Complexity of Constraint Satisfaction Problems over Finite Subsets of Natural Numbers

  title={Complexity of Constraint Satisfaction Problems over Finite Subsets of Natural Numbers},
  author={Titus Dose},
  journal={Electron. Colloquium Comput. Complex.},
  • Titus Dose
  • Published 2016
  • Mathematics, Computer Science
  • Electron. Colloquium Comput. Complex.
We study the computational complexity of constraint satisfaction problems that are based on integer expressions and algebraic circuits. On input of a finite set of variables and a finite set of constraints the question is whether the variables can be mapped onto finite subsets of N (resp., finite intervals over N) such that all constraints are satisfied. According to the operations allowed in the constraints, the complexity varies over a wide range of complexity classes such as L, P, NP, PSPACE… 
Balance Problems for Integer Circuits
  • Titus Dose
  • Mathematics, Computer Science
    Electron. Colloquium Comput. Complex.
  • 2018
The work shows that the balance problem for { ∖, ⋅ } -circuits is undecidable which is the first natural problem for integer circuits or related constraint satisfaction problems that admits only one arithmetic operation and is proven to be Undecidable.
Emptiness Problems for Integer Circuits
It turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity.
Circuit satisfiability and constraint satisfaction around Skolem Arithmetic


Satisfiability of algebraic circuits over sets of natural numbers
Complexity of Equations over Sets of Natural Numbers
The general membership problem for equations of the form Xi=φi (X1,…,Xn) (1≤i≤n) is proved to be EXPTIME-complete, and it is established that least solutions of all such systems are in EXPTime.
Constraint Satisfaction Problems around Skolem Arithmetic
This work studies interactions between Skolem Arithmetic and certain classes of Constraint Satisfaction Problems (CSPs) and proves the decidability of SkoleM Arithmetic.
Generation problems
On equations over sets of integers
The class of sets representable by unique solutions of equations using the operations of union and addition using ultimately periodic constants is exactly the class of hyper-arithmetical sets.
Word problems requiring exponential time(Preliminary Report)
A number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines are considered.
Univariate Equations Over Sets of Natural Numbers
It is shown that equations of the form p(X) = ψ(X), in which the unknown X is a set of natural numbers and p, ψ use the operations of union, intersection and addition of sets S + T = {m + n |, m ∈ S,
The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory
This paper isolates a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit a dichotomy, and explains the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy.
A Compendium of Problems Complete for P
A compilation of the known P-complete problems, including several unpublished or new P-completeness results, and many open problems is presented, mainly containing the problem list.
The Complexity of Membership Problems for Circuits Over Sets of Natural Numbers
The problem of testing membership in the subset of the natural numbers produced at the output gate of a combinational circuit is shown to capture a wide range of complexity classes, and results extend in nontrivial ways past work by Stockmeyer and Meyer, Wagner, Wagner and Yang.