Is BPP equal to P?
. As a result, P = NP leads to P = BPP since PH collapses to P in this case. Thus either P = BPP or P ≠ NP or both. Adleman’s theorem states that membership in any language in BPP can be determined by a family of polynomial-size Boolean circuits, which means BPP is contained in P/poly.
What is complexity in automata theory?
Complexity theory is a central topic in theoretical computer science. Complexity helps determine the difficulty of a problem, often measured by how much time and space (memory) it takes to solve a particular problem.
What are the different complexity classes?
Some common complexity classes are constant complexity (O(1)), logarithmic complexity (O(lgn)), linear complexity (O(n)), polynomial complexity (O(nk), for some fixed value of k and exponential complexity (e.g., O(2n)).
What does P stands for in complexity classes theory?
It is easy to see that the complexity class P (all problems solvable, deterministically, in polynomial time) is contained in NP (problems where solutions can be verified in polynomial time), because if a problem is solvable in polynomial time then a solution is also verifiable in polynomial time by simply solving the …
Is BPP a subset of RP?
Note that both RP and co-RP are subsets of BPP.
Is BPP closed under complement?
BPP and ZPP are both closed under complement.
Why is complexity theory important?
Complexity Theory allows us to better understand systems as diverse as cells, human beings, forest ecosystems, and organizations, that are only partially understood by traditional scientific methods (Zimmerman et al. 2001).
What are tractable and intractable problems?
Tractable Problem: a problem that is solvable by a polynomial-time algorithm. Intractable Problem: a problem that cannot be solved by a polynomial-time al- gorithm.
What is Reducibility in DAA?
Reducibility for any problem (NP-hard or any other) means the possibility to convert problem A into other problem B. If we know the complexity of problem B then the complexity of problem A is at least the same as the complexity of problem A.
What does BPP mean in Computer Science?
BPP (complexity) In computational complexity theory, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/3 for all instances.
What are the theoretical properties of BPP?
Complexity-theoretic properties. It is known that BPP is closed under complement; that is, BPP = co-BPP. BPP is low for itself, meaning that a BPP machine with the power to solve BPP problems instantly (a BPP oracle machine) is not any more powerful than the machine without this extra power.
What is the BPP of Mo?
By definition, BPP = BPP(1 / 3, 1 / 3), so every time M calls Mo it takes a probabilistic risk basing a deciion upon Mo answer. We’ve learned in class that BPP = BPP(1 / 3, 1, 3) = BPP( 1 3n, 1 3n) and I think I shall explain that even though Mo may return false answers, Mo is still in BPP.
What is the relationship between bppbpp and NP?
BPP is low for itself, meaning that a BPP machine with the power to solve BPP problems instantly (a BPP oracle machine) is not any more powerful than the machine without this extra power. In symbols, BPPBPP = BPP . The relationship between BPP and NP is unknown: it is not known whether BPP is a subset of NP, NP is a subset of BPP or neither.