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CS 367: Model-Based Reasoning Lecture 3 (01/17/2002)

CS 367: Model-Based Reasoning Lecture 3 (01/17/2002). Gautam Biswas. Today’s Lecture. Lecture 2 (01/15): Modeling Paradigms; Discrete Time versus Discrete Event Systems Topic 1(2-3 weeks): Discrete Event Modeling of Systems (ref: S. Lafortune, et al. – Automata based models)

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CS 367: Model-Based Reasoning Lecture 3 (01/17/2002)

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  1. CS 367: Model-Based ReasoningLecture 3 (01/17/2002) Gautam Biswas

  2. Today’s Lecture • Lecture 2 (01/15): Modeling Paradigms; Discrete Time versus Discrete Event Systems • Topic 1(2-3 weeks): Discrete Event Modeling of Systems (ref: S. Lafortune, et al. – Automata based models) • Discrete Event Languages • Automata

  3. State Evolution in DES • Sequence of states visited • Associated events cause the state transitions • Formal ways for describing DES behavior, i.e., what is a language for describing DES behavior? • Automata • Petri Nets

  4. Languages for DES behavior • Simplest: a timed language where timing information has been deleted • untimed modeling formalism defined by event sequence: e1 e2 …… en • Timed Language: set of all timed sequences of events that the DES can generate/execute • (e1,t1) (e2,t2) …… (en,tn) • Stochastic Timed Language: a timed langauge with a probability distribution function defined over it

  5. Discrete Event Modeling Formalisms • State-based: define a state space and specify a state-transition structure: (out_state, event, in_state) triples • e.g., Automata and Petri Nets • Trace-based: based on (recursive) algebraic expressions • e.g., Communicating Sequential Processes (CSPs) • We will study modeling, analysis, and supervisory control with untimed and timed automata

  6. Automaton Model for two philosophers Notion of parallel composition

  7. Recursive Equation Model: Two philosopher problem

  8. Language defined on event set • Starting point of DES: event set, E : alphabet of language E = {e1, e2, …., en} • s: finite sequence of events from E: word or string s1 = e2e3e1e1e10 (includes repititions)  = empty string • Operations: • Concatenation: s2 = e5e3 then s1 s2 = e2e3e1e1e10 e5e3 • s = t u v t is prefix; u is substring; v is suffix • Kleene-closure; Prefix-closure

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

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

  11. Operation of Automata • Start at x0. • Occurrence of event e  (x0)  E causes transition to state f(x0,e)  X • Process continues based on transitions for which f is defined

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

  13. Examples of Equivalent Automata

  14. Blocking Automata • Automata G can reach state x, but (x) =  but 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

  15. Example of Deadlock and Livelock

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