 ### M7 again

parent c5a71dc0
 ... ... @@ -70,7 +70,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 1) Write a random generator of Boolean values for a vector of 3 variables a, b, c" "## Question 1) Write a random generator of Boolean values for a vector of 3 variables a, b, c\n", "\n", "Write a stochastic CRN randomly sampling the Boolean values of a, b and c." ] }, { ... ... @@ -156,12 +158,37 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 4) Determine the phase transition threshold in 3-SAT\n", "## Question 4) Any idea to decide unsatisfiability?" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Question 5) Determine the phase transition threshold in 3-SAT\n", "\n", "As seen in the class\n", "* the density of a SAT instance is the ratio of the number of clauses divided by the number of variables\n", "* a phase transition phenomenon is an asymptotic result showing the existence of a density threshold\n", "* under the threshold the instances are almost surely satisfiable\n", "* above the threshold the instances are almost surely unsatisfiable\n", "* the hard instances are around the density threshold " "* the hard instances are around the density threshold \n", "\n", "Can we observe that phenomenon with our CRN? Why?" ] }, { ... ... @@ -205,7 +232,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 5) Write $K_j$\n", "## Question 6) Write $K_j$\n", "\n", "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.\n", "\n", ... ... @@ -230,7 +257,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 6) Obtain $dx_i/dt$\n", "## Question 7) Obtain $dx_i/dt$\n", "\n", "Obtain the formal expression for $\\displaystyle\\frac{dx}{dt}$ (if you have used $s_i$ just note that $\\frac{\\partial E}{\\partial x}=\\frac{\\partial E}{\\partial x}\\frac{\\partial s}{\\partial x}$)." ] ... ... @@ -253,7 +280,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 7) Implement on the above example (x⋁¬y)⋀(y⋁¬z)⋀(z⋁¬x)\n", "## Question 8) Implement on the above example (x⋁¬y)⋀(y⋁¬z)⋀(z⋁¬x)\n", "\n", "Using the commands:\n", "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 $dx_i/dt$)\n", ... ... @@ -287,7 +314,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 8) Improving the search\n", "## Question 9) Improving the search\n", "\n", "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 $$E = \\sum_{1\\leq j\\leq M}a_j K_j^2$$\n", "These are new variables that will have an exponential increase proportional to $K_j$.\n", ... ... @@ -314,7 +341,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 9) Complexity\n", "## Question 10) Complexity\n", "\n", "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?" ] ... ... @@ -337,7 +364,31 @@ "cell_type": "markdown", "metadata": {}, "source": [ "## Question 10) Biochemical interpretation\n", "## Question 11) Phase transition\n", "\n", "Do you think one can observe the phase-transition phenomenon?\n", "In other words are _hard instances_ the ones with an average clause density? why?" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Question 12) Biochemical interpretation\n", "\n", "What variables are always positive? What do you think about the associated chemical reaction network? [use biocham commands to obtain it!]" ] ... ... @@ -356,6 +407,36 @@ "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Question 13) Genericity?\n", "\n", "How generic is the method we use here for our continuous solver?" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, ... ...
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