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DISCRETE DYNAMICS

DISCRETE DYNAMICS. EEN 417 Fall 2013. Midterm I. In class on 10/4 Covered Material will be: Chapter 1 (Introduction) Chapters 2 & 3 (Continuous and Discrete Dynamics) Chapters 7 (Processors) Chapter 12 (Linear Temporal Logic). Processor Pipelining. Processor Pipelining.

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DISCRETE DYNAMICS

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  1. DISCRETE DYNAMICS EEN 417 Fall 2013

  2. Midterm I • In class on 10/4 • Covered Material will be: • Chapter 1 (Introduction) • Chapters 2 & 3 (Continuous and Discrete Dynamics) • Chapters 7 (Processors) • Chapter 12 (Linear Temporal Logic)

  3. Processor Pipelining

  4. Processor Pipelining

  5. If B relies on ALU results of A

  6. What if A can forward the result of writeback to the ALU?

  7. What if A can forward the result of ALU to the ALU?

  8. Recall Actor Model of a Continuous-Time System • Example: integrator: • Continuous-time signal: • Continuous-time actor:

  9. Discrete Systems • Example: count the number of cars that enter and leave a parking garage: • Pure signal: • Discrete actor:

  10. Reaction

  11. Input and Output Valuations at a Reaction

  12. State Space

  13. Garage Counter Finite State Machine (FSM) in Pictures

  14. Garage Counter Finite State Machine (FSM) in Pictures

  15. Garage Counter Finite State Machine (FSM) in Pictures Initial state

  16. Garage Counter Finite State Machine (FSM) in Pictures Output valuation

  17. Garage Counter Mathematical Model The picture above defines the update function.

  18. FSM Notation state initial state transition self loop

  19. Examples of Guards for Pure Signals

  20. Examples of Guards for Signals with Numerical Values

  21. Example: Thermostat • Exercise: From this picture, construct the formal mathematical model.

  22. More Notation: Default Transitions • A default transition is enabled if no non-default transition is enabled and it either has no guard or the guard evaluates to true. When is the above default transition enabled?

  23. Extended State Machines

  24. Traffic Light Controller

  25. Definitions • Stuttering transition: Implicit default transition that is enabled when inputs are absent and that produces absent outputs. • Receptiveness: For any input values, some transition is enabled. Our structure together with the implicit default transition ensures that our FSMs are receptive. • Determinism: In every state, for all input values, exactly one (possibly implicit) transition is enabled.

  26. Example: Nondeterminate FSM • Nondeterminate model of pedestrians arriving at a crosswalk: • Formally, the update function is replaced by a function

  27. Behaviors and Traces • FSM behavior is a sequence of (non-stuttering) steps. • A trace is the record of inputs, states, and outputs in a behavior. • A computation tree is a graphicalrepresentation of all possible traces. • FSMs are suitable for formalanalysis. For example, safetyanalysis might show that some unsafestate is not reachable.

  28. Modeling unknown aspects of the environment or system Such as: how the environment changes the iRobot’s orientation Hiding detail in a specification of the system We will see an example of this later (see notes) Any other reasons why nondeterministic FSMs might be preferred over deterministic FSMs? Uses of nondeterminism

  29. Non-deterministic FSMs are more compact than deterministic FSMs ND FSM  D FSM: Exponential blow-up in #states in worst case Size Matters

  30. For a fixed input sequence: A deterministic system exhibits a single behavior A non-deterministic system exhibits a set of behaviors Non-deterministic Behavior: Tree of Computations Deterministic FSM behavior for a particular input sequence: . . . Non-deterministic FSM behavior for an input sequence: . . . . . . . . . . . .

  31. What does receptiveness mean for non-deterministic state machines? Non-deterministic  Probabilistic Related points

  32. Example from Industry: Engine Control Source: Delphi Automotive Systems (2001)

  33. Elements of a Modal Model (FSM) initial state state input output transition Source: Delphi Automotive Systems (2001)

  34. Actor Model of an FSM This model enables composition of state machines.

  35. FSMs provide: A way to represent the system for: Mathematical analysis So that a computer program can manipulate it A way to model the environment of a system. A way to represent what the system must do and must not do – its specification. A way to check whether the system satisfies its specification in its operating environment. What we will be able to do with FSMs

  36. WRAP UP

  37. For next time • Read Chapter 3 – Discrete Dynamics • Assignment 3 – Chapter 3, problems 2, 3, 4, and 5. • Due 10/4 • You can turn them in early so yours is graded by the exam. Any turned in by class on Friday 9/27 will be ready for pickup in my office by Monday 9/30.

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