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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 6: Transition Graphs

CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 6: Transition Graphs. Chapter 6: Transition Graphs We introduce the first non-deterministic but simple theoretical machine: Transition Graph. a. a. +. b. –. b. a,b. b. a. a,b. Chapter 6: Transition Graphs.

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CSI 3104 /Winter 2006 : Introduction to Formal Languages Chapter 6: Transition Graphs

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  1. CSI 3104 /Winter 2006: Introduction to Formal Languages Chapter 6: Transition Graphs Chapter 6: Transition Graphs We introduce the first non-deterministic but simple theoretical machine: Transition Graph. Dr. Nejib Zaguia CSI3104-W06

  2. a a + b – b a,b b a a,b Chapter 6: Transition Graphs • An FA: {baa} • The word a? The word baabb? • The input fails, or the machine fails on the input. The input is rejected. Dr. Nejib Zaguia CSI3104-W06

  3. a ba b + – a,b aa, ab, bb a,b Chapter 6: Transition Graphs • A transition graph that accepts the language {baa} What it seems to be a More Powerful Machine Dr. Nejib Zaguia CSI3104-W06

  4. baa – + baa – + a,b all else a,b Chapter 6: Transition Graphs Two other equivalent Transition Graphs with fewer states • The word a? The word baabb? • The input crashes. The machine crashes. The input is rejected. (2 ways for an input to be rejected) Dr. Nejib Zaguia CSI3104-W06

  5. a,b a,b aa,bb – + Chapter 6: Transition Graphs • baa? • a choice, a decision 2 possible paths b|aa - accepted b|a|a - rejected 1 way to crash ba|a – rejected • The machine represents a language L. baa  L? • For all w, w  L if there exists a path that arrives at a final state. Dr. Nejib Zaguia CSI3104-W06

  6. 1 ab ba + – b baa 2 Chapter 6: Transition Graphs • baab? 2 possible paths, both end in a final state. Dr. Nejib Zaguia CSI3104-W06

  7. Chapter 6: Transition Graphs • A transition graph (TG) is the following 3 things: • a finite set of states, at least one of which is designated as the start state, and some (maybe none) of which are designated the final states (or accepting states) • an alphabet of input letters • a finite set of transitions that show how to go to a new state, for some pairs of state and substrings of letters (or Λ). (One pair can have 0, 1, or more next-states.) Dr. Nejib Zaguia CSI3104-W06

  8. Chapter 6: Transition Graphs • A successful path is a series of edges beginning at some start state and ending at a final state. • The concatenation of all the substrings that label the edges in the path is a word accepted by this machine. • The set of words accepted is the language of the transition graph. Dr. Nejib Zaguia CSI3104-W06

  9. a L 4+ abb 3 2 1– aa b 2 1 4 3 abb L aa b Chapter 6: Transition Graphs • Example: abbaab 1 abbab crashes. Dr. Nejib Zaguia CSI3104-W06

  10. 1 L a 1– a b L 2 + – b 2– + aba 3 L aba 3– Chapter 6: Transition Graphs • These two machines are clearly equivalent. • Remark: Every finite automaton is a transition graph. Many start states Dr. Nejib Zaguia CSI3104-W06

  11. – L abba – + baa – Chapter 6: Transition Graphs L = {L, abba, baa} + – L = {L} L =  Dr. Nejib Zaguia CSI3104-W06

  12. L + L L L – + – + Chapter 6: Transition Graphs L = {Λ} + Dr. Nejib Zaguia CSI3104-W06

  13. a,b b – a + b b – + a Chapter 6: Transition Graphs transition graph: Some words can fail, crash, and succeed: abab. All words ending in b: (a+b)*b FA Dr. Nejib Zaguia CSI3104-W06

  14. a,b b + a – a,b b a + aa b b b b + – aa Chapter 6: Transition Graphs Dr. Nejib Zaguia CSI3104-W06

  15. b 1+ 2 b a a a a b 3 4 aa,bb aa,bb b ab,ba + ab,ba Chapter 6: Transition Graphs Language EVEN-EVEN Transition Graph: FA: even a odd b even a even b balanced unbalanced odd a odd b odd a even b Dr. Nejib Zaguia CSI3104-W06

  16. b a,b a b + – bbb ab a a a bbb b bb bb a Chapter 6: Transition Graphs abbbabbbabba Dr. Nejib Zaguia CSI3104-W06

  17. L L a a + – Chapter 6: Transition Graphs • Infinitely many paths for aa • Is there an algorithm to determine if a word is accepted? Dr. Nejib Zaguia CSI3104-W06

  18. L a a – + L a, L a a – + b, L Chapter 6: Transition Graphs We can delete the L transition But not here Dr. Nejib Zaguia CSI3104-W06

  19. Chapter 6: Transition Graphs • A generalized transition graph (GTG) is the following 3 things: • a finite set of states, at least one of which is designated as the start state, and some (maybe none) of which are designated the final states (or accepting states) • an alphabet of input letters • a finite set of edges connecting some pairs of states, each labeled with a regular expression Dr. Nejib Zaguia CSI3104-W06

  20. a* a* b+L (ba+a)* + + Chapter 6: Transition Graphs Words without 2 b’s in a row: Dr. Nejib Zaguia CSI3104-W06

  21. a,b b a a,b + – (a+b)* b* a (a+b) + – Chapter 6: Transition Graphs Kleene Star Closure and Loops Dr. Nejib Zaguia CSI3104-W06

  22. … 4 abb … 3 abb … 5 … b 4 ab … 3 abb … 5 Chapter 6: Transition Graphs Choosing Transitions Dr. Nejib Zaguia CSI3104-W06

  23. … 8 a … … b 9 7 b … 10 … 8 a … … b 9 7 L … b 10 Chapter 6: Transition Graphs Choices even with restriction of 1 letter per edge Machine is nondeterministic Dr. Nejib Zaguia CSI3104-W06

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