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# M7 Chemical SAT Solver

* Deciding the satisfiability of a Boolean formula in conjunctive normal form is NP-complete

* How can we program a chemical SAT solver ?

F. Fages, S. Soliman, Apr. 2020

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# "Generate and Test" Algorithm with Stochastic CRN

* A stochastic CRN can be used to enumerate random Boolean values for a variable

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parameter(k=1).

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MA(k) for _/a => a. % reactions with inhibitors cannot fire in presence of the inhibitor, here a

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MA(k) for a => _.

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option(method:spn, stochastic_conversion:1).

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numerical_simulation.

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## Question 1) Write a random generator of Boolean values for a vector of 3 variables a, b, c

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## Question 2) Write a CRN to find values satisfying the formula (x⋁¬y)⋀(y⋁¬z)⋀(z⋁¬x)

* Use an event to stop the simulation when the formula is satisfied

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# Guided Search Algorithm with Stochastic CRN

## Question 3) Improve your CRN to decrease the probability of moves of the variables that belong to satisfied clauses

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## Question ) Any idea to decide unsatisfiability ? statistical test question ?

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## Question 4) Determine the phase transition threshold in 3-SAT

* the density of a SAT instance is the ratio of the number of clauses divided by the number of variables

* a phase transition phenomenon is an asymptotic result showing the existence of a density threshold

* under the threshold the instances are almost surely satisfiable

* above the threshold the instances are almost surely unsatisfiable

* the hard instances are around the density threshold

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## Question ) Evaluation on 2-SAT and Horn-SAT ?

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# Guided Search Algorithm by Continuous CRN

We can see the SAT solving problem as a (continous) global optimization problem and try to solve it by gradient descent.

The idea is to find an energy function $E$ that is minimal when all clauses are satisfied, and then to simply enforce that $$\frac{dx}{dt} = -\frac{\partial E}{\partial x}$$

For a SAT problem involving variables $x_i\in [0, 1], 1\leq i\leq N$ and clauses $C_j, 1\leq j\leq M$ (with $C_{ji} = 1$ if $x_i$ appears positively in $C_j$, $C_{ji} = -1$ if $x_i$ appears negatively, and $0$ otherwise), we will define our energy function as a sum of squares of sub-energies for each clause.

$$E = \sum_{1\leq j\leq M}K_j^2$$

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## Question 5) Write $K_j$

Define (formally) $K_j$ as a function of the $C_{ji}$ and of the $x_i$, such that $K_j = 0$ iff clause $j$ is satisfied, and $K_j = 2^N$ if all $N$ variables appear in clause $j$ and are currently at the _wrong_ value.

One might want to define $s_i\in[-1, 1]$ as a function of $x_i\in[0, 1]$ and then $K_j$ as a function of the $s_i$ for ease of writing.

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## Question 6) Obtain $dx_i/dt$

Obtain the formal expression for $\displaystyle\frac{dx}{dt}$ (if you have used $s_i$ just note that $\frac{\partial E}{\partial x}=\frac{\partial E}{\partial x}\frac{\partial s}{\partial x}$).

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## Question 7) Implement on the above example (x⋁¬y)⋀(y⋁¬z)⋀(z⋁¬x)

Using the commands:

`new_ode_system`, `init` (to set $x_1, x_2$ and $x_3$ initial state to 0.5), `ode_parameter` (to set the $c_ji$ corresponding to our 3 clauses), `ode_function` (for the $k_j$ and $s_i$) and `add_ode` (to add the above $dx_i/dt$)

Run a numerical simulation and plot the $x_i$ to see the result.

Experiment with different initial states, and variants of the problem (changing only the $c_{ji}$), what do you observe? [please leave all results and remarks *in* the notebook!]

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## Question 8) Improving the search

To avoid getting stuck in some local minima, we can add *Lagrange multipliers* $a_j, 1\leq j\leq M$ so that the energy becomes $$E = \sum_{1\leq j\leq M}a_j K_j^2$$

These are new variables that will have an exponential increase proportional to $K_j$.

Add the 3 new variables and their ODEs with `add_ode`, change also the existing model accordingly.

Do you notice any difference? What can you say about the steadiness and stability of the initial state with all $x_i$ equal to 0.5?

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## Question 9) Complexity

The authors of the paper observe on random SAT instances that the $x_i$ trajectories have a polynomial length, however they do **not** conclude that the algorithm is polynomial. What have you observed that might not remain polynomial?

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## Question 10) Biochemical interpretation

What variables are always positive? What do you think about the associated chemical reaction network? [use biocham commands to obtain it!]