Recently, a novel, MaxSAT-based method for error correction in quantum computing has been proposed that requires both incremental MaxSAT solving capabilities and support for XOR constraints, but no dedicated MaxSAT solver fulfilling these criteria existed yet. We alleviate that and introduce IGMaxHS, which is based on the existing solvers iMaxHS and GaussMaxHS, but poses fewer restrictions on the XOR constraints than GaussMaxHS. IGMaxHS is fuzz tested with xwcnfuzz, an extension of wcnfuzz that can directly output XOR constraints. As a result, IGMaxHS is the only solver that reported neither incorrect unsatisfiability verdicts nor invalid models nor incoherent cost model combinations in a final fuzz testing comparison of all three solvers with 10000 instances. We detail the steps required for implementing Gaussian elimination on XOR constraints in CDCL SAT solvers, and extend the recently proposed re-entrant incremental MaxSAT solver application program interface to allow for incremental addition of XOR constraints. Finally, we show that IGMaxHS is capable of decoding quantum color codes through simulation with the Munich Quantum Toolkit.

Approximate model counting is the task of approximating the number of solutions to an input Boolean formula. The state-of-the-art approximate model counter for formulas in conjunctive normal form (CNF), ApproxMC, provides a scalable means of obtaining model counts with probably approximately correct (PAC)-style guarantees. Nevertheless, the validity of ApproxMC's approximation relies on a careful theoretical analysis of its randomized algorithm and the correctness of its highly optimized implementation, especially the latter's stateful interactions with an incremental CNF satisfiability solver capable of natively handling parity (XOR) constraints. We present the first certification framework for approximate model counting with formally verified guarantees on the quality of its output approximation. Our approach combines: (i) a static, once-off, formal proof of the algorithm's PAC guarantee in the Isabelle/HOL proof assistant; and (ii) dynamic, per-run, verification of ApproxMC's calls to an external CNF-XOR solver using proof certificates. We detail our general approach to establish a rigorous connection between these two parts of the verification, including our blueprint for turning the formalized, randomized algorithm into a verified proof checker, and our design of proof certificates for both ApproxMC and its internal CNF-XOR solving steps. Experimentally, we show that certificate generation adds little overhead to an approximate counter implementation, and that our certificate checker is able to fully certify $84.7\%$ of instances with generated certificates when given the same time and memory limits as the counter.

Satisfiability Modulo Counting (SMC) encompasses problems that require both symbolic decision-making and statistical reasoning. Its general formulation captures many real-world problems at the intersection of symbolic and statistical Artificial Intelligence. SMC searches for policy interventions to control probabilistic outcomes. Solving SMC is challenging because of its highly intractable nature($\text{NP}^{\text{PP}}$-complete), incorporating statistical inference and symbolic reasoning. Previous research on SMC solving lacks provable guarantees and/or suffers from sub-optimal empirical performance, especially when combinatorial constraints are present. We propose XOR-SMC, a polynomial algorithm with access to NP-oracles, to solve highly intractable SMC problems with constant approximation guarantees. XOR-SMC transforms the highly intractable SMC into satisfiability problems, by replacing the model counting in SMC with SAT formulae subject to randomized XOR constraints. Experiments on solving important SMC problems in AI for social good demonstrate that XOR-SMC finds solutions close to the true optimum, outperforming several baselines which struggle to find good approximations for the intractable model counting in SMC.

We propose a new technique for sampling the solutions of combinatorial problems in a near-uniform manner. We focus on problems specified as a Boolean formula, i.e., on SAT instances. Sampling for SAT problems has been shown to have interesting connections with probabilistic reasoning, making practical sampling algorithms for SAT highly desirable. The best current approaches are based on Markov Chain Monte Carlo methods, which have some practical limitations. Our approach exploits combinatorial properties of random parity (X O R) constraints to prune away solutions near-uniformly.

The runtime performance of modern SAT solvers is deeply connected to the phase transition behavior of CNF formulas. While CNF solving has witnessed significant runtime improvement over the past two decades, the same does not hold for several other classes such as the conjunction of cardinality and XOR constraints, denoted as CARD-XOR formulas. The problem of determining the satisfiability of CARD-XOR formulas is a fundamental problem with a wide variety of applications ranging from discrete integration in the field of artificial intelligence to maximum likelihood decoding in coding theory. The runtime behavior of random CARD-XOR formulas is unexplored in prior work. In this paper, we present the first rigorous empirical study to characterize the runtime behavior of 1-CARD-XOR formulas. We show empirical evidence of a surprising phase-transition that follows a non-linear tradeoff between CARD and XOR constraints.

One form of characterizing the expressiveness of a piecewise linear neural network is by the number of linear regions, or pieces, of the function modeled. We have observed substantial progress in this topic through lower and upper bounds on the maximum number of linear regions and a counting procedure. However, these bounds only account for the dimensions of the network and the exact counting may take a prohibitive amount of time, therefore making it infeasible to benchmark the expressiveness of networks. In addition, we present a tighter upper bound that leverages network coefficients. We test both on trained networks. The algorithm for probabilistic lower bounds is several orders of magnitude faster than exact counting and the values reach similar orders of magnitude, hence making our approach a viable method to compare the expressiveness of such networks. The refined upper bound is particularly stronger on networks with narrow layers. Neural networks with piecewise linear activations have become increasingly more common along the past decade, in particular since Nair & Hinton (2010) and Glorot et al. (2011). The simplest and most commonly used among such forms of activation is the Rectifier Linear Unit (ReLU), which outputs the maximum between 0 and its input argument (Hahnloser et al., 2000; LeCun et al., 2015). In the functions modeled by these networks, we can associate each part of the domain in which the network corresponds to an affine function with a particular set of units having positive outputs.

Pattern sampling has been proposed as a potential solution to the infamous pattern explosion. Instead of enumerating all patterns that satisfy the constraints, individual patterns are sampled proportional to a given quality measure. Several sampling algorithms have been proposed, but each of them has its limitations when it comes to 1) flexibility in terms of quality measures and constraints that can be used, and/or 2) guarantees with respect to sampling accuracy. We therefore present Flexics, the first flexible pattern sampler that supports a broad class of quality measures and constraints, while providing strong guarantees regarding sampling accuracy. To achieve this, we leverage the perspective on pattern mining as a constraint satisfaction problem and build upon the latest advances in sampling solutions in SAT as well as existing pattern mining algorithms. Furthermore, the proposed algorithm is applicable to a variety of pattern languages, which allows us to introduce and tackle the novel task of sampling sets of patterns. We introduce and empirically evaluate two variants of Flexics: 1) a generic variant that addresses the well-known itemset sampling task and the novel pattern set sampling task as well as a wide range of expressive constraints within these tasks, and 2) a specialized variant that exploits existing frequent itemset techniques to achieve substantial speed-ups. Experiments show that Flexics is both accurate and efficient, making it a useful tool for pattern-based data exploration.

Maintaining landscape connectivity is increasingly important in wildlife conservation, especially for species experiencing the effects of habitat loss and fragmentation. We propose a novel approach to dynamically optimize landscape connectivity. Our approach is based on a mixed integer program formulation, embedding a spatial capture-recapture model that estimates the density, space usage, and landscape connectivity for a given species. Our method takes into account the fact that local animal density and connectivity change dynamically and non-linearly with different habitat protection plans. In order to scale up our encoding, we propose a sampling scheme via random partitioning of the search space using parity functions. We show that our method scales to real-world size problems and dramatically outperforms the solution quality of an expectation maximization approach and a sample average approximation approach.

Constrained sampling and counting are two fundamental problems in artificial intelligence with a diverse range of applications, spanning probabilistic reasoning and planning to constrained-random verification. While the theory of these problems was thoroughly investigated in the 1980s, prior work either did not scale to industrial size instances or gave up correctness guarantees to achieve scalability. Recently, we proposed a novel approach that combines universal hashing and SAT solving and scales to formulas with hundreds of thousands of variables without giving up correctness guarantees. This paper provides an overview of the key ingredients of the approach and discusses challenges that need to be overcome to handle larger real-world instances.

We propose a new technique for sampling the solutions of combinatorial problems in a near-uniform manner. We focus on problems specified as a Boolean formula, i.e., on SAT instances. Sampling for SAT problems has been shown to have interesting connections with probabilistic reasoning, making practical sampling algorithms for SAT highly desirable. The best current approaches are based on Markov Chain Monte Carlo methods, which have some practical limitations.