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Syntax Matters
Tom Henzinger (IST Austria)
with
Barbara Jobstmann (Verimag)
Maria Mateescu (EPFL)
Verena Wolf (Saarbruecken)
Science
Experiment
Theory
Mathematics
14
Mathematics
Semantics
Syntax
14
Mathematics
Semantics
Syntax
14
1110
XIV
Syntax Matters
1. Expressiveness
0
Syntax Matters
1. Expressiveness
0
2. Succinctness
IIII IIII IIII
Syntax Matters
1. Expressiveness
0
2. Succinctness
IIII IIII IIII
3. Operations
+1
addition
multiplication
14 x 34 =
42
56
476
Syntax is More than Notation
1. Expressiveness
0
2. Succinctness
IIII IIII IIII
3. Operations
+1
addition
multiplication
(* 14 34) =
14 x 34 =
42
56
476
Syntax is More than Notation
1. Expressiveness
0
2. Succinctness
IIII IIII IIII
3. Operations
+1
addition
multiplication
Two languages are equivalent if there is a linear translation
from each to the other (e.g. prefix – infix, binary – decimal).
Computer Science
Semantics
0
1
2
3
4
...
Syntax
1
1
2
6
24
...
(defun f (n)
(if (<= n 1) 1
(* n (f (- n 1)))))
Computer Science
Semantics
Syntax
1
1
2
6
24
...
0
1
2
3
4
...
(defun f (n)
(if (<= n 1) 1
(* n (f (- n 1)))))
finite
infinite
Computer Science
Semantics
Syntax
1
1
2
6
24
...
0
1
2
3
4
...
infinite
(defun f (n)
(if (<= n 1) 1
(* n (f (- n 1)))))
finite
recursively enumerable
Computer Science
Semantics
0
1
2
3
4
...
Syntax
1
1
2
6
24
...
(defun f (n)
(if (<= n 1) 1
(* n (f (- n 1)))))
Composition:
(defun h (n) (f (g n)))
Computer Science
Semantics
Syntax
x := 0;
y := 0;
while true do
x := x + 1;
y := y + 1
end
Computer Science
Semantics
2,2
1,1
0,0
1,0
2,1
Syntax
3,2
x := 0;
y := 0;
while true do
x := x + 1;
y := y + 1
end
Computer Science
Semantics
Syntax
(inc x)*
0,0
1,0
2,0
Computer Science
Semantics
Syntax
(inc x)*
0,2
(inc y) *
0,1
0,0
1,0
2,0
Computer Science
Semantics
Syntax
(inc x)*
0,2
1,2
2,2
0,1
1,1
2,1
0,0
1,0
2,0
(inc y) *
Composition:
(inc x)* || (inc y)*
Markovian Population Models
Markovian Population Models
+
Markovian Population Models
+
Markovian Population Models
+
Markovian Population Models
State: ( 8 , 6 , 6 )
Markovian Population Models
State: ( 9 , 7 , 5 )
Transition:
State: ( 8 , 6 , 6 )
Markovian Population Models
-discrete state
-location unaware
Markovian Population Models
-discrete state
-location unaware
-stochastic transition
-continuous time
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
8, 6, 6
discrete time
0.6
9, 7, 5
0.4
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
8, 6, 6
discrete time 0
0.6
9, 7, 5
1
0.4
0
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
8, 6, 6
discrete time 1
0.6
9, 7, 5
0.6
0.4
0.4
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
8, 6, 6
discrete time 2
0.6
9, 7, 5
0.36
0.4
0.64
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
8, 6, 6
discrete time 3
0.6
9, 7, 5
0.216
(9,7,5): 0.36
(8,6,6): 0.64
0.4
0.784
(9,7,5): 0.216
(8,6,6): 0.784
Markovian Population Models
9, 7, 5
8, 6, 6
deterministic
8, 6, 6
discrete time
8, 6, 6
continuous time
exit rate 0.5
exp residence time 2
0.6
9, 7, 5
9, 7, 5
0.4
0.5
Markovian Population Models
9, 7, 5
1
0.5
8, 6, 6
0
continuous time 0
exit rate 0.5
exp residence time 2
Markovian Population Models
9, 7, 5
0.6
0.5
8, 6, 6
0.4
continuous time 1
exit rate 0.5
exp residence time 2
Markovian Population Models
9, 7, 5
0.4
0.5
8, 6, 6
0.6
continuous time 1.8
exit rate 0.5
exp residence time 2
Markovian Population Models
9, 7, 5
8, 6, 6
nondeterministic
8, 6, 6
discrete time
8, 6, 6
continuous time
0.6
9, 7, 5
9, 7, 5
0.4
0.5
Markovian Population Models
9, 7, 5
8, 6, 6
nondeterministic
8, 6, 6
discrete time MDP
8, 6, 6
continuous time
0.6
9, 7, 5
0.3
0.4
0.7
9, 7, 5
0.5
Markovian Population Models
9, 7, 5
8, 6, 6
nondeterministic
8, 6, 6
discrete time MDP
8, 6, 6
continuous time
exit rate 2
exp residence time 0.5
0.6
0.4
9, 7, 5
0.3
0.7
0.5
9, 7, 5
1.5
Markovian Population Models
9, 7, 3
0.2
0.2
8, 6, 4
8, 6, 5
0.2
7, 5, 5
0.2
0.1
0.2
8, 6, 6
CTMC
7, 5, 6
0.2
0.1
0.1
0.1
9, 7, 5
0.2
0.1
0.1
9, 7, 4
0.1
0.1
0.1
0.2
7, 5, 7
0.2
Markovian Population Models
+
0.2
0.1
Syntax: set of transition classes (finite object)
Markovian Population Models
0.2
+
0.1
Syntax: set of transition classes (finite object)
0.1
0.1
9, 7, 5
0.2
8, 6, 6
0.2
Semantics: CTMC (infinite object)
Markovian Population Models
+
0.2
0.1
Syntax: set of transition classes (finite object)
0.5
9, 7, 5
0.6
8, 6, 6
Semantics: CTMC (infinite object)
Markovian Population Models
0.2
+
0.1
Syntax: set of transition classes (finite object)
0.6
0.5
9, 7, 5
12.6
8, 6, 6
9.6
Semantics: CTMC (infinite object)
Syntax: Transition Class Model (TCM)
Dimension n: state (x1, ..., xn) 2 S
state space S = Nn
Finite set of transition classes:
each transition class consists of
1. guard G µ S
2. injective update function u: G ! S
3. rate function : G ! R+
Syntax: Transition Class Model (TCM)
Dimension n: state (x1, ..., xn) 2 S
state space S = Nn
n=3
Finite set of transition classes:
each transition class consists of
1. guard G µ S
2. injective update function u: G ! S
3. rate function : G ! R+
G1: x1 ¸ 1 Æ x2 ¸ 1
u1(x1,x2,x3) = (x1-1, x2-1, x3+1)
1(x1,x2,x3) = 0.2 ¢ x1 ¢ x2
G2: x3 ¸ 1
u2(x1,x2,x3) = (x1, x2, x3-1)
2(x1,x2,x3) = 0.1 ¢ x3
Semantics: Continuous-Time Markov Chain (CTMC)
For all times t 2 R+, a random variable X(t) 2 S.
Semantics: Continuous-Time Markov Chain (CTMC)
For all times t 2 R+, a random variable X(t) 2 S.
Syntax ! Semantics: TCM ! CTMC
For each transition class (Gi,ui,i) and all times t 2 R+ and  ! 0,
Pr( X(t+) = ui(x) | X(t) = x ) = i(x) ¢ .
In addition, Pr( X(0) = x0 ) = 1 for some given initial state x0 2 S.
Semantics: Continuous-Time Markov Chain (CTMC)
For all times t 2 R+, a random variable X(t) 2 S.
Syntax ! Semantics: TCM ! CTMC
For each transition class (Gi,ui,i) and all times t 2 R+ and  ! 0,
Pr( X(t+) = ui(x) | X(t) = x ) = i(x) ¢ .
In addition, Pr( X(0) = x0 ) = 1 for some given initial state x0 2 S.
The Chemical Master Equation
Under certain technical conditions that hold for TCMs,
X(t) is the unique solution of the ODE system
dpt(x) / dt = i: x 2 Hi i(ui-1(x)) ¢ pt(ui-1(x)) - i: x 2 Gi i(x) ¢ pt(x)
inflow
where
outflow
pt(x) = Pr( X(t) = x)
Hi = { x 2 S | ui-1(x) is defined }.
x1 =
u1-1(x)
2(x)
1(x1)
u2(x)
x
x2 =
u2-1(x)
2(x2)
1(x)
u1(x)
The Chemical Master Equation
Under certain technical conditions that hold for TCMs,
X(t) is the unique solution of the ODE system
dpt(x) / dt = i: x 2 Hi i(ui-1(x)) ¢ pt(ui-1(x)) - i: x 2 Gi i(x) ¢ pt(x)
where
pt(x) = Pr( X(t) = x)
Hi = { x 2 S | ui-1(x) is defined }.
If S is finite, then
pt = p0 ¢ eQt = p0 ¢ k=0,1,2,... (Qt)k / k!
where Q is the generator matrix with
Q(x,x’) = i(x)
if ui(x) = x’
-i i(x) if x = x’
0
else.
The Chemical Master Equation
Under certain technical conditions that hold for TCMs,
X(t) is the unique solution of the ODE system
dpt(x) / dt = i: x 2 Hi i(ui-1(x)) ¢ pt(ui-1(x)) - i: x 2 Gi i(x) ¢ pt(x)
where
pt(x) = Pr( X(t) = x)
Hi = { x 2 S | ui-1(x) is defined }.
If S is finite, then
pt = p0 ¢ eQt = p0 ¢ k=0,1,2,... (Qt)k / k!
where Q is the generator matrix with
Q(x,x’) = i(x)
if ui(x) = x’
-i i(x) if x = x’
0
else.
Numerically
unstable!
Special Kinds of CTMC
Poisson process: n = 1
0
Q=

1

2

( -  0 0 0 ... )
0 -  0 0 ...
0 0 -  0 ...
...
3

Special Kinds of CTMC
Poisson process: n = 1
0

1

2

3

Birth process: n = 1
0
0
1
1
2
2
3
3
Solving the Chemical Master Equation
• Computing distributions versus individual parameters (e.g. mean)
-”molecular noise” important for small populations
-entire distributions are of interest
+distribution of time until protein concentration reaches
effective signaling level
+distribution of time for DNA replication initiation
+distribution of concentration of transcription factors
Solving the Chemical Master Equation
• Computing distributions versus individual parameters (e.g. mean)
-”molecular noise” important for small populations
-entire distributions are of interest
+distribution of time until protein concentration reaches
effective signaling level
+distribution of time for DNA replication initiation
+distribution of concentration of transcription factors
Deterministic ODE model inadequate
Solving the Chemical Master Equation
• Computing distributions versus individual parameters (e.g. mean)
• Computing transient distributions versus steady state
We want to compute pt for certain times t 2 [0,T] up to a given horizon T.
Let  = limt ! 1 pt.
If x 2 S (x) = 1, then x is called stationary distribution and can be
computed as the solution of the linear equation system
0=¢Q
x 2 S (x) = 1.
Solving the Chemical Master Equation
• Computing distributions versus individual parameters (e.g. mean)
• Computing transient distributions versus steady state
• Propagating distributions versus states (Monte-Carlo simulation)
Toggle Switch: Four Simulation Runs
Gillespie Simulation
-simulation algorithms are easy to understand and implement
-numerical handling of large (or infinite) matrices not feasible
-one trajectory is considered as the outcome of one “in-silico”
experiment
Gillespie Simulation
-simulation algorithms are easy to understand and implement
-numerical handling of large (or infinite) matrices not feasible
-one trajectory is considered as the outcome of one “in-silico”
experiment
-too expensive for computing event probabilities or distributions:
+double precision requires 4 times more simulation runs
+millions of runs for a precision of 10-5 [CMSB 09]
Solving the Chemical Master Equation
• Computing distributions versus individual parameters (e.g. mean)
• Computing transient distributions versus steady state
• Propagating distributions versus states (Monte-Carlo simulation)
Deterministic ODE model inadequate
Gillespie simulation inadequate
Solving the Chemical Master Equation
• Computing distributions versus individual parameters (e.g. mean)
• Computing transient distributions versus steady state
• Propagating distributions versus states (Monte-Carlo simulation)
Deterministic ODE model inadequate
Gillespie simulation inadequate
Numerical algorithms ! [Burrage, Munsky, Zhang, ...]
Toggle Switch: Fast Adaptive Uniformization
t = 5000
t = 15000
t = 30000
t = 50000
Toggle Switch: Four Simulation Runs
Decomposition of CTMC
Uniformization:
CTMC X = DTMC Y + Poisson process N (“clock”)
4
X
0.8
=
1
+
Y
0.2
N
5
Decomposition of CTMC
Uniformization:
CTMC X = DTMC Y + Poisson process N (“clock”)
 = supx 2 S x < 1
where x = x’ (x,x’)
Poisson process N:
nt(k) = Pr( N(t) = k ) = e ¢ (t)k / k!
Probability matrix of Y: P = -1¢ Q + I
pt(x)
= Pr( Y(N(t)) = x )
= k=0,1,2,... Pr( Y(k) = x ) ¢ Pr( N(t) = k )
Decomposition of CTMC
Uniformization:
CTMC X = DTMC Y + Poisson process N (“clock”)
 = supx 2 S x < 1
where x = x’ (x,x’)
Poisson process N:
nt(k) = Pr( N(t) = k ) = e ¢ (t)k / k!
Probability matrix of Y: P = -1¢ Q + I
pt(x)
= Pr( Y(N(t)) = x )
= k=0,1,2,... Pr( Y(k) = x ) ¢ Pr( N(t) = k )
pt
= k=0,1,2,... p0 ¢ Pk ¢ nt(k)
Decomposition of CTMC
Uniformization:
CTMC X = DTMC Y + Poisson process N (“clock”)
 = supx 2 S x < 1
where x = x’ (x,x’)
Poisson process N:
nt(k) = Pr( N(t) = k ) = e ¢ (t)k / k!
Probability matrix of Y: P = -1¢ Q + I
pt(x)
= Pr( Y(N(t)) = x )
= k=0,1,2,... Pr( Y(k) = x ) ¢ Pr( N(t) = k )
pt
= k=0,1,2,... p0 ¢ Pk ¢ nt(k)
Can be approximated to
any desired accuracy!
Decomposition of CTMC
Uniformization:
CTMC X = DTMC Y + Poisson process N (“clock”)
-clock rate is faster than all transition rates
-too expensive for stiff systems
(i.e., systems with very different transition rates)
Decomposition of CTMC
Uniformization:
CTMC X = DTMC Y + Poisson process N (“clock”)
-clock rate is faster than all transition rates
-too expensive for stiff systems
(i.e., systems with very different transition rates)
Adaptive Uniformization (van Moorsel & Sanders):
CTMC = DTMC + Birth process
-state-dependent clock rate
Decomposition of CTMC
Uniformization:
CTMC X = DTMC Y + Poisson process N (“clock”)
-clock rate is faster than all transition rates
-too expensive for stiff systems
(i.e., systems with very different transition rates)
Adaptive Uniformization (van Moorsel & Sanders):
CTMC = DTMC + Birth process
-state-dependent clock rate
-too expensive for large/infinite state spaces, because at
all times t > 0, every state has a positive probability
Fast Adaptive Uniformization
[Diedier, H, Mateescu & Wolf]
Transient distributions can be efficiently approximated by
combining adaptive uniformization with two ideas from verification:
1. on-the-fly state space exploration
2. abstraction of insignificant (low-probability) states
Fast Adaptive Uniformization
[Diedier, H, Mateescu & Wolf]
Transient distributions can be efficiently approximated by
combining adaptive uniformization with two ideas from verification:
1. on-the-fly state space exploration
2. abstraction of insignificant (low-probability) states
Executability
Sliding Window
Abstraction
Sliding Window Abstraction for DTMC: t = 0
Sliding Window Abstraction for DTMC: t = 1
high-probability
states
low-probability
states
Sliding Window Abstraction for DTMC: t = 1
lost probability mass
contributes to error
window of significantprobability states
Sliding Window Abstraction for DTMC: t = 2
Sliding Window Abstraction for DTMC: t = 2
Sliding Windows in Adaptive Uniformization
CTMC = DTMC + Birth process (state-dependent clock)
Sliding Windows in Adaptive Uniformization
CTMC = DTMC + Birth process (state-dependent clock)
DTMC + Poisson process
Sliding Windows in Adaptive Uniformization
CTMC = DTMC + Birth process (state-dependent clock)
DTMC + Poisson process
2 independent invocations of sliding windows !
Sliding Windows in Adaptive Uniformization
CTMC = DTMC + Birth process (state-dependent clock)
DTMC + Poisson process
2 independent invocations of sliding windows !
0
0
supk k - 0
1
1
2
2
3
3
4
4
Sliding Windows in Adaptive Uniformization
CTMC = DTMC + Birth process (state-dependent clock)
DTMC + Poisson process
2 independent invocations of sliding windows !
0
0
supk k - 0
1
1
2
2
3
3
4
4
Sliding Windows in Adaptive Uniformization
CTMC = DTMC + Birth process (state-dependent clock)
DTMC + Poisson process
2 independent invocations of sliding windows !
0
0
supk k - 0
1
1
2
2
3
3
4
4
Sliding Windows in Adaptive Uniformization
CTMC = DTMC + Birth process (state-dependent clock)
DTMC + Poisson process
2 independent invocations of sliding windows !
0
0
supk k - 0
1
1
2
2
3
3
4
4
Three Sources of Errors:
Pr( X(t) = x ) = k=0,1,2,... Pr( Y(k) = x ) ¢ Pr( B(t) = k )
truncation point k = R
sliding-window abstraction of Y
sliding-window abstraction of B
Three Sources of Errors:
All Monotonically Refinable Underapproximations
Pr( X(t) = x ) = k=0,1,2,... Pr( Y(k) = x ) ¢ Pr( B(t) = k )
truncation point k = R
sliding-window abstraction of Y
sliding-window abstraction of B
Fast Adaptive Uniformization
GOOD if probability mass forms a (moving) cloud.
BAD
if probability mass spreads.
Bacteriophage  Model
Bacteriophage  Model
Desired precision:
3 x 10-6
Standard uniformization:
timeout
Fast adaptive uniformization:
55 min runtime
Gillespie simulation ( = 0.95): 67 h runtime (3 x 108 runs)
94
94
Why Syntax Matters
Epidemic Process: Stoichiometric Equations
n = 2:
x1 ... number of healthy individuals
x2 ... number of infected individuals
;
x1
;
x2
x1 + x2
a
b
c
d
e
x1
;
x2
;
2x2
Epidemic Process: Rate Matrix
Q( (x1,x2), (x1+1,x2) )
Q( (x1,x2), (x1-1,x2) )
=a
= bx1
if x1 > 0
Q( (x1,x2), (x1,x2+1) )
Q( (x1,x2), (x1,x2-1) )
=c
= dx2
if x2 > 0
Q( (x1,x2), (x1-1,x2+1) ) = ex1x2
if x1,x2 > 0
All other nondiagonal entries are 0, and each diagonal
entry is the negative sum of the other row entries.
Epidemic Process: Stochastic Petri Net
a
x1
b ¢ m(x1)
e ¢ m(x1) ¢
2 m(x2)
c
x2
d ¢ m(x2)
Epidemic Process: Stochastic Process Algebra
Originally: one process per molecule
Epidemic Process: Stochastic Process Algebra
Originally: one process per molecule
Bio-PEPA [Hillston et al.]:
x1 = a>> + bx1<< + ex1x2<<
x2 = c>> + dx2<< + ex1x2<<
x1 <ex1x2> x2
Epidemic Process: Guarded Commands
[] true
[] x1 > 0
- a ->
- bx1 ->
x1 := x1 + 1
x1 := x1 - 1
[] true
[] x2 > 0
- c ->
- dx2 ->
x2 := x2 + 1
x2 := x2 – 1
[] x1 > 0 Æ x2 > 0
x2+1
- ex1x2 ->
x1 := x1-1; x2 :=
Reactive Modules, PRISM [Kwiatkowska et al.]
Properties of Languages:
Composition
Petri Nets:
components need modification
Rate Matrices: Kronecker product (stochastic automata)
x1
x1
II
x2
x3
Properties of Languages:
Composition
Petri Nets:
components need modification
Rate Matrices: Kronecker product (stochastic automata)
Process Algebras, Stoichiometric Equations, Guarded Commands
Properties of Languages:
Expressiveness and Succinctness
Rate Matrices:
not succinct
Process Algebras:
most have only constant rate functions
Stoichiometric Equations:
insufficient rate functions
Bistable toggle switch (a genetic regulatory network):
1(x1,x2) = a / (b + x22)
[2 species, 4 reactions]
Properties of Languages:
Expressiveness and Succinctness
Rate Matrices:
not succinct
Process Algebras:
most have only constant rate functions
Stoichiometric Equations:
insufficient rate functions
Bistable toggle switch (a genetic regulatory network):
1(x1,x2) = a / (b + x22)
[2 species, 4 reactions]
Petri Nets, Guarded Commands
Properties of Languages:
Executability
Petri Nets, Process Algebras:
compiled into matrices
Rate Matrices:
expensive (global)
Properties of Languages:
Executability
Petri Nets, Process Algebras:
compiled into matrices
Rate Matrices:
expensive (global)
Stoichiometric Equations, Guarded Commands:
-easy computation of successor distributions
-support efficient simulation and on-the-fly reachability analysis
Properties of Languages:
Executability
Petri Nets, Process Algebras:
compiled into matrices
Rate Matrices:
expensive (global)
Stoichiometric Equations, Guarded Commands:
-easy computation of successor distributions
-support efficient simulation and on-the-fly reachability analysis
Propagating States vs. Distributions = Testing vs. Verification
Properties of Languages:
Encapsulation
Petri Nets, Rate Matrices, Stoichiometric Equations
Process Algebras, Guarded Commands
Properties of Languages:
Encapsulation
Dynamic Reconfiguration
Petri Nets, Rate Matrices, Stoichiometric Equations
Process Algebras, Guarded Commands
Properties of Languages:
Encapsulation
Dynamic Reconfiguration
Petri Nets, Rate Matrices, Stoichiometric Equations
Process Algebras, Guarded Commands
... and the winner is: Guarded Commands !
Conclusions
1. Syntax matters
(composition, expressiveness, succinctness, executability,
encapsulation, dynamic reconfiguration)
2. Ideas from verification (on-the-fly, abstraction, data structures)
make distribution propagation possible
Conclusions
1. Syntax matters
(composition, expressiveness, succinctness, executability,
encapsulation, dynamic reconfiguration)
2. Ideas from verification (on-the-fly, abstraction, data structures)
make distribution propagation possible
3. Main limitation: large populations (not high dimensionality)
Hybrid approach
(discrete stochastic + continuous deterministic)
works well.