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# 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

- A97] F. Ablayev, Randomization and Nondeterminism are Incomparable for Ordered Read-Once Branching Programs, Proc. ICALP'97, LNCS 1256, Springer, 1997, pp. 195{202.
- F. Ablayev and M. Karpinski, On the Power of Randomized Branching Problems, Proc. ICALP'96, LNCS 1099, Springer, 1996, pp. 348{356 also available as ECCC TR95-054 (1995) at http://www.eccc.uni-trier.de/eccc/.
- AK98a] F. Ablayev and M. Karpinski, On the Power of Randomized Ordered Branching Programs, ECCC TR98-004 (1998), available at http://www.eccc.uni-trier.de/eccc/.
- AK98b] F. Ablayev and M. Karpinski, A Lower Bound for Integer Multiplication on Randomized Read-Once Branching Programs, ECCC TR98011 (1998), available at http://www.eccc.uni-trier.de/eccc/.
- AKM98] F. Ablayev, M. Karpinski and R. Mubarakzjanov, On BPP versus NP coNP for Ordered Read-Once Branching Programs and AC0 class, 1998, this volume.
- M. Agrawal and T. Thierauf, The Satis ability Problem for Probabilistic Ordered Branching Programs, ECCC TR97-060 (1997), available at http://www.eccc.uni-trier.de/eccc/.
- 3 (1993), pp. 1{18.
- BSSW93] B. Bollig, M. Sauerho, D. Sieling and I. Wegener, Read k-Times Ordered Binary Decision Diagrams-E cient Algorithms in the Presence of Null-Chains, Technical Report Nr. 474, Univ. Dortmund, 1993.
- B86] R. Bryant, Graph-Based Algorithms for Boolean Function Manipulation, IEEE Trans. Comput., C-35, (8), (1986), pp. 677{691.
- Multiplication, IEEE Trans. Comput. 40 (1991), pp. 205{213.
- Br92] R. Bryant, Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams, ACM Computing Surveys, 24, No. 3, (1992), pp. 293{
- 318. Bu92] R. Buss, The Graph of Multiplication is Equivalent to Counting, Information Processing Letters, 41, (1992), pp. 199{201.
- G68] R. Gallager, Information Theory and Reliable Communication, Wiley, New York, 1968.
- Letters 51 (1991), pp. 265{269.
- GKMS96] D. Grigoriev, M. Karpinski, F. Meyer auf der Heide and R. Smolensky, A Lower Bound for Randomized Algebraic Decision Trees, Proc. 28th ACM STOC (1996), pp. 612{619 also in Computational Complexity
- 6 (1997), pp. 357{375.
- J89] S.Jukna, On the E ect of Null-Chains on the Complexity of Contact Schemes, Proc. FCT'89, LNCS 380, Springer, 1989, pp. 246{256.
- J94] S. Jukna, A Note on Read-k-Times Branching Programs, RAIRO Theoretical Informatics and Applications, 29, No. 1 (1995), pp. 75{83.
- J95] S. Jukna, The Graph of Integer Multiplication is Hard for Read-kTimes Networks, TR 95-10 Mathematik/Informatik, Univ. of Trier, 1995.
- NP co;NP for Decision Trees and Read-Once Branching Programs, Proc. 22th MFCS'98, LNCS 1295, Springer, 1997, pp. 319{326.
- KM98] M. Karpinski, R. Mubarakzjanov, Some Separation Problems on Randomized OBDDs, Manuskript, 1998.
- 324, Springer, 1988, pp. 405{413.
- K97] E. Kushilevitz and N. Nisan, Communication Complexity, Cambridge University Press, 1997.
- P95a] S. Ponzio, A Lower Bound for Integer Multiplication with Read-Once Branching Programs, Proc. 27th ACM STOC (1995), pp. 130{139.
- P95b] S. Ponzio, Restricted Branching Programs and Hardware Veri cation, Technical Report, MIT/LCS-TR-633, MIT, 1995.
- R91] A. Razborov, Lower Bounds for Deterministic and Nondeterministic Branching Programs, Proc. FCT'91, LNCS 529, Springer, 1991, pp. 47{60.
- SZ96] P. Savicky and S. Zak, A Large Lower Bound for 1-Branching Programs, ECCC, Revision 01 of TR96-036 (1996), available at http://www.eccc.uni-trier.de/eccc/.
- S97a] M. Sauerho, A Lower Bound for Randomized Read-k-Times Branching Programs, ECCC TR97-019 (1997), available at http://www.eccc.uni-trier.de/eccc/.
- S97b] M. Sauerho, On Nondeterminism Versus Randomness for ReadOnce Branching Programs, ECCC, TR97-030, (1997), available at http://www.eccc.uni-trier.de/eccc/.
- S98] M. Sauerho, Randomness and Nondeterminism are Incomparable for Read-Once Branching Programs, ECCC TR98-018, 1998, also see the Correction available at http://www.eccc.uni-trier.de/eccc/.
- T98] J.S. Thathacher, On Seperating the Read-k-Times Branching Program Hierarchy, ECCC TR98-002 (1998), available at http://www.eccc.uni-trier.de/eccc/to appear in Proc.30th ACM STOC (1998).
- Mathematics 136 (1994), pp. 347{372.
- Y79] A.C. Yao, Some Complexity Questions Related to Distributive Computing, Proc. 11th Annual ACM Symposium on the Theory of Computing, (1979), pp. 209-213.
- Y83] A.C. Yao, Lower Bounds by Probabilistic Arguments, Proc. 27th Annual IEEE Symposium on Foundations of Computer Science (1983), pp. 420-428.

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