Hybrid Automata
Examples
Undecidability Results
Finite Precision Semantics
Conclusions
Discreteness, Hybrid Automata, and Biology
A. Casagrande1,2
1 DIMI,
C. Piazza1
A. Policriti1,2
University of Udine, Udine, Italy
2 Applied
Genomics Institute, Udine, Italy
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Hybrid Automata
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Which is Your Point of View?
The world is dense
The world is discrete
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Which is Your Point of View?
The world is dense
(R, +, ∗, <, 0, 1) first-order theory is decidable
The world is discrete
Diophantine equations are undecidable
What about their interplay?
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Outline
1
Hybrid Automata
2
Examples
3
Undecidability Results
4
Finite Precision Semantics
5
Conclusions
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Hybrid Systems
Many real systems have a double nature. They:
evolve in a continuous way
are ruled by a discrete system
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Hybrid Systems
Many real systems have a double nature. They:
evolve in a continuous way
are ruled by a discrete system
We call such systems hybrid systems and we can formalize
them using hybrid automata
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Hybrid Automata - Intuitively
Intuitively, a hybrid automaton is a finite state automaton H with
continuous variables X
v0
v
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Hybrid Automata - Intuitively
Intuitively, a hybrid automaton is a finite state automaton H with
continuous variables X
v0
v
Dyn(v)[X, X 0 , T ]
Dyn(v 0 )[X, X 0 , T ]
Inv (v)[X]
Inv (v 0 )[X]
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Hybrid Automata - Intuitively
Intuitively, a hybrid automaton is a finite state automaton H with
continuous variables X
Act(e)[X] Reset(e)[X, X 0 ]
v0
v
Dyn(v)[X, X 0 , T ]
Dyn(v 0 )[X, X 0 , T ]
Inv (v)[X]
Inv (v 0 )[X]
Act(e0 )[X] Reset(e0 )[X, X 0 ]
A state is a pair hv , r i where r is an evaluation for X
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Hybrid Automata - Semantics
v0
v
f (t0 )
s
r
Definition (Continuous Transition)
t
hv , r i →
− C hv , si
⇐⇒
there exists a continuous g : R+ 7→
Rk such that r = g(0), s = g(t),
and for each t 0 ∈ [0, t] the formulæ
Inv (v )[g(t 0 )] and Dyn(v )[r , g(t 0 ), t 0 ]
hold
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Hybrid Automata - Semantics
v0
v
r
s
Definition (Discrete Transition)
e
hv , r i −
→D hv 0 , si
⇐⇒
e ∈ E and Inv (v )[r ], Act(e)[r ],
Reset(e)[r , s], and Inv (v 0 )[s]
hold
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Example
Bouncing Ball
X0 = X
V 0 = −γV
Ẋ = V
V̇ = −g
X≥0
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Example
Zeno Behavior The automaton avoids time elapsing by
crossing edges infinitely often
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Example
Zeno Point The limit point of a Zeno behavior
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Delta-Notch
Delta and Notch are proteins involved in cell differentiation
(see, e.g., Collier et al., Ghosh et al.)
Notch production is triggered by high Delta levels in
neighboring cells
Delta production is triggered by low Notch concentrations in
the same cell
High Delta levels lead to differentiation
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Delta-Notch: Single Cell Automaton
q1
q2
0
XD
= fD (XD , T )
0
XD
= gD (XD , T )
0
XN
= fN (XN , T )
0
XN
= fN (XN , T )
0
XD
= fD (XD , T )
0
XD
= gD (XD , T )
0
XN
0
XN
= gN (XN , T )
= gN (XN , T )
q3
q4
fD and fN increase Delta and Notch, gD and gN decrease Delta
and Notch, respectively
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Delta-Notch: Two Cells Automaton
It is the Cartesian product of two “single cell” automata
The Zeno state can occur only in the case of
two cells with identical initial concentrations
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Verification
Question
Can we automatically verify hybrid automata?
Let us start from the basic case of Reachability
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Verification
Question
Can we automatically verify hybrid automata?
Let us start from the basic case of Reachability
Naive Reachability(H, Initial set)
Old ← ∅
New ← Initial set
while New 6= Old do
Old ← New
New ← Discrete Reach(H, Continuous Reach(H, Old))
return Old
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Bounded Sets and Undecidability
Even if the invariants are bounded, reachability is undecidable
Proof sketch
Encode two-counter machine by exploiting density:
each counter value, n, is represented in a continuous
variable by the value 2−n
each control function is mimed by a particular location
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Where is the Problem?
Keeping in mind our examples:
Question “Meaning”
What is the meaning of these undecidability results?
Question “Decidability”
Can we avoid undecidability by adding some natural hypothesis
to the semantics?
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Undecidability in Real Systems
Undecidability in our models comes from . . .
infinite domains: unbounded invariants
dense domains: the “trick” n as 2−n
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Undecidability in Real Systems
Undecidability in our models comes from . . .
infinite domains: unbounded invariants
dense domains: the “trick” n as 2−n
But which real system does involve . . .
unbounded quantities?
infinite precision?
Unboundedness and density abstract discrete large quantities
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Dense vs Discrete - Intuition
We do not really want to completely abandon dense domains
We need to introduce a finite level of precision in bounded
dense domains, we can distinguish two sets only if they differ of
“at least ”
Intuitively, we can see that something new has been reached
only if a reasonable large set of new points has been
discovered, i.e., we are myope
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Dense vs Discrete
Lemma (Convergence)
Let S ⊆ Rk be a bounded set such that S = ∪i∈N Di , with either
Di = Dj or Di ∩ Dj = ∅
If there exists > 0 such that for each i ∈ N there exists ai such
that B({ai }, ) ⊆ Di , then there exists j ∈ N such that
S = ∪i≤j Di
This is a trivial compactness-like result
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Finite Precision Semantics
Definition (-Semantics)
Let > 0. For each formula ψ:
() either {|ψ|} = ∅ or {|ψ|} contains an -ball
(∩) {|ψ1 ∧ ψ2 |} ⊆ {|ψ1 |} ∩ {|ψ2 |}
(∪) {|ψ1 ∨ ψ2 |} = {|ψ1 |} ∪ {|ψ2 |}
(¬) {|ψ|} ∩ {|¬ψ|} = ∅
It is a general framework: there exist many different -semantics
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Reachability
Eps-Reachability(H, ψ[Z ], {| · |} )
R[Z ] ← ψ[Z ]
May New R[Z 0 ] ← ∃Z (Reach1 (Z , Z 0 ) ∧ R[Z ])
New R[Z ] ← May New R[Z ] ∧ ¬R[Z ]
while({|New R[Z ]|} 6= ∅)
R[Z ] ← R[Z ] ∨ New R[Z ]
May New R[Z 0 ] ← ∃Z (Reach1 (Z , Z 0 ) ∧ R[Z ])
New R[Z ] ← May New R[Z ] ∧ ¬R[Z ]
return R[Z ]
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A Decidability Result
Theorem (Reachability Problem)
Using -semantics and assuming both bounded invariants and
decidability for specification language, we have decidability of
reachability problem for hybrid automata
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A Decidability Result
Theorem (Reachability Problem)
Using -semantics and assuming both bounded invariants and
decidability for specification language, we have decidability of
reachability problem for hybrid automata
Proof Sketch
Because of condition () of -semantics, continuous steps can
either:
increase the reached set by at least do not increase the reach set
(∩), (∪), and (¬) ensure that the sets New R are disjoint
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An Instance of -semantics
Definition
Let > 0. We define [|ψ|] by structural induction on ψ as
follows:
[|t1 ◦ t2 |] = B([|t1 ◦ t2 |], ), for ◦ ∈ {=, <}
[|ψ1 ∨ ψ2 |] = [|ψ1 |] ∪ [|ψ2 |]
[|ψ1 ∧ ψ2 |] = ∪B({p},)⊆[|ψ1 |] ∩[|ψ2 |] B({p}, )
[|∃Z ψ[Z , X ]|] = ∪p∈R [|ψ[p, X ]|]
[|∀Z ψ[Z , X ]|] = ∪B({p},)⊆∩Z ∈R [|ψ[Z ,X ]|] B({p}, )
[|¬ψ|] = ∪B({p},)∩[|ψ|] =∅ B({p}, )
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Conclusions
Hybrid automata are both powerful and natural in the
modeling of hybrid systems
May be a little bit too expressive . . .
Real systems always have finite precision
-semantics introduce a finite precision ingredient in hybrid
automata
Using -semantics we do not have Zeno behaviors
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Why not. . .
. . . modeling systems over discrete latices?
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No, because three main reasons:
modeling would became harder
we would increase computational complexity
we would still assume infinite precision!!! (e.g.,
0, 999 . . . 9 6= 1)
. . . using only < and > instead of =?
No, because reachability is still undecidable.
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Under, Over and Demorgan
Example
Consider the formula 1 < X < 5 and = 0.1
We have that [|1 < X < 5|] = [|1 < X ∧ X < 5|] = (0.9, 5.1),
Consider the formula ¬(1 < X < 5)
We get that [|¬(1 < X < 5)|] = (−∞, 0.9) ∪ (5.1, +∞)
Notice that this last formula is not equivalent to X ≤ 1 ∨ X ≥ 5
whose semantics is [|X ≤ 1 ∨ X ≥ 5|] = (−∞, 1.1) ∪ (4.9, +∞)
Casagrande, Piazza, Policriti
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Related Literature
R. Lanotte, S. Tini, “Taylor approximation for hybrid
systems” Inf. Comput. 2007
A. Girard and G. J. Pappas, “Approximation metrics for
discrete and continuous systems”, IEEE TAC 2007
M. Fränzle, “Analysis of hybrid systems: An ounce of
realism can save an infinity of states”, CSL 99
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Discreteness, Hybrid Automata, and Biology