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Quite recently some further exponential lower bounds on randomized read-ktimes branching programs were obtained by Thathacher T98]

# On the computational power of PP and (+)P

SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pp.514-519, (1989)

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Abstract

Two complexity classes, PP and (+)P, are compared with PH (the polynomial-time hierarchy). The main results are as follows: (1) every set in PH is reducible in a certain sense to a set in PP, an (2) every set in PH is reducible to a set in (+)P under randomized polynomial-time reducibility with two-sided bounded error probability. It foll...More

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Introduction
• The model of restricted branching programs has recently been found very useful in a number of applications.
• A deterministic branching program P for computing a boolean function f: f0 1gn !
• The authors de ne some explicit boolean functions for which the authors are going to prove computational upper and lower bounds on di erent types of branching programs.
• The authors are going to characterize the computational power of randomized OBDDs on the explicit boolean functions introduced in Section 3, and formulate corresponding lower bounds on the deterministic branching programs.
Highlights
• The model of restricted branching programs has recently been found very useful in a number of applications
• We are going to characterize the computational power of randomized OBDDs on the explicit boolean functions introduced in Section 3, and formulate corresponding lower bounds on the deterministic branching programs
• Quite recently some further exponential lower bounds on randomized read-ktimes branching programs were obtained by Thathacher T98]
• It is well known that this function cannot be computed by polynomial size deterministic read-once branching programs
• One can construct an explicit boolean function which is computable by polynomial size randomized OBDD but not computable in polynomial size by any nondeterministic or co-nondeterministic OBDD (cf
Results
• The size lower bound on any nondeterministic ordered read-k-times branching program computing fn is 2 (n=k) : Theorem 2.
• The size lower bound on any nondeterministic read-once branching program computing PERMn is 2 (n).
• 1. The test function for integer multiplication DMULT can be computed by an "(n)-error randomized OBDD of size n6 "5(n)
• The size lower bound on any nondeterministic syntactic read-k-times branching program computing DMULT is O(2 ).
• The following randomized lower bounds of A97], and AK98b] were established using the property of the entropy function, and the one-way probabilistic communication complexity arguments.
• 1. The size lower bound on any randomized OBDD computing the function gn is 2 .
• 2. The function gn can be computed by a nondeterministic ordered read-once branching program in size O(n3).
• The size lower bound on any randomized OBDD computing the integer multiplication function MULT is 2 .
• The following recent results of Agrawal and Thierauf AT97] relate the computational complexity of the satis ability problem for randomized OBDDs to their error probability.
• The size lower bound on any randomized 3-way (2-way) read-k-times branching program computing SIP (SIPB) is 2 (n=ckk3) for some constant c.
• Quite recently some further exponential lower bounds on randomized read-ktimes branching programs were obtained by Thathacher T98].
Conclusion
• It is well known that this function cannot be computed by polynomial size deterministic read-once branching programs.
• On another hand Karpinski and Mubarakzjanov KM98] proved using communication complexity techniques, that Las Vegas public coin OBDDs are equivalent to deterministic OBDDs. One can construct an explicit boolean function which is computable by polynomial size randomized OBDD but not computable in polynomial size by any nondeterministic or co-nondeterministic OBDD
• It remains an important open problem to develop new more powerful lower bound techniques for randomized read-once branching programs.
• An important open problem remains the status of the integer multiplication function MULT on randomized read-once and read-k-times branching programs on both types syntactic, and semantic programs.
Reference
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