Compute1 dfa
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The document discusses finite automata and regular languages. It begins by asking what is a computer and noting that real computers are too complex to model entirely, so simpler abstract mathematical models called finite automata are used. It then discusses the formal definition of finite automata, deterministic vs non-deterministic finite automata, regular languages, and operations on regular languages. Examples are provided of modeling an automatic door and a simple finite automaton over the binary alphabet.
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Compute1 dfa
- 1. Finite Automata Question: What is a computer? real computers too complex for any theory Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.1
- 2. Finite Automata Question: What is a computer? real computers too complex for any theory need manageable mathematical abstraction Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.1
- 3. Finite Automata Question: What is a computer? real computers too complex for any theory need manageable mathematical abstraction idealized models: accurate in some ways, but not in all details Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.1
- 4. Finite Automata – Important ideas formal definition of finite automata Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
- 5. Finite Automata – Important ideas formal definition of finite automata deterministic vs. non-deterministic finite automata Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
- 6. Finite Automata – Important ideas formal definition of finite automata deterministic vs. non-deterministic finite automata regular languages Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
- 7. Finite Automata – Important ideas formal definition of finite automata deterministic vs. non-deterministic finite automata regular languages operations on regular languages Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
- 8. Finite Automata – Important ideas formal definition of finite automata deterministic vs. non-deterministic finite automata regular languages operations on regular languages regular expressions Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
- 9. Finite Automata – Important ideas formal definition of finite automata deterministic vs. non-deterministic finite automata regular languages operations on regular languages regular expressions pumping lemma Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
- 10. Example: An Automatic Door front rear pad pad door open when person approaches Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.3
- 11. Example: An Automatic Door front rear pad pad door open when person approaches hold open until person clears Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.3
- 12. Example: An Automatic Door front rear pad pad door open when person approaches hold open until person clears don’t open when someone standing behind door Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.3
- 13. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 14. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: OPEN Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 15. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: OPEN CLOSED Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 16. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: OPEN CLOSED Sensor: Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 17. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: OPEN CLOSED Sensor: FRONT: someone on rear pad Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 18. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: OPEN CLOSED Sensor: FRONT: someone on rear pad REAR: someone on rear pad Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 19. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: OPEN CLOSED Sensor: FRONT: someone on rear pad REAR: someone on rear pad BOTH: someone on both pads Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 20. The Automatic Door as DFA REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER States: OPEN CLOSED Sensor: FRONT: someone on rear pad REAR: someone on rear pad BOTH: someone on both pads NEITHER no one on either pad. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
- 21. The Automatic Door as DFA DFA is Deterministic Finite Automata Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.5
- 22. The Automatic Door as DFA DFA is Deterministic Finite Automata REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.5
- 23. The Automatic Door as DFA DFA is Deterministic Finite Automata REAR FRONT FRONT BOTH REAR NEITHER BOTH closed open NEITHER neither front rear both closed closed open closed closed open closed open open open Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.5
- 24. DFA: Informal Definition 0 1 1 0 q1 q q3 2 0,1 The machine : states: , and . ¡ ¥¡ ¥¡ £ ¤ ¦ ¢ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
- 25. DFA: Informal Definition 0 1 1 0 q1 q q3 2 0,1 The machine : states: , and . ¡ ¥¡ ¥¡ £ ¤ ¦ ¢ start state: (arrow from “outside”). ¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
- 26. DFA: Informal Definition 0 1 1 0 q1 q q3 2 0,1 The machine : states: , and . ¡ ¥¡ ¥¡ £ ¤ ¦ ¢ start state: (arrow from “outside”). ¡ accept state: (double circle). ¤¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
- 27. DFA: Informal Definition 0 1 1 0 q1 q q3 2 0,1 The machine : states: , and . ¡ ¥¡ ¥¡ £ ¤ ¦ ¢ start state: (arrow from “outside”). ¡ accept state: (double circle). ¤¡ state transitions: arrows. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
- 28. DFA: Informal Definition (cont.) 0 1 1 0 q1 q q3 2 0,1 On an input string Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
- 29. DFA: Informal Definition (cont.) 0 1 1 0 q1 q q3 2 0,1 On an input string DFA begins in start state ¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
- 30. DFA: Informal Definition (cont.) 0 1 1 0 q1 q q3 2 0,1 On an input string DFA begins in start state ¡ after reading each symbol, DFA makes state transition with matching label. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
- 31. DFA: Informal Definition (cont.) 0 1 1 0 q1 q q3 2 0,1 On an input string DFA begins in start state ¡ after reading each symbol, DFA makes state transition with matching label. After reading last symbol, DFA produces output: Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
- 32. DFA: Informal Definition (cont.) 0 1 1 0 q1 q q3 2 0,1 On an input string DFA begins in start state ¡ after reading each symbol, DFA makes state transition with matching label. After reading last symbol, DFA produces output: accept if DFA is an accepting state. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
- 33. DFA: Informal Definition (cont.) 0 1 1 0 q1 q q3 2 0,1 On an input string DFA begins in start state ¡ after reading each symbol, DFA makes state transition with matching label. After reading last symbol, DFA produces output: accept if DFA is an accepting state. reject otherwise. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
- 34. Informal Definition - Example 0 1 1 0 q q q3 1 2 0,1 What happens on input strings ¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.8
- 35. Informal Definition - Example 0 1 1 0 q q q3 1 2 0,1 What happens on input strings ¡ ¡ ¡ ¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.8
- 36. Informal Definition - Example 0 1 1 0 q q q3 1 2 0,1 What happens on input strings ¡ ¡ ¡ ¡ ¡ ¡ ¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.8
- 37. Informal Definition 0 1 1 0 q q q3 1 2 0,1 This DFA accepts all input strings that end with a 1 all input strings that contain at least one 1, and end with an even number of 0’s no other strings Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.9
- 38. Languages and Alphabets An alphabet is a finite set of letters. £¡ ¨ ¤ – the English alphabet. ¢ ¢¦ ¥ § ¢ ¢ ¦ ¦ ¢ £¡ ¨ – the Greek alphabet. © ¢¦ ¢ ¢ ¦ ¦ ¢ ¡ ¨ – the binary alphabet. ¡ ¢ ¡ ¨ – the digital alphabet. ¡ ¢ ¢¦ ¦ ¦ ¢ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.10
- 39. Languages and Alphabets An alphabet is a finite set of letters. £¡ ¨ ¤ – the English alphabet. ¢ ¢¦ ¥ § ¢ ¢ ¦ ¦ ¢ £¡ ¨ – the Greek alphabet. © ¢¦ ¢ ¢ ¦ ¦ ¢ ¡ ¨ – the binary alphabet. ¡ ¢ ¡ ¨ – the digital alphabet. ¡ ¢ ¢¦ ¦ ¦ ¢ The collection of all strings over is denoted by . Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.10
- 40. Languages and Alphabets An alphabet is a finite set of letters. £¡ ¨ ¤ – the English alphabet. ¢ ¢¦ ¥ § ¢ ¢ ¦ ¦ ¢ £¡ ¨ – the Greek alphabet. © ¢¦ ¢ ¢ ¦ ¦ ¢ ¡ ¨ – the binary alphabet. ¡ ¢ ¡ ¨ – the digital alphabet. ¡ ¢ ¢¦ ¦ ¦ ¢ The collection of all strings over is denoted by . For the binary alphabet, , , , , ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ are all members of . ¡ ¡ ¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.10
- 41. Languages and Examples A language over is a subset . For example Modern English. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
- 42. Languages and Examples A language over is a subset . For example Modern English. Ancient Greek. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
- 43. Languages and Examples A language over is a subset . For example Modern English. Ancient Greek. All prime numbers, writen using digits. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
- 44. Languages and Examples A language over is a subset . For example Modern English. Ancient Greek. All prime numbers, writen using digits. ¡ ¡ ¨ has at most seventeen 0’s Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
- 45. Languages and Examples A language over is a subset . For example Modern English. Ancient Greek. All prime numbers, writen using digits. ¡ ¡ ¨ has at most seventeen 0’s ¡ ¡ ¨ ¡ ¡ ¡ Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
- 46. Languages and Examples A language over is a subset . For example Modern English. Ancient Greek. All prime numbers, writen using digits. ¡ ¡ ¨ has at most seventeen 0’s ¡ ¡ ¨ ¡ ¡ ¡ ¡ ¡ ¨ has an equal number of 0’s and 1’s Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
- 47. Languages and DFA ¡ Definition: , the language of a DFA , is the ¡ set of strings that accepts, . Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
- 48. Languages and DFA ¡ Definition: , the language of a DFA , is the ¡ set of strings that accepts, . Note that may accept many strings, but accepts only one language Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
- 49. Languages and DFA ¡ Definition: , the language of a DFA , is the ¡ set of strings that accepts, . Note that may accept many strings, but accepts only one language What language does accept if it accepts no strings? Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
- 50. Languages and DFA ¡ Definition: , the language of a DFA , is the ¡ set of strings that accepts, . Note that may accept many strings, but accepts only one language What language does accept if it accepts no strings? A language is called regular if some deterministic finite automaton accepts it. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
- 51. Formal Definitions A deterministic finite automaton (DFA) is a 5-tuple ¡ , where ¡ ¡ ¢ ¢ ¢ ¢ ¢ is a finite set called the states, is a finite set called the alphabet, ¤ is the transition function, ¥ £ is the start state, and ¦ ¡¡ is the set of accept states. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.13