Natural Problems Complete For Bpp?

Is BPP Closed Under Complement?

The complexity class BPP (Bounded-error Probabilistic Polynomial time) contains decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/2 for all instances. A key question is whether BPP is closed under complement – that is, if a problem is in BPP, is its complement also in BPP?

Defining complementation formally, if language L is in BPP, then the complement of L (written L’) contains those inputs not in L. The question of whether BPP is closed under complement thus asks: if L is in BPP, must L’ also be in BPP? This property would make BPP robust and allow randomized algorithms for a wider range of problems.

Defining the Complexity Class BPP

BPP consists of decision problems with a randomized polynomial-time algorithm such that for any input string x:

• If x is in L, the algorithm outputs “yes” with probability at least 2/3
• If x is not in L, the algorithm outputs “no” with probability at least 2/3

By running the algorithm multiple times and taking the majority vote, the error probability can be reduced as needed. So BPP captures the power of randomness to enable efficient probabilistic computation for decision problems.

BPP and Randomized Algorithms

Randomized algorithms access random bits in addition to the input and exploit randomness to gain efficiency. For many problems like primality testing, matrices with special properties, combinatorial search and counting, randomized approaches outperform deterministic ones.

BPP models such algorithms – success probability on any input may be less than 100% but is always reasonably high. This flexibility makes randomness a powerful tool for efficient computation. Understanding properties like complementation closure for BPP reveals deeper structure about the power of randomness.

The Question of Complementation

Complementation closure is linked to fundamental aspects of computation – it enables easier search for optimal algorithms and understanding problem structure. Also, complementation leads to full verification algorithms.

But permitting even small error probabilities poses challenges, as amplifying success chances involves running algorithms multiple times. Ensuring correctness for L’ via randomized reductions from L is tricky since L and L’ have opposing goals for error bounds.

Despite significant research, complementation for BPP remains an open question – neither positive nor negative answers are known. Resolving this will require new algorithmic techniques or complexity separations.

Reductions Between Problems

Reductions map decision problems to each other preserving underlying structure. They formalize notions of comparitive problem difficulty and enable re-use of solutions.

For BPP, randomized polynomial-time Turing reductions are used – the reduction itself is a probabilistic algorithm calling the oracle problem as a subroutine. This models well the randomness present.

Constructing such reductions between problems is challenging, as error bounds must be controlled across multiple algorithm invocations. Also, oracles have limited information – just yes/no answers without access to actual witnesses.

Despite difficulties, reductions provide a path to proving BPP closure laws. By reducing complement problems to base ones, inheritance of randomized efficiency can be demonstrated.

Open Questions on BPP Completeness

Complete problems characterize core difficulty of entire complexity classes. But unlike classes like NP, few natural complete problems are known for BPP.

候选完全BPP的问题应满足以下条件:

• 问题应该有内在的随机性质,这使得随机算法看起来更自然
• 问题应易于构建规约,因此易于证明其完全性
• 问题应广泛适用,同时捕捉BPP的本质困难

发现这样的完全问题将深入理解BPP的权力并指导新算法的设计。 解决BPP闭包问题的关键可能在于找到合适的完全问题。

Examples of BPP-complete Problems

While no natural complete problems are proven for BPP so far, some candidate issues believed to be hard for the class are:

• Approximate Counting: Estimating set size for combinatorial structures
• Formula Value: Evaluating {0,1}-valued expressions with AND/OR clauses
• Circuit Acceptance: Determining if constant-depth circuits output 1

These capture aspects of numerical computation, logic and circuit design amenable to probabilistic algorithms. Proving completeness requires constructing appropriate reductions from complement languages.

Conclusions on the Power of BPP

The capabilities of randomized computation make BPP a central complexity class. Properties like complementation closure have foundational importance. Despite significant progress, BPP retains intriguing mysteries due to subtleties arising from randomness.

Negative answers would indicate severe limitations on constructive power of probabilistic proofs. Positive resolutions may uncover new techniques for algorithm design and analysis. The questions pose deep mathematical challenges with practical implications in computer science.