Theory of computing presentation
- 1. Presentation Topic is: To Our Theory of Computing Presentation
- 2. Group Member List: Name ID Md. Touhidur Rahman 152-15-6232
- 3. Computational Models Computational Model are four type: Finite Automata - DFA & NFA Context Free Grammar Push Down Automata Turing Machine
- 4. Finite Automata - DFA & NFA Definition: A deterministic finite automaton (DFA) consists of • 1. a finite set of states (often denoted Q) • 2. a finite set of symbols (alphabet) • 3. a transition function that takes as argument a state and a • symbol and returns a state (often denoted ) • 4. a start state often denoted q0 • 5. a set of final or accepting states (often denoted F)
- 5. Finite Automata - DFA & NFA How to present a DFA? With a transition table • The ! indicates the start state: here q0 • The indicates the final state(s) (here only one final state q1) • This defines the following transition diagram
- 6. Finite Automata - DFA & NFA Formal Definition:
- 7. Finite Automata - DFA & NFA How to present a NFA? With a transition table
- 8. Context Free Grammar What is a grammar?: A grammar consists of one or more variables that represent classes of strings (i.e., languages) There are rules that say how the strings in each class are constructed. The construction can use : 1. symbols of the alphabet 2. strings that are already known to be in one of the classes 3. or both
- 9. Context Free Grammar Definition of Context-Free Grammar: A CFG (or just a grammar) G is a tuple G = (V, T, P, S) where 1. V is the (finite) set of variables (or nonterminal or syntactic categories). Each variable represents a language, i.e., a set of strings . 2. T is a finite set of terminals, i.e., the symbols that form the strings of the language being defined. 3. P is a set of production rules that represent the recursive definition of the language. 4. S is the start symbol that represents the language being defined. Other variables represent auxiliary classes of strings that are used to help define the language of the start symbol.
- 10. Context Free Grammar Examples 1(CFG): The grammar: 1. E → I 2. E → E + E 3. E → E ∗ E 4. E → (E) 5. I → a 6. I → b 7. I → Ia 8. I → Ib 9. I → I 0 10. I → I 1 The Grammar for expression is started formally as, Grammar G 1 = ( {E, I }, T, P, E) where: T = { +, ∗, (, ), a, b, 0, 1 } is the set of symbol and P is the set of productions
- 11. Context Free Grammar Derivation: The sequence of substitutions to generate a string is called a derivation. Derivation both are two types: 1.Left-Most Derivation. 2.Right-Most Derivation. Left-Most Derivation: A derivation which always replace the leftmost variable in each step is called a leftmost derivation. Examples of Left-Most derivation to Previous Examples1: Derivation of a ∗ (a + b00) by G 1 E ⇒ E ∗ E ⇒ I ∗ E ⇒ a ∗ E ⇒ a ∗ (E) ⇒ a ∗ (E + E) ⇒ a ∗ (I + E) ⇒ a ∗ (a + E) ⇒ a ∗ (a + I) ⇒ a ∗ (a + I0) ⇒ a ∗ (a + I00) ⇒ a ∗ (a + b00) We can conclude that E ∗ ⇒lm a ∗ (a + b00).
- 12. Context Free Grammar Right-Most Derivation: Rightmost derivation Always replace the rightmost variable by one of its rule-bodies. Examples of Right-Most derivation to Previous Examples 1: E ⇒E ∗ E ⇒ E ∗ (E) ⇒ E ∗ (E + E) ⇒ E ∗ (E + I) ⇒rm E ∗ (E + I0) ⇒ E ∗ (E + I00) ⇒ E ∗ (E + b00) ⇒ E ∗ (I + b00) ⇒ E ∗ (a + b00) ⇒ I ∗ (a + b00) ⇒ a ∗ (a + b00) We can conclude that E ∗ ⇒rm a ∗ (a + b00).
- 13. Push Down Automata Definition: A pushdown automata (PDA) is essentially an -NFA with a stack. • On a transition, the PDA: 1. Consumes an input symbol. 2. Goes to a new state (or stays in the old). 3. Replaces the top of the stack by any string (does nothing, pops the stack, or pushes a string onto the stack)
- 14. Push Down Automata Formal Definition:
- 16. Turing Machine Definition: We can give a formal description to a particular TM by specifying each of its seven components. This way a TM can become cumbersome. Note: To avoid this we use higher level descriptions which are precise enough for the purpose of understanding However, every higher level description is actually just a short hand for its formal counterpart.