# Equivalence Problems for Circuits over Sets of Natural Numbers

@article{Glaer2007EquivalencePF, title={Equivalence Problems for Circuits over Sets of Natural Numbers}, author={Christian Gla{\ss}er and Katrin Herr and Christian Reitwie{\ss}ner and Stephen D. Travers and Matthias Waldherr}, journal={Theory of Computing Systems}, year={2007}, volume={46}, pages={80-103} }

We investigate the complexity of equivalence problems for {∪,∩,−,+,×}-circuits computing sets of natural numbers. These problems were first introduced by Stockmeyer and Meyer (1973). We continue this line of research and give a systematic characterization of the complexity of equivalence problems over sets of natural numbers. Our work shows that equivalence problems capture a wide range of complexity classes like NL, C=L, P,Π2P, PSPACE, NEXP, and beyond. McKenzie and Wagner (2003) studied…

## 15 Citations

### The Complexity of Membership Problems for Circuits over Sets of Positive Numbers

- MathematicsFCT
- 2007

It is shown that the membership problem for the general case and for (∪∩,+,×) is PSPACE-complete, whereas it is NEXPTIME-hard if one allows 0, and several other cases are resolved.

### Balance Problems for Integer Circuits

- Mathematics, Computer ScienceElectron. 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

- MathematicsMFCS
- 2017

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.

### Complexity of Constraint Satisfaction Problems over Finite Subsets of Natural Numbers

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2016

The computational complexity of constraint satisfaction problems that are based on integer expressions and algebraic circuits, such as L, P, NP, PSPACE, NEXP, and even Sigma_1, the class of c.e. languages, is studied.

### Complexity of Equations over Sets of Natural Numbers

- MathematicsTheory of Computing Systems
- 2009

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.

### Functions Definable by Arithmetic Circuits

- Mathematics, Computer ScienceCiE
- 2009

Two negative results are proved: the first shows, roughly, that a function is not circuit-definable if it has an infinite range and sub-linear growth; the second shows,roughly, that it has a finite range and fails to converge on certain `sparse' chains under inclusion.

### Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2017

### Constraint Satisfaction Problems around Skolem Arithmetic

- Mathematics, Computer ScienceArXiv
- 2015

This work studies interactions between Skolem Arithmetic and certain classes of Constraint Satisfaction Problems (CSPs) and proves the decidability of SkoleM Arithmetic.

### Unary Pushdown Automata and Straight-Line Programs

- Computer ScienceICALP
- 2014

The results imply Π2 P-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards.

### Functions definable by numerical set-expressions

- Mathematics, Computer ScienceJ. Log. Comput.
- 2013

This article investigates the definability of sets and functions by means of additive and arithmetic circuits, occasionally augmented with additional operations.

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