Boolean Formula In CNF Research Articles

Learning Horn envelopes via queries from language models

We present an approach for systematically probing a trained neural network to extract a symbolic abstraction of it, represented as a Boolean formula. We formulate this task within Angluin's exact learning framework, where a learner attempts to extract information from an oracle (in our work, the neural network) by posing membership and equivalence queries. We adapt Angluin's algorithm for Horn formula to the case where the examples are labelled w.r.t. an arbitrary Boolean formula in CNF (rather than a Horn formula). In this setting, the goal is to learn the smallest representation of all the Horn clauses implied by a Boolean formula—called its Horn envelope—which in our case correspond to the rules obeyed by the network. Our algorithm terminates in exponential time in the worst case and in polynomial time if the target Boolean formula can be closely approximated by its envelope. We also show that extracting Horn envelopes in polynomial time is as hard as learning CNFs in polynomial time. To showcase the applicability of the approach, we perform experiments on BERT based language models and extract Horn envelopes that expose occupation-based gender biases.

New bounds for the Moser‐Tardos distribution

The Lovász local lemma (LLL) is a probabilistic tool to generate combinatorial structures with good “local” properties. The “LLL‐distribution” further shows that these structures have good global properties in expectation. The seminal algorithm of Moser and Tardos turned the simplest, variable‐based form of the LLL into an efficient algorithm; this has since been extended to other probability spaces including random permutations. One can similarly define an “MT‐distribution” for these algorithms, that is, the distribution of the configuration they produce. We show new bounds on the MT‐distribution in the variable and permutation settings which are significantly stronger than those known to hold for the LLL‐distribution. As some example illustrations, we show a nearly tight bound on the minimum implicate size of a CNF Boolean formula, and we obtain improved bounds on weighted Latin transversals and partial Latin transversals.

Exact location of the phase transition for random (1,2)-QSAT

The QSAT problem is the quantified version of the SAT problem. We show the existence of a threshold effect for the phase transition associated with the satisfiability of random quantified boolean CNF formulas of the form ∀X∃Yϕ(X,Y), where X has m variables, Y has n variables and each clause in ϕ has one literal from X and two from Y. For such formulas, we show that the threshold phenomenon is controlled by the ratio between the number of clauses and the number n of existential variables. Then we give the exact location of the associated critical ratio c∗: it is a decreasing function of α, where α is the limiting value of m/ log (n) when n tends to infinity. Thus we give a precise location of the phase transition associated with a coNP-complete problem.

Quantifier elimination by dependency sequents

We consider the problem of existential quantifier elimination for Boolean CNF formulas. We present a new method for solving this problem called Derivation of Dependency-Sequents (DDS). A Dependency-sequent (D-sequent) is used to record that a set of quantified variables is redundant under a partial assignment. We introduce the join operation that produces new D-sequents from existing ones. We show that DDS is compositional, i.e., if our input formula is a conjunction of independent formulas, DDS automatically recognizes and exploits this information. We introduce an algorithm based on DDS and present experimental results demonstrating its potential.

The complexity of approximating conservative counting CSPs

We study the complexity of the approximate weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known over the Boolean domain; we extend this to arbitrary finite domains. We show that if F is “weakly log-modular”, then #CSP(F) is in FP. Otherwise, it is at least as difficult to approximate as #BIS (counting independent sets in bipartite graphs). #BIS is complete for the complexity class #RHΠ1, and believed to be intractable. We further sub-divide the #BIS-hard case: if F is “weakly log-supermodular”, #CSP(F) is as easy as a Boolean log-supermodular weighted #CSP; otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case: #CSP(F) is in FP, is #BIS-equivalent, or is equivalent to #SAT (approximately counting the satisfying assignments of Boolean CNF formulas). We also discuss algorithmic aspects of our classification.

The Complexity of Approximately Counting Tree Homomorphisms

We study two computational problems, parameterised by a fixed tree H . #HOMSTO( H ) is the problem of counting homomorphisms from an input graph G to H . #WHOMSTO( H ) is the problem of counting weighted homomorphisms to H , given an input graph G and a weight function for each vertex v of G . Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in # P . We give a complete trichotomy for #WHOMSTO( H ). If H is a star, then #WHOMSTO( H ) is in FP. If H is not a star but it does not contain a certain induced subgraph J 3, then #WHOMSTO( H ) is equivalent under approximation-preserving (AP) reductions to #BIS, the problem of counting independent sets in a bipartite graph. This problem is complete for the class #RHΠ1 under AP-reductions. Finally, if H contains an induced J 3 , then #WHOMSTO( H ) is equivalent under AP-reductions to #SAT, the problem of counting satisfying assignments to a CNF Boolean formula. Thus, #WHOMSTO( H ) is complete for #P under AP-reductions. The results are similar for #HOMSTO( H ) except that a rich structure emerges if H contains an induced J 3 . We show that there are trees H for which #HOMSTO( H ) is # SAT -equivalent (disproving a plausible conjecture of Kelk). However, it is still not known whether #HOMSTO( H ) is #SAT-hard for every tree H which contains an induced J 3. It turns out that there is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model . In particular, we show that for a family of graphs Jq , parameterised by a positive integer q , the problem #HOMSTO( Jq ) is AP-interreducible with the problem of approximating the partition function of the q -state Potts model. It was not previously known that the Potts model had a homomorphism-counting interpretation. We use this connection to obtain some additional upper bounds for the approximation complexity of #HOMSTO( Jq ).

The Complexity of Computing the Sign of the Tutte Polynomial

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually \#P-hard. As a trivial consequence, approximating the polynomial is also \#P-hard in this case. Thus, approximately evaluating the Tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a CNF Boolean formula. For most other points in the parameter space, we show that computing the sign of the polynomial is in FP, whereas approximating the polynomial can be done in polynomial time with an NP oracle. As a special case, we completely resolve the complexity of computing the sign of the chromatic polynomial---this is easily computable at q=2 and when $q\leq 32/27$, and is NP-hard to compute for all other values of the parameter q.

A new upper bound for Max-2-SAT: A graph-theoretic approach

In MaxSat, we ask for an assignment to the variables which satisfies the maximum number of clauses for a boolean formula in CNF. We present an algorithm yielding a run time upper bound of O ⁎ ( 2 K 6.265 ) for Max-2-Sat (each clause contains at most 2 literals), where K is the number of clauses. The run time has been achieved by using heuristic priorities on the choice of the variable on which we branch. The implementation of these heuristic priorities is rather simple, though they have a significant effect on the run time. The analysis uses a non-standard measure. 1

Stochastic local search for incremental SAT

The boolean satisfiability problem (SAT) is stated as follows: given a boolean formula in CNF, find a truth assignment that satisfies its clauses. In this paper, we present a general framework based on stochastic local search and the structure of the CNF formula for solving incremental SAT problems. Given a satisfiable boolean formula in CNF, incremental SAT consists of checking whether the satisfiability is preserved when new clauses are added to the current clause set and if not, look for a new assignment that satisfies the old clauses and the new ones. The results of the experimentation we have conducted demonstrate the efficiency of our method to deal with large randomly generated incremental SAT problems.

Exact Algorithms for MAX-SAT

The maximum satisfiability problem (MAX-SAT) is stated as follows: Given a boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2n “steps”? Here, n is the number of distinct variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that a popular algorithm based on branch-and-bound is bounded by O(2n) in time, where n is the maximum number of occurrences of any variable in the input.When the input formula is in 2-CNF, that is, each clause has at most two literals, MAX-SAT becomes MAX-2-SAT and the decision version of MAX-2-SAT is still NP-complete. The best bound of the known algorithms for MAX-2-SAT is O(m2m/5), where m is the number of clauses. We propose an efficient decision algorithm for MAX-2-SAT whose time complexity is bound by O(n2n). This result is substantially better than the previously known results. Experimental results also show that our algorithm outperforms any algorithm we know on MAX-2-SAT.

The Complexity of Selecting Maximal Solutions

Many important computational problems involve finding a maximal (with respect to set inclusion) solution in some combinatorial context. We study such maximality problems from the complexity point of view, and categorize their complexity precisely in terms of tight upper and lower bounds. Our results give characterizations of coNP, DP, ΠP2, FPNP||, FNP//OptP [log n] and FPΣP||2 in terms of subclasses of maximality problems. An important consequence of our results is that finding an X-minimal satisfying truth assignment for a given CNF boolean formula is complete for FNP//OptP[log n], solving an open question by Papadimitriou [Proceedings of the 32nd IEEE Symposium on the Foundations of Computer Science, 1991, pp. 163-169].