# Integer circuit evaluation is PSPACE-complete

@article{Yang2000IntegerCE, title={Integer circuit evaluation is PSPACE-complete}, author={Ke Yang}, journal={Proceedings 15th Annual IEEE Conference on Computational Complexity}, year={2000}, pages={204-211} }

In this paper, we address the problem of evaluating the integer circuit (IC), or the {U,/spl times/,+}-circuit over the set of natural numbers. The problem is a natural extension to the integer expression by L.J. Stockmeyer and A.R. Mayer (1973); and is also studied by P. Mckenzie et al. (1999) in their "Polynomial Replacement System". We show a polynomial-time algorithm that reduces QBP (quantified Boolean formula) problem to the integer circuit problem. This complements the result of K.W…

## 20 Citations

### 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.

### Arithmetic Circuits and Polynomial Replacement Systems

- Mathematics, Computer ScienceSIAM J. Comput.
- 2000

This paper addresses the problems of counting proof trees and counting proof circuits, a related but seemingly more natural question, and contributes a classification of these systems, which are called polynomial replacement systems.

### 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.

### Satisfiability of algebraic circuits over sets of natural numbers

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2007

### Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

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

### Equivalence Problems for Circuits over Sets of Natural Numbers

- Mathematics, Computer ScienceTheory of Computing Systems
- 2008

This work gives a systematic characterization of the complexity of equivalence problems over sets of natural numbers and provides an improved upper bound for the case of {∪,∩,−,+,×}-circuits.

### The complexity of membership problems for circuits over sets of integers

- MathematicsTheor. Comput. Sci.
- 2004

### 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.

### 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.

### 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.

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