Information Science 2

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1 Information Science 2 Finite State Automata and Turing Machines Week 14 College of Information Science and Engineering Ritsumeikan University

2 Agenda Terms and concepts from Week 13 Computation and (abstract) computational devices Finite state automata Regular languages and expressions Regular grammars Turing machines Test 2

3 Recall concepts from Week 13 Random variables Descriptive statistics Mean, Variance PDF, PMF, Histogram (True) random numbers Pseudorandom numbers (PRN) PRN generator (PRNG) Evaluating PRNG Monte Carlo method Simulation 3

4 Today s class objective Learn the concepts of abstract computational devices After this lecture and study, you must be able to: Understand the concepts of finite state automata, regular languages and expressions, regular grammars, and Turing machines Understand and draw simple state diagrams of (finite) automata 4

5 Computation Computation is a mapping of a binary input to a binary output. Solving a problem with computers then is finding a particular ( desired, required, etc) mapping. For example, a decision problem is always a mapping from a given input to 0 or 1 (Yes or No) (Abstract) computational devices are simplified models of real computations. Automata theory is the study of abstract computational devices Finite (state) automata are devices used to model (analyze and compare) small (but not necessarily simple) computers. Turing machines are to model any computers Regular languages, expressions, and grammars provide mathematics for finite automata 5

6 Why do we need models? Consider a simple computer: a POWER b The lamp (printer, etc) is on if and only if both switches, a and b were flipped an odd number of times A simple model can be used to analyze this computer: a start 0 a 0 Inputs: switches a and b Actions: a for flip switch a, and b for flip switch b States: 0, 1 Accepting (final, good, desired, or Yes ) state: b b b b a 0 1 a 6

7 start Finite state automaton (FSA) We can now use the obtained model to analyze, for example, various inputs to our computer: a 0 a 0 b b b b a 0 1 a Input aabb State 0 Input abbb State 1 Input aaaaa State 0 Input aaaaab State 1 We can use similar diagrams (called state diagrams) to reason about and test much more complicated devices State diagrams represent finite state automata (FSA) a method to model computation 7

8 The linked image cannot be displayed. The file may have been moved, renamed, or deleted. Verify that the link points to the correct file and location. Formal definitions An alphabet Σ is a finite set of symbols A language L over an alphabet Σ (written as L(Σ )) is any subset of the set of all possible strings over Σ The set of all possible strings, including the empty string denoted ε, is written as Σ * Example: a language L 1 can be defined as the set of strings of length less than or equal to 2: L 1 = {A a, a, b, aa, ab, ba, bb}, given Σ = {a, b} and Σ * = {ε, a, b, ab, ba, aa, bb, aab, } 8

9 FSA formal definition A (deterministic) finite state automaton (FSA) is a 5-tuple M = Q, Σ, δ, q 0, F where Q is a finite set of states Σ is an alphabet (for the input set) δ : Q Σ Q is a transition function q 0 Q is the initial state F Q is a set of accepting states In state diagrams, states are denoted by circles, the accepting states by double circles, transitions by arrows, and the initial state by the arrow with no state on the left 9

10 Simple example Let M be the automaton with the input set (i.e., the alphabet) A = {c, d}, state set S = {s 0, s 1, s 2 }, and accepting state set Y = {s 1, s 2 }. Draw a state diagram D = D(M) of M c d or s 0 c s 2 s 1 d d c c s 1 s 0 s 2 d c, d or... draw yourself a few other possible diagrams 10

11 Regular languages The language L M of (or recognized by) FSA M = Q, Σ, δ, q 0, F is the set of all strings over Σ that, starting from q 0 and following the transitions as the string is read left to right, will reach an accepting state Example: M: a 0 a 0 b b b b a 0 1 a Not really b b b b L M = {(ab) n : n is odd} A regular language (over an alphabet Σ ) is any language for which there exists a finite automaton that recognizes this language 11

12 12 Regular expressions A regular expression is a mathematical model for defining a language. Regular expressions are similar to arithmetic expressions, but are defined recursively to work with symbols and strings Formally, for an alphabet Σ, regular expressions are ε,, and any r Σ. Other regular expressions for Σ can be constructed using only the following operations: Union: r 1 r 2 = L(r 1 ) L(r 2 ) for any r 1, r 2 Σ Concatenation: r 1 r 2 = L(r 1 ) L(r 2 ) Repetition: r 1 * = (L(r 1 ))* for any r 1 Σ

13 Simple examples Example 1: What language does 0*(0 1) represent? It represents zero or more 0's followed by either 0 or 1. Therefore, L = {0, 1, 00, 01, 000, 001, 0000, 0001, } Example 2: Find a regular expression for the language of the FSA defined as follows: a 0 a 0 b b b b a 0 1 a The language must include all strings over {a, b} that contain an odd number of a's and an odd number of b's. Therefore, L = (aa)*a(bb)*b + (bb)*b(aa)*a + ((aa)*a(bb)*b + (bb)*b(aa)*a)* 13

14 Practical regular expressions When writing regular expressions it is convenient to use two additional operations that can help make some expressions simpler: Optional: r? = L(r 1 ) L(ε) Multiple: r + = L(r) (L(r))* In terms of the basic operations, these are equivalent to: Optional: r? = r ε Multiple: r + = rr* (Remember that ε represents the empty string.) 14

15 15 Formal grammars Grammars provide a mathematical method for modeling languages A grammar is formally defined as a 4-tuple G = N, T, S, P where N is a finite set of non-terminal symbols T is a finite set of terminal symbols (alphabet) S is in N and is called the start symbol P is a finite set of productions, which are rules of the form α β (i.e., α can be replaced by β), where α and β are sequences of terminal and non-terminal symbols

16 Regular grammars Regular grammars provide an alternative method for modeling regular languages A regular grammar is a formal grammar in which the productions P are all of the form A a, A ab, or A ε where A N, B N, and a T * 16

17 Simple examples Example 1: Let S aa, A abs b, where a, b T, A N, and A abs b means A abs or A b. Draw a state diagram of the corresponding FSA. Solution: S a A b b a Example 2: Find a regular grammar for L = aab*a. Solution: S aa A ab B bb B a 17

18 FSA, expressions, and grammars Regular expressions FSA Regular grammars FSA is easy to program but is hard to analyze Regular expressions can be used to analyze FSA. It is, however, not always easy to find a regular expression of the language of a particular FSA. On the other hand, it is not obvious how to generate the language using a regular expression Regular grammars are convenient for language generation, but are often not easy to create, understand, and analyze 18

19 19 Turing machines Turing machines (first proposed by Alan Turing in 1937) are similar to FSA. Unlike FSA, they may be difficult to program but are (very) convenient to understand and analyze the complexity of computations Read/Write Head... Tape Rules: Current State: Control Device

20 Turing machine: The idea Read/Write Head... Tape Rules: Current State: Control Device A Turing machine can do the following: Write a character to the current tape cell Move the Read/Write Head one cell to the left or right Go into a new state Its behavior is defined by the rules and depends only on the input (i.e. the character in the current tape cell) and the current state 20

21 21 Turing machine: Example a b b a Rules: Read/Write Head Current State: A State Symbol Write Move State A a a R A A b a R B B b a R B... Tape Control Device

22 22 Example (cont-d) a b b a Rules: Read/Write Head Current State: A State Symbol Write Move State A a a R A A b a R B B b a R B... Tape Control Device

23 23 Example (cont-d) a b b a Rules: Read/Write Head Current State: A State Symbol Write Move State A a a R A A b a R B B b a R B... Tape Control Device

24 24 Example (cont-d) a a b a Rules: Read/Write Head Current State: B State Symbol Write Move State A a a R A A b a R B B b a R B... Tape Control Device

25 25 Example (cont-d) a a a a Read/Write Head... Tape Rules: Current State: B State Symbol Write Move State A a a R A A b a R B B b a R B Control Device

26 Example (cont-d) a a a a Read/Write Head... Tape Rules: Current State: B State Symbol Write Move State A a a R A A b a R B B b a R B Control Device Computation finishes (the machine stops) when there is no suitable rule 26

27 Summary of this lecture After this class, you are expected to know the following: What is computation What we need abstract computational devices for What is a FSA What are regular expressions and grammars What is a Turing machine (and how it works ) 27

28 28 Homework Read these slides There are no dedicated selfpreparation problems for Week 14, BUT review all the examples in these slides, as well as the last 2 self-preparation assignments! There will be similar problems in your final exam! Learn the English terms

29 29 Next class Course overview and preparations for the final exam Important terms and concepts of Information Science 2

30 Test 05 30