[Paper Review] Power Circuits, Exponential Algebra, and Time Complexity

Alexei Myasnikov, Alexander Ushakov+1 more|arXiv (Cornell University)|Jun 13, 2010

Computability, Logic, AI Algorithms5 references3 citations

TL;DR

This paper introduces power circuits—a novel compressed representation for integers that enables efficient computation with exponentially large numbers. By encoding integers via directed acyclic graphs with operations $x \mapsto x \cdot 2^y$, the authors show that basic arithmetic and comparison operations, as well as quantifier-free theories of exponential algebras, can be decided in polynomial time, overcoming super-exponential blowup in standard binary representations.

ABSTRACT

Motivated by algorithmic problems from combinatorial group theory we study computational properties of integers equipped with binary operations +, -, z = x 2^y, z = x 2^{-y} (the former two are partial) and predicates < and =. Notice that in this case very large numbers, which are obtained as n towers of exponentiation in the base 2 can be realized as n applications of the operation x2^y, so working with such numbers given in the usual binary expansions requires super exponential space. We define a new compressed representation for integers by power circuits (a particular type of straight-line programs) which is unique and easily computable, and show that the operations above can be performed in polynomial time if the numbers are presented by power circuits. We mention several applications of this technique to algorithmic problems, in particular, we prove that the quantifier-free theories of various exponential algebras are decidable in polynomial time, as well as the word problems in some "hard to crack" one-relator groups.

Motivation & Objective

• To address the computational intractability of manipulating very large integers in standard binary form, especially those arising from iterated exponentiation.
• To develop a compressed, unique, and efficiently computable representation for integers that supports fast arithmetic operations.
• To prove that quantifier-free theories of various exponential algebras are decidable in polynomial time.
• To apply the framework to solve hard algorithmic problems in combinatorial group theory, such as the word problem in certain one-relator groups.

Proposed method

• Represent integers using power circuits—acyclic directed graphs with labeled nodes for constants and operations $x \cdot 2^y$, $x \cdot 2^{-y}$, $+$, and $-$.
• Define standard, reduced, and normal forms of power circuits to ensure uniqueness and enable efficient evaluation.
• Introduce a reduction process based on geometric ordering and vertex equivalence to normalize circuits and minimize size.
• Use term evaluation and normal form computation to perform addition, subtraction, exponentiation, and comparison in polynomial time.
• Leverage the power circuit representation to encode and evaluate quantifier-free formulas in exponential algebras efficiently.
• Demonstrate that the size of circuits can grow exponentially under multiplication, revealing inherent computational limits.

Experimental results

Research questions

• RQ1 Can the word problem in certain one-relator groups be solved in polynomial time using power circuits?
• RQ2 Is the quantifier-free theory of the structure $\tilde{Z}$, which includes addition, subtraction, and exponentiation by powers of two, decidable in polynomial time?
• RQ3 What is the time complexity of solving linear equations such as $3x = \mathcal{P}$ in the power circuit model?
• RQ4 Can a minimal-size constant power circuit be computed efficiently for any given natural number?
• RQ5 Is the existential theory of the structure $\tilde{N} = \langle \mathbb{N}_{>0}; +, \cdot 2^y, \leq, 1 \rangle$ decidable?

Key findings

• The quantifier-free theory of $\tilde{Z}$ is decidable in polynomial time using power circuits.
• The quantifier-free theory of $\tilde{N} = \langle \mathbb{N}; +, \cdot 2^y, \cdot 2^{-y}, \leq, 0, 1 \rangle$ is decidable in polynomial time.
• The word problem in certain one-relator groups is solvable in polynomial time via power circuit representations.
• Solving linear equations like $3x = \mathcal{P}$ may require super-exponential time in the worst case due to circuit size blowup.
• The product of $n$ power circuits can require a circuit of size at least $2^{n-3}$, demonstrating inherent exponential growth in multiplication.
• The minimal power circuit representation of a given integer may require super-exponential time to compute, indicating a fundamental complexity barrier.