Fundamental Algorithms for System Modeling, Analysis, and Optimization

Transcription

1 Fundamental lgorithms for System Modeling, nalysis, and Optimization Stavros Tripakis, Edward. Lee UC erkeley EECS 144/244 Fall 2013 Copyright , E.. Lee, J. Roydhowdhury, S.. Seshia, S. Tripakis, ll rights reserved Synchronous Systems part 3 Compositionality Fundamental principle in system design. Yet often overlooked. lso sometimes difficult to achieve. lso many facets. Here, we will see whether our 3 approaches to handle feedback are compositional in the following sense: can a composite system be treated in the same way as an atomic system? EECS 144/244, UC erkeley: 3 1

2 Non-deterministic approach Composite system = parallel composition = conjunction: Set of variables: Initial state formula: Next state formula:,, Grouping and together results in a similar system non-deterministic approach is compositional. EECS 144/244, UC erkeley: 4 What about the constructive fixpoint approach? Can handle open circuits = circuits with free inputs. Can compute characteristic functions for such circuits. Can plug them into larger circuit context => approach is compositional. EECS 144/244, UC erkeley: 5 2

3 What about the strict approach? Forbid instantaneous feedback = Forbid feedback unless if broken by unit-delay components (Moore machines) OK Not OK EECS 144/244, UC erkeley: 6 Composition of Mealy machines Suppose and are Mealy machines. Is a Mealy machine? EECS 144/244, UC erkeley: 7 3

4 Composition of Mealy machines If is a Mealy machine, it has an output function:.out.out(, ) returns (, ) { :=.out(); :=.out(); return (,); } EECS 144/244, UC erkeley: 8 roblem with treating in a monolithic way If is a Mealy machine, it has an output function:.out.out(, ) returns (, ); Instantaneous feedback => forbidden! EECS 144/244, UC erkeley: 9 4

5 roblem with treating in a monolithic way monolithic machine => false I/O dependencies.out(, ) returns (, ); EECS 144/244, UC erkeley: 10 State of the art tools suffer from this problem Note: replacing the CodeReuseSubsystem with a normal Subsystem allows Simulink to run the model without problems. Why? Simulink flattens Subsystems. EECS 144/244, UC erkeley: 11 5

6 Solution: non-monolithic Mealy machines Mealy machine with multiple output functions.out1( ) returns { return.out( ); }.out2( ) returns { return.out( ); } EECS 144/244, UC erkeley: 12 Non-monolithic interfaces.out1( ) returns ;.out2( ) returns ; EECS 144/244, UC erkeley: 13 6

7 Non-monolithic interfaces.out1( ) returns ;.out2( ) returns ; EECS 144/244, UC erkeley: 14 Non-monolithic interfaces Non-monolithic interface does not restrict usage.out1( ) returns ;.out2( ) returns ; EECS 144/244, UC erkeley: 15 7

8 Non-monolithic interfaces Interface is generally a graph: no cyclic dependency! // output function: UD.out( ) returns output { return state; } in UD.up unit-delay what if we add feedback? // state update function:.up( input ) returns void { state := input; } UD.out INTERFCE GRH out EECS 144/244, UC erkeley: 16 ottom-up interface synthesis C Given interfaces for sub-blocks,, C, compute interface for composite block. EECS 144/244, UC erkeley: 17 8

9 Simple non-monolithic interface.out1( ) returns ;.out2( ) returns ; EECS 144/244, UC erkeley: 18 What about more complex diagrams? what about this? or this? EECS 144/244, UC erkeley: 19 9

10 Interface synthesis for block diagrams = graph clustering block diagram interface EECS 144/244, UC erkeley: 20 How it s done C D EECS 144/244, UC erkeley: 21 10

11 How it s done Interface for Interface for Interface for C Interface for D EECS 144/244, UC erkeley: 22 How it s done clustering EECS 144/244, UC erkeley: 23 11

12 How it s done Interface for.out1().out2().out3() EECS 144/244, UC erkeley: 24 Different clusterings => different interfaces.out1().out2() x3.out3() non-monolithic interface for trade-off: interface size vs. reusability.out() x3 monolithic interface for EECS 144/244, UC erkeley: 25 12

13 Different clustering algorithms = different tradeoffs Details: Lublinerman, T. et al. [DTE 08,RTS 08,OL 09] Clustering method Complexity chieves maximal reusability? chieves minimal interface size? Modularity bound? chieves minimal code size? step-get olynomial No lmost <=2 Yes functions dynamic olynomial Yes Yes <=N+1 No functions* disjoint Ncomplete Yes Yes? Yes greedy olynomial Yes No? Yes * N = number of block outputs EECS 144/244, UC erkeley: 26 13