept all regular languages. But do NFAs accept more? Consider the following DFA: 2 or more c s Q = {q 0, q 1, q 2 } Σ = {a, b, c} Start state is q 0 F = {q 2 }

22 22 An Equivalent NFA: Q = {q 0, q 1, q 2 } Σ = {a, b, c} Start state is q 0 F = {q 2 }

23 23 Lemma 1: Let M be an DFA. Then there exists a NFA M such that L(M) = L(M ). Proof: Every DFA is an NFA. Hence, if we let M = M, then it follows that L(M ) = L(M). The above is just a formal statement of the observation from the previous slide. Lemma 2: Let M be an NFA. Then there exists a DFA M such that L(M) = L(M ). Proof: (sketch) Let M = (Q, Σ, δ,q 0,F). Define a DFA M = (Q, Σ, δ,q 0,F ) as: Q = 2 Q Each state in M corresponds to a = {Q 0, Q 1,,} subset of states from M whereq u = [q i0, q i1, q ij ] F = {Q u Q u contains at least one state in F} q 0 = [q 0 ] δ (Q u, a) = Q v iff δ(q u, a) = Q v Example: empty string or start and end with 0 Q = {q 0, q 1 } Σ = {0, 1} Start state is q 0 F = {q 1 }

24 24 Construct DFA M as follows: δ({q 0 }, 0) = {q 1 } => δ ([q 0 ], 0) = [q 1 ] δ({q 0 }, 1) = {} => δ ([q 0 ], 1) = [ ] δ({q 1 }, 0) = {q 0, q 1 } => δ ([q 1 ], 0) = [q 0, q 1 ] δ({q 1 }, 1) = {q 1 } => δ ([q 1 ], 1) = [q 1 ] δ({q 0, q 1 }, 0) = {q 0, q 1 }=> δ ([q 0, q 1 ], 0) = [q 0, q 1 ] δ({q 0, q 1 }, 1) = {q 1 } => δ ([q 0, q 1 ], 1) = [q 1 ] δ({}, 0) = {} => δ ([ ], 0) = [ ] δ({}, 1) = {} => δ ([ ], 1) = [ ] Theorem: Let L be a language. Then there exists an DFA M such that L = L(M) iff there exists an NFA M such that L = L(M ). Proof: (if) Suppose there exists an NFA M such that L = L(M ). Then by Lemma 2 there exists an DFA M such that L = L(M). (only if) Suppose there exists an DFA M such that L = L(M). Then by Lemma 1 there exists an NFA M such that L = L(M ). Corollary: The NFAs define the regular languages.

25 25 Note: Suppose R = {} δ(r, 0) = δ(δ(r, ε), 0) = δ(r, 0) = δ(q, 0) = {} Since R = {} Exercise - Convert the following NFA to a DFA: NFAs with ε Moves An NFA-ε is a five-tuple: M = (Q, Σ, δ, q 0, F) Q A finite set of states Σ A finite input alphabet q 0 The initial/starting state, q 0 is in Q F A set of final/accepting states, which is a subset of Q δa transition function, which is a total function from Q x Σ U {ε} to 2 Q

26 26 δ: (Q x (Σ U {ε})) > 2 Q δ(q,s) -The set of all states p such that there is a transition labeled a from q to p, where a is in Σ U {ε} Sometimes referred to as an NFA-ε other times, simply as an NFA. Example:

27 27 Informal Definitions Let M = (Q, Σ, δ,q 0,F) be an NFA-ε. A String w in Σ * is accepted by M iff there exists a path in M from q 0 to a state in F labeled by w and zero or more ε transitions. The language accepted by M is the set of all strings from Σ * that are accepted by M. ε-closure Define ε-closure(q) to denote the set of all states reachable from q by zero or more ε transitions. Examples: (for the previous NFA) ε-closure(q 0 ) = {q 0, q 1, q 2 } ε-closure(q 2 ) = {q 2 } ε-closure(q 1 ) = {q 1, q 2 } ε-closure(q 3 ) = {q 3 } ε-closure(q) can be extended to sets of states by defining: ε-closure(p) = ε-closure(q) Examples: ε-closure({q 1, q 2 }) = {q 1, q 2 } ε-closure({q 0, q 3 }) = {q 0, q 1, q 2, q 3 } Extension of δ to Strings and Sets of States What we currently have: δ : (Q x (Σ U {ε})) > 2 Q What we want (why?): δ : (2 Q x Σ * ) > 2 Q As before, we will do this in two steps, which will be slightly different from the book, and we will make use of the following NFA.

28 28 Step #1: Given δ: (Q x (Σ U {ε})) > 2 Q define δ # : (2 Q x (Σ U {ε})) > 2 Q as follows: 1) δ # (R, a) = δ(q, a) for all subsets R of Q, and symbols a in Σ U {ε} Note that: δ # ({p},a) = δ(q, a) by definition of δ #, rule #1 above = δ(p, a) Hence, we can use δ for δ # δ({q 0, q 2 }, 0) δ({q 0, q 1, q 2 }, 0) These now make sense, but previously they did not. Examples: What is δ({q 0, q 1, q 2 }, 1)? δ({q 0, q 1, q 2 }, 1) = δ(q 0, 1) U δ(q 1, 1) U δ(q 2, 1) = { } U {q 0, q 3 } U {q 2 } = {q 0, q 2, q 3 } What is δ({q 0, q 1 }, 0)? δ({q 0, q 1 }, 0) = δ(q 0, 0) U δ(q 1, 0) = {q 0 } U {q 1, q 2 } = {q 0, q 1, q 2 }

29 29 Step #2: Given δ: (2 Q x (Σ U {ε})) > 2 Q define δ^: (2 Q x Σ * ) > 2 Q as follows: δ^(r,w) The set of states M could be in after processing string w, having starting from any state in R. Formally: 2) δ^(r, ε) = ε-closure(r) - for any subset R of Q 3) δ^(r,wa) = ε-closure(δ(δ^(r,w), a)) - for any w in Σ *, a in Σ, and Can we use δ for δ^? Consider the following example: δ({q 0 }, 0) = {q 0 } subset R of Q δ^({q 0 }, 0) = ε-closure(δ(δ^({q 0 }, ε), 0)) By rule #3 = ε-closure(δ(ε-closure({q 0 }), 0)) By rule #2 = ε-closure(δ({q 0, q 1, q 2 }, 0)) By ε-closure = ε-closure(δ(q 0, 0) U δ(q 1, 0) U δ(q 2, 0)) By rule #1 = ε-closure({q 0 } U {q 1, q 2 } U {q 2 }) = ε-closure({q 0, q 1, q 2 }) = ε-closure({q 0 }) U ε-closure({q 1 }) U ε-closure({q 2 }) = {q 0, q 1, q 2 } U {q 1, q 2 } U {q 2 } = {q 0, q 1, q 2 } So what is the difference? δ(q 0, 0) δ^(q 0, 0) - Processes 0 as a single symbol, without ε transitions. - Processes 0 using as many ε transitions as are possible.

30 30 Example: δ^({q 0 }, 01) = ε-closure(δ(δ^({q 0 }, 0), 1)) By rule #3 = ε-closure(δ({q 0, q 1, q 2 }), 1) Previous slide = ε-closure(δ(q 0, 1) U δ(q 1, 1) U δ(q 2, 1)) By rule #1 = ε-closure({ } U {q 0, q 3 } U {q 2 }) = ε-closure({q 0, q 2, q 3 }) = ε-closure({q 0 }) U ε-closure({q 2 }) U ε-closure({q 3 }) = {q 0, q 1, q 2 } U {q 2 } U {q 3 } = {q 0, q 1, q 2, q 3 } Definitions for NFA-ε Machines Let M = (Q, Σ, δ,q 0,F) be an NFA-ε and let w be in Σ *. Then w is accepted by M iff δ^({q 0 }, w) contains at least one state in F. Let M = (Q, Σ, δ,q 0,F) be an NFA-ε. Then the language accepted by M is the set: L(M) = {w w is in Σ * and δ^({q 0 },w) contains at least one state in F} Another equivalent definition: L(M) = {w w is in Σ * and w is accepted by M} Equivalence of NFAs and NFA-εs Do NFAs and NFA-ε machines accept the same class of languages? Is there a language L that is accepted by a NFA, but not by any NFA-ε? Is there a language L that is accepted by an NFA-ε, but not by any DFA? Observation: Every NFA is an NFA-ε. Therefore, if L is a regular language then there exists an NFA-ε M such that L = L(M). It follows that NFA-ε machines accept all regular languages. But do NFA-ε machines accept more?

31 31 Lemma 1: Let M be an NFA. Then there exists a NFA-ε M such that L(M) = L(M ). Proof: Every NFA is an NFA-ε. Hence, if we let M = M, then it follows that L(M ) = L(M). The above is just a formal statement of the observation from the previous slide. Lemma 2: Let M be an NFA-ε. Then there exists a NFA M such that L(M) = L(M ). Proof: (sketch) Let M = (Q, Σ, δ,q 0,F) be an NFA-ε. Define an NFA M = (Q, Σ, δ,q 0,F ) as: F = F U {q 0 } if ε-closure(q 0 ) contains at least one state from F F = F otherwise δ (q, a) = δ^(q, a) - for all q in Q and a in Σ Notes: δ : (Q x Σ) > 2 Q is a function M has the same state set, the same alphabet, and the same start state as M M has no ε transitions

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34 34 Theorem: Let L be a language. Then there exists an NFA M such that L= L(M) iff there exists an NFA-ε M such that L = L(M ). Proof: (if) Suppose there exists an NFA-ε M such that L = L(M ). Then by Lemma 2 there exists an NFA M such that L = L(M). (only if) Suppose there exists an NFA M such that L = L(M). Then by Lemma 1 there exists an NFA-ε M such that L = L(M ). Corollary: The NFA-ε machines define the regular languages.

35 35 Applications of Automata Theory Linguistics Automata theory is the basis for the theory of formal languages. A proper treatment of formal language theory begins with some basic definitions: A symbol is simply a character, an abstraction that is meaningless by itself. An alphabet is a finite set of symbols. A word is a finite string of symbols from a given alphabet. Finally, a language is a set of words formed from a given alphabet. The set of words that form a language is usually infinite, although it may be finite or empty as well. Formal languages are treated like mathematical sets, so they can undergo standard set theory operations such as union and intersection. Additionally, operating on languages always produces a language. As sets, they are defined and classified using techniques of automata theory. Formal languages are normally defined in one of three ways, all of which can be described by automata theory: regular expressions standard automata a formal grammar system Regular Expressions Example alphabeta1={a,b} alphabeta2={1,2} language L1= the set of all words over A1 = {a, aab,...} language L2= the set of all words over A2 = {2, 11221,...} language L3 = L1 L2 language L4 = {an n is even} = {aa, aaaa,...} language L5 = {anbn n is natural} = {ab, aabb,...} Languages can also be defined by any kind of automaton, like a Turing Machine. In general, any automata or machine M operating on an alphabet A can produce a perfectly valid language L. The system could be represented by a bounded Turing Machine tape, for example, with each cell representing a word. After the instructions halt, any word with value 1 (or ON) is accepted and becomes part of the generated language. From this idea, one can defne the complexity of a language, which can be classified as P or NP, exponential, or probabilistic, for example. Noam Chomsky extended the automata theory idea of complexity hierarchy to a formal language hierarchy, which led to the concept of formal grammar. A formal grammar system is

36 36 a kind of automata specifically defined for linguistic purposes. The parameters of formal grammar are generally defined as: a set of non-terminal symbols N a set of terminal symbols Σ a set of production rules P a start symbol S Grammar Example Start symbol = S non-terminals = {S} terminals = {a, b} production rules: S asb, S ba S asb abab S â asb aasbb aababb L = {abab, aababb,...} As in purely mathematical automata, grammar automata can produce a wide variety of complex languages from only a few symbols and a few production rules. Chomsky's hierarchy defines four nested classes of languages, where the more precise aclasses have stricter limitations on their grammatical production rules. The formality of automata theory can be applied to the analysis and manipulation of actual human language as well as the development of human-computer interaction (HCI) and artificial intelligence (AI). Biology To the casual observer, biology is an impossibly complex science. Traditionally, the intricacy and variation found in life science has been attributed to the notion of natural selection. Species become "intentionally" complex because it increases their chance for survival. For example, a camoflauge-patterned toad will have a far lower risk of being eaten by a python than a frog colored entirely in orange. This idea makes sense, but automata theory offers a simpler and more logical explanation, one that relies not on random, optimizing mutations but on a simple set of rules. Basic automata theory shows that simplicity can naturally generate complexity. Apparent randomness in a system results only from inherent complexities in the behavior of automata, and seemingly endless variations in outcome are only the products of different initial states. A simple mathematical example of this notion is found in irrational numbers. The square root of nine is just 3, but the square root of ten has no definable characteristics. One could compute the decimal digits for the lifetime of the universe and never find any kind of recurring patter or orderly progression; instead, the sequence of numbers seemse utterly random. Similar results are

37 37 found in simple two-dimensional cellular automaton. These structures form gaskets and fractals that sometimes appear orderly and geometric, but can resemble random noise without adding any states or instructions to the set of production rules. The most classic merging of automata theory and biology is John Conway's Game of Life. "Life" is probably the most frequently written program in elementary computer science. The basic structure of Life is a two-dimensional cellular automaton that is given a start state of any number of filled cells. Each time step, or generation, switches cells on or off depending on the state of the cells that surround it. The rules are defined as follows: All eight of the cells surrounding the current one are checked to see if they are on or not. Any cells that are on are counted, and this count is then used to determine what will happen to the current cell: 1. Death: if the count is less than 2 or greater than 3, the current cell is switched off. 2. Survival: if (a) the count is exactly 2, or (b) the count is exactly 3 and the current cell is on, the current cell is left unchanged. 3. Birth: if the current cell is off and the count is exactly 3, the current cell is switched on. Like any manifestation of automata theory, the Game of LIfe can be defined using extremely simple and concise rules, but can produce incredibly complex and intricate patterns. In addition to the species-level complexity illustrated by the Game of Life, complexity within an individual organism can also be explained using automata theory. An organism might be complex in its full form, but examining constituent parts reveals consistency, symmetry, and patterns. Simple organisms, like maple leaves and star fish, even suggest mathematical structure in their full form. Using ideas of automata theory as a basis for generating the wide variety of life forms we see today, it becomes easier to think that sets of mathematical rules might be responsible for the complexity we notice every day. Inter-species observations also support the notion of automata theory instead of the specific and random optimization in natural selection. For example, there are striking similarities in patterns between very different orgranisms: Mollusks and pine cones grow by the Fibonacci sequence, reproducible by math. Leopards and snakes can have nearly identical pigmentation patterns, reproducible by two-dimensional automata. With these ideas in mind, it is difficult not to imagine that any biolgical attribute can be simulated with abstract machines and reduced to a more manageable level of simplicity.

38 38 Key Terms: Automaton: Abstract model of a digital computer. Acceptor: Automaton whose output response is Yes or No DFA: Deterministic Finite Automata. NFA: Non-deterministic Finite Automata. Regular Language: Language that can be constructed from the set operations Union, Concatenation and Kleene star. Regular expression: Mathematical tool built from a set of primitives and operations. Two-way Finite Automata: Machines that can read input string in either direction. Multiple Choice Questions 1. The length of the string is= Can not be determined 2. Let A and B are two sets, an element in A but not in B can be denoted by 1. A-B 2. AUB 3. A*B 4. A+B 3. A relation which satisfies reflexive, symmetric, transitive properties then the relation is called as 1. Equivalent Relation 2. Non-equivalent 3. Transpose relation 4. None 4. The Finite Automata accepts which of the language 1. Regular language 2. Context free language 3. Context sensitive language 4. None 5. Identify which of the following is not a regular expression 1. (10+01)* 2. (1+0)*(1+01)* 3. (1+0) (10^11)* 6. Which of the following conversion is not possible 1. NFA to DFA

39 39 2. NFA- to NFA 3. RE to NFA 4. None 7. Finite automata can remember the previous input. 1. True 2. False 8. The transition function of a NFA- includes 1. The mapping from set of states to set of input symbols The mapping from set of states to set of input symbols except and power set of Q 3. The mapping from set of states to set of input symbols including and power set of Q 4. The mapping from set of states to set of input symbols except 9. For every NFA, there exists an equivalent DFA. 1. True 2. False 10. In Finite Automata there may be more than one final state 1. True 2. False 11. is a input symbol, on which we can move from one state to another state with out any effect. 12. The -closure of state q0 is 1. q0 2. q1 3. q2 4. Final state 13. Which of the following is the equivalent language for the regular expression 1) {0, } 2) {,0,01,001,0101.} 3) {,01,011,0111 } 4) {01,0111,..) 14. +RR*= +R*R= 1) R 2) R* 3) 4) which of the following is the method of converting R.E to NFA- 1. Thompson s Construction 2. Arden s Theorem 3. State elimination Technique 4. Mealy Machine

40 40 Review Questions: 1. Define Finite automaton. 2. What is a finite automaton? Give two examples. 3. What is a transition diagram? 4. Enumerate the differences between DFA and NFA. 5. Construct a DFA for the regular expression aa*/bb*. 6. Construct a finite automata for the language {0n n mod 3 =2,n>=0}. 7. Explain the extended transition function for NFA, DFA and - NFA. 8. Construct a DFA that accepts all the strings on {0,1} except those containing the substring Explain in detail with an example the conversion of NFA to DFA. 10. Construct a NFA accepting the set of strings over {a,b} ending in aba. Use it to construct a DFA accepting the same set of strings. 11. For the finite state machine M given in the following table, test whether the strings , are accepted by M. State Input 0 1 q0 q0 q1 q1 q3 q0 q2 q0 q3 q3 q1 q2 12. Prove the following by the principle of induction Σk2 = n(n+1)(2n+1)/6 13. Construct a DFA that will accept strings on {a, b} where the number of b s is divisible by 3.

41 41 2 Marks question and Answers

42 42 SUMMARY Finite automata involves states and transitions among states in response to inputs. The basic method of deductive proofs proceeds by listing statements that are either given to be true or that follow logically from some of the previous statements. A DFA has a finite set of states and a finite set of input symbols. Transition diagrams are convenient to represent automata by a graph in which the nodes are the states and arcs are labeled by input symbols, indicating the transitions of that automaton. NFA differs from the DFA in that the NFA can have any number of transitions