Commit e9e7a039 authored by SOLIMAN Sylvain's avatar SOLIMAN Sylvain
Browse files

M7 again

parent c5a71dc0
......@@ -70,7 +70,9 @@
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"## 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?"
]
},
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"## 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 @@
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"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 @@
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"## 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",
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{
"cell_type": "code",
"execution_count": null,
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{
"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 @@
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"## Question 13) Genericity?\n",
"\n",
"How generic is the method we use here for our continuous solver?"
]
},
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......
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