Output Stability and Semilinear Sets ... -...
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Output Stability and Semilinear Setsin
Chemical Reaction Networks and Deciders
Robert Brijder
Hasselt University, Belgium
DNA20, September 23, 2014
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Our question
Assuming a chemical reaction network computes something, hasthe chemical reaction network finished its computation in itscurrent state/configuration?
If yes, then read out its output and we are done.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Chemical Reaction Network (CRN)
Finite set Λ = {X1, . . . ,Xn} of species.
Reaction α over Λ:
X1 + 2X2 → 2X1 + X3.
Concisely, α = (r , p) with r , p ∈ NΛ, reactants and productsof α. Example:
r =
X1 1X2 2X3 0
p =
X1 2X2 0X3 1
.
Definition
A Chemical Reaction Network (CRN) is a tuple (Λ,R) with Λ afinite set of species and R a finite set of reactions over Λ.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Chemical Reaction Network (CRN) – Discrete
NΛ is the set of configurations. Model is discrete.
If c ∈ NΛ with c ≥ r (point-wise ≥), then we can apply α toc to obtain configuration c ′ = c − r + p, denoted c →α c ′.Example: for α : X1 + 2X2 → 2X1 + X3,
c =
X1 3X2 2X3 1
→α c ′ =
X1 4X2 0X3 2
.
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Reachability and Related models
For configurations c and c ′, if c →∗ c ′, then c ′ is reachablefrom c .
c
c ′
α1 α2
α1
Reachability questions can be equivalently stated in terms ofPetri Nets and Vector Addition Systems (VASs).
We consider a CRN as an “acceptor” of input instead of a“generator” of output (as done in Petri Net and VAS theory).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Reachability and Related models
For configurations c and c ′, if c →∗ c ′, then c ′ is reachablefrom c .
c
c ′
α1 α2
α1
Reachability questions can be equivalently stated in terms ofPetri Nets and Vector Addition Systems (VASs).
We consider a CRN as an “acceptor” of input instead of a“generator” of output (as done in Petri Net and VAS theory).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Chemical Reaction Decider (CRD)
Definition (≈ Population Protocol)
A (leaderless) Chemical Reaction Decider (CRD) is a tupleD = (Λ,R,Σ,Υ) with (Λ,R) a CRN, Σ ⊆ Λ the set of inputspecies, Υ : Λ→ {0, 1}.
Υ−1(0) NO voters, Υ−1(1) YES voters
c has output 1 (or YES) if nonzero and c contains only YESvoters; similar for 0 (or NO); undefined otherwise.
c is initial configuration if nonzero and all species in c are inΣ (i.e., c |Λ\Σ = 0).
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Chemical Reaction Decider (CRD)
Definition (≈ Population Protocol)
A (leaderless) Chemical Reaction Decider (CRD) is a tupleD = (Λ,R,Σ,Υ) with (Λ,R) a CRN, Σ ⊆ Λ the set of inputspecies, Υ : Λ→ {0, 1}.
Υ−1(0) NO voters, Υ−1(1) YES voters
c has output 1 (or YES) if nonzero and c contains only YESvoters; similar for 0 (or NO); undefined otherwise.
c is initial configuration if nonzero and all species in c are inΣ (i.e., c |Λ\Σ = 0).
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Chemical Reaction Decider (CRD)
Definition (≈ Population Protocol)
A (leaderless) Chemical Reaction Decider (CRD) is a tupleD = (Λ,R,Σ,Υ) with (Λ,R) a CRN, Σ ⊆ Λ the set of inputspecies, Υ : Λ→ {0, 1}.
Υ−1(0) NO voters, Υ−1(1) YES voters
c has output 1 (or YES) if nonzero and c contains only YESvoters; similar for 0 (or NO); undefined otherwise.
c is initial configuration if nonzero and all species in c are inΣ (i.e., c |Λ\Σ = 0).
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Output stable
c is output stable if c has output 0 or 1 and for all c ′ withc →∗ c ′, c ′ has the same output as c . Thus c is in a “trap”.D is output stable if for any initial configuration c , thereexists a o ∈ {0, 1}, such that for all configurations c ′ withc →∗ c ′, we have c ′ →∗ c ′′ for some output stableconfiguration c ′′ with output o.Any output stable D defines a Boolean predicateϕ : NΣ \ {0} → {0, 1}.
initial conf.
= no output
= output 0 (NO)
= output 1 (YES)
= output stable 0 (NO)
= output stable 1 (YES)
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Output stable
c is output stable if c has output 0 or 1 and for all c ′ withc →∗ c ′, c ′ has the same output as c . Thus c is in a “trap”.D is output stable if for any initial configuration c , thereexists a o ∈ {0, 1}, such that for all configurations c ′ withc →∗ c ′, we have c ′ →∗ c ′′ for some output stableconfiguration c ′′ with output o.Any output stable D defines a Boolean predicateϕ : NΣ \ {0} → {0, 1}.
initial conf.
= no output
= output 0 (NO)
= output 1 (YES)
= output stable 0 (NO)
= output stable 1 (YES)
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Deciding output stability
Are we done yet? In other words, can the output change?
Can we decide if a given configuration is output stable for agiven CRD?
Answer: YES. Using reachability results for VASs:
Theorem (Karp and Miller, 1969)
For given CRN R and configurations x , y of R, it is decidablewhether or not x →∗ y ′ for some configuration y ′ ≥ y .
Theorem (Mayr, 1981)
For given CRN R and configurations x , y of R, it is decidablewhether or not x →∗ y .
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Deciding output stability
Are we done yet? In other words, can the output change?
Can we decide if a given configuration is output stable for agiven CRD?
Answer: YES. Using reachability results for VASs:
Theorem (Karp and Miller, 1969)
For given CRN R and configurations x , y of R, it is decidablewhether or not x →∗ y ′ for some configuration y ′ ≥ y .
Theorem (Mayr, 1981)
For given CRN R and configurations x , y of R, it is decidablewhether or not x →∗ y .
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Deciding output stability efficiently
Can we efficiently decide if a given configuration is outputstable for a given output stable CRD? Answer: unknown.
We consider output stable CRDs with only bimolecularreactions, i.e., of the form X1 + X2 → X3 + X4 (Xi ’s are notnecessarily distinct).
Naive approach: for given c , check the output of allconfigurations reachable from c . There are at most((|Λ|‖c‖
))=(|Λ|+‖c‖−1
‖c‖)
such configurations (non-negative
integer points on hyperplane).
We give a
O(|Λ|2 · |R| · n log|Λ|−12 (n) + m log|Λ|−
12 (n))
method for testing m configurations for output stability foroutput stable CRDs with only bimolecular reactions. Thevalue n depends only on the CRD.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Deciding output stability efficiently
Can we efficiently decide if a given configuration is outputstable for a given output stable CRD? Answer: unknown.
We consider output stable CRDs with only bimolecularreactions, i.e., of the form X1 + X2 → X3 + X4 (Xi ’s are notnecessarily distinct).
Naive approach: for given c , check the output of allconfigurations reachable from c . There are at most((|Λ|‖c‖
))=(|Λ|+‖c‖−1
‖c‖)
such configurations (non-negative
integer points on hyperplane).
We give a
O(|Λ|2 · |R| · n log|Λ|−12 (n) + m log|Λ|−
12 (n))
method for testing m configurations for output stability foroutput stable CRDs with only bimolecular reactions. Thevalue n depends only on the CRD.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Deciding output stability efficiently
Can we efficiently decide if a given configuration is outputstable for a given output stable CRD? Answer: unknown.
We consider output stable CRDs with only bimolecularreactions, i.e., of the form X1 + X2 → X3 + X4 (Xi ’s are notnecessarily distinct).
Naive approach: for given c , check the output of allconfigurations reachable from c . There are at most((|Λ|‖c‖
))=(|Λ|+‖c‖−1
‖c‖)
such configurations (non-negative
integer points on hyperplane).
We give a
O(|Λ|2 · |R| · n log|Λ|−12 (n) + m log|Λ|−
12 (n))
method for testing m configurations for output stability foroutput stable CRDs with only bimolecular reactions. Thevalue n depends only on the CRD.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Theorem (Angluin et al., PODC ’06)
Let D be an output stable CRD. Then the set U of (nonzero)output unstable configurations of D is closed upward under ≤.
In other words, for all c , c ′ ∈ NΛ with c ≤ c ′, if c ∈ U, thenc ′ ∈ U.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Theorem (Dickson’s lemma)
Let Λ be a finite set. Then for every S ⊆ NΛ, the set of minimalelements min(S) of S under ≤ is finite.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Let U be the set of (nonzero) output unstable configurationsof D.
The finite set min(U) uniquely determines U.
Observation: nonzero configuration c is output stable iffu 6≤ c for all u ∈ min(U).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Main result
Theorem (Main result)
For output-stable bimolecular CRDs, the set min(U) can be
computed in O(|Λ|2 · |R| · n log|Λ|−12 (n)) time, where n = |min(U)|.
Check if u 6≤ c for all u ∈ min(U) takes O(log|Λ|−12 (n)) time.
k-fold tree Tb(k) of [Willard, SICOMP 1985]. Balanced treefor orthogonal range querying.
Consequently,
O(|Λ|2 · |R| · n log|Λ|−12 (n) + m log|Λ|−
12 (n)).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Algorithm
First compute vectors M2 of min(U) of size 2:
1 all configurations c containing one YES voter and one NOvoter,
2 all configurations r containing a YES voter and r →α p with pa NO voter (or vice versa) and α = (r , p), and
3 all configurations c such that c →∗ r with r as in Case 2(iterate backwards).
Compute vectors Mk of min(U) of size k > 2:
1 if c → c ′ + B for some c ′ ∈ min(U) and c incomparable withall of Mj (j < k), then c ∈ Mk .
2 for all configurations c ′ such that c ′ →∗ c with c as in Case 1and c ′ incomparable with all of Mj (j ≤ k), then c ′ ∈ Mk
(iterate backwards).
Iterate over k until Mk = ∅.
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Open problems
Generalization to arbitrary output-stable CRDs (not onlybimolecular)?
Is it even decidable for a given output-stable CRD and a finiteset of configurations C , whether or not C = min(U).
Estimates on n = |min(U)|? It seems there are a lot of“small” configurations in the antichain min(U) that restrict|min(U)|.Is |min(U)| a useful measure of “complexity” of the CRD?
What does, say, |min(U)| = 2 mean for the underlying CRD?
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Thanks!
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Example: modulo arithmetic
Compute: x ≡ y mod m with m ≥ 2.
Input: x = #X and y = #Y .
Solution: output-stable CRD with Λ = {X ,Y ,T ,F},Σ = {X ,Y }, T is YES voter and the other species NO voters,and reactions:
mX → T , mY → T (1)
X + Y → T (2)
Y + T → Y + F , X + T → X + F (3)
T + F → T (4)
Proof: if x ≡ y mod m, then eventually all X and Y areconsumed by (1) and (2). The last reaction introduces a Twhich eats all F ’s by (4).If x 6≡ y mod m, then eventually only X ’s or only Y ’sremain. These eat all T ’s by (3).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Example: modulo arithmetic
Compute: x ≡ y mod m with m ≥ 2.
Input: x = #X and y = #Y .
Solution: output-stable CRD with Λ = {X ,Y ,T ,F},Σ = {X ,Y }, T is YES voter and the other species NO voters,and reactions:
mX → T , mY → T (1)
X + Y → T (2)
Y + T → Y + F , X + T → X + F (3)
T + F → T (4)
Proof: if x ≡ y mod m, then eventually all X and Y areconsumed by (1) and (2). The last reaction introduces a Twhich eats all F ’s by (4).If x 6≡ y mod m, then eventually only X ’s or only Y ’sremain. These eat all T ’s by (3).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Example: modulo arithmetic
Compute: x ≡ y mod m with m ≥ 2.
Input: x = #X and y = #Y .
Solution: output-stable CRD with Λ = {X ,Y ,T ,F},Σ = {X ,Y }, T is YES voter and the other species NO voters,and reactions:
mX → T , mY → T (1)
X + Y → T (2)
Y + T → Y + F , X + T → X + F (3)
T + F → T (4)
Proof: if x ≡ y mod m, then eventually all X and Y areconsumed by (1) and (2). The last reaction introduces a Twhich eats all F ’s by (4).If x 6≡ y mod m, then eventually only X ’s or only Y ’sremain. These eat all T ’s by (3).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Example: modulo arithmetic
Compute: x ≡ y mod m with m ≥ 2.
Input: x = #X and y = #Y .
Solution: output-stable CRD with Λ = {X ,Y ,T ,F},Σ = {X ,Y }, T is YES voter and the other species NO voters,and reactions:
mX → T , mY → T (1)
X + Y → T (2)
Y + T → Y + F , X + T → X + F (3)
T + F → T (4)
Proof: if x ≡ y mod m, then eventually all X and Y areconsumed by (1) and (2). The last reaction introduces a Twhich eats all F ’s by (4).If x 6≡ y mod m, then eventually only X ’s or only Y ’sremain. These eat all T ’s by (3).
Robert Brijder Output Stability and Semilinear Sets in CRNs
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Complexity
Dominant for complexity: finite set M ⊆ NΛ and c ∈ NΛ,check if u 6≤ c for all u ∈ M.
Special case of orthogonal range querying: worst caseO(logk−1/2(|M|)) vector comparisons needed with k = |Λ|.Store vectors of M in k-fold tree Tb(k) of [Willard, SICOMP1985]. Balanced tree.
Adding a vector also takes O(logk−1/2(|M|)) time; notdominant.
Robert Brijder Output Stability and Semilinear Sets in CRNs