Hadi Shafei | University of Wyoming (original) (raw)

Papers by Hadi Shafei

Theory of computing systems, Jun 1, 2022

Nonuniformity is a central concept in computational complexity with powerful connections to circu... more Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP-completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP. Under various hypotheses, we obtain the following separations: 1. There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice. 2. There is a set complete for NP under nonuniform many-one reductions with polynomialsize advice, but not under uniform Turing reductions. That is, polynomial nonuniformity is stronger than a polynomial number of queries. 3. For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it more powerful than a nonuniform reduction with fixed polynomial advice. 4. There is a set complete for NP under nonuniform many-one reductions with polynomial advice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing. We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete. We show that under a derandomization hypothesis, the same statement for truth-table reductions and truth-table completeness also holds.

Electronic Colloquium on Computational Complexity, 2016

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under st... more We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: • For every k ≥ 2, there is a k-T-complete set for NP that is k-T autoreducible, but is not k-tt autoreducible or (k − 1)-T autoreducible. • For every k ≥ 3, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-tt autoreducible or (k − 2)-T autoreducible. • There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP ∩ coNP, we show: • For every k ≥ 2, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-Tautoreducibility from 2-tt-autoreducibility.

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under st... more We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: For every k ≥ 2, there is a k-T-complete set for NP that is k-T autoreducible, but is not k-tt autoreducible or (k − 1)-T autoreducible. For every k ≥ 3, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-tt autoreducible or (k − 2)-T autoreducible. There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP ∩ coNP, we show: For every k ≥ 2, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility. 1998 ACM Sub...

Electron. Colloquium Comput. Complex., 2016

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under st... more We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: - For every k >= 2, there is a k-T-complete set for NP that is k-T autoreducible, but is not k-tt autoreducible or (k-1)-T autoreducible. - For every k >= 3, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k-1)-tt autoreducible or (k-2)-T autoreducible. - There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP cap coNP, we show: - For every k >= 2, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k-1)-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility.

Bennett and Gill [1981] showed that PA ≠ NPA ≠ coNPA for a random oracle A, with probability 1. W... more Bennett and Gill [1981] showed that PA ≠ NPA ≠ coNPA for a random oracle A, with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem. (1) We first show that PA ≠ NPA for every oracle A that is p-betting-game random. Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation: (2) If PA ≠ NPA relative to every p-random oracle A, then BPP ≠ EXP. (3) If PA ≠ NPA relative to some p-random oracle A, then P ≠ PSPACE. Rossman, Servedio, and Tan [2015] showed that the poly...

Container types can be modeled as instances of the Haskell MonadPlus type class which support a f... more Container types can be modeled as instances of the Haskell MonadPlus type class which support a fold operation. In this paper we present subclasses that extend the MonadPlus type class to support a membership operator. The laws for the EMonadPlus type class specify how membership behaves with respect to the monad and monad plus operators. Using EMonads we are able write and prove properties of generic specifications of containers. In the second part of the paper we present an induction rule for monads we call bind induction. The computational content of the new induction rule is the Monad bind operator and the new proof rule is proved to be sound. Using this new proof rule we are able to extract monadic programs from proofs. We present an example that uses the rule to extract a simple monadic program from a proof of a specification. We have used the Coq theorem prover with the Coq Type Class mechanism to formalize the definitions presented here and to prove many properties of the fo...

Theory of computing systems, Jun 1, 2022

Electronic Colloquium on Computational Complexity, 2016

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under st... more We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: For every k ≥ 2, there is a k-T-complete set for NP that is k-T autoreducible, but is not k-tt autoreducible or (k − 1)-T autoreducible. For every k ≥ 3, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-tt autoreducible or (k − 2)-T autoreducible. There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP ∩ coNP, we show: For every k ≥ 2, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility. 1998 ACM Sub...

Electron. Colloquium Comput. Complex., 2016

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under st... more We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: - For every k >= 2, there is a k-T-complete set for NP that is k-T autoreducible, but is not k-tt autoreducible or (k-1)-T autoreducible. - For every k >= 3, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k-1)-tt autoreducible or (k-2)-T autoreducible. - There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP cap coNP, we show: - For every k >= 2, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k-1)-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility.