相关论文: R_{1-tt}^{SN}(NP) Distinguishes Robust Many-One an…
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…
The relationship between the complexity classes P and NP is a question that has not yet been answered by the Theory of Computation. The existence of a language in NP, proven not to belong to P, is sufficient evidence to establish the…
Pudl\'ak [Pud17] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are: - $\mathsf{DisjNP}$: The class of all disjoint…
The objective of this article is to formalize the definition of NP problems. We construct a mathematical model of discrete problems as independence systems with weighted elements. We introduce two auxiliary sets that characterize the…
This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential time at almost all lengths, then every many-one NP-complete set is complete under length-increasing…
We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table…
The complexity class $\exists\mathbb R$, standing for the complexity of deciding the existential first order theory of the reals as real closed field in the Turing model, has raised considerable interest in recent years. It is well known…
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered…
In this paper, we consider the rank-one tensor completion problem. We address the question of existence and uniqueness of the rank-one solution. In particular we show that the global uniqueness over the field of real numbers can be verified…
Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in…
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 $k \geq 2$, there is a $k$-T-complete set…
This paper discusses advances, due to the work of Cai, Naik, and Sivakumar and Glasser, in the complexity class collapses that follow if NP has sparse hard sets under reductions weaker than (full) truth-table reductions.
We construct an oracle relative to which $\mathrm{P} = \mathrm{NP} \cap \mathrm{coNP}$, but there are no many-one complete sets in $\mathrm{UP}$, no many-one complete disjoint $\mathrm{NP}$-pairs, and no many-one complete disjoint…
In this paper, we take a closer look at Czerwinski's "${\rm P}\neq{\rm NP}$ relative to a ${\rm P}$-complete oracle" [Cze23]. There are (uncountably) infinitely-many relativized worlds where ${\rm P}$ and ${\rm NP}$ differ, and it is…
We study reductions that limit the extreme adaptivity of Turing reductions. In particular, we study reductions that make a rapid, structured progression through the set to which they are reducing: Each query is strictly longer (shorter)…
A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from polynomial space by showing that all…
We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. 1. BQP is low for PP, i.e.,…
We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The…
In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and order. Afterwards, we combine the results from our study of…
We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…