# 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. [DROPS]The decision problem for linear arithmetic over Z with just equality (no inequalities) is as hard as for standard linear integer arithmetic. To prove this, 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).