# 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…

## Figures from this paper

## 20 Citations

### 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.

### 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.

## References

SHOWING 1-10 OF 10 REFERENCES

### Word problems requiring exponential time(Preliminary Report)

- Computer Science, MathematicsSTOC
- 1973

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.

### Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1995

### The Complexity of Problems Concerning Graphs with Regularities (Extended Abstract)

- MathematicsMFCS
- 1984

### Sparse Complex Polynomials and Polynomial Reducibility

- Mathematics, Computer ScienceJ. Comput. Syst. Sci.
- 1977

### Greatest common divisors of polynomials given by straight-line programs

- Computer Science, MathematicsJACM
- 1988

It is shown that most algebraic algorithms can be probabilistically applied to data that are given by a straight-line computation, and every degree-bounded rational function can be computed fast in parallel, that is, in polynomial size and polylogarithmic depth.

### 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.

### ‘Computational Complexity,’’ pp

- 455–491, Addison–Wesley, Reading, MA,
- 1994