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Publications on geometric problems

  • D. Chistikov, C. Haase, A. Mansutti. Geometric decision procedures and the VC dimension of linear arithmetic theories. LICS 2022. [WRAP]

    A decision procedure for Presburger arithmetic based on semilinear sets runs in triply exponential time. Also, the VC dimension of Presburger formulas is at most doubly exponential in their length.

  • D. Chistikov, C. Haase, A. Mansutti. Quantifier elimination for counting extensions of Presburger arithmetic. FoSSaCS 2022. [doi]

    We extend linear integer arithmetic with quantifiers of the form “there exists at least c values of x” and similar. It turns out such theories still support efficient quantifier elimination, even in bounded alternation depth.

  • D. Chistikov, C. Haase, Z. Hadizadeh, A. Mansutti. Higher-order Boolean satisfiability. MFCS 2022. To appear.

    We prove that the decision problem for linear arithmetic over Z with just equality (no inequalities) is as hard as for standard linear integer arithmetic. To do so, we define and study the complexity of a quantified version of SAT that supports higher-order Boolean functions.

  • S. Almagor, D. Chistikov, J. Ouaknine, J. Worrell. O-minimal invariants for discrete-time dynamical systems. ACM Transactions on Computational Logic (2022). Extended version of the ICALP’18 paper. [doi]

    For while loops of the form “while x in S do x:=A*x” (where x is initialized to a rational vector, A is a rational matrix, and S is a nice set), minimal invariants look like truncated cones.

  • D. Chistikov, O. Goulko, A. Kent, M. Paterson. Globe-hopping. Proceedings of the Royal Society A (2020). [WRAP] [arXiv] [doi]

    We look at a probabilistic puzzle on the sphere which has applications to quantum information theory (Bell inequalities).

  • D. Chistikov, C. Haase. On the power of ordering in linear arithmetic theories. ICALP 2020. [DROPS] [WRAP]

    Given a formula of linear arithmetic, can we decide if the same set can be defined by another formula that uses just equality, without inequalities?

  • S. Almagor, D. Chistikov, J. Ouaknine, J. Worrell. O-minimal invariants for linear loops. ICALP 2018. [DROPS] Extended version in ACM Transactions on Computational Logic (2022).
  • D. Chistikov, S. Kiefer, I. Marušić, M. Shirmohammadi, J. Worrell. Nonnegative matrix factorization reqires irrationality. SIAM Journal on Applied Algebra and Geometry (2017). Extended version of the SODA’17 paper. [WRAP] [arXiv]

    We find a matrix for which the nonnegative rank over the reals and over the rationals are different.

  • D. Chistikov, R. Dimitrova, R. Majumdar. Approximate counting in SMT and value estimation for probabilistic programs. ACTA Informatica (2017). Special issue for TACAS’15. [WRAP] [arXiv]

    Relying on ideas of Sipser and Stockmeyer, existing SMT solvers can do approximate model counting (discrete counting or volume estimation) for logical theories of arithmetic.

  • D. Chistikov, C. Haase. On the complexity of quantified integer programming. ICALP 2017. [DROPS] [pdf]

    If some variables in integer programs are quantified universally instead of existentially, then the decision problem becomes complete for the k-th level of the polynomial hierarchy, assuming k quantifier blocks.

  • D. Chistikov, S. Kiefer, I. Marušić, M. Shirmohammadi, J. Worrell. On rationality of nonnegative matrix factorization. SODA 2017. [doi]

    We find a matrix for which the nonnegative rank over the reals and over the rationals are different. Consequently, state minimization of hidden Markov models may require irrational probabilities.

  • D. Chistikov, S. Kiefer, I. Marušić, M. Shirmohammadi, J. Worrell. On restricted nonnegative matrix factorization. ICALP’16. [DROPS] [arXiv]

    There exists a pair of 3D polytopes with rational vertices, one inside the other, such that every intermediate polytope with 5 vertices must have a vertex with an irrational coordinate.

  • D. Chistikov, C. Haase. The taming of the semi-linear set. ICALP’16. [DROPS]

    To measure how semilinear sets “grow” under Boolean operations, we keep track of the maximum norm of generators.

  • D. Chistikov, R. Dimitrova, R. Majumdar. Approximate counting in SMT and value estimation for probabilistic programs. TACAS’15. Extended version in Acta Informatica (2017).