# Publications on arithmetic theories (logical theories of arithmetic)

**M. Benedikt, D. Chistikov, A. Mansutti. The complexity of Presburger arithmetic with power or powers.**ICALP 2023. [DROPS]Existence of solutions over N to systems of linear equations and constraints of the form

*y*=2^{x}can be decided in nondeterministic exponential time. Also, linear integer arithmetic extended with a predicate for powers of 2 can be decided in triply exponential time.

**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.

**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?

**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, 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).