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CS 367: Model-Based Reasoning Lecture 4 (01/24/2002)

CS 367: Model-Based Reasoning Lecture 4 (01/24/2002). Gautam Biswas. Today’s Lecture. Last Lecture: Deterministic Languages and Automata Concept of Blocking Today’s Lecture: Concept of Blocking Non deterministic Automata Operations on Automata. Automata.

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CS 367: Model-Based Reasoning Lecture 4 (01/24/2002)

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  1. CS 367: Model-Based ReasoningLecture 4 (01/24/2002) Gautam Biswas

  2. Today’s Lecture • Last Lecture: • Deterministic Languages and Automata • Concept of Blocking • Today’s Lecture: • Concept of Blocking • Non deterministic Automata • Operations on Automata

  3. Automata • Device for representing language as a set of well-defined rules • Represented as directed graph: nodes are states and labeled arcs represent transitions

  4. Deterministic Automata -- Which states to mark: modeling issue • Deterministic versus non deterministic • Note: f is a partially defined function • is derived from f

  5. Language generated by Automata Automata G1 and G2 are equivalent if

  6. Examples of Equivalent Automata

  7. Blocking Automata • Automata G can reach state x, but (x) =  and x  xm This is called deadlock because no further events can be executed System blocks when it enters deadlock state • When we have set of unmarked states that form strongly connected component – Livelock States are reachable from one another but there is no transition out of them

  8. Example of Deadlock and Livelock

  9. Blocking Prefix-closed by defn. prefix closure Deadlock  (x) =  and x  xm then Livelock  (can execute event, but not complete task)

  10. Blocking Automata G is blocking if Automata G is non blocking if

  11. Example: Machine Status Check : livelock occurs at state 5

  12. Nondeterministic Automaton Languages generated are similar to that of deterministic languages Extend the domain of fndto traces of events:

  13. Nondeterministic Automata: Example Do nondeterministic automata have more expressive power than deterministic automata? No: Any nondeterministic automaton can be transformed into an equivalent deterministic automaton.

  14. Operations on Automata • Every automata models two languages: • Notion of Accessibility and Co-accessibility • Accessible Part: notion of deleting states that are not reachable or accessible from x0 Ac has no effect on Therefore, always assume G = Ac(G)

  15. Operations on Automata • Coaccessible Part • A state is coaccessible if it can reach a marked state • Taking the coaccssible part, means building CoAc operation shrinks L(G) but not Lm(G)

  16. Operations on Automaton • An automaton that is accessible and coaccessible is said to be trim • Coaccessibility linked to concept of blocking Nonblocking  coaccessible

  17. Complement of Trim Automaton • 1. Complete the transition function f of G and make it a total • function -- done by adding dead or dump state • Change the marking status of all states.

  18. Example of Trim Automaton

  19. Composition of Automata • Two kinds • Product:   completely synchronous • Parallel:   synchronous

  20. Product Composition

  21. Example: Product Composition b

  22. Parallel Composition

  23. Parallel Composition: Example b

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