On operators of higher types

We discuss the use of operators of higher types in complexity theory. These are operators ranging over sets of words, i.e. over oracles. Depending on different oracle access mechanisms we consider two types of operators. In particular we examine existential, universal, and bounded-error probabilistic operators. We identify some of the emerging classes and we interpret recent results about interactive protocols in terms of these operators.

[1]  Hans Jürgen Prömel,et al.  Probabilistically checkable proofs and their consequences for approximation algorithms , 1994, Discret. Math..

[2]  Pekka Orponen,et al.  Complexity Classes of Alternating Machines with Oracles , 1983, ICALP.

[3]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[4]  Heribert Vollmer,et al.  Probabilistic Type-2 Operators and “Almost”-Classes , 1998, computational complexity.

[5]  Ronald V. Book.  Some Observations on Separating Complexity Classes , 1991, SIAM J. Comput..

[6]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[7]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[8]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[9]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[10]  John Gill,et al.  Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1 , 1981, SIAM J. Comput..

[11]  Carsten Lund,et al.  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[12]  Klaus Ambos-Spies,et al.  Randomness, Relativizations, and Polynomial Reducibilities , 1986, SCT.

[13]  László Babai,et al.  Transparent (Holographic) Proofs , 1993, STACS.

[14]  Heribert Vollmer,et al.  Measure One Results in Computational Complexity Theory , 1997, Advances in Algorithms, Languages, and Complexity.

[15]  Ding-Zhu Du,et al.  Advances in Algorithms, Languages, and Complexity , 1997 .

[16]  Albert R. Meyer,et al.  Word problems requiring exponential time(Preliminary Report) , 1973, STOC.

[17]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[18]  Jin-Yi Cai,et al.  PSPACE Survives Constant-Width Bottlenecks , 1991, Int. J. Found. Comput. Sci..

[19]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[20]  Heribert Vollmer,et al.  On Type-2 Probabilistic Quantifiers , 1996, ICALP.

[21]  Denis Thérien,et al.  Logspace and logtime leaf languages , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[22]  Thomas Schwentick,et al.  On the power of polynomial time bit-reductions , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[23]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[24]  Yongge Wang,et al.  Separations by Random Oracles and "Almost" Classes for Generalized Reducibilities , 1995, Math. Log. Q..

[25]  Celia Wrathall,et al.  Complete Sets and the Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[26]  L. Fortnow,et al.  On the power of multi-power interactive protocols , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[27]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[28]  Sanjeev Arora,et al.  Probabilistic checking of proofs; a new characterization of NP , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[29]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[30]  Clemens Lautemann,et al.  BPP and the Polynomial Hierarchy , 1983, Inf. Process. Lett..

[31]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[32]  Klaus W. Wagner,et al.  The Analytic Polynomial‐Time Hierarchy , 1998, Math. Log. Q..

[33]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[34]  Lance Fortnow,et al.  On the Power of Multi-Prover Interactive Protocols , 1994, Theor. Comput. Sci..

[35]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[36]  Uwe Schöning.  Probabilistic Complexity Classes and Lowness , 1989, J. Comput. Syst. Sci..

[37]  U. Schoning.  Probalisitic complexity classes and lowness , 1989 .

[38]  Noam Nisan,et al.  On read-once vs. multiple access to randomness in logspace , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[39]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.