Interchange Semantics For Hybrid System Models

Transcription

1 Interchange Semantics For Hybrid System Models Alessandro Pinto 1, Luca P. Carloni 2, Roberto Passerone 3 and Alberto Sangiovanni-Vincentelli 1 1 University of California at Berkeley apinto,alberto@eecs.berkeley.edu 2 Columbia University luca@cs.columbia.edu 3 University of Trento, Italy roby@dit.unitn.it March 3, 2006

2 Hybrid systems A hybrid system is a tuple

3 Hybrid system semantics A hybrid time basis τ is a finite or an infinite sequence of intervals I j = {t R : t j t t j } where t j t j and t j = t j+1

4 Hybrid systems execution

5 Applications Abstract uninteresting dynamics

6 Digitally controlled PLL

7 A car engine φ = 0 / q H u = 0 I j r φ = 180 / z = G * q * j; u = 0 φ = 180 / z = z * r; u = 0 u φ φ = 0 / E z = 0; u = z C

8 Need for an interchange format Hybrid Systems (HS) have proven to be a powerful design representation for system -level design. There has been a proliferation of tool for simulation, verification and synthesis of HS but all based on different models. The need for and interchange format (IF) is very much felt.

9 Need for an interchange format Hybrid Systems (HS) have proven to be a powerful design representation for system -level design. There has been a proliferation of tool for simulation, verification and synthesis of HS but all based on different models. The need for and interchange format (IF) is very much felt. We presented a proposal for an IF (HSCC2005). We have defined its semantics (HSCC2006). Here we summarize our findings and justify our choices.

10 Outline 1. Interchange Formats in EDA 2. Interchange Format Syntax and Abstract Semantics 3. Composition and Hierarchy 4. Conclusions

11 Interchange Formats in EDA

12 Interchange Formats in EDA: a Brief History Electronic Design Interchange Format (EDIF)

13 Interchange Formats in EDA: a Brief History Electronic Design Interchange Format (EDIF) Library Exchange Format / Design Exchange Format (LEF / DEF)

14 Interchange Formats in EDA: a Brief History Electronic Design Interchange Format (EDIF) Library Exchange Format / Design Exchange Format (LEF / DEF) Berkeley Logic Interchange Format (BLIF)

15 Interchange Formats in EDA: a Brief History Electronic Design Interchange Format (EDIF) Library Exchange Format / Design Exchange Format (LEF / DEF) Berkeley Logic Interchange Format (BLIF) Hybrid System Interchange Format (HSIF)

16 Interchange Format: Approaches An interchange format only defines the syntax of a common data structure (first version of EDIF): very flexible, ambiguous.

17 Interchange Format: Approaches An interchange format only defines the syntax of a common data structure (first version of EDIF): very flexible, ambiguous. An interchange Format defines the common model of computation (BLIF,HSIF): unambiguous. BLIF uses a model that is universal for logic but it is used for a restricted domain (boolean algebra). HSIF allows direct analysis of models, but in HS there is a great degree of semantic differences across tools. Not flexible. It reduces the degree of freedom of the tools that share the date using the format.

18 Interchange Format: Approaches An interchange format only defines the syntax of a common data structure (first version of EDIF): very flexible, ambiguous. An interchange Format defines the common model of computation (BLIF,HSIF): unambiguous. BLIF uses a model that is universal for logic but it is used for a restricted domain (boolean algebra). HSIF allows direct analysis of models, but in HS there is a great degree of semantic differences across tools. Not flexible. It reduces the degree of freedom of the tools that share the date using the format. An interchange format has a formal abstract semantics that can be refined into concrete semantics An interchange format must be capable of capturing the largest possible class of models in use today and even tomorrow At the same time has to have precise semantics to avoid ambiguity.

19 The Big Picture

20 Interchange Format Syntax and Abstract Semantics

21 Preliminary Definitions Valuations of variables : Given a variable with name v, its value is denoted by val(v). Valuation of tuples of variables : If V is the tuple (v 1,..., v n ) then val(v ) = (val(v 1 ),..., val(v n )). Valuation of sets of variables : If V is the set {v 1,..., v n } then its valuation is the multi-set val(v ) = {val(v 1 ),..., val(v n )}. Valuations domain : For a set of variables V, the set of all possible valuations of V is denoted by R(V ). Lifting : Given a subset D R(V ) and V V, the lifting of D to V is given by the operator L(V )(D) = {p R(V ) : p V RV }.

22 Definition of a Hybrid System: Syntax A hybrid system is a tuple H = (V, E, D, I, σ, ω, ρ) where:

23 Definition of a Hybrid System: Syntax A hybrid system is a tuple H = (V, E, D, I, σ, ω, ρ) where: V = {v 1,..., v n } is a set of variables,

24 Definition of a Hybrid System: Syntax A hybrid system is a tuple H = (V, E, D, I, σ, ω, ρ) where: V = {v 1,..., v n } is a set of variables, E = {e 1,..., e m } is a set of equations in the variables V,

25 Definition of a Hybrid System: Syntax A hybrid system is a tuple H = (V, E, D, I, σ, ω, ρ) where: V = {v 1,..., v n } is a set of variables, E = {e 1,..., e m } is a set of equations in the variables V, D 2 R(V ) is a set of domains, or regions, of the possible valuations of the variables V,

26 Definition of a Hybrid System: Syntax A hybrid system is a tuple H = (V, E, D, I, σ, ω, ρ) where: V = {v 1,..., v n } is a set of variables, E = {e 1,..., e m } is a set of equations in the variables V, D 2 R(V ) is a set of domains, or regions, of the possible valuations of the variables V, I N is a set of indexes, σ : 2 R(V ) 2 I is a function that associates a set of indexes to each domain ω : I 2 E is a function that associates a set of equations to each index,

27 Definition of a Hybrid System: Syntax A hybrid system is a tuple H = (V, E, D, I, σ, ω, ρ) where: V = {v 1,..., v n } is a set of variables, E = {e 1,..., e m } is a set of equations in the variables V, D 2 R(V ) is a set of domains, or regions, of the possible valuations of the variables V, I N is a set of indexes, σ : 2 R(V ) 2 I is a function that associates a set of indexes to each domain ω : I 2 E is a function that associates a set of equations to each index, ρ : 2 R(V ) 2 R(V ) R(V ) 2 R(V ) is a function to reset the values of the variables.

28 Definition of a Hybrid System: Syntax A hybrid system is a tuple H = (V, E, D, I, σ, ω, ρ) where: V = {v 1,..., v n } is a set of variables, E = {e 1,..., e m } is a set of equations in the variables V, D 2 R(V ) is a set of domains, or regions, of the possible valuations of the variables V, I N is a set of indexes, σ : 2 R(V ) 2 I is a function that associates a set of indexes to each domain ω : I 2 E is a function that associates a set of equations to each index, ρ : 2 R(V ) 2 R(V ) R(V ) 2 R(V ) is a function to reset the values of the variables. V t = {v t1,..., v tn } is a set of temporary variables, π : E {1, 2,..., E } is an equation ordering function.

29 Syntax Example A bouncing ball can be modeled as a hybrid system with V = {y, v}, E = { v = g, ẏ = v}. There are two domains: D 1 = {{val(y), val(v)} : val(y) 0 val(v) < 0} and D 2 = {{val(y), val(v)} : val(y) > 0}, hence D = {D 1, D 2 }; I = {1}, σ(d 1 ) = σ(d 2 ) = {1}, ω(1) = E. The reset function is defined as follows: ρ(d 2, D 1, val(v )) = {val(y), ɛval(v)}.

30 Syntax Example A bouncing ball can be modeled as a hybrid system with V = {y, v}, E = { v = g, ẏ = v}. There are two domains: D 1 = {{val(y), val(v)} : val(y) 0 val(v) < 0} and D 2 = {{val(y), val(v)} : val(y) > 0}, hence D = {D 1, D 2 }; I = {1}, σ(d 1 ) = σ(d 2 ) = {1}, ω(1) = E. The reset function is defined as follows: ρ(d 2, D 1, val(v )) = {val(y), ɛval(v)}.

31 Syntax Example A bouncing ball can be modeled as a hybrid system with V = {y, v}, E = { v = g, ẏ = v}. There are two domains: D 1 = {{val(y), val(v)} : val(y) 0 val(v) < 0} and D 2 = {{val(y), val(v)} : val(y) > 0}, hence D = {D 1, D 2 }; I = {1}, σ(d 1 ) = σ(d 2 ) = {1}, ω(1) = E. The reset function is defined as follows: ρ(d 2, D 1, val(v )) = {val(y), ɛval(v)}.

32 Syntax Example A bouncing ball can be modeled as a hybrid system with V = {y, v}, E = { v = g, ẏ = v}. There are two domains: D 1 = {{val(y), val(v)} : val(y) 0 val(v) < 0} and D 2 = {{val(y), val(v)} : val(y) > 0}, hence D = {D 1, D 2 }; I = {1}, σ(d 1 ) = σ(d 2 ) = {1}, ω(1) = E. The reset function is defined as follows: ρ(d 2, D 1, val(v )) = {val(y), ɛval(v)}.

33 Syntax Example A bouncing ball can be modeled as a hybrid system with V = {y, v}, E = { v = g, ẏ = v}. There are two domains: D 1 = {{val(y), val(v)} : val(y) 0 val(v) < 0} and D 2 = {{val(y), val(v)} : val(y) > 0}, hence D = {D 1, D 2 }; I = {1}, σ(d 1 ) = σ(d 2 ) = {1}, ω(1) = E. The reset function is defined as follows: ρ(d 2, D 1, val(v )) = {val(y), ɛval(v)}.

34 Syntax Example A bouncing ball can be modeled as a hybrid system with V = {y, v}, E = { v = g, ẏ = v}. There are two domains: D 1 = {{val(y), val(v)} : val(y) 0 val(v) < 0} and D 2 = {{val(y), val(v)} : val(y) > 0}, hence D = {D 1, D 2 }; I = {1}, σ(d 1 ) = σ(d 2 ) = {1}, ω(1) = E. The reset function is defined as follows: ρ(d 2, D 1, val(v )) = {val(y), ɛval(v)}.

35 Syntax Example A bouncing ball can be modeled as a hybrid system with V = {y, v}, E = { v = g, ẏ = v}. There are two domains: D 1 = {{val(y), val(v)} : val(y) 0 val(v) < 0} and D 2 = {{val(y), val(v)} : val(y) > 0}, hence D = {D 1, D 2 }; I = {1}, σ(d 1 ) = σ(d 2 ) = {1}, ω(1) = E. The reset function is defined as follows: ρ(d 2, D 1, val(v )) = {val(y), ɛval(v)}.

36 Definition of a Hybrid System: Semantics The semantics of a hybrid system H is defined by the tuple (H, B, T, resolve, init, update) where:

37 Definition of a Hybrid System: Semantics The semantics of a hybrid system H is defined by the tuple (H, B, T, resolve, init, update) where: H is a hybrid system;

38 Definition of a Hybrid System: Semantics The semantics of a hybrid system H is defined by the tuple (H, B, T, resolve, init, update) where: H is a hybrid system; B is a set of pairs (γ, t) where γ R(H.V ) is a multi-set of possible values of the hybrid system variables and t R + is a time stamp;

39 Definition of a Hybrid System: Semantics The semantics of a hybrid system H is defined by the tuple (H, B, T, resolve, init, update) where: H is a hybrid system; B is a set of pairs (γ, t) where γ R(H.V ) is a multi-set of possible values of the hybrid system variables and t R + is a time stamp; T is a time stamper: selects the next time stamp, decides whether a new pair (val, t) can be added to the set B;

40 Definition of a Hybrid System: Semantics The semantics of a hybrid system H is defined by the tuple (H, B, T, resolve, init, update) where: H is a hybrid system; B is a set of pairs (γ, t) where γ R(H.V ) is a multi-set of possible values of the hybrid system variables and t R + is a time stamp; T is a time stamper: selects the next time stamp, decides whether a new pair (val, t) can be added to the set B; resolve, init and update are three algorithms.

41 Time Stamper Automaton

42 Time Stamper Automaton Initialization: B = (V 0, 0).

43 Time Stamper Automaton Initialization: B = (V 0, 0). Set of actions: next, resolve, init, update.

44 Time Stamper Automaton Initialization: B = (V 0, 0). Set of actions: next, resolve, init, update. Set of conditions: true, false, error thresholds, domainchange.

45 Time Stamper Automaton Initialization: B = (V 0, 0). Set of actions: next, resolve, init, update. Set of conditions: true, false, error thresholds, domainchange. Valuation and time stamp acceptance: B = B (V(V ), t).

46 Resolve Algorithm resolve(t) D {D D val(v t ) D} I, E t I D D σ(d) for all i I do E t = E t ω(i) end for sort(e t, π) for all e i E t do solve(e i,t) end for D {D D val(v t ) D} markchange(d, D ) // Compute the set of active domains. // Collect all active dynamics and components. // Collect all active equations. // Order the equations. // Set of active domains after the computation. // Check if the set of active domains has changed.

47 Resolve Algorithm resolve(t) D {D D val(v t ) D} I, E t I D D σ(d) for all i I do E t = E t ω(i) end for sort(e t, π) for all e i E t do solve(e i,t) end for D {D D val(v t ) D} markchange(d, D ) // Compute the set of active domains. // Collect all active dynamics and components. // Collect all active equations. // Order the equations. // Set of active domains after the computation. // Check if the set of active domains has changed.

48 Resolve Algorithm resolve(t) D {D D val(v t ) D} I, E t I D D σ(d) for all i I do E t = E t ω(i) end for sort(e t, π) for all e i E t do solve(e i,t) end for D {D D val(v t ) D} markchange(d, D ) // Compute the set of active domains. // Collect all active dynamics and components. // Collect all active equations. // Order the equations. // Set of active domains after the computation. // Check if the set of active domains has changed.

49 Resolve Algorithm resolve(t) D {D D val(v t ) D} I, E t I D D σ(d) for all i I do E t = E t ω(i) end for sort(e t, π) for all e i E t do solve(e i,t) end for D {D D val(v t ) D} markchange(d, D ) // Compute the set of active domains. // Collect all active dynamics and components. // Collect all active equations. // Order the equations. // Set of active domains after the computation. // Check if the set of active domains has changed.

50 Resolve Algorithm resolve(t) D {D D val(v t ) D} I, E t I D D σ(d) for all i I do E t = E t ω(i) end for sort(e t, π) for all e i E t do solve(e i,t) end for D {D D val(v t ) D} markchange(d, D ) // Compute the set of active domains. // Collect all active dynamics and components. // Collect all active equations. // Order the equations. // Set of active domains after the computation. // Check if the set of active domains has changed.

51 Resolve Algorithm resolve(t) D {D D val(v t ) D} I, E t I D D σ(d) for all i I do E t = E t ω(i) end for sort(e t, π) for all e i E t do solve(e i,t) end for D {D D val(v t ) D} markchange(d, D ) // Compute the set of active domains. // Collect all active dynamics and components. // Collect all active equations. // Order the equations. // Set of active domains after the computation. // Check if the set of active domains has changed.

52 Refinement into Concrete Semantics Define the time stamper automaton: conditions and actions Multiple iterations for fixed point or event detection Define the next function Different algorithm to decide the next time stamp Define the domainchange function Different transition semantics Define the solve function Different integration methods

53 Composition and Hierarchy

54 Running Example (a) Half-wave rectifier (b) Block diagram of the half-wave rectifier

55 Composition of Hybrid Systems Given H 1 = (V 1, V t1, E 1, D 1, I 1, σ 1, ω 1, ρ 1, π 1 ) and H 2 = (V 2, V t2, E 2, D 2, I 2, σ 2, ω 2, ρ 2, π 2 ), H = H 1 H 2 is such that:

56 Composition of Hybrid Systems Given H 1 = (V 1, V t1, E 1, D 1, I 1, σ 1, ω 1, ρ 1, π 1 ) and H 2 = (V 2, V t2, E 2, D 2, I 2, σ 2, ω 2, ρ 2, π 2 ), H = H 1 H 2 is such that: V = V 1 V 2, V t = V t1 V t2, E = E 1 E 2, D = L(V )(D 1 ) L(V )(D 2 ) I = {1,..., I 1 + I 2 } D 2 R(V ), σ(d) = σ 1 (D V1 ) (σ 2 + I 1 + 1)(D V2 ) where (σ + k)(d) = {n + k : n σ(d)} is a shifting of the indexes;

57 Composition of Hybrid Systems Given H 1 = (V 1, V t1, E 1, D 1, I 1, σ 1, ω 1, ρ 1, π 1 ) and H 2 = (V 2, V t2, E 2, D 2, I 2, σ 2, ω 2, ρ 2, π 2 ), H = H 1 H 2 is such that: V = V 1 V 2, V t = V t1 V t2, E = E 1 E 2, D = L(V )(D 1 ) L(V )(D 2 ) I = {1,..., I 1 + I 2 } D 2 R(V ), σ(d) = σ 1 (D V1 ) (σ 2 + I 1 + 1)(D V2 ) where (σ + k)(d) = {n + k : n σ(d)} is a shifting of the indexes; ω(i) = ω 1(i), if 1 i I 1, ω(i) = ω 2 (i I 1 ), if I i I 1 + I 2

58 Composition of Hybrid Systems Given H 1 = (V 1, V t1, E 1, D 1, I 1, σ 1, ω 1, ρ 1, π 1 ) and H 2 = (V 2, V t2, E 2, D 2, I 2, σ 2, ω 2, ρ 2, π 2 ), H = H 1 H 2 is such that: V = V 1 V 2, V t = V t1 V t2, E = E 1 E 2, D = L(V )(D 1 ) L(V )(D 2 ) I = {1,..., I 1 + I 2 } D 2 R(V ), σ(d) = σ 1 (D V1 ) (σ 2 + I 1 + 1)(D V2 ) where (σ + k)(d) = {n + k : n σ(d)} is a shifting of the indexes; ω(i) = ω 1(i), if 1 i I 1, ω(i) = ω 2 (i I 1 ), if I i I 1 + I 2 π(e) = j π1 (e) if e E 1 π 2 (e) + I if e E 2

59 Composition of Hybrid Systems Given H 1 = (V 1, V t1, E 1, D 1, I 1, σ 1, ω 1, ρ 1, π 1 ) and H 2 = (V 2, V t2, E 2, D 2, I 2, σ 2, ω 2, ρ 2, π 2 ), H = H 1 H 2 is such that: V = V 1 V 2, V t = V t1 V t2, E = E 1 E 2, D = L(V )(D 1 ) L(V )(D 2 ) I = {1,..., I 1 + I 2 } D 2 R(V ), σ(d) = σ 1 (D V1 ) (σ 2 + I 1 + 1)(D V2 ) where (σ + k)(d) = {n + k : n σ(d)} is a shifting of the indexes; ω(i) = ω 1(i), if 1 i I 1, ω(i) = ω 2 (i I 1 ), if I i I 1 + I 2 j π1 (e) if e E π(e) = 1 π 2 (e) + I if e E 2 ρ(d i, D j, val(v )) = L(V )(ρ 1 (D i V1, D j V1, val(v 1 )) L(V )(ρ 2 (D i V2, D j V2, val(v 2 ))

60 Composition Example

61 Composition Example

62 Composition Example diode = R d I d diode.v = {v a, v k, i d }, diode.e = {e 1, e 2 }, diode.d = {D 1, D 2 }, I = {1, 2}, diode.σ(d 1 ) = {1}, diode.σ(d 2 ) = {2}, ω(1) = e 1, ω(2) = e 2,

63 Representing Hierarchy (c) Rect = vs (R d I 0 ) GND SUB (R C)

64 Representing Hierarchy (c) Rect = vs (R d I 0 ) GND SUB (R C) (d) Component Tree

65 Representing Hierarchy (c) Rect = vs (R d I 0 ) GND SUB (R C) (d) Component Tree (e) Scheduler Tree

66 Representing Hierarchy (c) Rect = vs (R d I 0 ) GND SUB (R C) (d) Component Tree (e) Scheduler Tree Let G : S N 2 S N be a function that associates to each scheduler the set of its children, and let Π : S N {1,..., S N } be a global ordering of the nodes. Let I : C S be a function that associates to each component it s scheduler.

67 Scheduler s resolve Algorithm resolve(t) children G(s) if children = then // s is a leaf, proceed to solve the equations and end recursion D {D I 1 (s).d val(i 1 (s).v t ) D} J D D s.σ(d) E t i J s.ω(i) E t sort(e t, s.π) for all e i E t do solve(e i, t) end for markchange ( D, val(i 1 (s).v t ) ) else // s is not a leaf, continue the recursion children sort(children, Π) for all s i children do s i.resolve(t) end for end if

68 Scheduler s resolve Algorithm resolve(t) children G(s) if children = then // s is a leaf, proceed to solve the equations and end recursion D {D I 1 (s).d val(i 1 (s).v t ) D} J D D s.σ(d) E t i J s.ω(i) E t sort(e t, s.π) for all e i E t do solve(e i, t) end for markchange ( D, val(i 1 (s).v t ) ) else // s is not a leaf, continue the recursion children sort(children, Π) for all s i children do s i.resolve(t) end for end if

69 Scheduler s resolve Algorithm resolve(t) children G(s) if children = then // s is a leaf, proceed to solve the equations and end recursion D {D I 1 (s).d val(i 1 (s).v t ) D} J D D s.σ(d) E t i J s.ω(i) E t sort(e t, s.π) for all e i E t do solve(e i, t) end for markchange ( D, val(i 1 (s).v t ) ) else // s is not a leaf, continue the recursion children sort(children, Π) for all s i children do s i.resolve(t) end for end if

70 Examples and Conclusions

71 Examples Structure CheckMate programs of Structure of HyTech programs Structure of HyVisual programs

72 Conclusions and Future Work We have presented an abstract semantics for hybrid systems. specifying: the time stamper automaton the functions domainchange, solve, next It can be refined by We have shown how the structure of hybrid systems can be captured in the interchange semantics. We have implemented a prototype of the half-wave rectifier in Metropolis and can be downloaded at Future work: Implementation of a Metropolis library for the interchange format; Integration of a DAE solver (in collaboration with Jaijeet Roychowdhury, University of Minnesota) Implementation of translators to/from known tools