Introduction to Embedded Systems

Transcription

1 Introduction to Embedded Systems Edward A. Lee & Sanjit A. Seshia UC Berkeley EECS 124 Spring 2008 Copyright 2008, Edward A. Lee & Sanjit A. Seshia, All rights reserved Lecture 7: Modeling Modal Behavior, Part II Recap of the last lecture Finite-state machine represented as a 5-tuple (States, Inputs, Outputs, update, initialstate determinacy, receptiveness Non-determinism, ND FSM (States, Inputs, Outputs, possibleupdates, initialstates Behavior & trace EECS 124, UC Berkeley: 2 1

2 Topic: Modeling with Finite-State Machines (FSMs Suppose that our only modeling formalism is the Finite-State Machine Four questions: How to represent the system for: Mathematical analysis So that a computer program can manipulate it How to model its environment? How to represent what the system MUST do its specification? How to check whether the system satisfies its specification in its operating environment? EECS 124, UC Berkeley: 3 Stuttering What happens when there is no input to the system? i.e. at step n, input signals x, x(n = absent The FSM s state remains unchanged. This is termed stuttering. EECS 124, UC Berkeley: 4 2

3 Stuttering What happens when there is no input to the system? i.e. at step n, input signals x, x(n = absent The FSM s state remains unchanged. This is termed stuttering. We model this using the special input symbol: absent an absent output can also be modeled this way update(s, absent = (s, absent Is this the only way to model a stuttering transition? EECS 124, UC Berkeley: 5 Example: Discretized irobot Hill Climber EECS 124, UC Berkeley: 6 3

4 FSM Controller for irobot EECS 124, UC Berkeley: 7 FSM Controller for irobot {level} / {drive_one_sq} tilt drive absent / {rotate} EECS 124, UC Berkeley: 8 4

5 Two FSM Controllers for the irobot {level} / {drive_one_sq} tilt drive Version 1: M 1 absent / {rotate} {level} / {drive_one_sq} Version 2: M 2 tilt drive0 drive1 drive2 absent / {rotate} Tries to track square s y-coordinate EECS 124, UC Berkeley: 9 Comparing state machines Let M 1 and M 2 be two FSMs M 2 obtained by implementing (adding more detail to M 1 Mathematically, what does it mean for M 2 to implement M 1? EECS 124, UC Berkeley: 10 5

6 Comparing state machines Let M 1 and M 2 be two FSMs M 2 obtained by implementing (adding more detail to M 1 Mathematically, what does it mean for M 2 to implement M 1? Two possible answers: 1. Trace containment 2. Simulation EECS 124, UC Berkeley: 11 Behavior/Trace containment Let M 1 = (S 1, U 1, s 10 and M 2 = (S 2, I 2, O 2, U 2 where I 1 I 2 and O 1 O 2 L(M = set of observable traces of M leave out states, only retain common inputs/outputs Defn 1: M 2 implements M 1 iff L(M 2 L(M 1 L(M 2 L(M 1 EECS 124, UC Berkeley: 12 6

7 Is L(M 2 L(M 1? {level} / {drive_one_sq} tilt drive Version 1: M 1 absent / {rotate} {level} / {drive_one_sq} Version 2: M 2 tilt drive0 drive1 drive2 absent / {rotate} EECS 124, UC Berkeley: 13 Simulation Intuition: a matching game between the two FSMs Let M 1 = (S 1, U 1, s 10 and M 2 = (S 2, I 2, O 2, U 2 where I 1 I 2 and O 1 O 2 M 1 M 2 i i U 1 U 2 n n+1 Correspondence: Common output signals take same values EECS 124, UC Berkeley: 14 7

8 Simulation Intuition: a matching game between the two FSMs Let M 1 = (S 1, U 1, s 10 and M 2 = (S 2, I 2, O 2, U 2 where I 1 I 2 and O 1 O 2 M 1 M 2 R U 1 i i U 2 R Defn2 : M 2 implements M 1 iff M 1 simulates M 2 EECS 124, UC Berkeley: 15 Formal definition of Simulation Let M 1 = (S 1, U 1, s 10 and M 2 = (S 2, I 2, O 2, U 2 where I 1 I 2 and O 1 O 2 We say M 1 simulates M 2 iff there exists a set R S 1 S 2 such that 1. R(s For all (s 1, s 2 R, the following condition holds: For all i I 2, and (t 2, o 2 = U 2 (s 2, i, there exists a (t 1, o 1 = U 1 (s 1, i I 1 s.t. (t 1, o 1 R and o 2 O 1 = o 1 R is called the simulation relation EECS 124, UC Berkeley: 16 8

9 Does M 1 simulate M 2? What s R? {level} / {drive_one_sq} tilt drive Version 1: M 1 absent / {rotate} {level} / {drive_one_sq} Version 2: M 2 tilt drive0 drive1 drive2 absent / {rotate} EECS 124, UC Berkeley: 17 Bisimulation (differences with simulation in red Let M 1 = (S 1, U 1, s 10 and M 2 = (S 2, I 2, O 2, U 2 where I = I 1 = I 2 and O = O 1 = O 2 We say M 1 bisimulates M 2 iff there exists a set R S 1 S 2 such that 1. R(s For all (s 1, s 2 R, the following conditions hold: For all i I, and (t 2, o 2 = U 2 (s 2, i, there exists a (t 1, o 1 = U 1 (s 1, i s.t. (t 1, o 1 R and o 2 = o 1 For all i I, and (t 1, o 1 = U 1 (s 1, i, there exists a (t 2, o 2 = U 2 (s 2, i s.t. (t 2, o 2 R and o 2 = o 1 EECS 124, UC Berkeley: 18 9

10 Simulation and Trace Containment Theorem: If M 1 simulates M 2, then L(M 2 L(M 1 Note: If L(M 2 L(M 1 then M 1 need not simulate M 2 EECS 124, UC Berkeley: 19 Modeling Systems with Communicating Finite-State Machines (FSMs Systems are made up of many communicating components How do we build an FSM model from smaller component FSM models? 1. Composition synchronous or asynchronous 2. Hierarchical modeling & Statecharts EECS 124, UC Berkeley: 20 10

11 Example: Traffic Light Controller EECS 124, UC Berkeley: 21 Environment of the Controller: Cars This FSM models the sensor input to the Controller; it has a single output signal iscar. As a modeling choice, we can introduce an input to this FSM as well. EECS 124, UC Berkeley: 22 11

12 Composition of FSMs based on Connectivity Side-by-side composition Cascade composition Feedback composition (see Lee & Varaiya, Ch. 4 for more kinds of composition EECS 124, UC Berkeley: 23 Side-by-Side Composition EECS 124, UC Berkeley: 24 12

13 Side-by-Side Composition EECS 124, UC Berkeley: 25 Cascade Composition EECS 124, UC Berkeley: 26 13

14 Feedback Composition Can apply composition hierarchically EECS 124, UC Berkeley: 27 Composition in the Traffic Light Example Controller Car Env Timer EECS 124, UC Berkeley: 28 14

15 Synchronous Composition M 1 = (S 1, U 1, s 10 and M 2 = (S 2, I 2, O 2, U 2 M is the synchronous composition of M 1 and M 2 = (S 1 S 2 I 2 O 2, U, (s 10 where U is defined exactly as in the side-by-side composition definition earlier (Note: Inputs and Outputs of M 1 and M 2 can be interconnected EECS 124, UC Berkeley: 29 Asynchronous Composition M 1 = (S 1, U 1, s 10 and M 2 = (S 2, I 2, O 2, U 2 M is the asynchronous composition of M 1 and M 2 = (S 1 S 2 I 2 O 2, U, (s 10 where U((s 1, s 2, (i 1, i 2 = ((s 1, s 2, (o 1, o 2 and OR (s 1, o 1 = U 1 (s 1, i 1 AND M 2 stutters (s 2, o 2 = U 2 (s 2, i 2 AND M 1 stutters EECS 124, UC Berkeley: 30 15

16 The Need for Hierarchical Representation Suppose we have a network of communicating FSMs We want to construct and analyze the synchronous composition of all of those FSMs Is it easy to simply construct the composition ( flat representation and work with that? EECS 124, UC Berkeley: 31 Problems with Flat FSM Representation Exponential blow-up possible during composition Too much detail for humans Not a natural way of representing parallel composition Does not reflect modular (top-down or bottom-up development Common structure cannot be easily extracted: e.g., what happens when an interrupt is received? EECS 124, UC Berkeley: 32 16

17 Hierarchical Modeling and Statecharts Modeling with Hierarchy Orthogonality (AND-states and OR-states Broadcast (for communication Example due to Reinhard von Hanxleden EECS 124, UC Berkeley: 33 17