System Modeling Introduction Rugby Meta-Model Finite State - PowerPoint PPT Presentation

System Modeling Introduction Rugby Meta-Model Finite State Machines Petri Nets Untimed Model of Computation Synchronous Model of Computation Timed Model of Computation Integration of Computational Models Tightly Coupled Process Networks Nondeterminism and Probability Applications

System Modeling Untimed MoC 1 Coexistence of Different MoC Domains A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 2 An Amplifier as a Network of Three Processes ✛✘ A 3 ✚✙ ✛✘ ✛✘ output input A 2 A 1 ✚✙ ✚✙ A 1 merges input data with a control signal; A 2 amplifies the input signal by applying the control signal; A 3 analyses the amplified signal and produces a new control value; A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 3 Events and Signals • Untimed events ˙ E = V ; • Synchronous events ¯ E = V ∪ {⊔} ; • Timed events ˆ E = ¯ E ; • Any kind of event E = ˙ E ∪ ¯ E ∪ ˆ E ; • Untimed signals ˙ e i ∈ ˙ S : ˙ s = � ˙ e 0 , ˙ e 1 , ˙ e 2 , ... � , ˙ E ; • Synchronous signals ¯ e i ∈ ¯ S : ¯ s = � ¯ e 0 , ¯ e 1 , ¯ e 2 , ... � , ¯ E ; • Timed signals ˆ e i ∈ ˆ S : ˆ s = � ˆ e 0 , ˆ e 1 , ˆ e 2 , ... � , ˆ E ; • Any kind of signal S : s = � e 0 , e 1 , e 2 , ... � , e i ∈ E ; • Empty signal �� ; A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 4 Processes Connected by Untimed, Synchronous and Timed Signals ✛✘ ✛✘ s 1 = � ˙ ˙ e 1 , ˙ e 2 , ˙ e 3 , ... � p 2 p 1 ✚✙ ✚✙ = � 6 , 3 , 1 , ... � ✛✘ ✛✘ s 2 = � ¯ ¯ e 1 , ¯ e 2 , ¯ e 3 , ¯ e 4 , ... � p 4 p 3 ✚✙ ✚✙ = � 6 , ⊥ , 3 , 1 , ... � ✛✘ ✛✘ s 3 = � ˆ ˆ e 1 , ˆ e 2 , ˆ e 3 , ... � p 6 p 5 ✚✙ ✚✙ = � 6 , ⊥ , ⊥ , 1 , ... � A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 5 Operations on Signals Concatenation of signals s 1 ⊕ ( s 2 ⊕ s 3 ) = ( s 1 ⊕ s 2 ) ⊕ s 3 , �� ⊕ s = s ⊕ �� = s . Length of a signal: # s , e.g. # � e, e ′ � = 2 , # �� = 0 ; Indexing of signals: [ ] : S × N → E , e.g. s = � e 1 , e 2 , e 3 � , s [2] = e 2 ; � � e 0 , ..., e n − 1 � if # s ≥ n take ( n, s ) = s otherwise � � e n , ..., e # s − 1 � if # s ≥ n drop ( n, s ) = �� otherwise � e 0 if s � = �� head ( s ) = undefined otherwise tail ( s ) = drop (1 , s ) A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 6 Signal Partitioning Definition: Let ν : N 0 → N 0 be a function on natural numbers and s ∈ S be a signal. The partition π ( ν, s ) is defined as follows. π ( ν, s ) = parts ( ν, 0 , s ) ∀ s ∈ S We write π ( ν, s ) = � r i � with i = 0 , 1 , 2 , ... . The remainder rem ( π, ν, s ) is � s = ( r i ) ⊕ rem ( π, ν, s ) , s ∈ S, i ∈ N 0 . � r i � = π ( ν,s ) � �� take ( ν ( i ) , s ) �� ⊕ parts ( ν, i + 1 , drop ( ν ( i ) , s )) if # s ≥ ν ( i ) parts ( ν, i, s ) = �� otherwise for all s ∈ S, i ∈ N 0 . A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 7 Signal Partitioning Example s = � e 0 , e 1 , e 2 , e 3 , e 4 , e 5 , ... � π ( ν, s ) = � r i � for ν ( i ) = 3 for all i � r 0 , r 1 , ... � = �� e 0 , e 1 , e 2 � , � e 3 , e 4 , e 5 � , ... � ✛✘ p ✚✙ s ′ = � e ′ 0 , e ′ 1 , e ′ 2 , e ′ 3 , e ′ 4 , e ′ 5 , ... � π ( ν ′ , s ′ ) = � r ′ i � for ν ′ ( i ) = 2 for all i � r ′ 0 , r ′ 1 , ... � = �� e ′ 0 , e ′ 1 � , � e ′ 2 , e ′ 3 � , ... � A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 8 Signal Partitioning in the Amplifier s ′ = � e ′ 0 , e ′ 1 , e ′ 2 , e ′ 3 , e ′ 4 , e ′ s = � e 0 , e 1 , e 2 , e 3 , e 4 , e 5 , ... � 5 , ... � ✛✘ π ( ν ′ , s ′ ) = � r ′ i � for ν ′ ( i ) = 5 ∀ i π ( ν, s ) = � r i � for ν ( i ) = 1 ∀ i � r ′ 0 , r ′ 1 , ... � = �� e ′ 0 , e ′ 1 , e ′ 2 , e ′ 3 , e ′ � r 0 , r 1 , ... � = �� e 0 � , � e 1 � , ... � 4 � , ... � ✚✙ A 2 ✛✘ ✚✙ A 3 s ′ = � e ′ s ′′′ = � e ′′′ 0 , e ′ 1 , e ′ 2 , e ′ 3 , e ′ 4 , e ′ 0 , e ′′′ 1 , e ′′′ 2 , e ′′′ 3 , e ′′′ 4 , e ′′′ 5 , ... � 5 , ... � π ( ν ′′ , s ′ ) = � r ′′ i � for ν ′′ ( i ) = 5 ∀ i π ( ν ′′′ , s ′′′ ) = � r ′′′ i � for ν ′′′ ( i ) = 1 ∀ i � r ′′ 0 , r ′′ 1 , ... � = �� e ′ 0 , e ′ 1 , ... � , � e ′ � r ′′′ 0 , r ′′′ 1 , ... � = �� e ′′′ 0 � , � e ′′′ 5 , ... � , ... � 1 � , ... � A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 9 Process Constructors Process constructors are templates to instantiate processes. Level Name Constructor Description 1 map Processes without internal state. mapU 2 scan Processes with an internal state scanU and a next-state function. The state is directly visible at the output. 3 moore Processes with a state; the mooreU output is a function of the state, but not directly of the input. 3 mealy Processes with a state; the mealyU output is a function of the state and the current input. A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 10 A Map-based Process ✛✘ A 3 ✚✙ ✛✘ ✛✘ output input A 1 A 2 ✚✙ ✚✙ A 2 = mapU ( c, f ) where c = 1 f (( � x � , � y 1 , y 2 , y 3 , y 4 , y 5 � )) = � xy 1 , xy 2 , xy 3 , xy 4 , xy 5 � A 2 ( � ( � 10 � , � 1 , 2 , 3 , 4 , 5 � ) , ( � 10 � , � 6 , 7 , 8 , 9 , 10 � ) � ) = � 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 � A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 11 The mapU Process Constructor mapU ( c, f ) = p s ′ where p ( ˙ s ) = ˙ a ′ f (˙ a i ) = ˙ i π ( ν, ˙ s ) = � ˙ a i � , ν ( i ) = c π ( ν ′ , ˙ s ′ ) = � ˙ a ′ i � , ν ′ ( i ) = # f (˙ a i ) A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 12 The scanU Process Constructor scanU ( γ, g, w 0 ) = p s ′ where p ( ˙ s ) = ˙ g (˙ a i , w i ) = w i +1 a ′ � w i +1 � = ˙ i π ( ν, ˙ s ) = � ˙ a i � , ν ( i ) = γ ( w i ) π ( ν ′ , ˙ s ′ ) = � ˙ a ′ i � , ν ′ ( i ) = # g (˙ a i ) = 1 i ∈ N 0 A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 13 A Scan-based Process ✛✘ A 3 ✚✙ ✛✘ ✛✘ output input A 1 A 2 ✚✙ ✚✙ A 3 = scanU ( γ, g, w 0 ) where w 0 = 10 γ ( w i ) = 5 ∀ i ∈ N 0  w i − 1 if x 1 + x 2 + x 3 + x 4 + x 5 > 500   g ( w i , � x 1 , x 2 , x 3 , x 4 , x 5 � ) = w i + 1 if x 1 + x 2 + x 3 + x 4 + x 5 < 400  w i otherwise  A 3 ( � 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 � ) = � 11 , 11 � . A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 14 The mealyU Process Constructor mealyU ( γ, g, f, w 0 ) = p s ′ where p ( ˙ s ) = ˙ a ′ f ( w i , ˙ a i ) = ˙ i g ( w i , ˙ a i ) = w i +1 π ( ν, ˙ s ) = � ˙ a i � , ν ( i ) = γ ( w i ) π ( ν ′ , ˙ s ′ ) = � ˙ a ′ i � , ν ′ ( i ) = # f ( w i , ˙ a i ) A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 15 The mooreU Process Constructor mooreU ( γ, g, f, w 0 ) = p s ′ where p ( ˙ s ) = ˙ a ′ f ( w i ) = ˙ i g ( w i , ˙ a i ) = w i +1 π ( ν, ˙ s ) = � ˙ a i � , ν ( i ) = γ ( w i ) π ( ν ′ , ˙ s ′ ) = � ˙ a ′ i � , ν ′ ( i ) = # f ( w i ) A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 16 The zipU Process Constructor zipU ( γ a , γ b ) = p s ′ where p ( ˙ s a , ˙ s b , ˙ s c ) = ˙ a i , ˙ e ′ � ˙ b i � = ˙ i π ( ν a , ˙ s a ) = � ˙ a i � , ν a ( i ) = γ a (˙ c i ) s b ) = � ˙ π ( ν b , ˙ b i � , ν b ( i ) = γ b (˙ c i ) π ( ν c , ˙ s c ) = � ˙ c i � , ν c ( i ) = 1 π ( ν ′ , ˙ s ′ ) = �� ˙ e ′ i �� , ν ′ ( i ) = 1 A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 17 The unzipU Process Constructor unzipU () = p s ′ , ˙ s ′′ � where p ( ˙ s ) = � ˙ a ′ a ′′ e i = � ˙ ˙ i , ˙ i � π ( ν, ˙ s ) = � ˙ e i � , ν ( i ) = 1 π ( ν ′ , ˙ s ′ ) = � ˙ a ′ i � , ν ′ ( i ) = #˙ a ′ i π ( ν ′′ , ˙ s ′′ ) = � ˙ a ′′ i � , ν ′′ ( i ) = #˙ a ′′ i A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 18 The zipUs Process Constructor zipUs ( c 1 , c 2 ) = p s ′ where p ( ˙ s a , ˙ s b ) = ˙ a i , ˙ e ′ (˙ b i ) = ˙ i π ( ν a , ˙ s a ) = � ˙ a i � , ν a ( i ) = c 1 s b ) = � ˙ π ( ν b , ˙ b i � , ν b ( i ) = c 2 π ( ν ′ , ˙ s ′ ) = �� ˙ e ′ i �� , ν ′ ( i ) = 1 A. Jantsch, KTH, Jan-Feb 2005

System Modeling Untimed MoC 19 The zipWithU Process Constructor zipWithU ( c 1 , c 2 , f ) = p s ′ where p ( ˙ s a , ˙ s b ) = ˙ a i , ˙ f ((˙ b i )) = ˙ c i π ( ν a , ˙ s a ) = � ˙ a i � , ν a ( i ) = c 1 s b ) = � ˙ π ( ν b , ˙ b i � , ν b ( i ) = c 2 π ( ν ′ , ˙ s ′ ) = � ˙ c i � , ν ′ ( i ) = #˙ c i A. Jantsch, KTH, Jan-Feb 2005