Broadly, our research group is focused on the development of a rigorous mathematical framework for provably verified, correct-by-construction, explainable probabilistic reasoning systems building on recent breakthroughs in formal methods and symbolic reasoning; and deployment of our algorithmic framework in the real world including applications in reliability estimation of critical infrastructure and computational sustainability. On a more technical level, we work at the intersection of formal methods and artificial intelligence. We seek to develop formal methods for AI techniques and also employ advances in AI techniques for development of formal method algorithmic and system frameworks.

Theme 1: Revisiting NP Revolution: New Paradigms for SAT/SMT/MaxSAT Solvers

This research theme is motivated by two observations:

  1. O1: The success of SAT/SMT/MaxSAT solvers has allowed researchers to employ them as “oracles” inside larger algorithmic components, which give rise to rich structure in the queries to the SAT solvers, and
  2. O2: Modern SAT/SMT solvers achieve scalability and robustness with complex heuristics that are challenging to understand and explain.

O1 necessitates design of solvers that can exploit the rich structures of queries without losing the generality offered by CNF. Our recent work has proposed new paradigms for CNF+XOR constraints [aaai19] and CNF+ANF constraints [date19].

A consequence of O2 is that 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 project, our focus is to enable data-driven design and understanding of SAT Solvers. In particular, we view modern conflict-driven clause learning (CDCL) solvers as a composition of classifiers and regressors for different tasks such as branching, clause memory management, and restarting. We will employ supervised learning and uses extensive, multi-gigabyte data extracted from runs of a single SAT solver to perform predictive analytics. Our [SAT-19] paper presents the first version of the system, called CrystalBall, that we are building.

Theme 2: Beyond NP Revolution: Constrained Sampling and Integration/Counting

Constrained sampling and counting are two fundamental problems in data analysis. In constrained sampling the task is to sample randomly, subject to a given weighting function, from the set of solutions to a set of given constraints. This problem has numerous applications, including probabilistic reasoning, machine learning, statistical physics, and constrained-random verification. A related problem is that of constrained counting, where the task is to count the total weight, subject to a given weighting function, of the set of solutions of the given constraints. This problem has applications in machine learning, probabilistic reasoning, and planning, among other areas. Both problems can be viewed as aspects of one of the most fundamental problems in artificial intelligence, which is to understand the structure of the solution space of a given set of constraints. This work focuses on the development of new algorithmic techniques for constrained sampling and counting, based on a universal hashing – a classical algorithmic technique in theoretical computer science. Many of the ideas underlying the proposed approach were go back to the 1980s, but they have never been reduced to practice. Recent progress in Boolean reasoning is enabling us to reduce these algorithmic ideas to practice, and obtain breakthrough results in constrained sampling and counting, providing a new algorithmic toolbox in design verification, machine learning, probabilistic reasoning, and the like.

We are focused on

  1. Algorithmic advances in sampling and counting.
  2. Application of sampling and counting to problems arising in different domains.
  3. Revisiting theoretical foundations in light of existence of solvers but not oracles.

Theme 3: Verification of AI Systems

The unprecedented advances in the machine learning techniques have led to a proliferation of Artificial Intelligence(AI)-based devices in our day to day life. The AI-based tools are increasingly employed to make consequential decisions for human subjects, in areas such as credit lending, policing, criminal justice, and medicine. Decisions made by these AI-based predictive models often have a long-lasting impact on people’s lives. As such there is a strong need for the development of methodology for verification of AI systems.

The modern AI systems significantly differ from traditional systems in their reliance on probabilistic reasoning. Often statistical tests are employed to argue for the accuracy/robustness of these systems, but such statistical tests are usually performed on a tiny number of samples for which no theoretical guarantees exist for their accuracy. In contrast to testing for traditional hardware and software systems, 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 non-conformity of the underlying systems.

Theme 4: Interpretable and Explainable Explanations for AI Systems

Interpretability has become a central thread in ML research. As ML algorithms continue to permeate critical application domains such as medicine, legal, and transportation, it becomes increasingly important to allow human domain experts to understand and interact with ML solutions. Providing meaningful explanations for automated decisions is an increasingly important task. In this research theme, we are investigating techniques for two categories of explanations:

  1. Interpretable explanations such as those specified using small set of rules/formulas , and
  2. In settings where the cognitive burden of complete explanations is too high, people often do not seek to know why a particular event happened, but rather why an alternative one didn’t . Providing a satisfactory answer to such contrastive questions is often much easier than giving a full causal path leading to the realized outcome.


More Publications

. GANAK: A Scalable Probabilistic Exact Model Counter. In Proceedings International Joint Conference on Artificial Intelligence (IJCAI), 2019.

. Phase Transition Behavior of Cardinality and XOR Constraints . In Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), 2019.

. Assessing Heuristic Machine Learning Explanations with Model Counting . In Proceedings of the International Conference on Theory and Applications of Satisfiability Testing (SAT), 2019.

. CrystalBall: Gazing in the Black Box of SAT Solving . In Proceedings of the International Conference on Theory and Applications of Satisfiability Testing (SAT), 2019.

. WAPS: Weighted and Projected Sampling . In Proceedings of Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 2019., 2019.

PDF Code


  • 15 May 2019

    Kuldeep recieved notification of the award of NRF Fellowship for AI for the project: Provably Verified and Explainable Probabilistic Reasoning.

    9 May 2019

    Two papers accepted to IJCAI.The first paper explores the phase transition behavior of conjunction of cardinality and XOR constraints. Authors: Yash Pote, Saurabh Joshi, Kuldeep Meel.
    The second paper describes a radically new approach to exact counting wherein we compute estimates that are probabilistically exact! Authors: Shubham Sharma, Kuldeep Meel. Combined with our invited paper on #DNF , this makes 3 papers that we will be presenting at IJCAI

    22 April 2019

    Two papers accepted to SAT 2019. The first paper introduces the first version of CrystalBall, a framework intended to allow gazing into the black box of SAT solving. Authors: Kuldeep, Mate Soos, Raghav Kulkarni.
    The second paper discusses how model counting can be used to analyze explanations provided by tools such as ANCHOR. Authors: Kuldeep, Nina Narodytska, Aditya Shrotri, Alexey Ignatiev, and Joao Marques Silva

    11 February 2019

    Kuldeep is appointed Visiting Assistant Professor in the Department of Computer Science and Engineering at IIT Bombay

    2 February 2019

    Paper on network reliability is accepted to The 13th International Conference on Applications of Statistics and Probability in Civil Engineering Authors: Roger Paredes, Leonardo Duenas-Osorio, Kuldeep S. Meel, and Moshe Y. Vardi

    26 January 2019

    Our paper on weighted and projected sampling is accepted at TACAS 2019. Authors: Shubham Sharma, Rahul Gupta,Subhajit Roy and Kuldeep S. Meel



Kuldeep Kuldeep Meel

PhD Students

Masters Students

Research Assistant

Mahi Mohimenul Kabir
Himanshu Himanshu Arora



Alexis Alexis de Colnet

Past visitors


Kuldeep S. Meel
Assistant Professor
Computer Science Department, School of Computing
National University of Singapore

Please consult Kuldeep’s calendar before suggesting a meeting time.


We are always looking for highly motivated Ph.D. students, research assistants and summer internship for exceptional undergraduate interns in our group. We work at the intersection of algorithmic design and systems; therefore, an ideal candidate should have deeper expertise in one area and willingness to learn the other. A strong background in statistics, algorithms/formal methods and prior experience in coding is crucial to make a significant contribution to our research.

  • We have multiple post-doc positions and long term (>= 6 months) internship positions available.
  • Looking for candidates with strong background in CS/Mathematics/Physics for post-doc for the project: Beyond NP Revolution. Read advertisement for more details.
  • Two post-doc positions available in the broad area of applying machine learning to SAT solvers, approximate counting techniques, and CP. Read advertisement for more details.

Application Procedure:

  • If you are a student at NUS, feel free to drop by Kuldeep’s office or schedule a meeting with him. (See his calendar).
  • Post-doc position: Email with a PDF of your CV, which must contain information of at least two references. Furthermore, you should indicate your interest in a particular theme. For research themes look here.
  • Internship (>=6 months): We strongly prefer candidates who want to use their internship as a way to apply for PhD programs (@NUS or elsewhere; of course, if you are good, we would like you to remain at NUS). Email with a PDF of your CV. Make sure your subject contains the word “olleh” and you should include reviews of two of the papers published in the previous 3 years at AAAI/IJCAI/CP/SAT/CAV conferences. The reviews should be in the body of the email (and not as pdf). Furthermore, the body of your email should contain the phrase: “Here are two papers that I have reviewed”. You should also provide reason for your choice of the papers.
  • Short term internship (=3 months): We may offer short term internship to exceptional undergraduates. Same process as above.
  • PhD positions: Admissions to School of Computing@NUS are handled via a central admission procedure: You can either apply via Department of Computer Science or the Institute of Data Science.