2021 
Approximate Model Counting
Supratik Chakraborty, Kuldeep S. Meel, and Moshe Y. Vardi Chapter 26 of Handbook of Satisfiability, 2nd Edition 
2017 
Constrained Counting and Sampling: Bridging the Gap between Theory and Practice
PhD Thesis, Rice University, September 2017 Winner of 2018 Ralph Budd Award for Best Engineering Thesis Honorable mention, 2018 ACP Doctoral Dissertation Award 
2014 
Sampling Techniques for Boolean Satisfiability
Masters Thesis, Rice University, April 2014 Winner of 2014 VCLA (Vienna Center of Logic and Algorithms) Outstanding Masters Thesis Announcement 
2021 
LogicEnabled Verification and Explanation of ML Models
. Copresented with Alexey Ignatiev, Joao MarquesSilva, and Nina Narodytska International Joint Conference on Artificial Intelligence (IJCAI 2020, held in Jan 2021) 
2020 
Rigorous Verification and Explanation of ML Models
. Copresented with Alexey Ignatiev, Joao MarquesSilva, and Nina Narodytska AAAI Conference on Artificial Intelligence (AAAI 2020) 
2018 
Scaling Discrete Integration and Sampling: Foundations and Challenges
. Copresented with Supratik Chakraborty International Joint Conference on Artificial Intelligence (IJCAI 2018) 
2017 
Discrete Sampling and Integration for the AI Practitioner
Copresented with Supratik Chakraborty and Moshe Y. Vardi AAAI Conference on Artificial Intelligence (AAAI 2017) 
2016 
Discrete Sampling and Integration in High Dimensional Spaces
CoPresented with Supratik Chakraborty and Moshe Y. Vardi Conference on Uncertainity in Artificial Intelligence (UAI 2016) 
2021  
[c55] 
Scalable Quantitative Verification For Deep Neural Networks
Proceedings of International Conference on Software Engineering (ICSE), 2021.
Verifying security properties of deep neural networks (DNNs) is becoming increasingly important. This paper introduces a new quantitative verification framework for DNNs that can decide, with userspecified confidence, whether a given logical property {\psi} defined over the space of inputs of the given DNN holds for less than a userspecified threshold,{\theta}. We present new algorithms that are scalable to large realworld models as well as proven to be sound. Our approach requires only blackbox access to the models. Further, it certifies properties of both deterministic and nondeterministic DNNs. We implement our approach in a tool called PROVERO. We apply PROVERO to the problem of certifying adversarial robustness. In this context, PROVERO provides an attackagnostic measure of robustness for a given DNN and a test input. First, we find that this metric has a strong statistical correlation with perturbation bounds reported by 2 of the most prominent whitebox attack strategies today. Second, we show that PROVERO can quantitatively certify robustness with high confidence in cases where the stateoftheart qualitative verification tool (ERAN) fails to produce conclusive results. Thus, quantitative verification scales easily to large DNNs.

[c54] 
The Power of Literal Equivalence in Model Counting
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2021.
The past two decades have seen the significant improvements of the scalability of practical model counters, which have been quite influential in many applications from artificial intelligence to formal verification. While most of exact counters fall into two categories, searchbased and compilationbased, Huang and Darwiche's remarkable observation ties these two categories: the trace of a searchbased exact model counter corresponds to a DecisionDNNF formula. Taking advantage of literal equivalences, this paper designs an efficient model counting technique such that its trace is a generalization of DecisionDNNF formula. We first propose a generalization of DecisionDNNF, called CCDD, to capture literal equivalences, then show that CCDD supports model counting in linear time, and finally design a model counter, called ExactMC, whose trace corresponds to CCDD. We perform an extensive experimental evaluation over a comprehensive set of benchmarks and conduct performance comparison of ExactMC visavis the state of the art counters, c2d, miniC2D, D4, ADDMC, and Ganak. Our empirical evaluation demonstrates ExactMC can solve 885 instances while the prior state of the art could solve only 843 instances, representing a significant improvement of 42 instances.

[c53] 
Symmetric Component Caching for Model Counting on Combinatorial Instances
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2021.
Given a propositional formula $\psi$, the model counting problem, also referred to as #SAT, seeks to compute the number of satisfying assignments (or models) of $\psi$. Modern searchbased model counting algorithms are built on conflictdriven clause learning, combined with the caching of certain subformulas (called components) encountered during the search process. Despite significant progress in these algorithms over the years, stateoftheart model counters often struggle to handle large but structured instances that typically arise in combinatorial settings. Motivated by the observation that these counters do not exploit the inherent symmetries exhibited in such instances, we revisit the component caching architecture employed in current counters and introduce a novel caching scheme that focuses on identifying symmetric components. We first prove the soundness of our approach, and then integrate it into the stateoftheart model counter GANAK. Our extensive experiments on hard combinatorial instances demonstrate that the resulting counter, SymGANAK, leads to improvements over GANAK both in terms of PAR2 score and the number of instances solved.

[c52] 
Counting Maximal Satisfiable Subsets
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2021.
Given an unsatisfiable set of constraints F, a maximal satisfiable subset (MSS) is a maximal subset of constraints C such that C is satisfiable. Over the past two decades, the steady improvement in runtime performance of algorithms for finding MSS has led to an increased adoption of MSSbased techniques in wide variety of domains. Motivated by the progress in finding an MSS, the past decade has witnessed a surge of interest in design of algorithmic techniques to enumerate all the MSSes, which has subsequently led to discovery of new applications utilizing enumeration of MSSes. The development of techniques for finding and enumeration of MSSes mirrors a similar phenomenon of finding and enumeration of SAT solutions in the early 2000s, which subsequently motivated design of algorithmic techniques for model counting. In a similar spirit, we undertake study to investigate the feasibility of MSS counting techniques. In particular, the focus point of our investigation is to answer whether one can design efficient MSS counting techniques that do not rely on explicit MSS enumeration. The primary contribution of this work is an affirmative answer to the above question. Our tool, CountMSS, uses a novel architecture of a wrapper W and a remainder R such that the desired MSS count can be expressed as W R. CountMSS relies on the advances in projected model counting to efficiently compute W and R. Our empirical evaluation demonstrates that CountMSS is able to scale to instances clearly beyond the reach of enumerationbased techniques.

[c51] 
Predicting Forest Fire Using Remote Sensing Data And Machine Learning
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2021.
Over the last few decades, deforestation and climate change have caused increasing number of forest fires. In SoutheastAsia, Indonesia has been the most affected country by tropical peatland forest fires. These fires have a significant impact on the climate resulting in extensive health, social and economic issues. Existing forest fire prediction systems, such as the Canadian Forest Fire Danger Rating System, are based on handcrafted features and require installation and maintenance of expensive instruments on the ground, which can be a challenge for developing countries such as Indonesia. We propose a novel, costeffective, machinelearning based approach that uses remote sensing data to predict forest fires in Indonesia.Our prediction model achieves more than 0.81 area under the receiver operator characteristic (ROC) curve, performing significantly better than the baseline approach which never exceeds 0.70 area under ROC curve on the same tasks. Our model's performance remained above 0.81 area under ROC curve even when evaluated with reduced data. The results support our claim that machine learning based approaches can lead to reliable and costeffective forest fire prediction systems.

[c50] 
Justicia: A Stochastic SAT Approach to Formally Verify Fairness
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2021.
As a technology ML is oblivious to societal good or bad, and thus, the field of fair machine learning has stepped up to propose multiple mathematical definitions, algorithms, and systems to ensure different notions of fairness in ML applications. Given the multitude of propositions, it has become imperative to formally verify the fairness metrics satisfied by different algorithms on different datasets. In this paper, we propose a stochastic satisfiability (SSAT) framework, Justicia, that formally verifies different fairness measures of supervised learning algorithms with respect to the underlying data distribution. We instantiate Justicia on multiple classification and bias mitigation algorithms, and datasets to verify different fairness metrics, such as disparate impact, statistical parity, and equalized odds. Justicia is scalable, accurate, and operates on nonBoolean and compound sensitive attributes unlike existing distributionbased verifiers, such as FairSquare and VeriFair. Being distributionbased by design, Justicia is more robust than the verifiers, such as AIF360, that operate on specific test samples. We also theoretically bound the finitesample error of the verified fairness measure.

2020  
[c49] 
Efficient Distance Approximation for Structured HighDimensional Distributions via Learning
Proceedings of Advances in Neural Information Processing Systems(NeurIPS), 2020.
We design efficient distance approximation algorithms for several classes of wellstudied structured highdimensional distributions. Specifically, we present algorithms for the following problems (where dTV is the total variation distance):  Given sample access to two Bayesian networks P1 and P2 over known directed acyclic graphs G1 and G2 having n nodes and bounded indegree, approximate dTV(P1, P2) to within additive error $\varepsilon$ using poly(n, 1/$\varepsilon$) samples and time.  Given sample access to two ferromagnetic Ising models P1 and P2 on n variables with bounded width, approximate dTV(P1 , P2 ) to within additive error $\varepsilon$ using poly(n, 1/$\varepsilon$) samples and time.  Given sample access to two ndimensional Gaussians P1 and P2, approximate dTV(P1, P2) to within additive error $\varepsilon$ using poly(n, 1/$\varepsilon$) samples and time.  Given access to observations from two causal models P and Q on n variables that are defined over known causal graphs, approximate dTV(Pa , Qa ) to within additive error $\varepsilon$ using poly(n, 1/$\varepsilon$) samples and time, where Pa and Qa are the interventional distributions obtained by the intervention do(A = a) on P and Q respectively for a particular variable A. The distance approximation algorithms immediately imply new tolerant closeness testers for the corresponding classes of distributions. Prior to our work, only nontolerant testers were known for both Bayes net distributions and Ising models, and no testers with quantitative guarantee were known for interventional distributions. To best of our knowledge, efficient distance approximation algorithms for Gaussian distributions were not present in the literature. Our algorithms are designed using a conceptually simple but general framework that is applicable to a variety of scenarios.

[c48] 
Taming Discrete Integration via the Boon of Dimensionality
Proceedings of Advances in Neural Information Processing Systems(NeurIPS), 2020.
Discrete integration is a fundamental problem in computer science that concerns the computation of discrete sums over exponentially large sets. Despite intense interest from researchers for over three decades, the design of scalable techniques for computing estimates with rigorous guarantees for discrete integration remains the holy grail. The key contribution of this work addresses this scalability challenge via an efficient reduction of discrete integration to model counting. The proposed reduction is achieved via a significant increase in the dimensionality that, contrary to conventional wisdom, leads to solving an instance of the relatively simpler problem of model counting. Building on the promising approach proposed by Chakraborty et al, our work overcomes the key weakness of their approach: a restriction to dyadic weights. We augment our proposed reduction, called DeWeight, with a state of the art efficient approximate model counter and perform detailed empirical analysis over benchmarks arising from neural network verification domains, an emerging application area of critical importance. DeWeight, to the best of our knowledge, is the first technique to compute estimates with provable guarantees for this class of benchmarks.

[c47] 
On Testing of Samplers
Proceedings of Advances in Neural Information Processing Systems(NeurIPS), 2020.
Given a set of items F and a weight function W, the problem of sampling seeks to sample an item proportional to its weight. Sampling is a fundamental problem in machine learning. The daunting computational complexity of sampling with formal guarantees leads designers to propose heuristicsbased techniques for which no rigorous theoretical analysis exists to quantify the quality of generated distributions. This poses a challenge in designing a testing methodology to test whether a sampler under test generates samples according to a given distribution. Only recently, Chakraborty and Meel (2019) designed the first scalable verifier, called Barbarik1, for samplers in the special case when the weight function W is constant, that is, when the sampler is supposed to sample uniformly from F . The techniques in Barbarik1, however, fail to handle general weight functions. The primary contribution of this paper is an affirmative answer to the above challenge: motivated by Barbarik1 but using different techniques and analysis, we design Barbarik2 an algorithm to test whether the distribution generated by a sampler is $\varepsilon$close or $\eta$far from any target distribution. In contrast to blackbox sampling techniques that require a number of samples proportional to F , Barbarik2 requires only $\mathcal{O}(titl(W,\varphi)2/\eta(\eta6\varepsilon)3)$ samples, where the tilt is the maximum ratio of weights of two satisfying assignments. Barbarik2 can handle any arbitrary weight function. We present a prototype implementation of Barbarik2 and use it to test three stateoftheart samplers.

[c46] 
Baital: An Adaptive Weighted Sampling Approach for Improved twise Coverage
Proceedings of ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering (ESEC/FSE), 2020.
The rise of highly configurable complex software and its widespread usage requires design of efficient testing methodology. Twise coverage is a leading metric to measure the quality of the testing suite and the underlying test generation engine. While uniform sampling based test generation is widely believed to be the state of the art approach to achieve twise coverage in presence of constraints on the set of configurations, uniform sampling fails to achieve high twise coverage in presence of complex constraints. In this work, we propose a novel approach Baital, based on adaptive weighted sampling using literal weighted functions, to generate test sets with high twise coverage. We demonstrate that our approach leads to significantly high twise coverage. The novel usage of literal weighted sampling leaves open several interesting directions, empirical as well as theoretical, for future research.

[c45] 
Phase Transition Behavior in Knowledge Compilation
Proceedings of International Conference on Constraint Programming (CP), 2020.
The success of SAT solvers has led to development of algorithmic paradigms for problems beyond NP such as knowledge compilation, which seeks to compile proposition theory into succinct and tractable representations. The study of phase transition behaviour in SAT has been subjected to intense theoretical and empirical investigations, and has led to deeper understanding and algorithmic improvements of modern SAT solvers. Motivated by the prior studies of phase transition in SAT, we seek to study the behaviour of size and compiletime behaviour for random kCNF formulas. The primary contribution of this paper is a rigorous empirical study and analysis of the size and runtime behavior for different knowledge compilation forms (and their corresponding compilation algorithms): dDNNFs, SDDs and OBDDs. We employ instances generated from random kCNF model with varying generation parameters to empirically reason about the expected and median behavior of size and compilationtime for these languages. A significant novel contribution of our work is the identification of a new control parameter, called solution density, defined as the ratio of the log of number of satisfying assignments to number of variables.

[c44] 
Sparse Hashing for Scalable Approximate Model Counting: Theory and Practice
Proceedings of Logic in Computer science (LICS), 2020.
Given a CNF formula F on n variables, the problem of model counting, also referred to as #SAT, is to compute the number of models or satisfying assignments of F. Model counting is a fundamental but hard problem in computer science with varied applications. Recent years have witnessed a surge of effort towards developing efficient algorithmic techniques that combine the classical 2universal hashing (from [Stockmeyer 1983]) with the remarkable progress in SAT solving over the past decade. These techniques augment the CNF formula F with random XOR constraints and invoke an NP oracle repeatedly on the resultant CNFXOR formulas. In practice, the NP oracle calls are replaced by calls to a SAT solver and it is observed that runtime performance of modern SAT solvers (based on conflictdriven clause learning) on CNFXOR formulas is adversely affected by the size of XOR constraints. However, the standard construction of 2universal hash functions chooses every variable with probability p =1/2 leading to XOR constraints of size n/2 in expectation. Consequently, the main challenge is to design sparse hash functions, where variables can be chosen with smaller probability and lead to smaller sized XOR constraints, which can then replace 2universal hash functions. In this paper, our goal is to address this challenge both from a theoretical and a practical perspective. First, we formalize a relaxation of universal hashing, called concentrated hashing, a notion implicit in prior works to design sparse hash functions. We then establish a novel and beautiful connection between concentration measures of these hash functions and isoperimetric inequalities on boolean hypercubes. This allows us to obtain tight bounds on variance as well as the dispersion index and show that p = O(log m /m ) suffices for the design of sparse hash functions from 2^n to 2^m belonging to concentrated hash family. Finally, we use sparse hash functions belonging to this concentrated hash family to develop new approximate counting algorithms. A comprehensive experimental evaluation of our algorithm on 1896 benchmarks with computational effort of over 20,000 computational hours demonstrates significant speedup compared to existing approaches. To the best of our knowledge, this work is the first study to demonstrate runtime improvement of approximate model counting algorithms through the usage of sparse hash functions, while still retaining strong theoretical guarantees (a la 2universal hash functions).

[c43] 
Designing New Phase Selection Heuristics
Proceedings of the International Conference on Theory and Applications of Satisfiability Testing (SAT), 2020.
CDCLbased SAT solvers have transformed the field of automated reasoning owing to their demonstrated efficiency at handling problems arising from diverse domains. The success of CDCL solvers is owed to the design of clever heuristics that enable the tight coupling of different components. One of the core components is phase selection, wherein the solver, during branching, decides the polarity of the branch to be explored for a given variable. Most of the stateoftheart CDCL SAT solvers employ phasesaving as a phase selection heuristic, which was proposed to address the potential inefficiencies arising from farbacktracking. In light of the emergence of chronological backtracking in CDCL solvers, we reexamine the efficiency of phase saving. Our empirical evaluation leads to a surprising conclusion: The usage of phase saving and random selection of polarity during chronological backtracking leads to indistinguishable runtime performance in terms of instances solved and PAR2 score. We introduce Decaying Polarity Score (DPS) to capture the trend of the polarities attained by the variable, and upon observing lack of performance improvement due to DPS, we turn to a more sophisticated heuristic seeking to capture the activity of literals and the trend of polarities: Literal State Independent Decaying Sum (LSIDS). We find the 2019 winning SAT solver, Maple_LCM_Dist_ChronoBTv3, augmented with LSIDS solves 6 more instances while achieving a reduction of over 125 seconds in PAR2 score, a significant improvement in the context of the SAT competition.

[c42] 
On the Sparsity of XORs in Approximate Model Counting
Proceedings of the International Conference on Theory and Applications of Satisfiability Testing (SAT), 2020.
Given a Boolean formula F, the problem of model counting, also referred to as #SAT, is to compute the number of solutions of F. The hashingbased techniques for approximate counting have emerged as a dominant approach, promising achievement of both scalability and rigorous theoretical guarantees. The standard construction of strongly 2universal hash functions employs dense XORs (i.e., involving half of the variables in expectation), which is widely known to cause degradation in the runtime performance of state of the art SAT solvers. Consequently, the past few years have witnessed an intense activity in the design of sparse XORs as hash functions. Such constructions have been proposed with beliefs to provide runtime performance improvement along with theoretical guarantees similar to that of dense XORs. The primary contribution of this paper is a rigorous theoretical and empirical analysis to understand the effect of the sparsity of XORs. In contradiction to prior beliefs of applicability of analysis for sparse hash functions to all the hashingbased techniques, we prove a contradictory result. We show that the bestknown bounds obtained for sparse XORs are still too weak to yield theoretical guarantees for a large class of hashing based techniques, including the state of the art approach ApproxMC3. We then turn to a rigorous empirical analysis of the performance benefits of sparse hash functions. To this end, we first design, to the best of our knowledge, the most efficient algorithm called SparseCount2 using sparse hash functions, which achieves at least up to two orders of magnitude performance improvement over its predecessor. Contradicting the current beliefs, we observe that SparseCount2 still falls short of ApproxMC3 in runtime performance despite the usage of dense XORs in ApproxMC3. In conclusion, our work showcases that the question of whether it is possible to use short XORs to achieve scalability while providing strong theoretical guarantees is still wide open.

[c41] 
Tinted, Detached, and Lazy CNFXOR solving and its Applications to Counting and Sampling
Proceedings of International Conference on ComputerAided Verification (CAV), 2020.
Given a Boolean formula, the problem of counting seeks to estimate the number of solutions of F while the problem of uniform sampling seeks to sample solutions uniformly at random. Counting and uniform sampling are fundamental problems in computer science with a wide range of applications ranging from constrained random simulation, probabilistic inference to network reliability and beyond. Despite intense theoretical and empirical investigations, development of scalable techniques for sampling and counting without sacrificing theoretical guarantees remains the holy grail. The past few years have witnessed the rise of hashingbased approaches that use XORbased hashing and employ SAT solvers to solve the resulting CNF formulas conjuncted with XOR constraints. Since over 99\% of the runtime of hashingbased techniques is spent inside the SAT queries, improving CNFXOR solvers has emerged as a key challenge. In this paper, we identify the key performance bottlenecks in the recently proposed BIRD architecture, and we focus on overcoming these bottlenecks by accelerating the XOR handling within the SAT solver and on improving the solver integration through a smarter use of (partial) solutions. We integrate BIRD2 with the state of the art approximate model counter, ApproxMC3, and the state of the art almostuniform model sampler UniGen2. Through an extensive evaluation over a large benchmark set of over 1896 instances, we observe that BIRD2 leads to consistent speed up for both counting and sampling, and in particular, we solve 77 and 51 more instances for counting and sampling respectively.

[c40] 
Manthan: A DataDriven Approach for Boolean Function Synthesis
Proceedings of International Conference on ComputerAided Verification (CAV), 2020.
Boolean functional synthesis is a fundamental problem in computer science with wideranging applications and has witnessed a surge of interest resulting in progressively improved techniques over the past decade. Despite intense algorithmic development, a large number of problems remain beyond the reach of the current state of the art techniques. Motivated by the progress in machine learning, we propose Manthan, a novel datadriven approach to Boolean functional synthesis. Manthan views functional synthesis as a classification problem, relying on advances in constrained sampling for data generation, and advances in automated reasoning for a novel proofguided refinement and provable verification. On an extensive and rigorous evaluation over 609 benchmarks, we demonstrate that Manthan significantly improves upon the current state of the art, solving 356 benchmarks in comparison to 280, which is the most solved by a state of the art technique; thereby, we demonstrate an increase of 76 benchmarks over the current state of the art. Furthermore, Manthan solves 60 benchmarks that none of the current state of the art techniques could solve. The significant performance improvements, along with our detailed analysis, highlights several interesting avenues of future work at the intersection of machine learning, constrained sampling, and automated reasoning.

[c39] 
Approximate Counting of Minimal Unsatisfiable Subsets
Proceedings of International Conference on ComputerAided Verification (CAV), 2020.
Given an unsatisfiable formula F in CNF, i.e. a set of clauses, the problem of Minimal Unsatisfiable Subset (MUS) seeks to identify the minimal subset of clauses N \subseteq F such that N is unsatisfiable. The emerging viewpoint of MUSes as the root causes of unsatisfiability has led MUSes to find applications in a wide variety of diagnostic approaches. Recent advances in finding and enumeration of MUSes have motivated researchers to discover applications that can benefit from rich information about the set of MUSes. One such extension is that of counting the number of MUSes, which has shown to describe the inconsistency metrics for general propositional knowledge bases. The current best approach for MUS counting is to employ a MUS enumeration algorithm, which often does not scale for the cases with a reasonably large number of MUSes. Motivated by the success of hashingbased techniques in the context of model counting, we design the first approximate counting procedure with (epsilon,delta) guarantees, called AMUSIC. Our approach avoids exhaustive MUS enumeration by combining the classical technique of universal hashing with advances in QBF solvers along with a novel usage of union and intersection of MUSes to achieve runtime efficiency. Our prototype implementation of AMUSIC is shown to scale to instances that were clearly beyond the reach of enumerationbased approaches.

[c38] 
Classification Rules in Relaxed Logical Form
Proceedings of European Conference on Artificial Intelligence(ECAI), 2020.
ML algorithms that produce rulebased predictions in Conjunctive Normal form (CNF) or in Disjunctive Normal form (DNF) are arguably some of the most interpretable ones. Although CNF/DNF rules are considered interpretable in practice, propositional logic has other very interpretable representations which are more expressive. In this paper, we generalize CNF/DNF rules and introduce relaxedCNF rules, which is motivated by the popular usage of checklists in the medical domain. We consider relaxed definitions of standard OR/AND operators which allow exceptions in the construction of a clause and also in the selection of clauses in a rule. While the combinatorial structure of relaxedCNF rules offers exponential succinctness, the naive learning techniques are computationally expensive. The primary contribution of this paper is to propose a novel incremental minibatch learning procedure, called CRR, that employs advances in the combinatorial solvers and efficiently learns relaxedCNF rules. Our experimental analysis demonstrates that CRR can generate relaxedCNF rules which are more accurate compared to CNF rules and sparser compared to decision lists.

[c37] 
Induction Models on N
Proceedings of International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR), 2020.
We generalize the definition of an Induction Model given by L. Henkin (1960). The main goal of the paper is to study reduction and equivalence between these Induction Models. We give a formal definition for these concepts and then prove a criterion which can be used to check when one Induction Model can be reduced to or is equivalent to another Induction Model. We also look at the base cases and generating functions which can give us an Induction Model. There are three cases which we look at depending on the structure of the generating functions (arbitrary, additive, multiplicative).

[c36] 
A Study of Symmetry Breaking Predicates and Model Counting
Proceedings of Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 2020.
Propositional model counting is a classic problem that has recently witnessed many technical advances and novel applications. While the basic model counting problem requires computing the number of all solutions to the given formula, in some important application scenarios, the desired count is not of all solutions, but instead, of all unique solutions up to isomorphism. In such a scenario, the user herself must try to either use the full count that the model counter returns to compute the count up to isomorphism, or ensure that the input formula to the model counter adequately captures the symmetry breaking predicates so it can directly report the count she desires. We study the use of CNFlevel and domainlevel symmetry breaking predicates in the context of the stateoftheart in model counting, specifically the leading approximate model counter ApproxMC and the recently introduced exact model counter ProjMC. As benchmarks, we use a range of problems, including structurally complex specifications of software systems and constraint satisfaction problems. The results show that while it is sometimes feasible to compute the model counts up to isomorphism using the full counts that are computed by the model counters, doing so suffers from poor scalability. The addition of symmetry breaking predicates substantially assists model counters. Domainspecific predicates are particularly useful, and in many cases can provide full symmetry breaking to enable highly efficient model counting up to isomorphism. We hope our study motivates new research on designing model counters that directly account for symmetries to facilitate further applications of model counting.

[c35] 
A MaxSATbased Framework for Group Testing
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2020.
The success of MaxSAT (maximum satisfiability) solving in recent years has motivated researchers to apply MaxSAT solvers in diverse discrete combinatorial optimization problems. Group testing has been studied as a combinatorial optimization problem, where the goal is to find defective items among a set of items by performing sets of tests on items. In this paper, we propose a MaxSATbased framework, called MGT, that solves group testing, in particular, the decoding phase of nonadaptive group testing. We extend this approach to the noisy variant of group testing, and propose a compact MaxSATbased encoding that guarantees an optimal solution. Our extensive experimental results show that MGT can solve group testing instances of 10000 items with 3% defectivity, which no prior work can handle to the best of our knowledge. Furthermore, MGT has better accuracy than the LPbased ap proach. We also discover an interesting phase transition behavior in the runtime, which reveals the easyhardeasy nature of group testing.

2019  
[c34] 
Embedding Symbolic Knowledge into Deep Networks
Proceedings of Advances in Neural Information Processing Systems(NeurIPS), 2019.
In this work, we aim to leverage prior symbolic knowledge to improve the performance of deep models. We propose a graph embedding network that projects propositional formulae (and assignments) onto a manifold via an augmented Graph Convolutional Network (GCN). To generate semanticallyfaithful embeddings, we develop techniques to recognize node heterogeneity, and semantic regularization that incorporate structural constraints into the embedding. Experiments show that our approach improves the performance of models trained to perform entailment checking and visual relation prediction. Interestingly, we observe a connection between the tractability of the propositional theory representation and the ease of embedding. Future exploration of this connection may elucidate the relationship between knowledge compilation and vector representation learning.

[c33] 
Quantitative Verification of Neural Networks And its Security Applications
Proceedings of ACM Conference on Computer and Communications Security (CCS), 2019.
Neural networks are increasingly employed in safetycritical domains. This has prompted interest in verifying or certifying logically encoded properties of neural networks. Prior work has largely focused on checking existential properties, wherein the goal is to check whether there exists any input that violates a given property of interest. However, neural network training is a stochastic process, and many questions arising in their analysis require probabilistic and quantitative reasoning, i.e., estimating how many inputs satisfy a given property. To this end, our paper proposes a novel and principled framework to quantitative verification of logical properties specified over neural networks. Our framework is the first to provide PACstyle soundness guarantees, in that its quantitative estimates are within a controllable and bounded error from the true count. We instantiate our algorithmic framework by building a prototype tool called NPAQ that enables checking rich properties over binarized neural networks. We show how emerging security analyses can utilize our framework in 3 concrete point applications: quantifying robustness to adversarial inputs, efficacy of trojan attacks, and fairness/bias of given neural networks.

[c32] 
Dual Hashingbased Algorithms for Discrete Integration
Proceedings of International Conference on Constraint Programming (CP), 2019.
Given a boolean formula F and a weight function $\rho$, the problem of discrete integration seeks to compute the weight of $F$, defined as the sum of the weights of satisfying assignments. Discrete integration, also known as weighted model counting, is a fundamental problem in computer science with wide variety of applications ranging from machine learning and statistics to physics and infrastructure reliability. Given the intractability of the exact variant, the problem of approximate weighted model counting has been subject to intense theoretical and practical investigations over the years. The primary contribution of this paper is to investigate development of algorithmic approaches for discrete integration. Our framework allows us to derive two different algorithms, which can be seen as dual to each other: \textsf{WISH}, which was already discovered by Ermon et al~\cite{EGSS13c}, and a new algorithm: \textsf{SWITCH}.We argue that these algorithms can be seen as dual to each other, in the sense that their complexities differ only by a permutation of certain parameters. Indeed we show that, for $F$ defined over $n$ variables, a weight function $\rho$ that can be represented using $p$ bits, and a confidence parameter $\delta$, there is a function $f$ and an NP oracle such that \textsf{WISH} makes $\mathcal{O} \left(f(n,p,\delta)\right)$ calls to NP oracle while \textsf{SWITCH} makes $\mathcal{O}\left(f(p,n,\delta)\right)$ queries. We find $f(x,y,\delta)$ polynomial in $x$, $y$ and $1/\delta$, more specifically $f(x,y,\delta) = x\log(y)\log(x/\delta)$. We first focus on striking similarities of both the design process and structure of the two algorithms but then show that despite this quasisymmetry, the analysis yields time complexities dual to each other. Another contribution of this paper is the use of 3wise property independence of XOR based hash functions in the analysis of WISH and SWITCH. To the best of our knowledge, this is the first usage of 3wise independence in deriving stronger concentration bounds.

[c31] 
Not All FPRASs are Equal: Demystifying FPRASs for DNFCounting (Extended Abstract)
Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2019. Sister Conference Best Papers Track
The problem of counting the number of solutions of a DNF formula, also called #DNF, is a fundamental problem in artificial intelligence with applications in diverse domains ranging from network reliability to probabilistic databases. Owing to the intractability of the exact variant, efforts have focused on the design of approximate techniques for #DNF. Consequently, several Fully Polynomial Randomized Approximation Schemes (FPRASs) based on Monte Carlo techniques have been proposed. Recently, it was discovered that hashingbased techniques too lend themselves to FPRASs for #DNF. Despite significant improvements, the complexity of the hashingbased FPRAS is still worse than that of the best Monte Carlo FPRAS by polylog factors. Two questions were left unanswered in previous works: Can the complexity of the hashingbased techniques be improved? How do the various approaches stack up against each other empirically? In this paper, we first propose a new search procedure for the hashingbased FPRAS that removes the polylog factors from its time complexity. We then present the first empirical study of runtime behavior of different FPRASs for #DNF. The result of our study produces a nuanced picture. First of all, we observe that there is no single best algorithm that outperforms all others for all classes of formulas and input parameters. Second, we observe that the algorithm with the worst time complexity, solves the largest number of benchmarks.

[c30] 
Ganak: A Scalable Probabilistic Exact Model Counter
Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2019.
Given a Boolean formula F, the problem of model counting, also referred to as #SAT, seeks to compute the number of solutions of F. Model counting is a fundamental problem with a wide variety of applications ranging from planning, quantified information flow to probabilistic reasoning and the like. The modern #SAT solvers tend to be either based on static decomposition, dynamic decomposition, or a hybrid of the two. Despite dynamic decomposition based #SAT solvers sharing much of their architecture with SAT solvers, the core design and heuristics of dynamic decompositionbased #SAT solvers has remained constant for over a decade. In this paper, we revisit the architecture of the stateoftheart dynamic decompositionbased #SAT tool, sharpSAT, and demonstrate that by introducing a new notion of probabilistic component caching and the usage of universal hashing for exact model counting along with the development of several new heuristics can lead to significant performance improvement over stateoftheart modelcounters. In particular, we develop GANAK, a new scalable probabilistic exact model counter that outperforms stateoftheart exact and approximate model counters sharpSAT and ApproxMC3 respectively, both in terms of PAR2 score and the number of instances solved. Furthermore, in our experiments, the model count returned by GANAK was equal to the exact model count for all the benchmarks. Finally, we observe that recently proposed preprocessing techniques for model counting benefit exact model counters while hurting the performance of approximate model counters.

[c29] 
Phase Transition Behavior of Cardinality and XOR Constraints
Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2019.
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 CARDXOR formulas. The problem of determining satisfiability of CARDXOR formulas is a fundamental problem with 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 CARDXOR formulas is unexplored in prior work. In this paper, we present the first rigorous empirical study to characterize the runtime behavior of 1CARDXOR formulas. We show empirical evidence of a surprising phasetransition that follows a nonlinear tradeoff between CARD and XOR constraints.

[c28] 
CrystalBall: Gazing in the Black Box of SAT Solving
Proceedings of the International Conference on Theory and Applications of Satisfiability Testing (SAT), 2019.
Boolean satisfiability is a fundamental problem in computer science with a wide range of applications including planning, configuration management, design and verification of software/hardware systems. The annual SAT competition continues to witness impressive improvements in the performance of the winning SAT solvers largely thanks to the development of new heuristics arising out of intensive collaborative research in the SAT community. Modern SAT solvers achieve scalability and robustness with complex heuristics that are challenging to understand and explain. Consequently, the development of new algorithmic insights has been largely restricted to expert intuitions and evaluation of the new insights have been restricted to performance measurement in terms of the runtime of solvers or a proxy for the runtime of solvers. In this context, one may ask: whether it is possible to develop a framework to provide whitebox access to the execution of SAT solver that can aid both SAT solver developers and users to synthesize algorithmic heuristics for modern SAT solvers? The primary focus of our project is precisely such a framework, which we call CrystalBall More precisely, we propose to view modern conflictdriven clause learning (CDCL) solvers as a composition of classifiers and regressors for different tasks such as branching, clause memory management, and restarting. The primary objective of this paper is to introduce a framework to peek inside the SAT solvers  CrystalBall  to the AI and SAT community. The current version of CrystalBall focuses on deriving a classifier to keep or throw away a learned clause. In a departure from recent machine learning based techniques, CrystalBall, employs supervised learning and uses extensive, multigigabyte data extracted from runs of a single SAT solver to perform predictive analytics.

[c27] 
Assessing Heuristic Machine Learning Explanations with Model Counting
Proceedings of the International Conference on Theory and Applications of Satisfiability Testing (SAT), 2019.
Machine Learning (ML) models are widely used in decision making procedures in finance, medicine, education, etc. In these areas, ML outcomes can directly affect humans, e.g.\ by deciding whether a person should get a loan or be released from prison. Therefore, we cannot blindly rely on black box ML models and need to explain the decisions made by them. This motivated the development of a variety of MLexplainer systems, concrete examples of which include LIME and its successor ANCHOR. Due to the heuristic nature of explanations produced by existing tools, it is necessary to validate them. In this work, we propose a SATbased method to assess the quality of explanations produced by ANCHOR We encode a trained ML model and an explanation for a given prediction as a propositional formula. Then, by using a stateoftheart approximate model counter, we estimate the quality of the provided explanation as the number of solutions supporting it.

[c26] 
WAPS: Weighted and Projected Sampling
Proceedings of Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 2019.
Previous work on applying Knowledge compilation has focused on uniform sampling over all the variables. Since the constraints are written in high level languages such as Verilog, the popular CNF encoding schemes as Tseitin encoding introduces additional auxiliary variables. The resulting CNF formulas are not equivalent but equisatisfiable. In particular, for a formula $G$ specified in high level language we obtain a CNF formula F such that $G (X) = \exists Y F(X,Y)$. This makes one wonder if it is possible to extend Knowledge compilation based techniques to sample over a subset of variables. Furthermore, languages such as Verilog allow specification of weights to userdefined constraints, so there is a need to sample according to the posterior distribution. In this paper, we provide affirmative question to the above two questions: We propose KUS that samples over user defined subset of variables from posterior distribution for a given prior distribution defined over product spaces.

[c25] 
Bosphorus: Bridging ANF and CNF Solvers
Proceedings of Design, Automation, and Test in Europe(DATE), 2019.
Algebraic Normal Form (ANF) and Conjunctive Normal Form (CNF) are commonly used to encode problems in Boolean algebra. ANFs are typically solved via Gr"obner basis algorithms, often using more memory than is feasible; while CNFs are solved using SAT solvers, which cannot exploit the algebra of polynomials naturally. We propose a paradigm that bridges between ANF and CNF solving techniques: the techniques are applied in an iterative manner to emph{learn facts} to augment the original problems. Experiments on over 1,100 benchmarks arising from four different applications domains demonstrate that learnt facts can significantly improve runtime and enable more benchmarks to be solved.

[c24] 
BIRD: Engineering an Efficient CNFXOR SAT Solver and its Applications to Approximate Model Counting
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2019.
Given a Boolean formula F, the problem of model counting, also referred to as \#SAT is to compute the number of solutions of F. Model counting is a fundamental problem in artificial intelligence with a wide range of applications including probabilistic reasoning, decision making under uncertainty, quantified information flow, and the like. Motivated by the success of SAT solvers, there has been surge of interest in the design of hashingbased techniques for approximate model counting for the past decade. We profiled the state of the art approximate model counter ApproxMC3 and observed that over 99.99\% of time is consumed by the underlying SAT solver, CryptoMinisat. This observation motivated us to ask: Can we design an efficient underlying CNFXOR SAT solver that can take advantage of the structure of hashingbased algorithms and would this lead to an efficient approximate model counter? The primary contribution of this paper is an affirmative answer to the above question. We present a novel architecture, called BIRD, to handle CNFXOR formulas arising from hashingbased techniques. The resulting hashingbased approximate model counter, called ApproxMC3, employs the BIRD framework in its underlying SAT solver, CryptoMinisat. To the best of our knowledge, we conducted the most comprehensive study of evaluation performance of counting algorithms involving 1896 benchmarks with computational effort totaling 86400 computational hours. Our experimental evaluation demonstrates significant runtime performance improvement for ApproxMC3 over ApproMC2. In particular, we solve 648 benchmarks more than ApproMC2, the state of the art approximate model counter and for all the formulas where both ApproMC2 and ApproxMC3 did not timeout and took more than 1 seconds, the mean speedup is 284.40  more than two orders of magnitude.

[c23] 
On testing of Uniform Samplers
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2019.
Recent years have seen an unprecedented adoption of artificial intelligence in wide variety of applications ranging from medical diagnosis, automobile, security to aircraft collision avoidance. Probabilistic reasoning is a key component of such modern artificial intelligence systems. Sampling techniques form the core of the state of the art probabilistic reasoning systems. In contrast to testing for deterministic programs, where one trace is sufficient to prove the existence of a bug; such is not the case for samplers as one sample is typically not sufficient to prove nonconformity of the sampler to the desired distribution. This makes one wonder: whether it is possible to design testing methodology to test whether a sampler under test generates samples close to a given distribution. The primary contribution of this paper is a positive answer to the above question: We design, to the best of our knowledge, the first algorithmic framework, Barbarik, to test whether the distribution generated by $\varepsilon$close or $\eta$far from the uniform distribution. In contrast to the sampling techniques that require an exponential or subexponential number of samples for sampler whose support can be represented by $n$ bits, Barbarik requires only $\mathcal{O}(1/(\eta\varepsilon)^2)$ samples. We present a prototype implementation of Barbarik and use it to test three state of the art uniform samplers over the support defined by combinatorial constraints. Barbarik is able to provide a certificate of uniformity to one sampler and demonstrate nonuniformity for the other two samplers.

[c22] 
On the Hardness of Probabilistic Inference Relaxations
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2019.
Probabilistic inference is increasingly being used in applications that compute with uncertain data. A promising approach to inference that has attracted recent attention exploits its interreducibility with model counting. Since probabilistic inference and model counting are #Pcomplete, various relaxations are used in practice, with the hope that these relaxations lend themselves to efficient computation while also providing rigorous approximation guarantees. In this paper, we show that contrary to commonly held belief, several relaxations used in the probabilistic inference literature do not really lead to computational efficiency in a complexity theoretic sense. Our arguments proceed by showing the corresponding relaxed notions of counting to be computationally hard. We argue that approximate counting (and hence, inference) with multiplicative tolerance and probabilistic guarantees of correctness is the only class of relaxations that provably simplifies the problem, given access to an NPoracle. Finally, we show that for applications that compare probability estimates with a threshold, a new notion of relaxation with gaps between low and high thresholds can be used. This new relaxation allows efficient decision making in practice, given access to an NPoracle, while also bounding the approximation error.

[c21] 
IMLI: An Incremental Framework for MaxSATBased Learning of Interpretable Classification Rules
Proceedings of AAAI/ACM Conference on AI, Ethics, and Society(AIES), 2019.
The wide adoption of machine learning in the critical domains such as medical diagnosis, law, education had propelled the need for interpretable techniques due to the need for end user to understand the reasoning behind decisions due to learning systems. The computational intractability of interpretable learning led practitioners to design heuristic techniques, which fail to provide sound handles to tradeoff accuracy and interpretability. Motivated by the success of MaxSAT solvers over the past decade, recently MaxSATbased approach, called MLIC, was proposed that seeks to reduce the problem of learning interpretable rules expressed in Conjunctive Normal Form (CNF) to a MaxSAT query. While MLIC was shown to achieve accuracy similar to that of other state of the art blackbox classifiers while generating small interpretable CNF formulas, the runtime performance of MLIC is significantly lagging and renders approach unusable in practice. In this context, authors raised the question: Is it possible to achieve the best of both worlds, i.e. a sound framework for interpretable learning that can take advantage of MaxSAT solvers while scaling to realworld instances? In this paper, we take a step towards answering the above question in affirmation. We propose an incremental approach to MaxSAT based framework that achieves scalable runtime performance via partitionbased training methodology. Extensive experiments on benchmarks arising from UCI repository demonstrate that IMLI achieves up to three orders of magnitude runtime improvement without loss of accuracy and interpretability.

2018  
[c20] 
Knowledge Compilation meets Uniform Sampling
Proceedings of International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR), 2018.
Uniform sampling has drawn diverse applications in programming languages and software engineering, like in constrainedrandom verification (CRV), constrainedfuzzing and bug synthesis. The effectiveness of these applications depend on the uniformity of test stimuli generated from a given set of constraints. Despite significant progress over the past few years, the performance of the state of the art techniques still falls short of those of heuristic methods employed in the industry which sacrifice either uniformity or scalability when generating stimuli. In this paper, we propose a new approach to the uniform generation that builds on recent progress in knowledge compilation. The primary contribution of this paper is marrying knowledge compilation with uniform sampling: our algorithm, KUS, employs the stateoftheart knowledge compilers to first compile constraints into dDNNF form, and then, generates samples by making two passes over the compiled representation. We show that KUS is able to significantly outperform existing stateoftheart algorithms, SPUR and UniGen2, by up to 3 orders of magnitude in terms of runtime while achieving a geometric speedup of 1.7 and 8.3 over SPUR and UniGen2 respectively. Also, KUS achieves a lower PAR2 score, around 0.82x that of SPUR and 0.38x that of UniGen2. Furthermore, KUS achieves speedups of up to 3 orders of magnitude for incremental sampling. The distribution generated by KUS is statistically indistinguishable from that generated by an ideal uniform sampler. Moreover, KUS is almost oblivious to the number of samples requested.

[c19] 
MLIC: A MaxSATBased framework for learning interpretable classification rules
Proceedings of International Conference on Constraint Programming (CP), 2018.
The wide adoption of machine learning approaches in the industry, government, medicine and science has renewed the interest in interpretable machine learning: many decisions are too important to be delegated to blackbox techniques such as deep neural networks or kernel SVMs. Historically, problems of learning interpretable classifiers, including classification rules or decision trees, have been approached by greedy heuristic methods as essentially all the exact optimization formulations are NPhard. Our primary contribution is a MaxSATbased framework, called MLIC, which allows principled search for interpretable classification rules expressible in propositional logic. Our approach benefits from the revolutionary advances in the constraint satisfaction community to solve largescale instances of such problems. In experimental evaluations over a collection of benchmarks arising from practical scenarios, we demonstrate its effectiveness: we show that the formulation can solve large classification problems with tens or hundreds of thousands of examples and thousands of features, and to provide a tunable balance of accuracy vs. interpretability. Furthermore, we show that in many problems interpretability can be obtained at only a minor cost in accuracy. The primary objective of the paper is to show that recent advances in the MaxSAT literature make it realistic to find optimal (or very high quality nearoptimal) solutions to largescale classification problems. The key goal of the paper is to excite researchers in both interpretable classification and in the CP community to take it further and propose richer formulations, and to develop bespoke solvers attuned to the problem of interpretable ML.

[c18] 
Not All FPRASs are Equal: Demystifying FPRASs for DNFCounting
Proceedings of International Conference on Constraint Programming (CP), 2018. Invited to Constraints journal
The problem of counting the number of solutions of a DNF formula, also called #DNF, is a fundamental problem in artificial intelligence with applications in diverse domains ranging from network reliability to probabilistic databases. Owing to the intractability of the exact variant, efforts have focused on the design of approximate techniques for #DNF. Consequently, several Fully Polynomial Randomized Approximation Schemes (FPRASs) based on Monte Carlo techniques have been proposed. Recently, it was discovered that hashingbased techniques too lend themselves to FPRASs for #DNF. Despite significant improvements, the complexity of the hashingbased FPRAS is still worse than that of the best Monte Carlo FPRAS by polylog factors. Two questions were left unanswered in previous works: Can the complexity of the hashingbased techniques be improved? How do the various approaches stack up against each other empirically? In this paper, we first propose a new search procedure for the hashingbased FPRAS that removes the polylog factors from its time complexity. We then present the first empirical study of runtime behavior of different FPRASs for #DNF. The result of our study produces a nuanced picture. First of all, we observe that there is no single best algorithm that outperforms all others for all classes of formulas and input parameters. Second, we observe that the algorithm with the worst time complexity, solves the largest number of benchmarks.

[c17] 
Scalable Approximation of Quantitative Information Flow in Programs
Proceedings of International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI), 2018.
Quantitative information flow measurement techniques have been proven to be successful in detecting leakage of confidential information from programs. Modern approaches are based on formal methods, relying on program analysis to produce a SAT formula representing the program's behavior, and model counting to measure the possible information flow. However, while program analysis scales to large codebases like the OpenSSL project, the formulas produced are too complex for analysis with precise model counting. In this paper we use the approximate model counter ApproxMC2 to quantify information flow. We show that ApproxMC2 is able to provide a large performance increase for a very small loss of precision, allowing the analysis of SAT formulas produced from complex code. We call the resulting technique ApproxFlow and test it on a large set of benchmarks against the state of the art. Finally, we show that ApproxFlow can evaluate the leakage incurred by the Heartbleed OpenSSL bug, contrarily to the state of the art.

2017  
[c16] 
On HashingBased Approaches to Approximate DNFCounting
Proceedings of IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2017.
Propositional model counting is a fundamental problem in artificial intelligence with a wide variety of applications, such as probabilistic inference, decision making under uncertainty, and probabilistic databases. Consequently, the problem is of theoretical as well as practical interest. When the constraints are expressed as DNF formulas, Monte Carlobased techniques have been shown to provide a fully polynomial randomized approximation scheme (FPRAS). For CNF constraints, hashingbased approximation techniques have been demonstrated to be highly successful. Furthermore, it was shown that hashingbased techniques also yield an FPRAS for DNF counting without usage of Monte Carlo sampling. Our analysis, however, shows that the proposed hashingbased approach to DNF counting provides poor time complexity compared to the Monte Carlobased DNF counting techniques. Given the success of hashingbased techniques for CNF constraints, it is natural to ask: Can hashingbased techniques provide an efficient FPRAS for DNF counting? In this paper, we provide a positive answer to this question. To this end, we introduce two novel algorithmic techniques: Symbolic Hashing and Stochastic Cell Counting, along with a new hash family of RowEchelon hash functions. These innovations allow us to design a hashingbased FPRAS for DNF counting of similar complexity as that of prior works. Furthermore, we expect these techniques to have potential applications beyond DNF counting.

[c15] 
The Hard Problems Are Almost Everywhere For Random CNFXOR Formulas
Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2017.
Recent universalhashing based approaches to sampling and counting crucially depend on the runtime performance of \SAT~solvers on formulas expressed as the conjunction of both CNF constraints and variablewidth XOR constraintsints (known as CNFXOR formulas). In this paper, we present the first study of the runtime behavior of \SAT~solvers equipped with XORreasoning techniques on random CNFXOR formulas. We empirically demonstrate that a stateoftheart \SAT~solver scales exponentially on random CNFXOR formulas across a wide range of XORclause densities, peaking around the empirical phasetransition location. On the theoretical front, we prove that the solution space of a random CNFXOR formula `shatters' at \emph{all} nonzero XORclause densities into wellseparated components, similar to the behavior seen in random CNF formulas known to be difficult for many \SATsolving algorithms.

[c14] 
CountingBased Reliability Estimation for PowerTransmission Grids
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2017.
Modern society is increasingly reliant on the functionality of infrastructure facilities and utility services. Consequently, there has been surge of interest in the problem of quantification of system reliability, which is known to be #Pcomplete. Reliability also contributes to the resilience of systems, so as to effectively make them bounce back after contingencies. Despite diverse progress, most techniques to estimate system reliability and resilience remain computationally expensive. In this paper, we investigate how recent advances in hashingbased approaches to counting can be exploited to improve computational techniques for system reliability. The primary contribution of this paper is a novel framework, RelNet, that provides provably approximately correct (PAC) estimates for arbitrary networks. We then apply RelNet to ten real world power transmission grids across different cities in the U.S. and are able to obtain, to the best of our knowledge, the first theoretically sound a priori estimates of reliability between several pairs of nodes of interest. Such estimates will help managing uncertainty and support rational decision making for community resilience.

2016  
[c13] 
Algorithmic Improvements in Approximate Counting for Probabilistic Inference: From Linear to Logarithmic SAT Calls
Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2016.
Probabilistic inference via model counting has emerged as a scalable technique with strong formal guarantees, thanks to recent advances in hashingbased approximate counting. Stateoftheart hashingbased counting algorithms use an {\NP} oracle, such that the number of oracle invocations grows linearly in the number of variables n in the input constraint. We present a new approach to hashingbased approximate model counting in which the number of oracle invocations grows logarithmically in $n$, while still providing strong theoretical guarantees. Our experiments show that the new approach outperforms stateoftheart techniques for approximate counting by 12 orders of magnitude in running time.

[c12] 
Combining the kCNF and XOR PhaseTransitions
Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2016.
The runtime performance of modern SAT solvers on random kCNF formulas is deeply connected with the `phasetransition' phenomenon seen empirically in the satisfiability of random kCNF formulas. Recent universal hashingbased approaches to sampling and counting crucially depend on the runtime performance of SAT solvers on formulas expressed as the conjunction of both kCNF and XOR constraints (known as kCNFXOR formulas), but the behavior of random kCNFXOR formulas is unexplored in prior work. In this paper, we present the first study of the satisfiability of random kCNFXOR formulas. We show empirical evidence of a surprising phasetransition that follows a linear tradeoff between kCNF and XOR constraints. Furthermore, we prove that a phasetransition for kCNFXOR formulas exists for k = 2 and (when the number of kCNF constraints is small) for k > 2.

[c11] 
Approximate Probabilistic Inference via WordLevel Counting
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2016.
Hashingbased model counting has emerged as a promising approach for largescale probabilistic inference on graphical models. A key component of these techniques is the use of xorbased 2universal hash functions that operate over Boolean domains. Many counting problems arising in probabilistic inference are, however, naturally encoded over fi nite discrete domains. Techniques based on bitlevel (or Boolean) hash functions require these problems to be propositionalized, making it impossible to leverage the remarkable progress made in SMT (Satisfiability Modulo Theory) solvers that can reason directly over words (or bitvectors). In this work, we present the first approximate model counter that uses wordlevel hashing functions, and can directly leverage the power of sophisticated SMT solvers. Empirical evaluation over an extensive suite of benchmarks demonstrates the promise of the approach.

[c10] 
Automatic Data Layout Generation and Kernel Mapping for CPU+GPU Architectures
Proceedings of International Conference on Compiler Construction (CC), 2016.
The ubiquity of hybrid CPU+GPU architectures has led to renewed interest in automatic data layout generation owing to the fact that data layouts have a large impact on performance, and that different data layouts yield the best performance on CPUs vs. GPUs. Unfortunately, current programming models still fail to provide an effective solution to the problem of automatic data layout generation for CPU+GPU processors. Specifically, the interaction among wholeprogram data layout optimizations, data movement optimizations, and mapping of kernels across heterogeneous cores poses a major challenge to current programming systems. In this paper, we introduce a novel twolevel hierarchical formulation of the data layout and kernel mapping problem for modern heterogeneous architectures. The top level formulation targets data layouts and kernel mapping for the entire program for which we provide a polynomial time solution using a graphbased shortest path algorithm that uses the data layouts for the code regions (sections) for a given processor computed in the bottom level formulation. The bottom level formulation deals with the data layout problem for a parallel code region on a given processor, which is NPHard, and we provide a greedy algorithm that uses an affinity graph to obtain approximate solutions. We have implemented this data layout transformation in the new Heterogeneous HabaneroC (H2C) parallel programming framework and propose performance models to characterize the data layout impact on both the CPU and GPU. Our data layout framework shows significant performance improvements of up to 2.9 (geometric mean 1.5) on a multicore CPU+GPU compared to the manually specified layouts for a set of parallel programs running on a heterogeneous platform consisting of an Intel Xeon CPU and a NVIDIA GPU. Further, our framework also shows performance improvements of up to 2.7 (geometric mean 1.6) on just the multicore CPU, demonstrating the applicability of our approach to both heterogeneous and homogeneous hardware platforms.

[c9] 
Design and Verification of Distributed Phasers
Proceedings of International European Conference on Parallel and Distributed Computing (EuroPar), 2016.
A phaser is an expressive synchronization construct that unifies collective and pointtopoint coordination with dynamic registration of parallel tasks. Each task can participate in a phaser as a signaler, a waiter, or both. The participants in a phaser may change over time as tasks are added and deleted. In this paper, we present a highly concurrent and scalable design of phasers for a distributed memory environment. Our design for a distributed phaser employs a pair of skip lists augmented with the ability to collect and propagate synchronization signals. To enable a high degree of concurrency, addition and deletion of participant tasks are performed in two phases: a "fast singlelinkmodify" step followed by multiple handoverhand "lazy multilinkmodify" steps. Verifying highlyconcurrent protocols is difficult. We analyze our design for a distributed phaser using the SPIN model checker. A straightforward approach to model checking a distributed phaser operation requires an infeasibly large state space. To address this issue, we employ a novel "messagebased" model checking scheme to enable a non approximate complete model checking of our phaser design. We guarantee the semantic properties of phaser operations by ensuring that a set of linear temporal logic formulae are valid during model checking. We also present complexity analysis of the cost of synchronization and structural operations.

2015  
[c8] 
On Computing Minimal Independent Support and Its Applications to Sampling and Counting
Proceedings of International Conference on Constraint Programming (CP), 2015. Best Student Paper Award
Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in stateoftheart constrained sampling and counting tools, while still retaining theoretical guarantees.

[c7] 
From Weighted to Unweighted Model Counting
Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2015.
The recent surge of interest in reasoning about probabilistic graphical models has led to the development of various techniques for probabilistic reasoning. Of these, techniques based on weighted model counting are particularly interesting since they can potentially leverage recent advances in unweighted model counting and in propositional satisfiability solving. In this paper, we present a new approach to weighted model counting via reduction to unweighted model counting. Our reduction, which is polynomialtime and preserves the normal form (CNF/DNF) of the input formula, allows us to exploit advances in unweighted model counting to solve weighted model counting instances. Experiments with weighted model counters built using our reduction indicate that these counters performs much better than a stateoftheart weighted model counter.

[c6] 
On Parallel Scalable Uniform SAT Witness Generator
Proceedings of Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 2015.
Constrainedrandom verification (CRV) is widely used in industry for validating hardware designs. The effectiveness of CRV depends on the uniformity of test stimuli generated from a given set of constraints. Most existing techniques sacrifice either uniformity or scalability when generating stimuli. While recent work based on random hash functions has shown that it is possible to generate almost uniform stimuli from constraints with 100,000+ variables, the performance still falls short of today's industrial requirements. In this paper, we focus on pushing the performance frontier of uniform stimulus generation further. We present a random hashingbased, easily parallelizable algorithm, UniGen2, for sampling solutions of propositional constraints. UniGen2 provides strong and relevant theoretical guarantees in the context of CRV, while also offering significantly improved performance compared to existing almostuniform generators. Experiments on a diverse set of benchmarks show that UniGen2 achieves an average speedup of about 20X over a stateoftheart sampling algorithm, even when running on a single core. Moreover, experiments with multiple cores show that UniGen2 achieves a nearlinear speedup in the number of cores, thereby boosting performance even further.

2014  
[c5] 
ADHA: Automatic Data layout framework for Heterogeneous Architectures
Proceedings of Parallel Architecture and Compilation Techniques (PACT), 2014.
Data layouts play a crucial role in determining the performance of a given application running on a given architecture. Existing parallel programming frameworks for both multicore and heterogeneous systems leave the onus of selecting a data layout to the programmer. Therefore, shifting the burden of data layout selection to optimizing compilers can greatly enhance programmer productivity and application performance. In this work, we introduce ADHA: a twolevel hierarchal formulation of the data layout problem for modern heterogeneous architectures. We have created a reference implementation of ADHA in the Heterogeneous HabaneroC (H2C) parallel programming system. ADHA shows significant performance benefits of up to 6.92X compared to manually specified layouts for two benchmark programs running on a CPU+GPU heterogeneous platform.

[c4] 
DistributionAware Sampling and Weighted Model Counting for SAT
Proceedings of AAAI Conference on Artificial Intelligence (AAAI), 2014.
Given a CNF formula and a weight for each assignment of values to variables, two natural problems are weighted model counting and distributionaware sampling of satisfying assignments. Both problems have a wide variety of important applications. Due to the inherent complexity of the exact versions of the problems, interest has focused on solving them approximately. Prior work in this area scaled only to small problems in practice, or failed to provide strong theoretical guarantees, or employed a computationallyexpensive maximum a posteriori probability (MAP) oracle that assumes prior knowledge of a factored representation of the weight distribution. We present a novel approach that works with a blackbox oracle for weights of assignments and requires only an {\NP}oracle (in practice, a SATsolver) to solve both the counting and sampling problems. Our approach works under mild assumptions on the distribution of weights of satisfying assignments, provides strong theoretical guarantees, and scales to problems involving several thousand variables. We also show that the assumptions can be significantly relaxed while improving computational efficiency if a factored representation of the weights is known.

[c3] 
Balancing Scalability and Uniformity in SATWitness Generator
Proceedings of Design Automation Conference (DAC), 2014.
Constrainedrandom simulation is the predominant approach used in the industry for functional verification of complex digital designs. The effectiveness of this approach depends on two key factors: the quality of constraints used to generate test vectors, and the randomness of solutions generated from a given set of constraints. In this paper, we focus on the second problem, and present an algorithm that significantly improves the stateoftheart of (almost)uniform generation of solutions of large Boolean constraints. Our algorithm provides strong theoretical guarantees on the uniformity of generated solutions and scales to problems involving hundreds of thousands of variables.

2013  
[c2] 
A Scalable Approximate Model Counter
Proceedings of International Conference on Constraint Programming (CP), 2013.
Propositional model counting (#SAT), i.e., counting the number of satisfying assignments of a propositional formula, is a problem of significant theoretical and practical interest. Due to the inherent complexity of the problem, approximate model counting, which counts the number of satisfying assignments to within given tolerance and confi dence level, was proposed as a practical alternative to exact model counting. Yet, approximate model counting has been studied essentially only theoretically. The only reported implementation of approximate model counting, due to Karp and Luby, worked only for DNF formulas. A few existing tools for CNF formulas are bounding model counters; they can handle realistic problem sizes, but fall short of providing counts within given tolerance and confidence, and, thus, are not approximate model counters. We present here a novel algorithm, as well as a reference implementation, that is the first scalable approximate model counter for CNF formulas. The algorithm works by issuing a polynomial number of calls to a SAT solver. Our tool, ApproxMC, scales to formulas with tens of thousands of variables. Careful experimental comparisons show that ApproxMC reports, with high confidence, bounds that are close to the exact count, and also succeeds in reporting bounds with small tolerance and high confidence in cases that are too large for computing exact model counts.

[c1] 
A Scalable and Nearly Uniform Generator of SAT Witnesses
Proceedings of International Conference on ComputerAided Verification (CAV), 2013.
Functional verification constitutes one of the most challenging tasks in the development of modern hardware systems, and simulationbased verification techniques dominate the functional verification landscape. A dominant paradigm in simulationbased verification is directed random testing, where a model of the system is simulated with a set of random test stimuli that are uniformly or nearuniformly distributed over the space of all stimuli satisfying a given set of constraints. Uniform or nearuniform generation of solutions for large constraint sets is therefore a problem of theoretical and practical interest. For Boolean constraints, prior work offered heuristic approaches with no guarantee of performance, and theoretical approaches with proven guarantees, but poor performance in practice. We offer here a new approach with theoretical performance guarantees and demonstrate its practical utility on large constraint sets.

2019  
[j3] 
Principled network reliability approximation: A countingbased approach.
Journal of Reliability Engineering and System Safety(RESS), 2019.
As engineered systems expand, become more interdependent, and operate in realtime, reliability assessment is indispensable to support investment and decision making. However, network reliability problems are known to be #Pcomplete, a computational complexity class largely believed to be intractable. The computational intractability of network reliability motivates our quest for reliable approximations. Based on their theoretical foundations, available methods can be grouped as follows: (i) exact or bounds, (ii) guaranteeless sampling, and (iii) probably approximately correct (PAC). Group (i) is well regarded due to its useful byproducts, but it does not scale in practice. Group (ii) scales well and verifies desirable properties, such as the bounded relative error, but it lacks error guarantees. Group (iii) is of great interest when precision and scalability are required, as it harbors computationally feasible approximation schemes with PACguarantees. We give a comprehensive review of classical methods before introducing modern techniques and our developments. We introduce KRelNet, an extended countingbased estimation method that delivers PACguarantees for the Kterminal reliability problem. Then, we test methods' performance using various benchmark systems. We highlight the range of application of algorithms and provide the foundation for future resilience engineering as it increasingly necessitates methods for uncertainty quantification in complex systems.

[j2] 
Not All FPRASs are Equal: Demystifying FPRASs for DNFCounting
Constraints, 2019.
The problem of counting the number of solutions of a DNF formula, also called #DNF, is a fundamental problem in artificial intelligence with applications in diverse domains ranging from network reliability to probabilistic databases. Owing to the intractability of the exact variant, efforts have focused on the design of approximate techniques for #DNF. Consequently, several Fully Polynomial Randomized Approximation Schemes (FPRASs) based on Monte Carlo techniques have been proposed. Recently, it was discovered that hashingbased techniques too lend themselves to FPRASs for #DNF. Despite significant improvements, the complexity of the hashingbased FPRAS is still worse than that of the best Monte Carlo FPRAS by polylog factors. Two questions were left unanswered in previous works: Can the complexity of the hashingbased techniques be improved? How do the various approaches stack up against each other empirically? In this paper, we first propose a new search procedure for the hashingbased FPRAS that removes the polylog factors from its time complexity. We then present the first empirical study of runtime behavior of different FPRASs for #DNF. The result of our study produces a nuanced picture. First of all, we observe that there is no single best algorithm that outperforms all others for all classes of formulas and input parameters. Second, we observe that the algorithm with the worst time complexity, solves the largest number of benchmarks.

2016  
[j1] 
On Computing Minimal Independent Support and Its Applications to Sampling and Counting
Constraints 21(1), 2016.
Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in stateoftheart constrained sampling and counting tools, while still retaining theoretical guarantees.

2016  
[w1] 
Constrained Sampling and Counting: Universal Hashing meets SAT Solving
Proceedings of Workshop on Beyond NP(BNP), 2016.
Constrained sampling and counting are two fundamental problems in artificial intelligence with a diverse range of applications, spanning probabilistic reasoning and planning to constrainedrandom 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 realworld instances.
