Monotone CNF Research Articles

Connecting Knowledge Compilation Classes Width Parameters

The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width bounds, e.g., they have bounded treewidth or pathwidth. In this work, we give a systematic picture of many circuit classes considered in knowledge compilation and show how they can be systematically connected to width measures, through upper and lower bounds. Our upper bounds show that bounded-treewidth circuits can be constructively converted to d-SDNNFs, in time linear in the circuit size and singly exponential in the treewidth; and that bounded-pathwidth circuits can similarly be converted to uOBDDs. We show matching lower bounds on the compilation of monotone DNF or CNF formulas to structured targets, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable): any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth, and the same holds for uOBDDs (resp., n-OBDDs) when considering pathwidth. Unlike most previous work, our bounds apply to any formula of this class, not just a well-chosen family. Hence, we show that pathwidth and treewidth respectively characterize the efficiency of compiling monotone DNFs to uOBDDs and d-SDNNFs with compilation being singly exponential in the corresponding width parameter. We also show that our lower bounds on CNFs extend to unstructured compilation targets, with an exponential lower bound in the treewidth (resp., pathwidth) when compiling monotone CNFs of constant arity and degree to DNNFs (resp., nFBDDs).

Recognizing Read-Once Functions from Depth-Three Formulas

Consider the following decision problem: for a given monotone Boolean function $f$ decide, whether $f$ is read-once. For this problem, it is essential how the input function $f$ is represented. Our contribution consists of the following two results. We show that we can test in polynomial-time whether a given expression $C\lor D$ computes a read-once function, provided that $C$ is a read-once monotone CNF and $D$ is a read-once monotone DNF and all the variables of $C$ occur also in $D$ (recall that due to Gurvich, the problem is coNP-complete when $C$ is read-2). The second result states that this is a coNP-complete problem to decide whether the expression $\Psi\land D_n$ computes a read-once function, where $D_n$ is as above and $\Psi$ is the $\bigwedge-\bigvee-\bigwedge$ depth-3 read-once monotone Boolean formula (so that the entire expression $\Psi\land D_n$ is depth-3 read-2). This result improves the result of \cite{elbassioni2011readability} in the depth and the result of \cite{gurvich2010it} in the readability of the input formula.

New Results on Monotone Dualization and Generating Hypergraph Transversals

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph) whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time, respectively, output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log2n /\log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.

Efficient Read-Restricted Monotone CNF/DNF Dualization by Learning with Membership Queries

We consider exact learning monotone CNF formulas in which each variable appears at most some constant k times ("read-k" monotone CNF). Let f : {0,1}n → {0,1} be expressible as a read-k monotone CNF formula for some natural number k. We give an incremental output polynomial time algorithm for exact learning both the read-k CNF and (not necessarily read restricted) DNF descriptions of f. The algorithm's only method of obtaining information about f is through membership queries, i.e., by inquiring about the value f(x) for points x ∈ {0,1}n. The algorithm yields an incremental polynomial output time solution to the (read-k) monotone CNF/DNF dualization problem. The unrestricted versions remain open problems of importance.