 # Topics Automata Theory Grammars and Languages Complexities

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Topics Automata Theory Grammars and Languages Complexities
Formal Languages and Automata Theory

Why Automata Theory? To study abstract computing devices which are closely related to today’s computers. A simple example of finite state machine: 1 on start off 1 There are many different kinds of machines. Formal Languages and Automata Theory

Another Example 1 off on start off 1 1 When will this be on?
off on start off 1 1 When will this be on? Try 100, 1001, 1000, 111, 00, … Formal Languages and Automata Theory

Grammar and Languages Grammars and languages are closely related to automata theory and are the basis of many important software components like: Compilers and interpreters Text editors and processors Search engines System verification components Formal Languages and Automata Theory

Complexities Study the limits of computations. What kinds of problems can be solved with a computer? What kinds of problems can be solved efficiently? Formal Languages and Automata Theory

Preliminaries Alphabets Strings Languages Problems
Formal Languages and Automata Theory

Alphabets An alphabet is a finite set of symbols.
Usually, use  to represent an alphabet. Examples:  = {0,1}, the set of binary digits.  = {a, b, … , z}, the set of all lower-case letters.  = {(, )}, the set of open and close parentheses. Formal Languages and Automata Theory

Strings A string is a finite sequence of symbols from an alphabet.
Examples: 0011 and 11 are strings from  = {0,1} abc and bbb are strings from  = {a, b, … , z} (()(())) and )(() are strings from  = {(, )} Formal Languages and Automata Theory

Strings Empty string:  Length of string: |0010| = 4, |aa| = 2, ||=0
Prefix of string: aaabc, aaabc, aaabc Proper prefix of string: aaabc, aaabc Suffix of string: aaabc, aaabc, aaabc Proper suffix of string: aaabc, aaabc Substring of string: aaabc, aaabc, aaabc Formal Languages and Automata Theory

Strings What is 1? Is 1 different from ? How?
Concatenation: =abd, =ce, =abdce Exponentiation: =abd, 3=abdabdabd, 0= Reversal: =abd, R = dba k = set of all k-length strings formed by symbols in  e.g., ={a,b}, 2={ab, ba, aa, bb}, 0={} What is 1? Is 1 different from ? How? Formal Languages and Automata Theory

Strings Kleene Closure * = 012… = k0 k
e.g., ={a, b}, * = {, a, b, ab, aa, ba, bb, aaa, aab, abb, … } is the set of all strings formed by a’s and b’s. + = 123… = k>0 k i.e., * without the empty string. Formal Languages and Automata Theory

Languages A language is a set of strings over an alphabet. Examples:
={(, )}, L1={(), )(, (())} is a language over . ={a, b, c, … , z}, the set L of all legal English words is a language over . The set {} is a language over any alphabet. What is the difference between  and {}? Formal Languages and Automata Theory

Languages Other Examples:
={0, 1}, L={0n1n | n1} is a language over  consisting of the strings {01, 0011, , … } ={0, 1}, L = {0i1j | ji0} is a language over  consisting of the strings with some 0’s (possibly none) followed by at least as many 1’s. Formal Languages and Automata Theory

Problems In automata theory, a problem is to decide whether a given string is a member of some particular language. This formulation is general enough to capture the difficulty levels of all problems. Formal Languages and Automata Theory

Finite Automata ( or Finite State Machines)
This is the simplest kind of machine. We will study 3 types of Finite Automata: Deterministic Finite Automata (DFA) Non-deterministic Finite Automata (NFA) Finite Automata with -transitions (-NFA) Formal Languages and Automata Theory

Deterministic Finite Automata (DFA)
We have seen a simple example before: 1 on start off 1 There are some states and transitions (edges) between the states. The edge labels tell when we can move from one state to another. Formal Languages and Automata Theory

Definition of DFA A DFA is a 5-tuple (Q, , , q0, F) where
Q is a finite set of states  is a finite input alphabet  is the transition function mapping Q   to Q q0 in Q is the initial state (only one) F  Q is a set of final states (zero or more) Formal Languages and Automata Theory

Definition of DFA Q is the set of states: {on, off}
For example: 1 on start off 1 Q is the set of states: {on, off}  is the set of input symbols: {1}  is the transitions: off  1  on; on  1  off q0 is the initial state: off F is the set of final states (double circle): {on} Formal Languages and Automata Theory

Definition of DFA What are Q, , , q0 and F in this DFA?
Another Example: 1 q2 q1 q0 start 1 1 What are Q, , , q0 and F in this DFA? Formal Languages and Automata Theory

Transition Table For the previous example, the DFA is (Q,,,q0,F) where Q = {q0,q1,q2},  = {0,1}, F = {q2} and  is such that States Inputs 1 q0 q1 *q2 q2 Note that there is one transition only for each input symbol from each state. Formal Languages and Automata Theory

Language of a DFA Given a DFA M, the language accepted (or recognized) by M is the set of all strings that, starting from the initial state, will reach one of the final states after the whole string is read. For example, the language accepted by the previous example is the string that ends with 00 Formal Languages and Automata Theory

DFA Example Consider the DFA M=(Q,,,q0,F) where Q = {q0,q1,q2,q3},  = {0,1}, F = {q0} and  is: Inputs 1 Start q0 q1 States 1 1 q0 q2 q1 q1 q3 q0 OR 1 q2 q0 q3 q2 q3 1 q3 q1 q2 We can use a transition table or a transition diagram to specify the transitions. What input can take you to the final state in M? Formal Languages and Automata Theory