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The goal is to solve SAT problems with weak supervision: In that case a model is trained only to predict ***the satisfiability*** of a formula in conjunctive normal form. As a byproduct, when the formula is satisfiable, an actual satisfying assignment can be worked out by clustering the network's activations in most cases. * **Pros (+):** Weak supervision, interesting structured architecture, seems to generalize nicely to harder problems by increasing the number message passing iterations. * **Cons ():** Limited practical applicability since it is outperfomed by classical SAT solvers.  # NeuroSAT ## Inputs We consider Boolean logic formulas in their ***conjunctive normal form*** (CNF), i.e. each input formula is represented as a conjunction ($\land$) of **clauses**, which are themselves disjunctions ($\lor$) of litterals (positive or negative instances of variables). The goal is to learn a classifier to predict whether such a formula is satisfiable. A first problem is how to encode the input formula in such a way that it preserves the CNF invariances (invariance to negating a litteral in all clauses, invariance to permutations in $\lor$ and $\land$ etc.). The authors use an ***undirected graph representation*** where: * $\mathcal V$: vertices are the litterals (positive and negative form of variables, denoted as $x$ and $\bar x$) and the clauses occuring in the input formula * $\mathcal E$: Edges are added to connect (i) the litterals with clauses they appear in and (ii) each litteral to its negative counterpart. The graph relations are encoded as an ***adjacency matrix***, $A$, with as many rows as there are litterals and as many columns as there are clauses. In particular, this structure does not constrain the vertices ordering, and does not make any preferential treatment between positive or negative litterals. However it still has some caveats, which can be avoided by preprocessing the formula. For instance when there are disconnected components in the graph, the averaging decision rule (see next paragraph) can lead to false positives. ## Messagepassing model In a highlevel view, the model keeps track of an embedding for each vertex (litterals, $L^t$ and clauses, $C^t$), updated via ***messagepassing on the graph***, and combined via a Multi Layer perceptrion (MLP) to output the model prediction of the formula's satisfiability. The model updates are as follow: $$ \begin{align} C^t, h_C^t &= \texttt{LSTM}_\texttt{C}(h_C^{t  1}, A^T \texttt{MLP}_{\texttt{L}}(L^{t  1}) )\ \ \ \ \ \ \ \ \ \ \ (1)\\ L^t, h_L^t &= \texttt{LSTM}_\texttt{L}(h_L^{t  1}, \overline{L^{t  1}}, A\ \texttt{MLP}_{\texttt{C}}(C^{t }) )\ \ \ \ \ \ (2) \end{align} $$ where $h$ designates a hidden context vector for the LSTMs. The operator $L \mapsto \bar{L}$ returns $\overline{L}$, the embedding matrix $L$ where the row of each litteral is swapped with the one corresponding to the litteral's negation. In other words, in **(1)** each clause embedding is updated based on the litteral that composes it, while in **(2)** each litteral embedding is updated based on the clauses it appears in and its negated counterpart. After $T$ iterations of this messagepassing scheme, the model computes a ***logit for the satisfiability classification problem***, which is trained via sigmoid crossentropy: $$ \begin{align} L^t_{\mbox{vote}} &= \texttt{MLP}_{\texttt{vote}}(L^t)\\ y^t &= \mbox{mean}(L^t_{\mbox{vote}}) \end{align} $$  # Training and Inference ## Training Set The training set is built such that for any satisfiable training formula $S$, it also includes an unsatisfiable counterpart $S'$ which differs from $S$ ***only by negating one litteral in one clause***. These carefully curated samples should constrain the model to pick up substantial characteristics of the formula. In practice, the model is trained on formulas containing up to ***40 variables***, and on average ***200 clauses***. At this size, the SAT problem can still be solved by stateoftheart solvers (yielding the supervision) but are large enough they prove challenging for Machine Learning models. ## Inferring the SAT assignment When a formula is satisfiable, one often also wants to know a ***valuation*** (variable assignment) that satisfies it. Recall that $L^t_{\mbox{vote}}$ encodes a "vote" for every litteral and its negative counterpart. Qualitative experiments show that thoses scores cannot be directly used for inferring the variable assignment, however they do induce a nice clustering of the variables (once the message passing has converged). Hence an assignment can be found as follows: * (1) Reshape $L^T_{\mbox{vote}}$ to size $(n, 2)$ where $n$ is the number of litterals. * (2) Cluster the litterals into two clusters with centers $\Delta_1$ and $\Delta_2$ using the following criterion: \begin{align} \x_i  \Delta_1\^2 + \\overline{x_i}  \Delta_2\^2 \leq \x_i  \Delta_2\^2 + \\overline{x_i}  \Delta_1\^2 \end{align} * (3) Try the two resulting assignments (set $\Delta_1$ to true and $\Delta_2$ to false, or viceversa) and choose the one that yields satisfiability if any. In practice, this method retrieves a satistifiability assignment for over 70% of the satisfiable test formulas.  # Experiments In practice, the ***NeuroSAT*** model is trained with embeddings of dimension 128 and 26 message passing iterations using standard MLPs: 3 layers followed by ReLU activations. The final model obtains 85% accuracy in predicting a formula's satisfiability on the test set. It also can generalize to ***larger problems***, requiring to increase the number of message passing iterations, although the classification performance decreases as the problem size grows (e.g. 25% for 200 variables). Interestingly, the model also generalizes well to other classes of problems that were first ***reduced to SAT***, although they have different structure than the random formulas generated for training, which seems to show that the model does learn some general structural characteristics of Boolean formulas.
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